SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE
By
NICOLETA ALINA APETRE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
Copyright 2005
by
Nicoleta Alina Apetre
This document is dedicated to my loving family: my mother, Elisabeta Apetre, my sister, Dana and my brother, Nicu and to my devoted boyfriend Laurence as they have offered their support and love throughout this endeavor. No words said can express how indebted I am to them for what I am today and for what I shall accomplish in the future.
ACKNOWLEDGMENTS
I am profoundly grateful to my advisor, Dr. Bhavani Sankar, for scientific guidance, moral support and endless patience. I would like to thank Dr. Peter Ifju for his help and ideas throughout this enterprise.
I would also like to thank my friend, mentor and committee member, Dr. Oana
Cazacu, for her guidance and support on my research and my life. Her comments and
advice made this dissertation a much more useful document.
At the same time, I wish to thank Dr. Raphael T. Haftka and Dr. Bjorn Birgisson
for serving on my committee and for their assistance and guidance.
I express sincere appreciation to a friend and mentor, Dr. Satchi Venkataraman, for his continuous scientific guidance and moral support during the years.
I would like to thank all my professors from whom I learned so much, especially the late Dr. E. Soos, who showed me the path I am walking now.
Special thanks go to all my friends, colleagues and family for their continual love, support, and encouragement throughout my time in graduate school and my life.
iv
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ...... iv
LIST OF TABLES...... vii
LIST OF FIGURES ...... viii
ABSTRACT...... xii
CHAPTER
1 INTRODUCTION AND BACKGROUND ...... 1
Introduction...... 1 Literature Review ...... 7 Sandwich Plate Theories ...... 7 Contact on Composite Materials ...... 14 Cellular Solids and Functionally Graded Materials ...... 28 Objectives and Approaches ...... 36
2 ANALYTICAL MODELING OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE ...... 38
Exact Method for Sandwich Structures with Functionally Graded Core ...... 38 Fourier – Galerkin Method for Sandwich Structures with Functionally Graded Core...... 42 Analytical Models for Sandwich Structures with Functionally Graded Core ...... 47 An Equivalent Single-Layer First-Order Shear Deformation Theory...... 47 An Equivalent Single-Layer Third-Order Shear Deformation Theory ...... 50 A Higher-Order Shear Deformation Theory ...... 52 Results for Sandwich Beam with Functionally Graded Core under a Distributed Load ...... 57 Summary and Conclusions ...... 67
3 CONTACT AND LOW-VELOCITY IMPACT Of SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE ...... 69
Assumed Stress Distribution Method ...... 71 Method of Point Matching...... 74
v Quasi-Static Impact of a Sandwich Beam with FG Core ...... 76 Results...... 78 Summary and Conclusions ...... 103
4 MASS OPTIMIZATION FOR A SANDWICH BEAM WITH FUNCTIONALLY GRADED CORE...... 105
Linear Model ...... 105 Case 1: Same Flexural Rigidity...... 109 Case 2: Same Mass and Flexural Rigidity...... 112 Quadratic Model ...... 118 Case 1: Same Flexural Rigidity...... 121 Case 2: Same Total Mass ...... 130 Summary and Conclusions ...... 138
5 SUMMARY AND CONCLUSIONS...... 140
LIST OF REFERENCES...... 144
BIOGRAPHICAL SKETCH ...... 153
vi
LIST OF TABLES
Table page
3-1 Maximum normal and shear strains for a given impact energy ...... 94
3-2 Core thicknesses for different materials with same flexural stiffness, D11 and same global stiffness ...... 97
4-1 Variations of Young’s modulus, density, mass and flexural rigidity...... 107
4-2 Total mass of sandwich structures with different density variations and different core thicknesses...... 111
4-3 Maximum normal and shear strains for a given impact energy ...... 117
4-4 Variations of Young’s modulus, density, mass and flexural rigidity...... 120
4-5 Total mass of sandwich structures with same geometry and different density variations ...... 123
4-6 Maximum normal and shear strains normal and interfacial shear stresses for a given impact energy ...... 127
4-7 Flexural rigidity of sandwich structures with same geometry and different density variations...... 132
4-8 Maximum normal and shear strains and stresses for a given impact energy ...... 134
vii
LIST OF FIGURES
Figure page
1.1 The behavior of (a) flexible face sheet and (b) rigid face sheet...... 4
2.1 Sandwich beam with functionally graded core with schematic of the analysis elements...... 39
2.2 Traction forces and displacements at the interfaces of each element in the FGM sandwich beam ...... 40
2.3 The beam geometry...... 48
2.4 The beam geometry...... 53
2.5 Through-the-thickness variations of core modulus...... 58
2.6 Variation of transverse shear stresses in the sandwich structure with FGM core....59
2.7 Variation of σzz stresses in the sandwich structure with FGM core ...... 60
2.8 Variation of bending stresses through the thickness of the FGM core ...... 61
2.9 Variation of u displacements through the thickness of the sandwich structure with FGM core ...... 61
2.10 Non-dimensional core modulus ...... 62
2.11 Comparison of deflections ...... 64
2.12 Comparison of longitudinal displacement...... 64
2.13 Comparison of axial stress in the core...... 65
2.14 Comparison of shear strain in the core...... 66
2.15 Comparison of shear stress in the core...... 67
3.1 Dimensions of the sandwich panel and the contact length, 2c...... 72
3.2 The stress distribution under the indenter...... 72
viii 3.3 Illustration of relation between w deflection and radius of curvature of the deformed surface...... 73
3.4 Discretization of contact load for the method of point matching...... 74
3.5 Low-velocity impact model ...... 77
3.6 Through the thickness variations of core modulus...... 80
3.7 Variation of the horizontal displacement ...... 81
3.8 Deflection, w, in the second half of the sandwich beam ...... 81
3.9 Variation of the deflection...... 82
3.10 Contour plot of normal strain in the functionally graded asymmetric core ...... 83
3.11 Contour plot of normal strain in the functionally graded symmetric core ...... 83
3.12 Contour plot of normal strain in the homogeneous core...... 83
3.13 Contour plot of shear strain in the functionally graded asymmetric core ...... 84
3.14 Contour plot of shear strain in the functionally graded symmetric core...... 84
3.15 Contour plot of shear strain in the homogeneous core...... 85
3.16 Variation of the axial stress...... 85
3.17 Variation of the axial stress through the thickness in functionally graded core...... 86
3.18 Variation of the normal (compressive) stress...... 87
3.19 Face sheet thickness influence on variation of the normal (compressive) stress .....88
3.20 Indentor radius influence on variation of the normal (compressive) stress ...... 89
3.21 Face sheet Young’s modulus influence on variation of the normal (compressive) stress...... 89
3.22 Shear stress in functionally graded asymmetric core ...... 90
3.23 Contour plot of shear stress in linear asymmetric core ...... 91
3.24 Contour plot of shear stress in linear symmetric core...... 91
3.25 Contour plot of shear stress in homogeneous core...... 91
3.26 Variation of contact force with indentation depth in functionally graded beams ....92
ix 3.27 Relation between contact force and the vertical displacement...... 93
3.28 Contact force vs. maximum normal strain ...... 95
3.29 Contact force vs. maximum shear strain ...... 96
3.30 Variation of contact force with global deflection...... 97
3.31 Comparison of maximum normal strains for different core materials ...... 99
3.32 Comparison of maximum shear strains for different core materials...... 99
3.33 Finite element simulation for half of the beam ...... 100
3.34 Finite element simulation of the contact between the functionally graded sandwich beam and the rigid spherical indentor ...... 101
3.35 Contact pressure under indenter and total load for three models...... 102
3.36 Deflections at core - top face sheet interface for same contact length and different contact load...... 102
3.37 Deflections at core - top face sheet interface for different contact length and same contact load ...... 103
4.1 Geometry and notations for the three cases investigated ...... 108
4.2 Through the thickness non-dimensional variations of core modulus and density .110
4.3 Through the thickness non-dimensional variations of core modulus and density .113
4.4 Relation between contact force and vertical displacement ...... 113
4.5 Variation of contact force with indentation depth...... 114
4.6 Variation of contact force with contact length ...... 115
4.7 Variation of contact force with contact length ...... 115
4.8 Contact force vs. maximum normal strain ...... 116
4.9 Contact force vs. maximum shear strain ...... 117
4.10 Young’s modulus variation ...... 119
4.11 Core Young’s modulus...... 123
4.12 Core densities variation...... 124
x 4.13 Relation between contact force and vertical displacement ...... 124
4.14 Relation between contact force and contact length...... 125
4.15 Relation between contact force and contact length...... 125
4.16 Relation between indentation (core compression) and contact length...... 126
4.17 Contact force vs. core maximum normal strain ...... 128
4.18 Contact force vs. core maximum shear strain ...... 129
4.19 Contact force vs. core maximum normal stress ...... 129
4.20 Contact force vs. maximum interfacial shear stress...... 130
4.21 Core Young’s modulus...... 132
4.22 Core densities variation...... 133
4.23 Relation between contact force and vertical displacement ...... 134
4.24 Relation between contact force and contact length...... 135
4.25 Contact force vs. core maximum normal strain ...... 136
4.26 Contact force vs. core maximum shear strain...... 136
4.27 Contact force vs. core maximum normal stress ...... 137
4.28 Contact force vs. maximum interfacial shear stress...... 137
xi
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
SANDWICH PANELS WITH FUNCTIONALLY GRADED CORE
By
Nicoleta Alina Apetre
December 2005
Chair: Bhavani V. Sankar Major Department: Mechanical and Aerospace Engineering
Although sandwich structure offer advantages over other types of structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). One alternative is represented by functionally graded materials (defined as materials with properties that vary with location within the material in order to optimize a prescribed function).
This study presents different analytical and finite element models for sandwich structures with functionally graded core. The trade-off between weight and stiffness as well as a comparison between these structures and sandwich structures with homogenous core is also presented. The problem of low-velocity impact between a sandwich structure with functionally graded core and a rigid spherical indentor is solved. A few advantages and disadvantages of these types of structure are presented.
With the new developments in manufacturing methods these materials can be used for a large number of applications ranging from implant teeth to rocket frames.
xii CHAPTER 1 INTRODUCTION AND BACKGROUND
Introduction
A sandwich structure consists of two thin, stiff, and strong face sheets connected by
a thick, light and low-modulus core using adhesive joints in order to obtain very efficient
lightweight structure (Zenkert, 1997; Vinson, 2001). In most of the cases the faces carry the loading, both in-plane and bending, while the core resists transverse shear loads. A sandwich operates in the same way as an I-beam with the difference that the core of a sandwich is of a different material and is stretched out as a continuous support for the face sheets. The main advantage of a sandwich structure is its exceptionally high flexural stiffness-to-weight ratio compared to other architectures. As a consequence, sandwich construction results in lower lateral deformations, higher buckling resistance, and higher natural frequencies than do other structures. Thus, for a given set of mechanical and environmental loads, sandwich construction often results in a lower structural weight than do other configurations. Few of the drawbacks of sandwich structures are: manufacturing methods, quality control and joining difficulties.
The idea to use two co-operating faces separated by a distance emerged at the beginning of 19th century. The concept was first applied commercially during World War
II in airplanes. Some of the first theoretical and experimental works on sandwich
constructions were published in the late 1940’s (Williams et al., 1941, and March, 1948).
Since then, the use of sandwich structures has increased rapidly. The need for high- performance, low-weight structures assures that the sandwich construction will continue
1 2
to be in demand. In view of increasing the use of sandwich structures in the aerospace
industry, critical issues such as impact resistance, optimization, fatigue and fracture
should be addressed. Lately, due to the difficulty to obtain closed-form solutions for the
governing differential equations, finite element method has become the most used design
tool for panel analysis.
The core material is perhaps the most important component of a sandwich structure. The following properties are required for the core material: low density in order
to add as little as possible to the total weight; Young’s modulus perpendicular to the face
sheet should be fairly high to prevent excessive deformation in the thickness direction
and therefore a rapid decrease in flexural rigidity. Even though the transverse force
creating normal stresses perpendicular to the core are usually low, even a small decrease
in core thickness would create a large decrease in flexural rigidity. This is crucial in the
case of impact loads wherein the contact force acts over a small area. The core is mainly
subjected to transverse shear stresses. The core shear strains contribute to global deformation and core shear stresses. Thus, a core must be chosen such that it would not fail under the applied transverse load and with a shear modulus high enough to give the required shear stiffness. Other functions such as thermal and acoustical insulation depend mainly on the core material and its thickness.
The core materials have been fashioned in various forms and developed for a range of applications. Few examples are: foam or solid cores (where properties can be tailored by orienting the cell structure), honeycomb cores, web cores and truss cores. Honeycomb cores offer the greatest shear strength and stiffness to weight ratio but require care in ensuring adequate bonding to the face-sheets. Also the core is highly anisotropic and
3
forming complex curved shapes is difficult. High-density cellular thermoplastic foams,
such as poly (vinyl chloride) (PVC) and polyurethane (PUR) do not have the same high
stiffness to weight ratio as honeycomb materials but they have more advantages: they are
less expensive, they are continuous at the macroscopic level, the foam surface is easier to
bond, and they ensure high thermal and acoustic insulation (cellular cores have a very
low thermal conductivity, e.g., Gibson and Ashby, 1997). A metallic foam core sandwich structure offers also a number of advantages. They can be made with integral skin, eliminating the need for adhesive layer, they can be formed in curved shapes, and their properties are nearly isotropic (Gibson and Ashby, 1997).
Foreign objects impact (such as tool drops, hail, bird strikes, and runway debris)
can introduce damage and significantly reduce the strength of the sandwich structure.
Unlike their solid metallic counterparts, predictions of the effects of low-velocity impact
damage on sandwich structures are difficult because significant internal damage is
achieved at impact energy levels lower than those required to create visible damage at the
surface.
Low-velocity impact, one of the topics of present research is described by low projectile velocity; therefore the damage can be analyzed by assuming that the structure is under a quasi-static loading. Impact/contact produces both overall deformation and local deformation. Usually, if the load is applied on top face sheet, the bottom face sheet experience only global deformation so many models assume a clamped bottom face sheet in order to eliminate the global deflection and focus the research on local effects.
4
There are different factors that can influence the local effect of indentation stress
field: face sheet elastic modulus and its thickness, the core density and its thickness. One
of the factors that influence the structure behavior under an indentor is the face sheet
stiffness: for flexible face-sheets there is a local deformation under the indentor/load
whereas the very rigid face-sheets spread the load (Figure 1.1). Usually the local failure
starts in the core and results in core crushing, delamination and therefore significant
reduction of the sandwich strength.
(a)
(b)
Figure 1.1: The behavior of (a) flexible face sheet and (b) rigid face sheet
Contact laws (defined as relation between the contact force and indentation or core compression) or relative displacement of the indentor and the target for the indentation of a sandwich structure are significantly different from those for monolithic laminates. They are usually nonlinear. With sandwich structures the indentation is dominated by the behavior of the core material, which crushes as the shear stress and compressive stress become large. Compressive stress in transverse direction is not uniform through the thickness and is not the only stress component to be considered in order to determine the onset of core failure. Few of the contact problem difficulties are: forced surface displacement, initially unknown contact length/surface and pressure distribution.
5
Although sandwich structure offer advantages over other types of structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). These new sandwiches should be compared with other sandwich construction and with alternative structures in order to select the best configuration. One of the new alternatives is a sandwich structure with functionally graded core.
Functionally graded materials (FGM) possess properties that vary gradually with location within the material; consequently they are inhomogeneous materials on the macroscopic scale. New developments in manufacturing methods offer the designer the opportunity to tailor the material’s microstructure in order to enhance their structural performance. The variation of the properties can be achieved by gradually changing the volume fraction of the constituents’ materials, by using reinforcements with different properties, sizes and shapes, by interchanging the roles of reinforcement and matrix phases in a continuous manner as well as by adjusting the cell structure. The result is a microstructure that produces continuously changing thermal and mechanical properties at the macroscopic or continuum level.
FGMs differ from composites wherein the volume fraction of the reinforcing inclusion is uniform throughout the composite. The closest analogy of FGM is a laminated composite face sheets, but the latter possess distinct interfaces across which properties change abruptly. FGMs have a large spectrum of applications. They can be used as thermal barrier coatings (Koizumi, 1997), interfacial zones to improve the bonding strength and to reduce the residual stresses (Hsueh and Lee, 2003). Cellular materials with continuously varying density or pore size (porosity) that can be used as a
6
core for a sandwich structure is an example of new FGMs. Although FGMs are highly
heterogeneous, it will be useful to idealize them as continua with properties changing
smoothly with respect to the spatial coordinate. This will enable obtaining closed-form
solutions to some fundamental solid mechanics problems, and also will help in
developing finite element models of structures made of FGMs.
The major objective of this dissertation is to develop new analytical models for sandwich structures with functionally graded core. Particular attention will be paid to contact and impact problems involving sandwich structures. This chapter provides a framework and background for sandwich structures and functionally graded materials in order to give a perspective for present research. Chapter 2 introduces analytical models for a sandwich beam with functionally graded core and compares them with analytical models presented in the literature. The governing equations for sandwich structures with
FG cores are solved for two types of cores by six different methods: exact solutions are presented for the exponential variation of core Young Modulus; a combination of Fourier series and Galerkin method for a polynomial variation of core Young Modulus; two equivalent single-layer theories based on assumed displacements, a higher-order theory and a finite-element analysis. Chapter 3 solves the problems of contact and low-velocity impact between a rigid cylindrical indentor and a sandwich beam with functionally graded core. A comparison of the total mass of different panels is discussed in Chapter 4, where different optimization problems for sandwich structure with FG core based on two models presented in the literature. They are the linear relation between relative density and Young’s modulus (Choi and Sankar, 2005) and quadratic relation between relative density and Young’s modulus (Gibson and Ashby, 1997). The results indicate that the
7 functionally graded cores can be used to mitigate or prevent impact damage in sandwich composites, so it is shown that materials with gradients in mechanical properties offer better opportunities than traditional homogenous materials.
Literature Review
The equations describing the behavior of sandwich structures are similar to the shear deformable plate theory for composite laminates (e.g., Reddy, 2003) if each of the core, face sheets and adhesive layers is modeled as a continuum. The main difference between the traditional and modern sandwich models is the transverse flexibility of the core. A flexible/soft core is a core for which the ratio of Young’s moduli of the face sheets to the core is high (for structural panels this ratio can be between 500 and 1000). If a transversely flexible core is used a higher order theory is necessary.
Typically sandwich composites are subjected to out of plane loadings where the primary loads are applied perpendicular to the panel surface. Localized loads are one of the major causes of failure in sandwich structures, because in most of the cases the face sheets are thin. The load, transmitted through the face sheet causes the core to be subjected to significant deformations locally, which can cause high shear and normal stresses that can exceed the allowable stress for the flexible, weak core material.
In order to give a framework to the present research, this chapter presents a literature review on three fields: composite plates theories, contact on composite plates models and functionally graded materials.
Sandwich Plate Theories
A considerable amount of literature exists on sandwich panels as they are used in large number of applications varying form high-performance composites in aerospace to low-cost materials for building constructions. The literature includes monographs,
8
conference proceedings, books and bibliographies. Noor et al. (1996) provide an
excellent literature review on computational models for sandwich structures that includes
over 800 references. The paper reviews the literature, classify the models and identify
future direction for research. Similar, Vinson (2001) offers a general introduction to the
structural mechanics involved in the field of sandwich structures and a sufficient number
of references for further investigation. A large number of books on composite materials
were published: they range from physical and mathematical analyses or mechanics of
structural components composed of composite materials (Vinson and Chou, 1975; Kollar and Springer, 2003; Vinson and Sierakowski, 1986), finite element modelling (Matthews et al., 2001) to reference for practical engineering and designing (Reddy and Miravete,
1995).
The limitations of classical plate theory in describing complex problems (e.g.
contact/impact problems, cutouts or laminate plates) necessitated the development of
higher-order theories. The higher-order refers at the level of truncation of terms in a
power series expansion about the thickness coordinate for displacements.
The models investigated here can be classified into: single-layer theories (where the
assumed displacement components represent the weight-average through the thickness of
sandwich panel), layer-wise theories (where separate assumptions for displacements in
each layer are made) and exact theories (equilibrium equations are solved without
displacements assumptions). Although discrete-layer theories are more representative for
sandwich construction than the single-layer theories, they experience computations
difficulties from a large number of field variables in proportion with number of layers.
9
First, a literature review on equivalent single-layer theory is presented in order to give details of the field development.
Reissner (1945) and Midlin (1951) were the first to propose a theory that included the effect of shear deformation. The displacements were assumed as:
ux(,y,z)= u0 (,xy)+ zψ x (x,y)
vx(,y,z)=+v0 (,xy) zψ y (x,y) (1.1)
wx(,y,z)= w0 (,xy) where z is the coordinate normal to the middle plane, u and v are in-plane displacements
and w is out-of-plane displacement and where ψ xy,,ψ w0 are weighted averages. Midlin introduced the correction factor into the shear stress to account for the fact that (1.1) predicts a uniform shear stress through the thickness of the plate. Considering the terms in (1.1) as first terms in a power series expansion in z, it is seen that the classical theory and the shear theory are of the same order of approximation. Yang et al. (1966) extended
Mindlin’s theory for homogeneous plate to laminates consisting of an arbitrary number of bonded layers. Yang et al. (1966) used the theory to solve the frequency equations for propagation of harmonic waves in a two-layer isotropic plate of infinite extent. Based on the same model (Mindlin’s theory), Whitney and Pagano (1970) developed a theory for anisotropic laminates plates consisting of an arbitrary number of bonded anisotropic layers that includes shear deformation and rotary inertia. Displacement field is assumed to be linear in thickness coordinate.
The next order theory was proposed by Essenburg (1975) and was based on the following assumptions:
10
ux(,y,z)= u0 (,xy)+ zψ x (x,y)
vx(,y,z)=+v0 (,xy) zψ y (x,y) (1.2) 2 wx(,y,z)=+w0 (,xy) zψζzz(x,y)+z (,xy)
Essenburg (1975) presented the advantages of the above expansion for the contact problem.
In order to analyze the bending of a plate with a hole, Reissner (1975) included third order terms in the in-plane displacements z-expansion:
3 ux(,y,z)=+zψφxx(,xy) z (x,y) 3 vx(,y,z)=+zψy(,xy) zφy(x,y) (1.3) 2 wx(,y,z)=+w0 (,xy) zζ z (x,y)
It should be noted that above assumptions neglect the contribution of in-plane modes of deformation.
Lo et al. (1977) theory included both in-plane and out-of-plane modes of deformation was based on:
23 u(,xy,z)=+u0 (,xyz) ψζxx(x,y)+z(,xyz)+φx(x,y) 23 v(,xy,z)=+v0 (,xy) zψζyy(xy, )+z (,xy)+zφy(xy, ) (1.4) 2 wx(,y,z)=+w0 (,xy) zψζzz(x,y)+z (,xy)
The principle of potential energy is used to derive the governing equations and boundary conditions and the theory is compared with lower-order plate theories and with exact elasticity solution in order to assess the importance of the displacements expansions. The theory was developed to model the behavior of infinite laminated plates.
Chomkwak and Avula (1990) modified previous theory in order to be applied to finite rectangular laminate composite plates. The Lagrange multipliers are used in order to constrain the displacement functions to satisfy the stress boundary conditions.
11
Reddy (1984) developed a third-order Shear Deformation theory (TSDT) for composite laminates, based on the following assumed displacement fields:
23 ux(,y,z)=+u0 (,xy) zψζxx(x,y)+z (,xy)+zφx(x,y) 23 v(,xy,z)=+v0 (,xy) zψζyy(xy, )+z (,xy)+zφy(xy, ) (1.5)
wx(,y,z)= w0 (,xy)
Functions ζ xx,φζ, y,φy are eliminated based on the condition that the transverse shear stresses vanish on the plate top and bottom surfaces. As in the previous model, the principle of virtual displacements is used to derive the equilibrium equations. Reddy
(1990) reviewed a number of other third-order theories and showed that they are special cases of his theory. Reddy (2000) and Reddy and Cheng (2001) expanded TSDT for analysis of functionally graded plates.
Tessler (1993) developed a two-dimensional laminate plate theory for the linear elastostatic analysis of thin and thick laminated composite plates, which utilize independent assumptions for the displacements, transverse shear strain and transverse normal stress. The displacements expansions are similar with those assumed by
Essenburg (1975). The transverse shear strains and transverse normal stress are assumed to be quadratic and cubic respectively through the thickness; they are expressed in terms of the kinematic variables of the theory by means of a least-squares compatibility requirements for the transverse strains and explicit enforcements of exact traction boundary conditions on top and bottom plate surfaces. The theory was extended to orthotropic shells by Tessler et al. (1995).
Barut et al. (2002) developed a higher-order theory based on cubic expansion for in-plane displacements and quadratic expansion for out-of-plane displacement and used
Reissner’s definition for kinematics of thick plates, to approximate the plate
12 displacements with weighted-average quantities that are functions of the in-plane coordinates. Principle of virtual displacement is employed to write the equilibrium equations and boundary conditions and Fourier series expansion is used to determine the solutions. The accuracy of their model is demonstrated by comparison with exact solution and previous single-layer theories.
Discrete-modeling (or layer wise) approach is based on separate assumptions for displacements in each layer: typically classical plates bending for the face sheets and shear deformation resistance for the core.
Pagano (1970) provides an exact solution to the problem of a rectangular orthotropic laminated subjected to a laterally distributed load. Governing field equations for each layer are written in terms of displacement components and an assumed exponential-trigonometric solution is substituted into the equations. This provides a set of equations with six unknowns that are tractable. The solution unknowns are determined by the boundary conditions and the interface-continuity conditions.
Ogorkiewicz and Sayigh (1973) calculated and compared the central deflection of the sandwich beam with a foam core using three basic mathematical models: one is obtained by transforming beam actual section into an equivalent section of one material;
(the core is transformed into a thin web of the same material as the face sheet); second model is based on the strain energy approach; third is based on stress functions. They compared the analytical result with experiments and concluded that the third model is the closest with the experiment observations.
Burton and Noor (1994) developed three-dimensional analytical thermoelasticity solutions for static problems of simply supported sandwich panels and cylindrical shells
13 subjected to mechanical and thermal loads. Each of the layers is modeled as a three- dimensional continuum and double Fourier series expansions are used for displacements and stresses.
Frostig et al. (1990, 1992) developed a high-order theory for behavior of a sandwich beam with transversely flexible core based on variational principles. The main feature of the method is the higher order displacement fields in the thickness coordinate.
The longitudinal and the transverse deformations of the core determined with the aid of constitutive equations of an isotropic material and the compatibility conditions at the interfaces consist of non-linear expressions in the thickness coordinate. Their formulation is based on the beam theory for face sheets and a two-dimensional elasticity theory for the core. The sandwich behavior is presented in terms of internal resultants and displacements in face sheets, peeling (i.e. normal stresses between the faces and the core that do not impose severe restrains on the performance of the sandwich; e.g. a load applied above the support) and shear stresses at face sheet-core interfaces, and stresses and displacements in the core, even in the vicinity of concentrated loads. Their model can be applied to sandwich structures with honeycomb or foam cores. The model was applied to the vibration analysis (Frostig and Baruch, 1994) and to sandwich structures with nonparallel skins (Peled and Frostig, 1994). Sokolinsky and Nutt (2004) improved
Frostig model by accounting for nonlinear acceleration fields in the transverse flexible core essential for the analysis because the inertia loads exerted on the sandwich beams are nonuniformly distributed along the span. A discretized formulation based on an implicit finite difference scheme is presented and validated.
14
Hohe and Librescu (2003) presented a higher-order geometrically nonlinear sandwich shell theory for sandwich panels that accounts for the transverse compressibility of the core and initial geometric imperfections. Tangential displacement is approximate by a second order polynomial and transverse displacement is approximated by a linear polynomial in thickness coordinate.
Among the large number of sandwich structures models present in the literature, the models reviewed here can be summarized as: assumed displacements methods
(single-layer or layer-wise) and exact models (when the displacements are not assumed but derived). Few of the above models are modified for a sandwich structure with functionally graded core and the results are presented in Chapter 2.
Contact on Composite Materials
Foreign objects impact (such as tool drops, hail, bird strikes, and runway debris) can introduce damage and significantly reduce the strength of the sandwich structure.
Unlike their solid metallic equivalent, predictions of the effects of impact damage on sandwich structures are difficult because significant internal damage can be achieved at impact energy levels lower than those required to create visible damage at the surface.
The first step in understanding the impact dynamics is to study, qualitatively and quantitatively, the contact problem, that describes the contact force - contact area relation. In this section two important research paths are reviewed: hertzian and non- hertzian contact problem.
From the contact area point of view, contact models can be classified into:
• Conforming contacts (e.g. flat punch) when the contact area is a constant and is independent of the load.
• Nonconforming contacts (e.g. sphere) when the contact area is not constant and is depending on the load.
15
The first contact analysis is due to Heinrich Hertz (1881) who investigated the quasi-static impact of spheres as a theoretical background for the localized deformation of the surface of two glass lenses due to the contact pressure between them. Hertz formulated the following boundary conditions for the contact surface (Johnson, 1985):
• The displacements and stresses must satisfy the differential equations of equilibrium, and the stresses must vanish at a great distance from the contact. Therefore, the radii of curvature of the contacting bodies need to be large compared with the radius of the contact surface (each surface is treated as an elastic half- space).
• The bodies are in frictionless contact (only normal pressure is transmitted between the indenter and the specimen).
• At the bodies’ surfaces, the normal pressure is zero outside and equal and opposite inside the surface of contact.
• The distance between the surfaces of the two bodies is zero inside and greater than zero outside the contact surface.
• The integral of the pressure distribution within the contact surface with respect to the area of the contact surface gives the force acting between the two bodies.
Hertz (1881) represented the contact surface by a quadratic function, and assumed a semi-elliptical distribution of pressure:
2 1/2 ⎛⎛⎞r ⎞ pr()=−p 1 (1.6) 0 ⎜⎜⎟⎟ ⎝⎠⎝⎠a where a is the radius of the contact area, and r is the coordinate in radial direction. For the contact between a rigid sphere and a flat surface, Hertz derived a relation among the radius of the circle of the contact a, the indentor total load F, the indenter radius R and the elastic properties:
4 kFR a3 = (1.7) 3 E where
16
9 ⎡ E ⎤ k =−()1ν2+(1−ν′2) (1.8) 16 ⎣⎢ E′ ⎦⎥ and where E, ν and E´, ν´ are the Young’s modulus and Poisson’s ratio for the specimen and indentor, respectively.
Also Hertz found that the maximum tensile stress:
F σ=−(1 2ν) (1.9) max 2π a2 is the radial stress that occurs at very edge of the contact and is the stress responsible for producing the Hertz cone cracks formed in a brittle material.
Willis (1966) derived formulas for the contact between a rigid sphere and a transversely isotropic half-space. During the loading phase, the modified Hertz law (also known as Meyer’s law) that relates the total load F and the indentation α as:
Fk= α n (1.10) where n = 3/2 for elastic contact and n = 1 for a fully plastic contact and where:
4 kE= R1/2 3 111 =+ (1.11) RR12R 111−ν 2 =+1 EE12E and where R1 and R2 are the radius of curvature of the indentor and target respectively and E1, ν1 are the elastic modulus and Poisson’s ratio of the indentor and E2 is the
Young’s modulus of the target. For the case where the indentor is much stiffer than the target, the above formula for E can be simplified as:
11 ≈ (1.12) EE2
17
Even for a small amount of load there could be a significant permanent indentation so the unloading path is different from loading path. For the unloading phase, Crook
(1952) proposed the following contact law:
q ⎛⎞αα− 0 FF= m ⎜⎟ (1.13) ⎝⎠ααm − 0 where q = 2.5, Fm is the contact load at which unloading begins, αm is indentation corresponding to Fm and α0 is the permanent indentation. The permanent indentation, α0 is zero when the maximum indentation αm remains below a critical value αcr during the loading phase, so the permanent indentations is given by:
⎡⎤2/5 ⎛⎞αcr α0 =−αm⎢⎥1⎜⎟ when αm>αcr (1.14) α ⎣⎦⎢⎥⎝⎠m
Problems concerning the contact between bodies have been solved since the work of Hertz in 1880’s. A large number of books have been devoted to the mathematical tools used to solve contact problems. Gladwell (1980) summarized mathematical methods
(based on complex variable, integral transformation and elliptic functions) used to find the solution for the contact between elastic bodies. An extensive review of the Russian literature for a large number of contact problems is detailed. Johnson (1985) derived expressions for stresses and displacements based on Hertz theory and generalized the theory to include different indentor profiles, interfacial friction, anisotropic and inhomogeneous materials or layered plates. Khludnev and Sokolowski (1997) monograph covers qualitative properties of solutions, optimization problem (e.g., finding the constructions which are of maximal strength and satisfy weight limitations), determination of crack shapes by optimization methods, existence theorems and
18 sensitivity analysis of solutions. Fischer-Cripps (2000) presented a review of displacements and stresses for different types of indenters (uniform, spherical, flat punch, roller, rigid cone) in order to develop a practical tool for investigating the mechanical properties of engineering materials. The mathematical modeling, variational analysis, and numerical solutions of contact problems involving viscoelastic and viscoplastic materials as well frictional boundary conditions, which lead to time dependent models was presented by Han sand Sofonea (2002). Alexandrov and Pozharskii (2001) developed analytical and numerical methods for nonclasical three-dimensional linear elasticity contact problems. Their methods focused on integral equations include fracture mechanics.
When a composite laminate with finite dimensions is in contact with a rigid indentor, the behavior may drastically deviate from that predicted for a half-space.
Various methods were developed to describe the contact problem for isotropic, orthotropic, monolithic or laminated beams and plates under different indentors profiles.
Most problems are based on a combination between a solution describing the local contact phenomenon and a beam/plate solution for the global response. Different models for predicting the elastic response of a beam/plate under a general load were presented in the previous section. Here a literature review on contact and low-velocity impact problems is presented.
Abrate (1998) studied the contact between a sandwich beam and a cylindrical indentor. He assumed that a rigid plate supported the bottom face sheet. Treating the top face-sheet as a beam of rigidity EI on an elastic foundation (as long as compressive stress
19 in the core does not exceed a maximum value), the following linear contact law is derived:
F= kα (1.15) where the contact stiffness is given by:
3/4 3/2 ⎛⎞Ebc 1/4 k= 2⎜⎟(EI) (1.16) ⎝⎠hc where Ec is the core modulus in the transverse direction, hc is the core thickness and b is the beam width.
Lee and Tsotsis (2000) studied the indentation failure behavior of honeycomb core sandwich panels by examining the effects of skin, core and indenter size on load transfer from the top skin to the core. They treat the problem as a plate (top face sheet) on an elastic foundation (core). A good agreement was found between the theoretical and FE model. Also comparison of maximum shear and normal stress with experimental data showed that the core shear failure dominates the onset of indentation failure.
Yang and Sun (1982) and Tan and Sun (1985) conducted static indentation tests and finite elements models on laminated plates to validate the contact laws (1.10) - (1.13)
They found a good agreement with the above laws.
The study of the effects of foreign object impact on sandwich structures has been a subject for a large number of researches in recent years. Understanding the mechanic of contact between the impactor and the panel is very important in the study of the impact dynamics in order to accurately predict the contact-force history and to predict how the damage will develop. Many solutions deals with statically, low-velocity impact when the duration of the impact (the interval of time from the first contact to the complete
20 detachment) is much larger than the time required by the elastic wave, generated at the first point of contact, to propagate through the whole body.
The use of static load-deflection behavior of the sandwich beam in the impact analysis needs some justification. In general the wave propagation effects, especially through the thickness of the core, should be considered in impact response of sandwich panels. This will be crucial when spalling type damage occurs in the panels. However, a study by Sankar (1992) showed that for very large impactor mass compared to that of the target plate and for very low impact velocities compared to the wave velocity in the target medium, quasi-static assumptions yield sufficiently accurate results for impact force history and ensuing stresses in the impacted plate.
Ambur and Cruz (1995) model the sandwich plate using the first order shear deformation theory (FSDT) and combining the equation of motion of the plate with the equation of motion of the projectile (given by Newton’s law) a set of differential equations were obtained. The solution is obtained by numerical integration. A highly nonlinear contact law is derived.
Lee et al. (1993) developed a theory based on the assumption that each face sheet deforms as a first order shear deformation plate and the core deformation is obtained by the difference between the transverse displacements of the skins.
Sun and Wu (1991) theory is based on the same assumption for the face sheets
(first order shear deformation plate) but the core is model by using springs to simulate the transverse and shear rigidities.
Meyer-Piening (1989) modeled the sandwich plate as a three-layer plate with independent linear in plane displacement in each layer.
21
Koissin et al. (2004) determined the closed-form solution of the elastic response of a foamed-core sandwich structure under a local loading, using static Lame equations for the core and thin plate Kickoff-Love dynamic theory for the faces. The integral transformation technique was used to solve the differential equations. A good agreement was found between their analytical solution, experimental results and FE analysis.
Choi and Hong (1994) derived a method for prediction the impact force history on composite subjected to low-velocity impact, applicable to isotropic and orthotropic plates with unknown contact laws. The first order shear deformation theory and von Karman’s large deflection theory were used to describe the dynamic response of the plate and Yang and Sun (1982) modified Hertz law to describe contact problem. The impact duration is computed from the eigenvalue analysis of the lumped mass system, and the impulse- momentum conservation law is used with the concept of spring-mass model to predict the impact force history. A good agreement was found between their analytical solution and experimental results.
A model for small mass impact on monolithic, transversely isotropic Midlin plate with a Hertzian contact law was given by Mittal (1987). An approximate analytical model for small mass impact on specially-orthotropic Kirchhoff plates based on Hertzian contact law was given by Olsson (1992). It is shown that transversal Young’s modulus has a small influence on the contact force, whereas the impact energy, impactor radius and plate thickness have a significant influence. The results are compared with numerical and experimental analyses. Olsson and Mc Manus (1996) included core crushing and large face-sheet deflections in a theory for contact indentation of sandwich panels. Olsson
22
(1997) used Olsson and Mc Manus (1996) to modify Mittal (1987) for specially- orthotropic sandwich panels with linear load-indentation relation.
Fischer-Cripps (1999) used the finite-element method to compute the radius of curvature of the contact surface for both elastic and elastic-plastic contacts. It is shown that indentors involving elastic-plastic deformations are equivalent with a perfectly rigid spherical whose radius is somewhat smaller than calculated using the Hertz equation for elastic contact. An experiment is used to validate his results.
The range of applicability of Hertz’s theory was found to be limited to very small indentors which also mean low loads. Many non-Hertzian contact problems do not permit analytical solutions in a closed form. This fact leads to various numerical models in order to determine the distribution of normal and tangential tractions, which satisfy the normal and tangential boundary conditions at the interface, inside and outside the contact area.
The continuous distribution of traction is replaced by a discrete set of tractions elements: and the boundary conditions are satisfied at a discrete number of points called the matching points. Few examples are:
• Array of concentrated forces (gives an infinite surface displacements);
• Stepwise distribution given by uniform tractions acting on discrete segments/areas of the surface (the displacements are finite but the displacements gradients are infinite between the adjacent elements), e.g., Singh and Paul (1974) and Wu and Yen (1994);
• Superposition of a finite number of symmetrical rectangular loadings of unknown magnitudes distributed over known lengths (e.g., Sankar and Sun, 1985);
• Piecewise-linear distribution of tractions given by superposition of overlapping triangular elements (for a three dimensional problem, regular pyramid on a hexagonal base), e.g., Johnson, 1985.
In order to find the values of the traction elements, which best satisfy the boundary conditions two different methods were developed:
23
• Direct/matrix inversion in which the boundary conditions are satisfied exactly at the matching points
• Variational method in which the tractions are chosen such that to minimize an appropriate energy function.
Abrate (1997, 1994, 1991) classified the impact models into: spring-mass models
(that accounts for the dynamics of the structure in a simplified manner); energy-balance models (when the structure behaves quasi-statically); assumed force distribution and complete models (when the dynamics of the structure is fully modeled).
Shivakumar et al. (1985) developed spring-mass and energy-balance models to calculated impact force and duration during low-velocity impact of transversely isotropic circular plates. Energy-balance model is based on equilibrium between the kinetic energy of the impactor and the sum of strain energy due to contact, bending, and transverse shear and membrane deformation at maximum deflection. The solution obtained using numerical methods gives only the maximum force so a spring-mass model is developed to describe the force history. The impactor and sandwich were represented by two rigid masses and their deformations were represented by springs (the general case includes four springs: two to represent linear stiffness of the structure (both bending and shear), one for the nonlinear membrane stiffness and the forth for the nonlinear contact stiffness).
The analyses were verified by experimental results.
Contact problems for finite thickness layers can be solved using numerical methods. Sankar and Sun (1983) used two types of numerical methods, point matching technique and assumed stress distribution method. The point matching method is essentially a numerical technique to solve the integral equations of the contact problem.
However, this method fails when the contact area is too small because of numerical difficulties. In the assumed contact stress distribution method, the contact stresses are
24 assumed to be of Hertzian form, i.e., similar to that of contact of a half-plane. The contact stresses take the shape of a semi-ellipse. A contact length is assumed and the contact stresses are expressed in terms of only one unknown, the peak contact stress. Requiring that the deflections beneath the contact region match the indentor profile one can solve for the peak contact stress.
Another popular method for beams was to use a Green’s function and integral equation to correlate the applied force and the beam response. Sankar (1989) superimposed and orthotropic half-space solution on a beam solution to construct an approximate Green’s function. To avoid the limitation of previous method, Wu and Yen
(1994) developed an analytical method, based on exact Green’s function derived from three-dimensional linear anisotropic elasticity theory. Effects of changing the material properties, stacking sequence, span and thickness of the plate and indentor size on the force-indentation relation are studied.
Anderson (2003) presented an analytical three-dimensional elasticity solution for a sandwich plate with functionally graded core subjected to transverse loading by a rigid spherical indenter. Governing equations are derived based on Reissner’s functional and are solved by enforcing continuity of tractions and displacement between the adjacent layers. The sphere unknown contact area and pressure distribution are obtained based on iterative solution method (developed by Singh and Paul, 1974 and modified by Chen and
Frederick, 1992): a large initial contact region is divided into a number of rectangular patches with assumed constant pressure. The unknown pressure is discretized based on conforming the contact region to the indentor surface. The contact region is also determined on an iterative process: tensile patches are removed from the contact region
25 and the procedure is repeated until only compressive pressure patches remain. Results of this analysis demonstrate that interfacial normal and transverse shear stresses are not necessary reduced by the incorporation of functionally graded core.
Abot et al. (2002) presented a combined experimental and analytical study of the contact behavior of sandwich structures with foam cores. The experimental results were used to model the load-deflection curves that include a linear range (derived from
Winkler foundation theory) followed by a nonlinear portion (derived from fitting experiments).
Sburlati (2002) determined the indentation produced by rigid indentors (flat punch and spherical) on sandwich plate with high-density closed cellular foam based on a three- layer classical sandwich theory and a distribution of surface pressure reproducing the contact law. The relevant influence of boundary condition on the elastic response of the sandwich plate is shown (a 20% difference in contact laws for simply supported plate and clamped plate was found). Another conclusion of the paper is that the assumption of distribution of surface pressure reproducing the contact of a rigid indentor over a half- space is correct also for a sandwich beam.
McCormack et al. (2001) estimated the initial failure load, corresponding to the first deviation from linearity in the load-deflection curves as well as peak load for several failure modes (face yielding, face wrinkling, core yielding and core indentation) of sandwich beam with metallic foam cores. They used finite-element method to compute the critical load in the core indentation case. The results agree with three-point bending experimental results.
26
Liao (2001) investigated the smooth contact between a rigid cylindrical indentor and a composite laminate with arbitrary ply-orientation (homogenized as an orthotropic medium). An exact Green’s function for the surface displacement of an anisotropic beam under cylindrical bending is derived. By matching the displacements of the top surface of the plate and those of the rigid indentor within the contact region, the contact stress distribution and the indentation is obtained.
Swanson (2004) calculated the core compressive stress in a sandwich beam with orthotropic face sheets based on elasticity equations for transverse loading of layered orthotropic materials. By systematically varying the contact pressure and comparing the computed surface displacements with the indentor profile the contact pressure distribution was determined. The results showed that the pressure distribution for an orthotropic half-space is applicable to a sandwich beam over a large range of variables.
Petras and Sutcliffe (1999) defined a spreading length to consider the way in which different wavelengths of sinusoidal pressure loading on top face sheet are transmitted to the core. This length characterizes the length over which a load is spread out by the face sheet. For a very flexible skin there is a large deformation under the load which can lead to core failure. For very rigid skins indentation failure will be relative hard, as the skins spread out the load. Their solution is obtained based on a combination between the higher order theory developed by Frostig et al. (1992) and contact overlapping triangular elements developed by Johnson (1985).
Anderson and Madenci (2000) conducted impact tests in order to characterize the type and extent of the damage observed in a variety of sandwich configurations with graphite/epoxy face sheets and foam or honeycomb cores. They found that the surfaces of
27 both the honeycomb and foam samples revealed very little damage at levels of impact energy that produced significant core damage. As the impact energy was increased, the samples experienced one of two types of damage: a tear or crack from the center of the laminate to the edge, or significant damage consisting of a dent localized in the region of impact.
Wang and Yew (1989) studied the damage produced by low velocity impact in graphite fiber composite plates based on the principle of virtual work. They took into account the time history of impact force, the dynamic deformation of the system and the distribution of damage in the target plate. The material was considered damaged when its designed strength was reduced by the failure of the constituents (matrix cracking, fiber breakages or/and delamination). It was demonstrated that fibers in the consecutive layers must be at an angle with respect to each other in order to achieve an effective containment of the impact induced damage.
In order to give a research perspective to the work on impact on sandwich plates with functionally graded core a literature review was presented. Among the large number of analytical models presented in the literature, the models reviewed here can be summarized as: plate on elastic foundation (Abrate, 1998; Lee and Tsotsis, 2000), laminate plates (Yang and Sun, 1982; Tan and Sun, 1985) and sandwich plates (e.g.
Ambur and Cruz, 1995; Sun and Wu, 1991), non-hertzian contact/numerical methods
(e.g. Sankar and Sun, 1983) and experiments on impact damage (e.g., Anderson and
Madenci, 2000). Two numerical methods and a finite element solution for impact on sandwich plates with functionally graded core are presented in Chapter 3.
28
Cellular Solids and Functionally Graded Materials
Cellular solids appear widely in nature in different shapes and with different functions. They can be found in leaves and stalks of plants, in corals and woods or in any types of bones. They can be made of closed cells, regular or non-regular, or can be made of open network of struts. They can be isotropic or can have cells oriented in a particular direction. Recently, engineered cellular materials are made using polymers, metals, and ceramics and it is believed that any solids can be foamed. They are widely used as thermal, and acoustic insulations, as absorbers of kinetic energy from impacts.
The cellular/foamed materials are qualitatively and quantitatively investigated in a large number of publications. Among those, two books worth to be mentioned: Ashby et al (2000) design guide for metal foams (contains processing techniques, characterization methods, design and applications) and Gibson and Ashby (1997) monograph of cellular solids.
Functionally graded materials (FGMs) are materials or structures in which the material properties vary with location in such a way as to optimize some function of the overall FGM.
Nature provides many examples of functionally graded materials. In many of the cases the nature functionally graded structures were evolved based on some mechanical function: bones give a light, stiff frame to the body, wood support the tree under environmental loadings, leaves transport fluids. Bamboo is one of the examples of a structurally smart plant (Amada et. al. 1997, 2001). Bamboo structure, which resembles that of a unidirectional, fiber-reinforced composite, is described by a macroscopically graded geometry that is adapted to environmental wind loads while the fiber distribution exhibits a microscopically graded architecture, which leads to smart properties of
29 bamboo. Amada et al. (1997, 2001) demonstrated experimentally and analytically how the functionally graded microstructure of a bamboo has been optimized through evolution to maximize its load-bearing capabilities under severe environmental loading conditions.
An interesting application of FGMs is dental implants (Watari et al, 1997). It is confirmed that the fabrication technique can be used to produce controlled functionally graded microstructures, and that the biocompatibility of the titanium/ hydroxyapatite implants is superior to that of pure titanium implants.
The variation of the properties of FGMs can be achieved by gradually changing the volume fraction of the constituents’ materials, by using reinforcements with different properties, sizes and shapes, by interchanging the roles of reinforcement and matrix phases in a continuous manner as well as by adjusting the cell structure or the material density. The result is a microstructure that produces continuously or discretely changing thermal and mechanical properties at the macroscopic or continuum scale.
The concept of functionally graded materials was proposed in the eighties by materials scientists in Japan as a way to create thermal barrier materials. Koizumi (1997) summarize the first projects in this field. The idea proposed was to combine in a gradual manner heat-resistance ceramics and tough metals with high thermal conductivity. The result is a panel with a high heat-resistance on the high-temperature side and high mechanical strength on the other side. The new materials were obtained by four methods:
Chemical Vapor Deposition (CVD), powder metallurgy, plasma sprays, and self- propagating combustion synthesis (SHS).
As the use of FGMs increases, for example, in aerospace, automotive, telecommunications and biomedical applications, new methodologies have to be
30 developed to characterize FGMs, and also to design and analyze structural components made of these materials. Although fabrication technology of FGMs is in its infancy, they offer many advantages. Few of the manufacturing methods are presented here. Fukui et al. (1991, 1997) developed a high-speed centrifugal casting method, in which the layers are formed in the radial direction due to different mass densities. An melted Al-Ni alloy cast into a thick-walled tube was rotated at a speed such that the molten metal experienced an acceleration, thereby producing two kinds of composition gradient, (i.e. phase gradient). The volume fraction of the A13Ni phase was determined by quantitative optical microscopy.
Poly (ethylene co-carbon monoxide) (ECO) was chosen to make the FGMs by ultraviolet radiations (Lambros et. al., 1999), because of its rapid degradation under UV light. Irradiated ECO becomes stiffer, stronger and more brittle with increasing time. The exposure time and, thus, the properties or the FGM can be adjusted truly continuous based on the relationship between modulus of elasticity and irradiation time.
Another manufacturing method is electrophoretic deposition (Sarkar, 1997) EPD is a combination of two processes, electrophoresis and deposition. Electrophoresis is the motion of charged particles in a suspension under the influence of an electric field gradient. The second process is deposition, i.e. the coagulation of particles into a dense mass.
El-Hadek and Tippur (2003) developed functionally graded syntactic foam sheets by dispersing micro balloons (with linear graded volume fraction) in epoxy. They determined Young’s modulus and the density by using the wave speed and density for syntactic foams having homogenous dispersion of the micro balloons. The resulting foam
31 sheets have a nearly constant Poison’s ratio. Other FGM manufacturing methods presented in literature are powder metallurgy, plasma sprays and self-propagating combustion synthesis.
Suresh and Mortensen (1998) provide an excellent introduction to FGMs. Suresh
(2001) presents a motivation for the use of graded materials and also enumerates few challenges in the field. Literature contains many analytical models: mechanical and thermal loads, fracture and optimization solutions for different structures made of FGM were published. Here, few of the significant papers are reviewed.
When using analytical methods to solve problems involving FGMs, a functional form for the variation of thermo-elastic constants has to be assumed. For example,
Aboudi et al. (1995a, 1995b, 1999) assumed a simple polynomial approximation for the elastic constants. Another useful approximation is the exponential variation, where the
0 λz elastic constants vary according to formulas of the type ccij = ij e. Many researchers have found this functional form of property variation to be convenient in solving elasticity problems. For example, Delale and Erdogan (1983) derived the crack-tip stress fields for an inhomogeneous cracked body with constant Poisson ratio and a shear
()αβx+ y modulus variation given by µµ= 0e .
Giannakopoulos and Suresh (1997a, 1997b) present two analytical models for indentation of a half-space with gradient in elastic properties, for a point force and an axisymmetric indentor. The analytical solutions, based on elasticity methods for axisymmetric equilibrium problems are compared with finite element simulations. It is found that the influence of the Poisson’s ratio is strong whenever the elastic modulus is increasing with depth, and is weak whenever the elastic modulus is decreasing with
32 depth. Also a decreasing elastic modulus with depth results in a spreading of stresses towards the surface rather than to the interior, whereas an increasing elastic modulus results in diffusing the stresses towards the interior of the half-space.
Markworth et al. (1995) review analytical models for microstructure-dependent thermo physical properties, and models for design, processing of FGMs. They also present few recommendations relative to areas in which additional work is needed.
Functionally graded materials as composite materials are generally described from two points of view: micromechanics and macromechanics. Micromechanics analysis recognizes the basic constituents of FGM but does not consider the internal structure of the constituents. Micro-constituents are treated as homogeneous continua. Although the material at the macro-scale is heterogeneous, it is assumed that the micro-heterogeneities are distributed in such a way such that the material volumes beyond some representative minimum have comparable macroscopic or overall properties. The hypothesis concerning the existence of an representative volume element (RVE) is fundamental: it means that there exist a relatively small sample of the heterogeneous material that is structurally typical of the whole mixture on average and contains sufficient numbers of inclusions, defects, pores such that the overall behavior (the behavior of an equivalent homogeneous material) would not depend on the surface values of constraints and loading as long as those are homogeneous. Properties are prescribed pointwise in the spirit of the mechanics of continua. The transition from the micro to the macroscale consists in finding the relationship between the macroscopic fields (stress, strain) and the volume averages of the same micro fields over RVEs. In mixture-type Continuum-Macromechanics analyses, the material properties are expressed in terms of volume fraction and individual
33 properties of constituent phases. Macromechanics models can be subdivided in: direct
(closed-form analytical solution for the averaged properties), variational (lower and upper bound for the overall properties in terms of the phase volume fraction) and approximation methods.
Aboudi et al. (1994a, 1994b 1999) developed a higher order (HOTFGM), temperature-dependent micro-mechanical theory for thermo-elastic response of functionally graded composites. A higher-order representation of the temperature and displacement fields is necessary in order to capture the local effects created by the thermomechanical field gradients, the microstructure of the composite and the finite dimensions in the functionally graded directions. Their model based on the concept of a representative volume element explicitly couples local (microstructural) and global
(macrostructural) effects. RVE is not unique in the presence of continuously changing properties due to non-uniform inclusion spacing. Pindera and Dunn (1995) evaluated the higher order theory by performing a detailed finite element analysis of the FGM. They found that the HOTFGM results agreed well with the FE results.
Abid Mian and Spencer (1998), present an exact solution for 3D elasticity equations for isotropic linearly elastic, inhomogeneous materials. Lame elastic moduli λ and µ, the expansion coefficient α and the temperature T depend in an arbitrary specified manner on the coordinate z. The 3D elasticity solutions are generalized from solutions for stretching and bending of symmetrically inhomogeneous plates. In order to completely describe these solutions an equivalent plate is introduced. This equivalent plate is a homogeneous plate (in the sense of a classical thin plate) with the same overall geometry as the inhomogeneous plate and elastic moduli that are suitable weighted averages of the
34 moduli of the inhomogeneous plate. It is shown that the exact three-dimensional solutions are generated by two-dimensional solutions of the thin-plate equations for the equivalent plate.
Reddy et al. (2000, 2001) use an asymptotic method to determine three- dimensional thermomechanical deformations of FG rectangular plates. The locally material properties are estimated by the Mori-Tanaka homogenization scheme for which the volume fraction of the ceramic phase is of the power law type. Reddy et al. (2003) developed an FGM beam finite element by deriving the approximation functions from the exact general solution to the static part of the governing equations. These solutions are then used to construct accurate shape functions which result in exact stiffness matrix and a mass matrix that captures mass distribution more accurately compared to any other existing beam finite elements. Thus, the element is an efficient tool for modeling structural systems to study wave propagation phenomena that results due to high frequency and low duration forcing (impact loading).
Rooney and Ferrari (1999) developed solutions for tension, bending and flexure of an isotropic prismatic bar with elastic moduli varying across the cross-section. It is demonstrated that the elastic moduli are convex functions of the volume fraction. If a structural member is required to be just in tension, any grading of the phases will result in a improved in the performance.
Vel and Batra (2002, 2003) present an exact solution for simply supported functionally graded rectangular thick or thin plates. The material has a power-law through-the-thickness variation of the volume fractions of the constituents. Suitable displacement functions that identically satisfy boundary conditions are used to reduce
35 equations governing steady state vibrations of a plate to a set of coupled ordinary differential equations, which are then solved by employing the power series method.
Woo and Meguid (2000) provide an analytical solution for the coupled large deflection of plates and shallow shells made of FGMs under transverse mechanical load and temperature field. The material properties of the shell are assumed to vary continuously through the thickness of the shell, according to a power law of volume fraction of the constituents. The equations obtained using von Karman theory for large transverse deflection, are solved by Fourier series method.
In a series of papers Sankar and his coworkers (Sankar and Tzeng, 2002; Sankar,
2001; Venkataraman and Sankar, 2001; Apetre, et al. 2002) reported analytical methods for the thermo-mechanical and contact analysis of FG beams and also sandwich beams with FG cores. In these studies the thermo-mechanical properties of the FGM were
λz assumed to vary through the thickness in an exponential fashion, e.g., E(z)=E0e . The material was assumed to be isotropic and the Poisson’s ratio was assumed to be constant.
The exponential variation of elastic stiffness coefficients allows exact elasticity solution via Fourier transform methods. Later Apetre et al. (2003) and Zhu and Sankar (2004) used Galerkin method to analyze cores with polynomial variation of mechanical properties.
Although a relatively new field, the study of functionally graded materials contains a large number of research areas: publications on fabrication techniques, experiments methods, analytical and finite element solutions were widely published later.
Developments in manufacturing methods now allow controlled spatial variation in material properties so analytical results can be validated by comparisons with
36 experimental results. In order to give a research perspective to the present work this section summarized few of the developments in this area. The main objective of present work is to develop analytical methods for sandwich structure with FG cores and to validate them by comparison with finite element solutions.
Objectives and Approaches
Learning from nature, engineers created new manufacturing methods to produce functionally graded materials. Few of them are presented in previous section and are a good research motivation for the present work. Analytical methods have to be developed in order to enable obtaining closed-form solutions to some fundamental solid mechanics problems, and also to extend finite element models to structures made of FGMs. Impact surfaces and interfaces between different materials where impact damage occur are regions of interest.
The main objectives of this dissertation can be summarized as:
1. Develop analytical models for sandwich structures with functionally graded core.
2. Solve contact and impact problems involving sandwich structures with FG core and compare the trade-off between using a functionally graded core as opposed to the conventional sandwich design.
3. Compare the trade-off between the total mass and stiffness in functionally graded materials and homogenous materials by solving optimizations problems.
Even with the sandwich structure advantages over the other structures, it is important to develop new types of materials in order to obtain the absolute minimum weight for given conditions (e.g., structural geometry and loadings). These new sandwiches should be compared with other sandwich constructions and with alternative structures in order to select the best configuration. One of the new alternatives is a sandwich structure with functionally graded core. A sandwich beam with soft,
37 transversely flexible core must be approached with a higher-order theory rather than the classical theory. The core gets compressed and the thickness of the beam is changed under concentrated loads. Plane sections do not remain plane after deformation, so displacement field is not a linear function of the thickness coordinate. Conditions imposed on one face sheet do not necessary hold for the other. These are few of the reasons that make higher-order theory necessary for a sandwich panel with FG core even if this theory involves more computational effort.
This research presents analytical solutions based on different models for sandwich structures with FG core when the core Young’s modulus is given by exponential and polynomial variation. A finite element model validates analytical solutions. Low-velocity impact on sandwich structures with FG core based on springs’ model is also solved. A comparison of the total mass of different panels based on different relations between
Young’s modulus and density is present. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.
CHAPTER 2 ANALYTICAL MODELING OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE
Although FGMs are highly heterogeneous, it will be useful to idealize them as continua with properties that change smoothly with respect to spatial coordinates. This will enable closed-form solutions for some fundamental solid mechanics problems, and will aid the development of finite element models for structures made of FGMs. This chapter investigates different analytical models available in literature for a sandwich beam and applies them to a sandwich beam with functionally graded core.
In the first two sections, the governing equations for sandwich structures with FG cores are solved for two types of core Young Modulus by two different methods: exact solutions are presented for the exponential variation of core Young Modulus and a combination of Fourier series and Galerkin method for a polynomial variation of core
Young Modulus. Those methods are compared with two equivalent single-layer theories based on assumed displacements, a higher-order theory and a finite-element analysis. A very good agreement among the Fourier series-Galerkin method, the higher-order theory and the finite-element analysis is found.
Exact Method for Sandwich Structures with Functionally Graded Core
Venkataraman and Sankar (2001) derived an elasticity solution for stresses in a sandwich beam with a functionally graded core. They used Euler-Bernoulli beam theory for analysis of face sheets and plane elasticity equations for the core. In the present work,
38 39 the solution of the sandwich problem was improved by using elasticity equation for face sheets also.
The dimensions of the sandwich beam are shown in Figure 2.1.The length of the beam is L, the core thickness is h and the face sheet thicknesses are hf. The beam is divided into 4 parts or elements: the top face sheet, top half of the core, bottom half of the core and the bottom face sheet.
hf z Element 1: Top facesheet Node (1) Element 2: Top half of sandwich core h x Node (2) Element 3: Bottom half of sandwich core Node 3 L Element 4: Bottom facesheet
Figure 2.1: Sandwich beam with functionally graded core with schematic of the analysis elements.
In general, this model can be applied for sandwich structures with core and face sheets orthotropic materials at every point and the principal material directions coincide with the x and z-axes. Consequently, the constitutive relations for each layer are:
⎧σ xx ⎫ ⎡c11 c13 0 ⎤ ⎧ε xx ⎫ ⎪ ⎪ ⎪ ⎪ σ = ⎢c c 0 ⎥ ε , for ith element, i = 1,2,3,4 ⎨ xx ⎬ ⎢ 13 33 ⎥ ⎨ xx ⎬ (1.17) ⎪τ ⎪ ⎢ 0 0 c ⎥ ⎪γ ⎪ ⎩ xz ⎭i ⎣ 55 ⎦i ⎩ xz ⎭i or {}σ = [c(z)]{ε}
(1.18)
The face sheets are assumed to be homogeneous and isotropic. The core is functionally graded but symmetric about the mid-plane given by z=0. The elastic coefficients (cij) of the core are assumed to vary according to:
40
0 λ z cij = cij e (1.19)
This exponential variation of elastic stiffness coefficients allows exact elasticity solution.
The tractions and displacements at the interface between each element are shown in
Figure 2.2. Each element has its own coordinate systems. The coordinate systems of each element are chosen at the interface because displacements and traction compatibility between elements will have to be enforced at these nodes.
pa z t ,U , p , W Top face sheet x 1 1 1 1 t2 ,U2 p2 , W2 p2 , W2 Top half of z x sandwich t3 , U3 p3 ,W3 z p3 ,W3 Bottom half of x t , U sandwich core 3 3 t ,U 4 4 p4 ,W4 z p4 , W4 Bottom face sheet x t4 , U4 t ,U , p , W 5 5 5 5
Figure 2.2: Traction forces and displacements at the interfaces of each element in the FGM sandwich beam
The governing equations are formulated separately for each element, and compatibility of displacements and continuity of tractions are enforced at each interface
(node) to obtain the displacement and stress field in the sandwich beam. This procedure is analogous to assembling element stiffness matrices to obtain global stiffness matrix in finite element analysis.
The top face sheet is subjected to normal tractions such that:
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σ zz (x,0) = pa sin(ξx) (1.20) where
nπ ξ = , n = 1,3,5,.... (1.21) L and pa is known. Since n is assumed to be odd, the loading is symmetric about the center of the beam. The loading given by equation (1.20) is of practical significance because any arbitrary loading can be expressed as a Fourier series involving terms of the same type.
The displacement field for each layer is assumed of the form:
u (x, z) = U (z)cos(ξx) i =1, 2, 3, 4 i i (1.22) wi (x, z) = Wi (z)sin(ξx) where u is horizontal displacement and w is vertical displacement and where it is assumed that:
(U i ,Wi ) = (ai ,bi )exp(α i z) (1.23) where ai, bi, αi are constants to be determined.
In order to satisfy equilibrium the contributions of the different tractions at each interface should sum to zero. Enforcing the force balance and the compatibility of force and displacements at the interfaces enables us to assemble the stiffness matrices of the four elements to obtain a global stiffness matrix K:
[K]{UWUWUWUWUW}T 1122334455 (1.24) T = {}0 pa 00000000 where K is the global stiffness matrix, a summation of the stiffness matrices for each element. The displacements U1, W1…W5, are obtained by solving (1.24). The obtained displacement field along with the constitutive relations is used to obtain the stress field in each element.
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Fourier – Galerkin Method for Sandwich Structures with Functionally Graded Core
Zhu and Sankar (2004) derived an analytical model for a FG beam with Young’s modulus expressed as a polynomial in thickness coordinate using a combined Fourier
Series-Galerkin method. In the present work, the model is applied to a sandwich beam with FG core. The geometry (Figure 2.1), the load (1.20), and the constitutive relations
(1.17) are the same as in previous model.
In this section, a brief description of the procedures to obtain the stiffness matrix of top half of the core is provided. The derivation of stiffness matrices of other elements follows the same procedures.
The differential equations of equilibrium for the top half of the core are:
∂σ ∂τ xx + xz = 0 ∂x ∂z (1.25) ∂τ ∂σ xz + zz = 0 ∂x ∂z
The variation of Young’s modulus, E in the thickness direction is given by a polynomial in z. e.g.,
4 3 2 ⎛ ⎛ z ⎞ ⎛ z ⎞ ⎛ z ⎞ ⎛ z ⎞ ⎞ E(z) = E ⎜a + a + a + a +1⎟ 0 ⎜ 1⎜ ⎟ 2 ⎜ ⎟ 3 ⎜ ⎟ 4 ⎜ ⎟ ⎟ (1.26) ⎝ ⎝ h ⎠ ⎝ h ⎠ ⎝ h ⎠ ⎝ h ⎠ ⎠ where E0 is the Young’s modulus at z=0 and a1, a2, a3 and a4 are material constants. The thickness in y direction is large and plain strain assumption can be used. The elasticity matrix [C] is related to the material constants by:
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⎛⎞ ⎜⎟10−νν Ez() ⎜⎟ []C =⎜νν1−0⎟ (1.27) (1 +−νν)(1 2 ) ⎜⎟12− ν ⎜⎟00 ⎝⎠2
The following assumptions are made for displacements:
u(x, z) = U (z) cosξx (1.28) w(x, z) = W (z) sin ξx
Substituting equation (1.28) into (1.27), the following constitutive relation is obtained:
⎛σ ⎞ ⎛c c 0 ⎞⎛ − ξU sinξx ⎞ ⎜ xx ⎟ ⎜ 11 13 ⎟⎜ ⎟ ⎜σ zz ⎟ = ⎜c13 c33 0 ⎟⎜ W'sinξx ⎟ (1.29) ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝τ xz ⎠ ⎝ 0 0 G⎠⎝U'+ξW cosξx⎠
A prime (′) after a variable denotes differentiation with respect to z. Boundary conditions of the beam at x=0 and x=L are w(0,z)=w(L,z)=0, and σ xx (0, z) = σ xx (L, z) = 0 , which corresponds to simple support conditions in the context of beam theory. Equations
(1.29) can be written as:
⎛σ xx ⎞ ⎛ S x ⎞ ⎜ ⎟ = ⎜ ⎟ sin ξx ⎝σ zz ⎠ ⎝ S z ⎠ (1.30) τ xz = Tz cosξx where
⎛ Sx ⎞ ⎛c11 c13 ⎞⎛−ξU ⎞ ⎜ ⎟ = ⎜ ⎟⎜ ⎟ ⎝ Sz ⎠ ⎝c13 c33 ⎠⎝ W ' ⎠ (1.31) T = G(U '+ξW ) z
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Substituting for σ xx ,σ zz ,τ xz from equations (1.31) into equilibrium equations
(1.25), a set of ordinary differential equations in U(z) and W(z) are obtained:
ξS x + Tz ' = 0 (1.32) S z '−Tzξ = 0
In order to solve equations (1.32) the Galerkin Method is employed: solutions in the form of polynomials in z are assumed:
U(z) = c1φ1(z) + c2φ2 (z) + c3φ3(z) + c4φ4 (z) + c5φ5 (z) (1.33) W(z) = b1φ1(z) + b2φ2 (z) + b3φ3(z) + b4φ4 (z) + b5φ5(z) where φs are basis functions, and b’s and c’s are coefficients to be determined. For simplicity 1, z, z2, z3, z4 are chosen as basis functions:
2 φ1(z) = 1; φ2 (z) = z; φ3(z) = z ; 3 4 (1.34) φ4 (z) = z ; φ5 (z) = z
Substituting the approximate solution in the governing differential equations, the residuals are obtained. The residuals are minimized by equating their weighted averages to zero:
h ()ξS x +Tz ' φi (z)dz = 0, i = 1,5 ∫0 (1.35) h ' ()S z −Tzξ φi (z)dz = 0, i = 1,5 ∫0
Using integration by parts (1.35) can be rewritten as:
hh φξSdz+−T()hφ()h T(0)φ(0)−Tφ'dz=0 ∫∫ix z i z i zi 00 hh (1.36) Sdφξ' z+−Tφdz S()hφ()h−S(0)φ(0)=0, i=1,5 ∫∫zi z i ()z i z i 00
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Substituting for Sx(z), Sz(z) and Tz(z) from equations (1.31) into (1.36) and using the approximate solution for U(z) and W(z) in (1.33) it is obtained:
(1) (2) ⎛ K K ⎞⎛b⎞ ⎛ f (1) ⎞ ⎜ ij ij ⎟⎜ ⎟ = ⎜ i ⎟ ⎜ (3) (4) ⎟⎜c⎟ ⎜ (2) ⎟ (1.37) ⎝ Kij Kij ⎠⎝ ⎠ ⎝ fi ⎠ or
(1) ⎛⎞b ⎛⎞fi []K ⎜⎟= ⎜(2) ⎟ (1.38) ⎝⎠c ⎝⎠fi where:
h h (1) ' ' Kij = ξ ∫c13φiφ jdz − ξ ∫Gφiφ jdz 0 0 h h (2) ' ' 2 Kij = −∫Gφiφ jdz − ξ ∫c11φiφ jdz 0 0 h h (3) 2 ' ' Kij = −ξ ∫Gφiφ jdz − ∫c33φiφ jdz i = 1,5 0 0 (1.39) h h (4) ' ' Kij = ξ ∫c13φiφ jdz − ξ ∫Gφiφ jdz 0 0 (1) fi = φi (0)Tz (0) − φi (h)Tz (h) (2) fi = φi (0)Sz (0) − φi (h)Sz (h)
T ⎛b⎞ ⎜ ⎟ = ()b1 b2 b3 b4 b5 c1 c2 c3 c4 c5 (1.40) ⎝c⎠
Let U2, W2, U3 and W3 be the displacements at top and bottom surface of top half of the element (top half of the core). Evaluating the expressions for U(z) and W(z) at the top and bottom surfaces and equating them to the surface displacements results in the expression:
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234 ⎛⎞U 2 ⎛⎞00 0 0 0 1hh h h ⎜⎟⎜⎟234 Wb10hhhh 0000⎛⎞ ⎜⎟2 = ⎜⎟ ⎜⎟⎜⎟ (1.41) ⎜⎟Uc3 00 0 0 0 10 0 0 0⎝⎠ ⎜⎟⎜⎟ ⎝⎠W 3 ⎝⎠10 0 0 0 00 0 0 0
This can be compactly expressed as:
⎛⎞Ub21⎛⎞ ⎜⎟ ⎜⎟ W ... ⎜⎟2 = []A ⎜⎟ ⎜⎟U ⎜...⎟ (1.42) ⎜⎟3 ⎜⎟ ⎝⎠Wc35⎝⎠
The tractions T2, P2, T3 and P3 acting on the surface can be related to the functions fi as follows:
⎛ (1) ⎞ f1 ⎡− φ1(h) 0 φ1(0) 0 ⎤ ⎜ ⎟ ⎢ ⎥ ⎜ (1) ⎟ − φ h 0 φ 0 0 f2 ⎢ 2 () 2 () ⎥ ⎜ (1) ⎟ f ⎢− φ3 ()h 0 φ3 (0) 0 ⎥ ⎜ 3 ⎟ ⎢ ⎥ ⎜ (1) ⎟ f ⎢− φ4 ()h 0 φ4 (0) 0 ⎥⎛T2 ⎞ ⎜ 4 ⎟ ⎜ ⎟ f (1) ⎢− φ ()h 0 φ (0) 0 ⎥⎜ S ⎟ ⎜ 4 ⎟ ⎢ 5 5 ⎥ 2 (2) = ⎜ ⎟ ⎜ f ⎟ ⎢ 0 − φ ()h 0 φ (0)⎥ T (1.43) ⎜ 1 ⎟ 1 1 ⎜ 3 ⎟ (2) ⎢ 0 − φ ()h 0 φ (0)⎥⎜ S ⎟ ⎜ f2 ⎟ ⎢ 2 2 ⎥⎝ 3 ⎠ ⎜ (2) ⎟ f ⎢ 0 − φ3 ()h 0 φ3 (0)⎥ ⎜ 3 ⎟ ⎢ ⎥ ⎜ (2) ⎟ 0 − φ h 0 φ 0 f4 ⎢ 4 () 4 ()⎥ ⎜ ⎟ ⎜ (2) ⎟ ⎢ 0 − φ h 0 φ 0 ⎥ ⎝ f5 ⎠ ⎣ 5 () 5 ()⎦ or
(1) ⎛⎞f1 ⎛⎞T2 ⎜⎟ ⎜⎟ ... S ⎜⎟= []B ⎜⎟2 (1.44) ⎜⎟... ⎜⎟T ⎜⎟ 3 ⎜⎟(2) ⎜⎟ ⎝⎠f5 ⎝⎠S3
From (1.37), (1.42) and (1.44) follows:
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⎛U2 ⎞ ⎛⎞TT11⎛⎞ ⎜ ⎟ ⎜⎟ ⎜⎟ W −1 SS ⎜2 ⎟=[]AK[][B]⎜⎟1=⎡⎤K* ⎜1⎟ (1.45) ⎜⎟U ⎜⎟T⎣⎦⎜T⎟ ⎜⎟3 ⎜⎟22⎜⎟ ⎝⎠W3 ⎝⎠SS22⎝⎠
Finally, the stiffness matrix of the top half of the FGM core [S(2)] that relates the surface tractions to the surface displacements is obtained as:
⎛⎞T1 ⎛⎞UU22⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ S −1 WW ⎜⎟1 ==⎡⎤K*(⎜⎟2⎡S2)⎤⎜⎟2 (1.46) ⎜⎟T ⎣⎦⎜⎟UU⎣⎦⎜⎟ ⎜⎟2 ⎜⎟33⎜⎟ ⎝⎠S2 ⎝WW33⎠ ⎝ ⎠
In order to satisfy equilibrium, the contributions of the different tractions at each interface should sum to zero. Enforcing the balance and the compatibility of force and displacements at the interfaces enables us to assemble the stiffness matrices of the four elements to obtain a global stiffness matrix [S]:
T []S ⎣⎢UW11U2W2U3W3U4W4U5W5 ⎦⎥ = (1.47) T = ⎣⎦⎢⎥0 pa 00000000
The displacements U1, W1…W5, are obtained by solving equation (1.47). The displacement field along with the constitutive relations is used to obtain the stress field in each element.
Analytical Models for Sandwich Structures with Functionally Graded Core
An Equivalent Single-Layer First-Order Shear Deformation Theory
The simplest model investigated in the present study is an equivalent single-layer model that includes a transverse shear deformation (First-order Shear Deformation theory, FSDT). The following kinematics assumptions are made:
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u(x, z) = u (x) + zψ (x) 0 (1.48) w(x, z) = w0 (x) where u is displacement in horizontal direction, x and w is displacement in vertical direction, z. u0, w0, ψ are unknown functions to be determined using the equilibrium equations of the first order theory. The dimensions of the sandwich beam are shown in
Figure 2.3. The length of the beam is L, the core thickness is h, the top face sheet thickness is ht and the bottom face sheet thickness is hb.
z
ht
h/2 z0 x h/2
hb Figure 2.3: The beam geometry
If the core material is orthotropic at every point and the principal material directions coincide with the x and z-axes, the plane strain constitutive relations are:
⎛⎞ ⎜⎟ ⎧⎫σ xx ⎡cz11 () c13 ()z 0 ⎤⎧ενxx ⎫ 1− ν0 ⎧εxx ⎫ ⎪⎪⎢⎥⎪⎪ Ez() ⎜⎟⎪⎪ ⎨⎬σ zz ==cz13 () c33 ()z 0 ⎨ενzz ⎬ ⎜⎟1−ν0 ⎨εzz ⎬ ⎢⎥(1 +−νν)(1 2 ) ⎪⎪τ ⎢⎥00cz()⎪γν⎪ ⎜⎟1− 2⎪γ⎪ ⎩⎭xz ⎣ 55 ⎦⎩xz ⎭ ⎜⎟00 ⎩xz ⎭ ⎝⎠2 (1.49)
or {σ} = [c()z ]{ε} (1.50)
The variation of Young’s modulus E in the thickness direction is given by a polynomial in z. e.g.,
49
4 3 2 ⎛ ⎛ z ⎞ ⎛ z ⎞ ⎛ z ⎞ ⎛ z ⎞ ⎞ E(z) = E ⎜a + a + a + a +1⎟ 0 ⎜ 1⎜ ⎟ 2 ⎜ ⎟ 3 ⎜ ⎟ 4 ⎜ ⎟ ⎟ (1.51) ⎝ ⎝ h ⎠ ⎝ h ⎠ ⎝ h ⎠ ⎝ h ⎠ ⎠ where E0 is the Young’s modulus at z=0 and a1, a2, a3 and a4 are material constants.
In order to calculate the flexural rigidity of the cross-section, the position of the neutral axis z0 must be found. It is given by the coordinate system for which the first moment of area is zero when integrated over the entire cross-section:
ht +h / 2−z0 B = zc (z)dz = 0 ⇒ z (1.52) 11 ∫ 11 0 −(hb +h / 2+z0 )
Equilibrium equations of the first order theory take the following form:
⎧ du0 ⎪NAx = 11 ⎪ dx ⎪ dψ ⎨Mx = D11 (1.53) ⎪ dx
⎪ ⎡ dw0 ⎤ ⎪VSz =+⎢ψ ⎥ ⎩ ⎣ dx ⎦ where A11 is the axial rigidity, D11 is the bending stiffness and S is the extensional stiffness:
hht +−/2 z0 Ac= ()zdz 11 ∫ 11 −+(/hhb 2+z0 )
hht +−/2 z0 D= z2c()zdz (1.54) 11 ∫ 11 −+(/hhb 2+z0 )
hz/2− 0 Sc= ()zdz ∫ 55 −+(/hz2 0 )
For a given set of external loads and boundary conditions, axial force resultant Nx, bending moment resultant Mx, and shear resultant Vz can be calculated. Then using system (1.53) the displacements are obtained.
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This model is applied for a sandwich beam with functionally graded core. The main drawback of the model is given by the fact that the transverse shear strain is constant through the thickness of the beam. Results and discussions are presented in the last section.
An Equivalent Single-Layer Third-Order Shear Deformation Theory
Reddy (1984) developed a third-order Shear Deformation theory (TSDT) for composite laminates, based on assumed displacement fields and using the principle of virtual displacements. Reddy (1990) reviewed a number of other third-order theories and showed that they are special cases of his theory. Reddy (2000) and Reddy and Cheng
(2001) expanded TSDT for analysis of functionally graded plates.
Here, a third-order equivalent single-layer model based on Reddy’s assumption of vanishing of transverse shear stresses on the bounding planes is investigated. The displacement field is assumed to be:
u(x, z) = u (x) + zψ (x) + z 2 χ(x) + z 3φ(x) 0 (1.55) w(x, z) = w0 (x) where u0 and w0 are displacements along middle axis, z=0. Functions χ and Φ are eliminated using the assumption of zero shear stress at top and bottom:
h − h ⎡ dw ⎤ χ x = t b ψ x + 0 () ∗ ⎢ () ⎥ h ⎣ dx ⎦ (1.56) 2 ⎡ dw ⎤ φ x = − ψ x + 0 () ∗ ⎢ () ⎥ 3h ⎣ dx ⎦ where
2 ∗ h h = 2ht hb + ht h + hb h + 2 (1.57)
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If the top face sheet thickness ht equals the bottom face sheet thickness hb then
χ(x)=0.
Axial force resultant, bending moment resultant and shear resultant are calculated as follows:
h ⎧ +ht ⎪ 2 Nb= σ x, zdz ⎪ xx∫ x() ⎪ ⎛⎞h −+⎜⎟hb ⎪ ⎝⎠2 h ⎪ +h ⎪ 2 t M = bzσ x, zdz (1.58) ⎨ x∫ xx() ⎪ ⎛⎞h −+⎜⎟hb ⎪ ⎝⎠2 h ⎪ +h ⎪ 2 t Vb= τ x, zdz ⎪ zx∫ z() ⎪ ⎛⎞h −+⎜⎟hb ⎩ ⎝⎠2
du0 where b is the beam width, ε = and constants AN through CV are given by: 0 dx
hh ++hhtt 22⎡⎤hh− 2 A==b czdz B b czz+z23tb−z dz NN∫∫11 () 11 ()⎢⎥∗∗ ⎛⎞hh⎛⎞ ⎣⎦hh3 −+⎜⎟hhbb−⎜+⎟ ⎝⎠22⎝⎠ h +ht 2 ⎡⎤hh− 2 Cb=−czz23tbz dz N ∫ 11 ()⎢⎥∗∗ ⎛⎞h ⎣⎦hh3 −+⎜⎟hb ⎝⎠2 hh ++hhtt 22⎡⎤hh− 2 A==b zczdz B b zczzz+23tb−z dz MM∫∫11 () 11 ()⎢⎥∗ ∗ ⎛⎞hh⎛⎞ ⎣⎦h 3h −+⎜⎟hhbb−+⎜⎟ ⎝⎠22⎝⎠ h +ht 2 ⎡⎤hh− 2 Cb=−zczz23tbz dz M ∫ 11 ()⎢⎥∗∗ ⎛⎞h ⎣⎦hh3 −+⎜⎟hb ⎝⎠2
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h +ht 2 ⎡⎤hh− 2 Bb=+cz12ztb−z2 dz V ∫ 55 ()⎢⎥∗∗ ⎛⎞h ⎣⎦hh −+⎜⎟hb ⎝⎠2 (1.59) h +ht 2 ⎡⎤hh− 2 Cb=−cz2ztbz2 dz V ∫ 55 ()⎢⎥∗∗ ⎛⎞h ⎣⎦hh −+⎜⎟hb ⎝⎠2
The following system is obtained:
⎧⎫du0 ( x) ⎪⎪ ⎪⎪dx ⎡⎤ABNNCN ⎧ Nx()x⎫ ⎢⎥⎪⎪dxψ () ⎪ ⎪ ABMMCM⎨ ⎬⎨= Mx()x⎬ (1.60) ⎢⎥dx ⎢⎥0 B CV⎪ ⎪⎪()x⎪ ⎣⎦VV⎪⎪dw()x ⎩⎭z ⎪⎪ ⎩⎭dx where c1 and c2 are constants to be determined from boundary conditions. For a given set of external loads and boundary conditions axial force resultant Nx, bending moment resultant Mx, and shear resultant Vz can be calculated. Then using system (1.60) the displacements are obtained.
This model is applied for a sandwich beam with functionally graded core. The main drawback of the model is given by the fact that the transverse shear strain has a quadratic variation with respect to the thickness coordinate. Results and discussions are presented in the last section.
A Higher-Order Shear Deformation Theory
Frostig et al. (1992) developed a higher-order theory for a sandwich beam with a transversely flexible core that uses a beam theory for the face-sheets and two-dimensional elasticity equations for the core. Swanson (1999) addressed details of implementation of
Frostig model and presented solutions for several cases. The main feature of the method
53 is the higher order displacement fields in the thickness coordinate: second order for vertical displacement and third order for the longitudinal displacement. Another advantage of this model, as a model based on variational principle is that the boundary conditions are obtained uniquely as a part of derivation. The main difference between this model and the previous one is that now, the higher order displacements are derived and not assumed. The equations developed by Frostig et al. (1992) and modified for a functionally graded core are briefly presented here.
The dimensions of the sandwich beam and the coordinate systems are shown in
Figure 2.4. The length of the beam is L, the core thickness is h, the top face sheet
h x, t x, uc zt, wt z, wc h
x, uob hb
zb, thickness is ht and the bottom face sheet thickness is hb.
Figure 2.4: The beam geometry
Constitutive relations (which assume isotropic elastic behavior) for the face-sheets and for the transversely flexible core (i.e. zero longitudinal stress) are given by:
tt bb Top face-sheet: NAxx ==11u0t,x Bottom face-sheet: Nxx A11u0b,x tt bb M xx =−Dw11 t,xx Mxx =−D11wb,xx (1.61) Core: τγ()xz, = Gc (z)()xz, (1 −ν ) Ez() σ ()xz,,= c w ()xz zz (1 +−νν)(1 2 ) c,z
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i i where Nxx are the resultant axial forces in the face-sheets, M xx are the bending
i i moments in the face-sheets, A11 and D11 are axial and respectively flexural rigidities for the face-sheets, u0i, wi are face sheets longitudinal and vertical displacements at the centroid (i=t for top face-sheet and i=b for bottom face-sheet):
hi / 2 Ai = ci (z)dz 11 ∫ 11 −h / 2 i (1.62) hi / 2 Di = z 2ci (z)dz 11 ∫ 11 −hi / 2
Linear variation for Young’s modulus and shear modulus are assumed:
Ez()= az+ b (1.63) Gz()= a11z+ b
Governing differential equations, boundary conditions and continuity conditions are derived based on variational principle. Using the equilibrium equations for the core:
σ +τ = 0 x,x xz,z (1.64) σ z,z +τ xz,x = 0 the compatibility of the displacement at the core-face-sheets interfaces and the assumption that for a transversely flexible core (i.e. made of a material with much lower modulus relative to the face-sheets), the shear stress is nearly constant through the core the following core displacement fields are derived:
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11⎛⎞a1 2 ⎛⎞ht uxcx(),lz=+ττ(x)n⎜⎟z1+,,x()x z−wtx()x⎜⎟z++uot+ ab11⎝⎠ 22a ⎝⎠ ⎛⎞ba⎛ ⎞ ⎜⎟zz++ln ⎜1⎟−z ⎛⎞h ⎝⎠ab⎝ ⎠ +−⎜⎟wwbx,,tx +τ ,xx()x (1.65) ⎝⎠a ⎛⎞a ln ⎜⎟h +1 ⎝⎠b ⎛⎞a ln ⎜⎟z +1 zh⎝⎠b ⎛⎞ wxcx(), z=− ττ,,()x+ ⎜⎟wbx−wt,x+ ,xx()x +wt aa⎛⎞a ⎝⎠ ln ⎜⎟h +1 ⎝⎠b where u0i and wi are longitudinal and vertical displacements of the centroid of each face- sheet (i=t for top face-sheet and i=b for bottom face-sheet); ν is core Poisson’s ratio. In the above expression a linear core Young’s modulus was assumed: Ec(z)=az+b. Similar relations can be obtained for any Young’s modulus variation expressed by a differentiable function.
Governing equations, written in terms of transverse displacements wt and wb of the top and bottom face-sheets and shear stress of the core, τ are:
⎧ ⎪ α1wt,xxxx ()x + β1τ ,x ()x +η1wb ()x + µ1wt ()x = qt (x)− mt,x ()x ⎪ ⎨α 2 wb,xxxx ()x + β 2τ ,x ()x +η2 wb ()x + µ 2 wt ()x = qb ()x − mb,x ()x (1.66) ⎪ ⎪ 1 1 α 3wt,xxx ()x + β3wb,xxx ()x +η3τ ,xxxx ()x + µ3τ ,xx ()x +ω3τ ()x = t nt ()x − b nb ()x ⎩⎪ A11 A11 where qi, mi and ni are distributed pressures, moments and axial forces applied on top
(i=t) and bottom (i=b) face-sheets and the coefficients are given by:
56
⎡⎤ ⎢⎥h h b ad ad αβ==Ddt −⎢⎥t + −η=− µ= 1111 ⎢⎥2 ⎛⎞aaa 1 ⎛⎞1 ⎛⎞a ⎢⎥ln ⎜⎟hh++1 ln ⎜⎟1 ln ⎜⎟h+1 ⎣⎦⎝⎠bb⎝⎠ ⎝⎠b ⎡⎤ ⎢⎥h h b ad ad αβ==Ddb ⎢⎥−h−b + −η= µ=− 2112 ⎢⎥2 ⎛⎞aaa 2 ⎛⎞2 ⎛⎞a ⎢⎥ln ⎜⎟hh+1 ln ⎜⎟++1 ln ⎜⎟h1 ⎣⎦⎝⎠bb⎝⎠ ⎝⎠b ⎛⎞ba⎛ ⎞ ⎛⎞ba⎛ ⎞ ⎜⎟hh++ln ⎜1⎟−hh⎜⎟++ln ⎜h1⎟−h h ab h ab α =−h − t +⎝⎠⎝ ⎠ β =− b −⎝⎠⎝ ⎠ 3 2 ⎛⎞aa3 2 ⎛⎞ ln ⎜⎟hh++1 ln ⎜⎟1 ⎝⎠bb⎝⎠ ⎛⎞ba⎛ ⎞ ⎜⎟hh++ln ⎜1⎟−h 112 h ⎝⎠ab⎝ ⎠ ⎛⎞a1 ⎛⎞11 ηµ33=−hh=ln ⎜⎟+1 ω3=−d⎜⎟tb+ 2aa ⎛⎞a a11⎝⎠b ⎝⎠AA11 11 ln ⎜⎟h +1 ⎝⎠b
(1.67) where d is the beam width.
In order to obtain the homogeneous solution for (1.66), the following characteristic equation is derived:
4 αλ11+ µ η1 β1λ 4 µα22λ+η2 β2λ = 0 (1.68) 3342 αλ33βλ η3λ ++µ3λ ω3
Denoting with λj, j = 1,…,12, the roots of characteristic equation, the following rescaling of constants needs to be done in order to avoid numerical difficulties (given by
the case when Real(λj) > 0 ⇒⎯exp(λ j x) xL⎯→ ⎯→∞):
Re(λj )x (ic) If Re(λjj) <⇒0 solution = e
−λj L ⎡⎤ (1.69) e Re(λj )x ⎣⎦−λj (Lx−) (ii) If Re(λjj) >⇒0 solution = c e =cje e−λj L
57
Forstig’s theory is clear but needs the above described rescaling of constants in order to avoid numerical difficulties. For a given set of external loads and boundary conditions the differential governing system (1.66) can be solved. Results and discussions are presented in the next section.
Results for Sandwich Beam with Functionally Graded Core under a Distributed Load
In this section includes two objectives: validate analytical methods for sandwich structure with FG core and use these methods to compare FG and homogenous cores.
The results of the modified Venkataraman and Sankar (2001) and Zhu and Sankar
(2004) methods were compared (Figures 2.5 – 2.9) and found to be identical. A one dimensional sandwich plate of length 0.2 m, core height 20×10-3 m and face-sheet thickness 0.3×10-3 m is considered as an example to investigate the effects of varying core properties through the thickness. The face sheet elastic modulus Ef is chosen as 10
GPa. The sandwich core modulus E0 at the mid-plane is kept fixed at Ef/1000 while the core modulus at the face sheet interface (Eh) is varied. The ratio of Eh/E0 varies from 1 to
100. When Eh/E0=1, core properties are constant through the thickness, and hence the problem is identical to that of a conventional sandwich plate. The value of Eh is gradually increased until it reaches the value of the face sheet, for Eh/E0=100. The different profiles of the elastic modulus variations in the sandwich core are shown in Figure 2.5. They display the profile of Young’s modulus for FG materials with Eh/E0=1 (a1=0, a2=0, a3=0, a4=0, a5=1), Eh/E0=10 (a1= 3.9372, a2= -.94217, a3= 3.8705, a4= 2.1215, a5=1) and
Eh/E0=100 (a1=238.50, a2=-258.92, a3=132.18, a4-14.327, a5=1.6335), respectively.
Poisson’s ratios for the core and face-sheet, respectively, were 0.35 and 0.25. The results
58 are restricted to the case where the plate is loaded in the transverse direction by a
sinusoidal load given by pa sin(πx / L) with pa=1.
An interesting result from analysis of the FGM sandwich beam is the transverse shear stress in the sandwich core at the face-sheet/core interface. Conventional design of sandwich panels restricts the shear stress at the core/ face sheet interface to the bond
(adhesive) shear strength, which is typically lower than the shear strength of the core material. Therefore, the core material is not fully utilized. It is hence desirable to reduce the interfacial shear stress while carrying a high shear stress in the core. It appears that this is possible with a functionally graded core. The shear stress variations in the core, given by both methods (exact and Fourier-Galerkin) are plotted in Figure 2.6. The interface shear stress reduces as Eh/E0 ratio is increased.
Figure 2.5: Through-the-thickness variations of core modulus considered for the
functionally graded sandwich panel (the load is p(xp) = a sin(π x/ L) )
59
Figure 2.6: Variation of transverse shear stresses (given by Fourier-Galerkin method) in the sandwich structure with FGM core for different ratios (Eh/E0) and Ef/E0 =
1000 at x = L/4 (the load is p(xp) = a sin(π x/ L) ).
The normal stress σ zz in the core (Figure 2.7) varies linearly and is independent of the variation in core properties. The value of normal stress varies linearly
(approximately) from the applied surface load at the top core to zero at the bottom of the core. This is an interesting result, because it simplifies the calculations required in order to analyze the core-crushing problem. It must be noted that the example considered here, used a smoothly varying (sinusoidal) surface pressure load.
60
Figure 2.7: Variation of σzz stresses (given by Fourier-Galerkin method) in the sandwich structure with FGM core for different values of Eh/ E0 and Ef/E0 = 1000 at x =
L/4 (the load is p(xp) = a sin(π x/ L) ).
The bending stress variation (Figure 2.8) in the core is as expected. The linear variation in strains results in small levels of bending stress in the core near the mid-plane.
The stresses increase near the face sheet. This is particularly pronounced as the value of the core modulus is increased to match the value of the face sheet thickness.
The u-displacement variation through the thickness is linear (plane sections remain plane) when the gradient in the core properties is small. However, for highly graded cores significant warping of the cross section occurs.
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Figure 2.8: Variation of bending stresses (given by Fourier-Galerkin method) through the thickness of the FGM core for different values of Eh/E0 and Ef/E0 = 1000 at x
= L/4 (the load is p(xp) = a sin(π x/ L) ).
Figure 2.9: Variation of u displacements (given by Fourier-Galerkin method) through the thickness of the sandwich structure with FGM core for different ratios (Eh/E0)
and Ef/E0 = 1000 at x = L/4 (the load is p(xp) = a sin(π x/ L) ).
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Next step is to compare the five models: the equivalent single-layer first-order and third-order shear deformation theories, Fourier series-Galerkin method, Frostig model and a finite-element model.
A simply supported sandwich beam, with length L = 0.3 m, core thickness h =
-3 -3 20×10 m and face-sheets thickness hf = 0.3×10 m is considered to investigate the effects of varying core properties through the thickness. The face-sheet Young’s modulus was chosen as 50 GPa. The face-sheets Poisson’s ratio is νf = 0.25 and the core Poisson’s ratio is ν = 0.35.
(a) E(z)
pasin(πx/L)
(b) E(z) Figure 2.10: (i) Non-dimensional core modulus and (ii) loading for: (a) symmetric core about the centerline and (b) asymmetric core about the centerline
Two cases are considered: (a) symmetric core about the centerline under uniform distributed load p = 1 N/m2 and (b) asymmetric core about the centerline under a distributed load given by p()xx= sin(π L) N/m2. Figure 2.10 presents core elastic moduli: for (a), the core Young’s modulus E has a linear symmetric variation with respect to thickness coordinate, z: E = 50 MPa at the middle core and E = 50×10 MPa at the core-top face-sheet interface (as well as at the core-bottom face-sheet interface); for
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(b), the core Young’s modulus E has a linear asymmetric variation with respect to thickness coordinate, z: E = 50 MPa at the core-bottom face-sheet interface and E=50×10
MPa at the core-top face-sheet interface.
Using the commercial finite element software ABAQUS a 2D finite element model was created to model problem (a). The FG core was partitioned through the thickness into twenty strips with constant properties. Four elements were considered through the thickness of each strip and two elements were considered through the thickness of the face-sheets. The elements considered were 2D, quadratic, plane strain elements.
Boundary conditions assume wz(0, ) = w(L, z) = 0
Figures 2.11 – 2.15 present a comparison for five models: the equivalent single- layer first-order and third order shear deformation theories, Fourier series-Galerkin method, Frostig model and an ABAQUS finite-element model. For problem (a), which is the symmetric core under a uniform distributed load, a very good agreement among the
Fourier series-Galerkin method and the ABAQUS model was found. Because the core is symmetric about the mid-plane, the variation of displacements, strains and stresses are symmetric with respect to the thickness coordinate.
Figure 2.11 presents deflections at bottom face sheet – core interface. The FSDT and the TSDT beam are stiffer than the Galerkin beam.
64
(a) (b)
Figure 2.11: Comparison of deflections: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by p()xx= sin(π L) N/m2.
Comparison of longitudinal displacements in the core at the same cross-section
(L/4) for the two cases are presented in Figures 2.12 (a) and (b). A perfect agreement was found between Galerkin model and the finite element model for (a) and between Galerkin model and the Frostig model for (b). For both problems FSDT gives a linear variation as expected.
(a) (b)
Figure 2.12: Comparison of longitudinal displacement in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by p()x= sin(π xL) N/m2.
65
Bending stress (Fig. 2.13 (a) and (b)) in the core at the same cross-section (x=L/4) is almost the same for all models. For the asymmetric core, a larger compression value is found at the top core where the load in applied and where Young’s modulus value is larger.
(a) (b)
Figure 2.13: Comparison of axial stress in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by p()x= sin(π xL) N/m2.
Comparison of shear strain in the core at the same cross-section (L/4) is presented in Figure 2.14 (a) and (b). Fourier-Galerkin method, Frostig model and the finite-element solution present a 1/z type variation for the core shear strain. The first order shear deformation theory gives a constant shear strain while the third order shear deformation theory gives a quadratic shear strain with respect to the thickness coordinate.
The same conclusion was reached for the shear stress in the core at the same cross- section (Figure 2.15): Fourier-Galerkin method, Frostig model and the finite-element solution present an almost constant shear stress whereas the first order shear deformation theory gives a linear shear stress while the third order shear deformation theory gives a
66 third order shear stress with respect to the thickness coordinate. The equivalent single- layer theories are not accurate for shear because they are not based on two dimensional equilibrium equations (1.64). In order to obtain accuracy for shear (strain and stress) in single-layer theories (both first order and third order) the shear stress was obtained using equilibrium equations (1.64) for the core and the bending stress previously derived
(Figure 2.13):
z στ+=0(⇒τx,zx)=−σ(,ξ)dξ+τ(x,0) (1.70) xx,,x xz z xz ∫ xx,x xz 0
Figure 2.15 includes both shear stresses: obtained from single-layer theories and obtained from equilibrium equations (1.70). The latest are identical with those obtained based on
Fourier-Galerkin and finite element model.
(a) (b)
Figure 2.14: Comparison of shear strain in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by p()x= sin(π xL) N/m2.
67
(a) (b)
Figure 2.15: Comparison of shear stress in the core at x = L/4: (a) symmetric core about the centerline under a constant uniform distributed load p = 1 N/m2 and (b) asymmetric core about the centerline under a sinusoidal distributed load given by p()x= sin(π xL) N/m2.
Summary and Conclusions
In this chapter, analytical models for sandwich structures with FG core are introduced, validated by comparison with finite element solutions and used to compare homogeneous and FG core sandwich plates.
The study presents analytical models for sandwich structures with two types of core
Young’s modulus variation through the thickness: exact solutions are presented for the exponential variation and a combination of Fourier series and Galerkin method for a polynomial variation of core Young’s modulus. Those methods are compared with models presented in literature and modified for this type of structure: two equivalent single-layer theories based on assumed displacements and a higher-order theory. A finite- element analysis is presented in order to validate the models. A very good agreement among the Fourier - Galerkin method, the higher-order theory and the finite-element analysis is found.
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A comparison among homogenous and FG cores is presented: sandwiches with same geometry, same sinusoidal distributed load and different core properties are analyzed. As the core becomes stiffer at the core-top face-sheet interface, the interface shear stress reduces as Eh/E0 ratio is increased.
The chapter deals only with distributed loads (uniform and sinusoidal). Next chapter introduces a more common load, the localized contact between a rigid indentor and a sandwich plate. The low-velocity impact solution is based on the solution developed here.
CHAPTER 3 CONTACT AND LOW-VELOCITY IMPACT OF SANDWICH BEAMS WITH FUNCTIONALLY GRADED CORE
One of the important problems in sandwich structures is damage due to low velocity impact. The interfacial shear stresses due to contact forces can be large enough to cause debonding of the face sheet from the core. Unlike for their solid metallic counterparts, predictions of the effects of low-velocity impact damage in sandwich structures are difficult because significant internal damage is achieved at impact energy levels lower than those required to create visible damage at the surface. One way of reducing the shear stresses is to use functionally graded core so that the abrupt change in stiffness between the face sheet and the core can be eliminated or minimized. The stresses that arise due to low-velocity impact can be easily understood by analyzing the static contact between the impactor and the structure (Sankar and Sun, 1985).
The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young’s modulus is represented by a polynomial in the thickness coordinate, but the Poisson’s ratio is kept constant. In the previous chapter, the two- dimensional elasticity equations for the plane sandwich structure were solved using different methods. Here, the contact problem is solved using two methods: the assumed
69 70 contact stress distribution method and method of point matching; and the results are compared with a finite element model. For the impact problem a simple dynamic model based on quasi-static behavior of the panel was used - the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects.
Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.
Contact problems for finite thickness layers can be solved using numerical methods. Sankar and Sun (1983) used two types of numerical methods, point matching technique and assumed stress distribution method. The main difference between the two methods is the way the contact load under the indenter is estimated: assumed stress distribution method is based on the assumption of a semi-elliptical contact stress distribution whereas the method of point matching attempts to capture the actual contact stress distribution approximating it as a superposition of several uniformly distributed loads. The point matching method is essentially a numerical technique to solve the integral equations of the contact problem. However this method fails when the contact area is too small because of numerical difficulties. In the assumed contact stress
71 distribution method, the contact stresses are assumed to be of Hertzian form, i.e., similar to that of contact between a rigid cylinder and a half-plane. The contact stresses take the shape of a semi-ellipse. A contact length is assumed and the contact stresses are expressed in terms of only one unknown, the peak contact stress. Requiring that the deflections beneath the contact region match the indenter profile, one can solve for the peak contact stress. Here, the two methods are used for a sandwich beam with functionally graded core.
Assumed Stress Distribution Method
Previous chapter presented analytical models for sandwich structures with FG cores as a tool to be used for different problems. This section presents one of the possible applications of the above-discussed models. Indentation of a surface with an object
(sharp, blunt or spherical) is a common engineering problem. If the properties of the material under the indentor are known, the analysis of the contact load versus indentation can provide valuable information about the contact damage. It is shown that spatially grading mechanical properties (e.g. Young’s modulus) it is possible to minimize the contact damage.
Sankar and Sun (1985) solved the problem of smooth indentation of a beam by a rigid circular cylindrical indenter using the method of assumed stress distribution. The method is used here for indentation of a sandwich beam with functionally graded core.
The indenter is a cylinder of radius R0 and unit length. The contact length 2c is considered as known and the other quantities (stresses, deflection) are calculated for a given contact length. Indentation is defined as the difference in the vertical displacement of the indenter and the corresponding point on the bottom side of the beam. The
72 dimensions of the sandwich beam are shown in Figure 3.1. The length of the beam is L, the core thickness is h and the face sheet thickness is hf.
2h
Figure 3.1: Dimensions of the sandwich panel and the contact length, 2c.
The stress distribution under the indenter (Figure 3.2) is assumed to be of the semi- elliptical form:
x 2 p(x) = p 1− (3.1) max c 2 where pmax is maximum value of the stress at center (unknown).
R0
pmax
c
Figure 3.2: The stress distribution under the indenter.
A similar stress distribution with some arbitrary pmax, say pM is assumed. Writing p(x) in the form of a Fourier sine series, it is obtained:
∞ nπ x (3.2) px()= ∑ pn sin n=1,3... L where
73
2 pM ⎛ nπ ⎞ ⎛ nπc ⎞ pn = sin⎜ ⎟J1⎜ ⎟ (3.3) n ⎝ 2 ⎠ ⎝ L ⎠ and J1 is the Bessel function of first order. The vertical displacements of the points in the contact zone are computed using the Fourier series - Galerkin Method described in the previous chapter. From geometrical considerations (Figure 3.3), for contact lengths smaller in comparison with the radius of indenter, the following approximate relation is used:
x 2 δ (x) = w(x) − w(x ) = (3.4) 0 2R′(x) where w(x0) is vertical displacement at the middle point (see Figure 3.3). In (3.4) R' is the radius of curvature at x. It is found that R′ is almost constant (usually, more then 70% of
R′ distribution lies within one standard deviation of the mean value). Then the average radius of curvature of the deformed top face sheet can be derived as:
1 c R = ∫ R′(x)dx (3.5) c 0
R′(x)
x w(x) w(x0) δ(x)
Figure 3.3: Illustration of relation between w deflection and radius of curvature of the deformed surface.
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Generally R is different from the radius of the intender R0. But, the displacements vary linearly with the load and hence the peak stress pmax required producing a radius R0 is given by:
R pmax = pM (3.6) R0
Once pmax is known, vertical displacements and the indentation can be calculated.
Plots for several examples are presented and discussed in the Results section.
Method of Point Matching
The main difference between method of point matching and the method of assumed stress distribution is the assumption of the load. If, for the previous method the load is of a semi-elliptical form, here it is assumed that the stress distribution under the indenter is a superposition of a finite number of symmetrical rectangular loadings of unknown magnitudes (denoted by qj, j=1,m) distributed over known lengths (denoted by xj, j=1,m) as depicted in Figure 3.4.
qj
2c x0 xj= j m
Figure 3.4: Discretization of contact load for the method of point matching
th So, the j uniform distributed load over the span 2xj is:
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⎧ ⎡⎤L x j ⎪0, x ∈−⎢⎥0, ⎪ ⎣⎦22 ⎪ ⎪ ⎡ LLx jx j⎤ pxjj()=∈⎨q, x ⎢ −, +⎥ (3.7) ⎪ ⎣ 2222⎦ ⎪ ⎡⎤L x j ⎪0, xL∈+⎢⎥, ⎩⎪ ⎣⎦22
Using Fourier series the above load can be written as:
N j ⎛kπ ⎞ p jk()xp= ∑ sin⎜x⎟ (3.8) k =1 ⎝⎠L where
j 4q j ⎛⎞kπ⎛kπ⎞ pk= sin ⎜⎟sin ⎜x j⎟ (3.9) kLπ ⎝⎠22⎝ ⎠
j For each coefficient pk , the vertical displacements of the points in the contact zone are computed using the Fourier series - Galerkin Method described in the previous
th th chapter: wji is the vertical displacement of j reference point due to i uniform distribute
load of unit magnitude; w0i is the vertical displacement of mid point reference point due
th th to i uniform distribute load of unit magnitude; v j is the vertical displacement of j
reference point due to indentation; v0 is the vertical displacement of mid point due to indentation. Using the same geometrical relation described by (3.4) and writing the displacements at each point as superposition of displacements due to each rectangular load, it is obtained:
m vx()00==v ∑ w0iiq i=1 (3.10) m vx()jj==v ∑ wjiqi i=1
76
Then, the following algebraic system is obtained:
m 2 x j ∑()ww0ij−=iqi ,j=1,m (3.11) i=1 2R
From (3.11) the unknown load magnitudes qi are obtained. The total load is obtained as a summation of all rectangular loadings over the contact area. This method was applied for a sandwich beam with functionally graded core and the results are presented in the Results Section.
Quasi-Static Impact of a Sandwich Beam with FG Core
After the static contact problem was solved, the method is applied to the problem of low-velocity impact of functionally graded sandwich panels. Solving the static contact problem first and combining the solution with the dynamic response of the sandwich panel obtained via simple spring-mass models (quasi-static assumption) accomplish this.
The use of static load-deflection behavior of the sandwich beam in the impact analysis needs some justification. In general the wave propagation effects, especially through the thickness of the core, should be considered in impact response of sandwich panels. This will be crucial when spalling type damage occurs in the panels. However, a study by
Sankar (1992) showed that for very large impactor mass compared to that of the target plate and for very low impact velocities compared to the wave velocity in the target medium, quasi-static assumptions yield sufficiently accurate results for impact force history and ensuing stresses in the impacted plate.
Because the problem deals with the low-velocity impact (applied force vary slowly in time, therefore the system response is relatively slow so that the inertial terms in the equations of motion can be neglected; the structure responds quasistatically), the sandwich beam is modeled as a combination of two springs (Shivakumar et al. 1985), a
77 linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects as depicted in Figure 3.5. Also, the problem assumes only a rigid indentor (wave effects in the impactor can be neglected) with negligible dynamics.
F
ki – nonlinear
ks - linear
Figure 3.5: Low-velocity impact model
Using the numerical results from the contact problem the spring constants ki and ks and the exponent n are determined such that:
n F = kiα (3.12)
F = ks wb (3.13) where F is the total load, α is the core indentation, wb is the vertical displacement of the core at the at bottom face sheet interface.
The displacement of the impactor is calculated as the sum of indentation depth
(core compression) and the global deflection of the sandwich beam:
1 / n F ⎛ F ⎞ ⎜ ⎟ w = α + w b = + ⎜ ⎟ (3.14) k s ⎝ k i ⎠
The work done by the impactor during the impact event can be expressed as:
1/ n w F F ⎛ F ⎛ F ⎞ ⎞ F 2 1 F 1+1/ n W = Fdw = Fw − wdF = Fw − ⎜ + ⎜ ⎟ ⎟ dF = + (3.15) ∫ ∫ ∫⎜ k ⎜ k ⎟ ⎟ 2k n +1 2k 1/ n 0 0 0 ⎝ s ⎝ i ⎠ ⎠ s i
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The kinetic energy stored in the target is assumed negligible. The maximum deflection is reached when the velocity of the projectile becomes zero. At that time, the initial kinetic energy of the projectile has been converted into two parts: strain energy stored into the target by global (bending and shear) deformation and the energy used to create local deformations in the contact region. Hence, equating the impactor kinetic energy to the work done or the strain energy stored in the springs, the maximum contact force can be calculated from:
F 2 1+1/ n max Fmax 1 2 + 1/ n = mv0 (3.16) 2ks ()n +1 ki 2 where m and v0 are, respectively, the mass and impact velocity of the impactor.
The results of the static contact problem are used to determine the constants that described the stiffness and compression of FG sandwich beam ((3.12) and (3.13)). Using these material properties and the quasi-static model the maximum contact force in the case of low-velocity impact of FG sandwich beam is determined. Using this maximum value the maximum normal and shear strains in the core were determined and compared.
Results for several examples are presented in the next section.
Results
A sandwich beam, with length L = 0.2 m, core thickness h = 20×10-3 m and face
-3 sheet thickness hf = 0.3×10 m is considered to investigate the effects of varying core properties through the thickness. The face-sheet Young’s modulus was chosen as 50 GPa.
Although these methods can be applied to a general form of E, in the present work, for core Young’s modulus, two cases are considered (Figure 3.6): linear symmetric about midplane and linear asymmetric. The variation of E with respect to z for the two cases is given below and also depicted in Figure 3.6.
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⎛ E sym − E sym z ⎞ E sym = E sym ⎜ h 0 +1⎟ if z ∈ − h, h (3.17) 0 ⎜ sym ⎟ [] ⎝ E0 h ⎠
⎛ E asym − E asym z E asym + E asym ⎞ E asym = E asym ⎜ h 0 + 0 h ⎟ if z ∈ − h, h (3.18) 0 ⎜ asym asym ⎟ [] ⎝ 2E0 h 2E0 ⎠
sym where 2h is the core thickness; E0 is the Young’s modulus at the midplane for
asym symmetric case; E0 is the Young’s modulus at the bottom surface for asymmetric
sym case. Eh is the sandwich core Young’s modulus at face sheet interfaces for symmetric
asym case; Eh is the sandwich core Young’s modulus at top face sheet interface for asymmetric case. Three different variations, such that Eh= E0×(1, 5, 10) are considered.
Eh is the core modulus at face sheet interfaces for symmetric case and is the sandwich core modulus at top face sheet interfaces for asymmetric case.
Poisson’s ratios are ν = 0.35 for the core and νf = 0.25 for the face-sheet material.
The intender is a cylinder with a radius of 10×10-3 m. The width of the cylinder and the width of the sandwich panel in y-direction are assumed to be unity. The cylinder is made of steel and its mass is m=15.7 kg. The impact velocity of the impactor is v0=6 m/s and the kinetic energy is K=282.2 J.
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Figure 3.6: Through the thickness variations of core modulus considered for the functionally graded sandwich beam. (H.C denotes homogeneous core; S.C. denotes symmetric core, A.C. denotes asymmetric core)
Figures 3.7 through 3.33 present different displacements, strains and stresses based on assumed stress distribution method and provide a comparison of different sandwich structures. This method is compared with the second numerical method and with the finite element solution at the end of the chapter (Figures 3.33 – 3.37)
Figures 3.7 thought 3.25 present displacements, strains and stresses in sandwich beam with linear symmetric and asymmetric functionally core. For the two cases
(symmetric and asymmetric), same Young’s modulus ratio, Eh/E0 of 10 and same semi contact length c = 3 mm are considered.
Figure 3.7 presents horizontal displacement in the second half of sandwich structure with asymmetric core. It can be noticed that the displacement is zero at the middle cross-section and has a higher order variation with respect to the thickness coordinate. The half-beam deflection is plotted for the same asymmetric core beam in
Figure 3.8, whereas Figure 3.9 shows deflection at core-top face sheet interface and core-
81 bottom face sheet interface near the contact in order to indicate the localized core compression.
Figure 3.7: Variation of the horizontal displacement, u, in the second half of the sandwich beam (Linear asymmetric core with Eh/E0 = 10)
Figure 3.8: Deflection, w, in the second half of the sandwich beam (Linear asymmetric core with Eh/E0 = 10)
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Figure 3.9: Variation of the deflection, w, in the second half of the sandwich beam at core - top face sheets interfaces, in the vicinity of contact (Linear asymmetric core with Eh/E0 = 10)
In order to visualize the location of the core maximum strains and to compare symmetric and asymmetric core maximum strains with the homogenous core maximum strains, contour plots of the strains (both normal and shear) for the half beam are presented in Figures 3.10 through 3.15. In the same figures the location of semi contact length (c = 3 mm ) is plotted. Maximum normal strain occurs at the middle core (x = L/2
= 100 mm) and slightly above the region with small Young’s modulus (bottom half of the beam for asymmetric core – Figure 3.10 and top half of beam for symmetric beam –
Figure 3.11). For the homogenous core maximum normal strain (Figure 3.12) occurs at the middle core (x = L/2 = 100 mm) and close to the core – top face sheet interface.
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Figure 3.10: Contour plot of normal strain in the functionally graded asymmetric core (with Eh/E0 = 10) normalized with respect to the maximum normal strain in the core, εxx max = 0.071 m/m, (contact length, c = 3 mm).
Figure 3.11: Contour plot of normal strain in the functionally graded symmetric core (with Eh/E0 = 10) normalized with respect to the maximum normal strain in the core, εxx max = 0.070, (contact length, c = 3 mm).
Figure 3.12: Contour plot of normal strain in the homogeneous core (with Eh/E0 = 1) normalized with respect to the maximum normal strain in the core, εxx max = 0.052, (contact length, c = 3 mm).
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As presented in Figure 2.14, FG core shear strain has a 1/z variation. Maximum shear strain in FG core is spread out along the beam where Young’s modulus is small (core bottom face sheet interface for asymmetric core – Figure 3.13 and middle beam for symmetric beam – Figure 3.14) and away of contact region. Maximum shear strain in the homogenous core is closed to the contact region, below the core – top face sheet interface
(Figure 3.15). Maximum stains are used in order to compare different cores.
Figure 3.13: Contour plot of shear strain in the functionally graded asymmetric core normalized with respect to the maximum shear strain in the core, γxz max = 0.44 (contact length, c = 3 mm).
Figure 3.14: Contour plot of shear strain in the functionally graded symmetric core normalized with respect to the maximum shear strain in the core, γxz max = 0.37. (contact length, c = 3 mm)
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Figure 3.15: Contour plot of shear strain in the homogeneous core normalized with respect to the maximum shear strain in the core, γxz max = 0.14 (contact length, c = 3 mm).
Figure 3.16 confirms the fact that axial stresses are larger in the face sheets than in the core (an assumption used in Chapter 2 for solving equilibrium equations). Also it can be noted that the axial stress σxx is higher on the topside of the core where contact occurs
(Figure 3.17). The stress concentration is not only due to contact but also due to the fact that the core Young’s modulus is higher near the interface.
Figure 3.16: Variation of the axial stress through the thickness in sandwich beam with functionally graded core at different cross-sections (Linear asymmetric core with Eh/E0 = 10)
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Figure 3.17: Variation of the axial stress through the thickness in functionally graded core (Linear asymmetric core with Eh/E0 = 10)
The normal stress σzz in the core for both asymmetric core and symmetric core is plotted in Figure 3.18 ((a) and respectively (b)). It can be seen that, although the maximum normal stress is slightly larger in the asymmetric core, in both cases it has same region of influence, very closed to the contact region (contact length is c = 3 mm).
These stresses are useful in determining if the core will get crushed due to impact or not.
Since the core density is higher in the contact region, the FG core will be able to withstand higher compressive stresses compared to uniform density core. In this case the skins are flexible so the load remains localized (it has large values for near the contact and is zero away of contact).
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(a) (b)
Figure 3.18: Variation of the normal (compressive) stress in (a) asymmetric and (b) symmetric functionally graded core (with Eh/E0 = 10) in the vicinity on the contact (contact length, c = 3 mm)
Usually the local failure starts in the core and results in core crushing, delamination and therefore significant reduction of the sandwich strength. There are different factors that can influence the local effect of indentation stress field. Here, the face sheets thickness, hf and indentor radius, R and the face sheet Young’s modulus, Ef are varied in order to study their influence on contact. The normal (compressive) stress in asymmetric functionally graded core (with Eh/E0 = 10) in the vicinity of the contact, at core - top face sheet interface is plotted for all cases (different hf, R and Ef). Figure 3.19 presents the face sheet thickness influence on impact: for the same asymmetric functionally graded core
(with Eh/E0 = 10) and same contact length, c, three cases with respect to the face sheet thickness are considered: hf = (0.3, 1, 2) mm. It can be noted that as the face sheet thickness increases not only the maximum compressive stress is increased, but because the face sheet becomes rigid, it spreads out the load. The variation of normal stress with indentor radius is plotted in Figure 3.20: for the same asymmetric functionally graded core (with Eh/E0 = 10) and same contact length, c, three cases with respect to the indentor radius are considered: R= (5, 10, 20) mm. In this case the maximum normal stress
88 decreases as the radius increases, but the non-zero stress region remains the same for different R-values.
Figure 3.19: Face sheet thickness influence on variation of the normal (compressive) stress in asymmetric functionally graded core (with Eh/E0 = 10) in the vicinity on the contact (c = 0.003 m), at core top face sheet interface (indenter radius, R = 10 mm).
The same type of collusions can be reached for the face sheet Young’s modulus influence on impact. Keeping the same geometry, indenter and the same type of functionally graded core and increasing the face sheet Young’s modulus results in an increase in the maximum compressive stress but the non-zero region remains the same
(Figure 3.21)
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Figure 3.20: Indentor radius influence on variation of the normal (compressive) stress in asymmetric functionally graded core (with Eh/E0 = 10) in the vicinity on the contact (c = 0.003 m), at core top face sheet interface (face sheet thickness, hf = 0.3 mm).
Figure 3.21: Face sheet Young’s modulus influence on variation of the normal (compressive) stress in asymmetric functionally graded core (with Eh/E0 = 10) in the vicinity on the contact (c = 0.003 m), at core top face sheet interface (face sheet thickness, hf = 0.3 mm).
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The transverse shear stresses in the core are plotted in Figures 3.22-3.23. Figure
3.22 gives a three dimensional variation in the asymmetric core whereas Figures 3.23 -
3.25 give contour plots for symmetric, asymmetric and homogenous core, respectively, in order to visualize the location of maximum shear stress. It is very interesting to see that the maximum shear stress occurs slightly below the core – top face sheet interface, and because of contact there is a severe stress concentration in the vicinity of contact (for graded core the maximum shear stress is at the end of the contact interval, Figures 3.23 and 3.24 whereas for homogenous core the maximum shear stress is outside the contact interval, Figure 3.25). These stresses will be useful in predicting the shear failure of the core due to impact.
Figure 3.22: Shear stress in functionally graded asymmetric core (with Eh/E0 = 10).
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Figure 3.23: Contour plot of shear stress (MPa) in linear asymmetric (with Eh/E0 = 10) core in the contact vicinity (c = 3 mm).
Figure 3.24: Contour plot of shear stress (MPa) in linear symmetric core (with Eh/E0 = 10) in the contact vicinity (c = 3 mm).
Figure 3.25: Contour plot of shear stress (MPa) in homogeneous core (with Eh/E0 = 1) in the contact vicinity (c = 3 mm).
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Figure 3.26 depicts the contact load-indentation (beam thickness compression) relations for various beams. It may be noted that the contact stiffness is higher when the core density is also higher near the face sheet. For an indentation of 0.2 mm the contact force increases by a factor of about 8 when the core modulus at the interface increases by a factor of 10. Evidently there is a shielding effect due to the stiff face sheet and that is why the increase in contact force is not in the same order as the core stiffness. The load- indentation relations are needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam.
Figure 3.26: Variation of contact force with indentation depth in functionally graded beams. (H.C.: homogeneous core; S.C.: symmetric core, A.C.: asymmetric core)
The central deflection of the beam as a function of applied force is plotted in Figure
3.27. The relations are approximately linear. The FG core affects the stiffness of the beam as the high-density core near the face sheet contributes significantly to the flexural stiffness of the beam. Also, it can be noted that the asymmetric beam is stiffer than the
93 symmetric one because near the top face sheet (where the contact occurs) the Young’s modulus for the asymmetric case is larger than that for symmetric case. Using these results the contact stiffness that describe the relation between the contact force and indentation (3.12) and the global stiffness that relates the contact force to the vertical displacement (3.13) were determined. These constants are needed for the quasi-static model. The total loads and maximum vertical displacements for various contact lengths are presented in Figure 3.27 in order to present the trend. Although the range of deflection in Figure 3.27 are much larger (~40 mm) for small deflection theory, the impactor mass and velocity in the example impact problems are taken such that the maximum deflection (~ 8 mm) does not exceed the limits for linear elastic approach.
Figure 3.27: Relation between contact force and the vertical displacement at bottom midpoint in functionally graded sandwich beams. (H.C.: homogeneous core; S.C.: symmetric core, A.C.: asymmetric core)
In order to determine if the core can withstand the contact loads due to impact not only the maximum stresses in the core needs to be computed but also it needs to be compared with corresponding strength values. Since the strength of a FG core is expected
94 to vary with location and also we do not have sufficient data on strength, it was decided to calculate the maximum strains. We assume that the maximum strain theory will hold well at all densities and thus strains can be used to determine the efficacy of FG cores compared to uniform cores. The variation of maximum normal strain εxx and maximum shear strain γxz in the core for a given contact force are plotted in the Figures 3.28 and
3.29, respectively. Using the maximum contact force values for a given impact energy
(282 J) in various panels we determined the maximum strains for that impact event.
These results are summarized in Table 3-1.
Table 3-1: Maximum normal and shear strains for a given impact energy of 282 J. FG denotes functionally graded core. The % reduction in strain in FG cores is with respect to the maximum strain in uniform core. Core type Eh/E0 Fmax (N) εx γxz % % Maximum Maximum Reduction Reduction Uniform 1 5.45×104 0.0300 - 0.0978 - FG, 5 7.03×104 0.0257 14.3 0.0830 15.1 Symmetric FG, 10 7.89×104 0.0194 35.3 0.0700 28.4 Symmetric FG, 5 7.39×104 0.0195 35.0 0.0500 48.9 Asymmetric FG, 10 8.31×104 0.0176 41.3 0.0368 62.3 Asymmetric
From Table 3-1 one can notice that maximum normal strain corresponding to the maximum impact load decreases by approximately 40% as the Young’s modulus of material at the top of the core increases by a factor of 10 for symmetric case, and by approximately 35% for the asymmetric cases. Also the maximum shear strain corresponding to the maximum impact load decreases by approximately 60% as the
Young’s modulus of the asymmetric material at top of the core is increased by a factor of
10. For the symmetric core case the reduction is with approximately 30% when Young’s
95 modulus of the symmetric material at top of the core is increased by a factor of 10. An interesting conclusion is that the maximum strain corresponding to the maximum impact load for asymmetric core is smaller than maximum strain for symmetric core. The reason for this is that the asymmetric core is stiffer in a larger region in the vicinity of contact compared to the symmetric core.
Figure 3.28: Contact force vs. maximum normal strain for beams with FGM core and homogeneous core. (H.C denotes homogeneous core; S.C. denotes symmetric core, A.S denotes asymmetric core)
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Figure 3.29: Contact force vs. maximum shear strain for beams with FGM core and homogeneous core. (H.C.: homogeneous core; S.C.: symmetric core, A.C.: asymmetric core)
From Figure 3.27 one can notice that the FG core affects the stiffness of the beam as the high-density core near the face sheet contributes significantly to the flexural stiffness of the beam. Hence, to maintain the same flexural stiffness for different core materials, the core thickness is varied as described below. The flexural stiffness is defined as:
h +h 2 f D = C z 2dz ij ∫ ij (3.19) ⎛ h ⎞ −⎜ +hf ⎟ ⎝ 2 ⎠ where Cij are stiffness coefficients of the constituents, face sheets and the core, h is the core thickness and hf is the face sheets thickness. The core thicknesses of the FG cores were chosen such that the flexural stiffness of the sandwich beam will be equal to that of the beam with homogeneous core. That is
FG hom D11 = D11 (3.20)
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Table 3-2: Core thicknesses for different materials with same flexural stiffness, D11 and same global stiffness. H.C.: homogeneous core; S.C.: symmetric core, A.S.: asymmetric core.
For constant D11 For constant global stiffness Core Global Core Core type Eh/E0 Global stiffness, thickness, stiffness, k thickness, h s k (MN/m) h (mm) (MN/m) (mm) s Uniform 1 20 5.5 20 5.65 Symmetric 5 12.58 4.4 14.44 5.65 Symmetric 10 10.08 3.67 12.88 5.65 Asymmetric 5 13.82 5.5 13.82 5.65 Asymmetric 10 11.28 4.78 12.32 5.65
The results for core thickness obtained using this method are presented in Table 3-
2. All the thicknesses for FGM obtained in this way are smaller than that for homogeneous core.
Figure 3.30: Variation of contact force with global deflection at bottom midpoint in functionally graded beams with different core thickness but the same flexural stiffness D11. (H.C: homogeneous core; S.C.: symmetric core, A.S.: symmetric core)
The load-deflection behavior of the sandwich beams having the same flexural stiffness is presented in Figure 3.30 and in Table 3-2. The results indicate that the beam
98 stiffness is not the same for all beams since the shear deformation effects wasn’t considered in Eq. (3.19).
Since the beam stiffness include both flexural and shear stiffness, an approximate method was used to obtain the thicknesses of FGM cores such that these beams exhibit the same stiffness as the homogenous core beam: the thicknesses of FGM were chosen such that the slopes of load-deflection curves to be the same as the slope of load- deflection curve for the homogeneous core. The results are presented in Table 3-2. One can notice that the values for the core thicknesses in the case of same global stiffness are slightly larger than those for the case of same flexural stiffness.
The impact analysis was repeated for the two sets of beams: having the same flexural stiffness and having the same total stiffness, and the results for maximum normal and shear strains in the core for all cases, for a given impact energy are presented in Figs.
3.31 and 3.32. From those plots some important results can be inferred. For both symmetric and asymmetric material as the ratio Eh/E0 increases the material become stiffer and the maximum strains decrease. Also the maximum strains corresponding to the maximum impact load for asymmetric core are smaller than maximum strains for symmetric core as the asymmetric core material is stiffer in the region of contact. For the same material (symmetric or asymmetric, with same ratio Eh/E0) as the core thickness decreases, the total stiffness also decreases and the maximum shear strain corresponding to the maximum impact load increases. Overall the “most graded” (with the largest ratio
Eh/E0) asymmetric core gives the smaller maximum strains. As a final conclusion, the results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.
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Figure 3.31: Comparison of maximum normal strains for different core materials.Eh/E0=1 represents the homogeneous core.
Figure 3.32: Comparison of maximum shear strains for different core materials. Eh/E0=1 represents the homogeneous core.
All the above results are derived and plotted based on assumed distribution method.
In what follows three contact methods are compared: the two numerical methods
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(assumed stress distribution and method of point matching) are compared with a finite element solution.
Using the commercial finite element software ABAQUS™ a 2D finite element model was created to model the problem of contact between a rigid indentor and a sandwich structure with functionally graded core (Figures 3.33 and 3.34). The FG core was partitioned through the thickness into a number of strips with constant properties, each strip containing more elements through the thickness. The elements considered were 2D, quadratic, plane strain elements. Boundary conditions assume wz(0, ) =w(L, z) =0 in order to be identical with those from Fourier series - Galerkin method. The load was applied by moving the rigid indenter with a specified distance. The contact length was calculated based on contact pressure at surface nodes: the contact length is the distance with non-zero contact pressure. In order to compare the analytical models with the finite element solution, the finite element contact length is calculated first. Then, the analytical solutions are obtained using this contact length.
Figure 3.33: Finite element simulation for half of the beam
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Figure 3.34: Finite element simulation of the contact between the functionally graded sandwich beam and the rigid spherical indentor
The contact pressure under indenter and the total load for the three models are calculated and plotted in Figure 3.35. As assumed, the contact pressure for the assumed stress distribution has a semi elliptical form. A striking feature of the contact pressure distribution for both methods of point matching and finite element is the sharp peaks in pressure at the edges of the contact. As noted by Johnson (1985), these peaks are not transmitted thought the skins as they are of short wave length than the spreading length scales of the beam. Another important observation is that the finite element gives a smaller contact force, although it has the same shape.
Figure 3.36 presents the deflections at core - top face sheet interface for the same contact length and different total forces (given in Figure 3.33). Because the total load differs among the models, the displacements differ also. For a better comparison the contact length was changed such that to obtained the same total load, and the displacements are plotted in Figure 3.37. As expected, they are matching.
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Figure 3.35: Contact pressure under indenter for a given contact length c = 3 mm and total load for three models
Figure 3.36: Deflections at core - top face sheet interface for same contact length and different contact load in three models
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Figure 3.37: Deflections at core - top face sheet interface for different contact length and same contact load in three models
Summary and Conclusions
Unlike for their solid metallic counterparts, predictions of the effects of low- velocity impact damage on sandwich structures are difficult because significant internal damage is achieved at impact energy levels lower than those required to create visible damage at the surface. So, it is important to understand the evolution of strains and stresses created inside a sandwich structure with functionally graded core in order to predict the impact damage. The first step in solving the dynamic impact is to understand the effects of a static contact.
This chapter solves the problems of contact and low-velocity impact between a rigid spherical indenter and a sandwich structure with functionally graded core. Two numerical methods are compared with a finite element solution. For the impact problem a simple dynamic model based on quasi-static behavior of the panel was used - the
104 sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects.
The beam geometry and face sheet properties influence on the contact are studied.
It is shown that changing the indenter radius or the face sheet Young’s modulus results in a change of maximum normal stress in the core – top face sheet interface over the same region whereas changing the face sheet thickness results in spreading the load over a larger region in the top core.
Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. Overall the “most graded” (with the largest ratio Eh/E0) asymmetric core gives the smaller maximum strains. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.
CHAPTER 4 MASS OPTIMIZATION FOR A SANDWICH BEAM WITH FUNCTIONALLY GRADED CORE
For a better understanding of the advantages of sandwich panels with FG cores over those with uniform cores, a comparison of the total mass of different panels is also needed. This chapter discuses three optimization problems for sandwich structure with
FG core based on two models presented in the literature: Choi and Sankar’s (2005) linear relation between relative density and Young’s modulus and Gibson and Ashby’s (1997) quadratic relation between relative density and Young’s modulus. The core dimensions and properties are changed such that to optimize the sandwich beam total mass and/or its flexural rigidity. The results show that sandwich panels with FG cores are more efficient than the one with uniform core.
Linear Model
Choi and Sankar (2005) assumed a cellular medium consisting of struts of square cross-section in a rectangular array. Based on mechanics of materials type calculation, and considering that the cellular solid is a homogeneous orthotropic material, they derived the following linear relation between the relative density of the foam material and its Young’s modulus:
ρ ∗ E∗ = 3 (4.1) ρs Es
105 106
where ρ* is the foam density, ρs is the solid material density, E* is foam Young’s modulus and Es is the solid material Young’s modulus. Their results were verified by finite element models.
Using the above formula, a relation between the density and Young’s modulus as functions of thickness coordinate, z can be derived:
3ρs ⎫ ρ(z) = E(z) ⎪ Es ⎪ ρ(z ) ⎬ ⇒ ρ(z) = 0 E(z) (4.2) 3ρ E(z ) ρ(z ) = s E(z )⎪ 0 0 0 ⎪ Es ⎭ where z0 is the thickness coordinate of a given/particular sandwich core point. As presented in Table 4-1 and also in Figure 4.1, three Young’s modulus variations are investigated: homogeneous core, linear asymmetric and linear symmetric about the core middle line. Based on equation (4.2), the linearity and the symmetry/asymmetry of the cores are preserved for the density also. The first step is to choose a Young’s modulus profile and a particular point z0. Then using equation (4.2) a formula for the densities is derived and then the sandwich total mass and sandwich flexural rigidities are calculated, using the formulas:
Lb hhf + /2 M ==∫∫ρρdV ∫∫()z dxdydz Vh00−−h/2 f (4.3) hhf + /2 Dz= ∫ 2 E()zdz −−hhf /2
Table 4-1: Variations of Young’s modulus, density, mass and flexural rigidity for the three cases investigated. Homogeneous core Linear asymmetric core Linear symmetric core
⎛⎞EEtb−+z EtEb ⎛⎞EEhm− z Core Young’s modulus EH (z) = E0 EzAb()=+E⎜⎟ EzSm()= E⎜⎟+1 ⎝⎠2/EhbA22Eb ⎝⎠2/EhmS2
⎛⎞Etb−+EEz tEb ⎛⎞EEhm− z Core density ρ H (z) = ρ0 ρρAb()z =+⎜⎟ρρSm()z = ⎜⎟+1 ⎝⎠2/EhbA22Eb ⎝⎠2/EhmS2
⎡ ⎤ ⎡ ⎤ ⎛⎞Et hA ⎛⎞EhhS
Total mass MH =+Lb(ρρ0hHf2 hf) M Ab=+Lb ⎢ρρ⎜⎟12+fhf⎥ M =+Lb ρρ12+h 107 E 2 Sm⎢ ⎜⎟ ff⎥ ⎣ ⎝⎠b ⎦ ⎣ ⎝⎠Em 2 ⎦
3 3 3 2 ⎛⎞h EE+ ⎛⎞h 3EE+ ⎛⎞h Flexural rigidity H btA hmS DEHf=+0 ⎜⎟D DA =+⎜⎟D f DDSf=+⎜⎟ 32⎝⎠ 32⎝⎠ 62⎝⎠
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In the Table 4-1, L is the beam length, b is the beam width, hs are the core thicknesses, hf
is the face-sheet thickness, Es and ρs are, respectively the Young’s modulus and density
of core for the three cases (see Figure 4.1), and Df is the face-sheets flexural rigidity
(same for all the cases):
2 ⎡ ⎛⎞hh⎛⎞232 ⎤ DEff=+⎢22⎜⎟hf⎜⎟hf+hf⎥ (4.4) ⎣⎢ ⎝⎠22⎝⎠ 3⎦⎥
hf hf hf
z z z
h hS hH 0 x A 0 x 0 x
(a) Homogeneous core: (b) Asymmetric core: (c) Symmetric core:
EH(z) = E0 EA(-hA/2) = Eb ES(-hS/2)= ES(hS/2)=Eh
ρH(z) = ρ0 EA(hA/2) = Et ES(0)=Em
ρA(-hA/2) = ρb ρS(-hS/2)=ρ(hS/2)=ρh
ρA(hA/2) = ρt ρS(0)= ρm
Figure 4.1: Geometry and notations for the three cases investigated: (a) sandwich beam with homogeneous core; (b) sandwich beam with linear asymmetric core and (c) sandwich beam with linear symmetric core. (The linearity and symmetry/asymmetry is referred at both elastic modulus and density.)
Because the three sandwich structures have identical face-sheets, only the cores
contribute to the difference in mass. In order to better investigate the cores mass, the
following ratios between the core masses are derived:
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⎡⎤EhSS( /2) ⎢⎥+1 hS M core ρ (0) E ()0 SS= ⎣⎦S 2 core h M H ρ0 2 H 2 (4.5) ⎡⎤EhAA()/2 ⎢⎥+1 hA Mhcore ρ (/− 2)Eh()− /2 AA= ⎣⎦AA 2 core h M H ρ0 2 H 2 where S denotes asymmetric core, A denotes asymmetric core and H denotes homogeneous core. The right hand sides of equations (4.5) contain all the parameters that can be changed in order to optimize the total mass and flexural rigidities of sandwich beams. The following two problems were solved:
4. Keeping Young’s moduli and densities fixed, determine the core thicknesses such that to obtain the same flexural rigidity.
5. Keeping core thicknesses fixed, determine Young’s moduli and densities such that to obtain the same mass and the same flexural rigidity.
Case 1: Same Flexural Rigidity
A sandwich beam with length L = 0.3 m, unit width b = 1 m, and facesheet
-3 thickness hf = 0.3×10 m is considered to investigate the effects of varying core properties through the thickness. The face-sheet Young’s modulus was chosen as 50 GPa and its density as 1,400 kg/m3.
The non-dimensional variations of core Young’s modulus with respect to z, as well as the non-dimensional variations of the core density given in Table 4-1 are plotted in
Figure 4.2 (as described by equation (4.2) they are equal). The core Young’s modulus is varied from 50 MPa at bottom core (asymmetric) or middle core (symmetric) to 50 × 10
MPa at core - top face-sheet interface. In the same trend the core density is varied from
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60 kg/m3 at bottom core (asymmetric) or middle core (symmetric) core to 60 × 10 kg/m3 at core - top face-sheet interface.
Figure 4.2: Through the thickness non-dimensional variations of core modulus and density considered for the homogeneous and functionally graded sandwich beam with same core thicknesses.
As described in Chapter 3, FG core affects the stiffness of the beam as the high- density core near the face sheet contributes significantly to the flexural stiffness of the beam. Hence, to maintain the same flexural stiffness for different core materials, the core thickness needs to be varied. For each of the three cases (homogeneous, linear symmetric and linear asymmetric) three cases from core thickness point of view were compared:
6. Same core thickness, h = 20×10-3 m;
7. The core thicknesses of the FG cores were chosen such that the flexural stiffness, D11 of the sandwich beam will be equal to that of the beam with homogeneous core.
8. The core thicknesses of the FG cores were chosen such that the global stiffness of the sandwich beam will be equal to that of the beam with homogeneous core.
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The results for the total mass are presented in Table 4-2. As it was expected (based on (4.5)) the total mass of a sandwich structure with functionally graded core that is stiffer/denser at core-top face-sheet interface is larger that the mass of a sandwich structure with homogeneous core.
ρSA(0) =−ρρ( h / 2) =0
EhSS()/2 ==EA(hA/2)10ES(0)=10EA(−hA/2) (4.6) core ⎧MhSS ⎪ core = 5.5 ⎪MhHH ⇒ ⎨ Mhcore ⎪ AA= 5.5 ⎪ core ⎩MhHH
For the same core thickness, FG core mass is 5.5 larger than the uniform core mass.
For different thickness, the homogeneous core is thicker (almost double) and spreads the mass outward to provide maximum moment of inertia (D11). Hence, the FG cores mass for different thicknesses (given in Table 4-2) and same stiffness is around 3.15 larger than the homogenous core.
Table 4-2: Total mass of sandwich structures with different density variations and different core thicknesses (face sheets mass, the same for all cases is Mf = 0.252 kg) Homogeneous core Symmetric core Asymmetric core -3 Same h = 20×10 m MH = 0.612 kg MS = 2.232 kg MA= 2.232 kg -3 -3 -3 Different h, same hH = 20×10 m hS = 10.08×10 m hA = 11.28×10 m flexural rigidity, D11 MH = 0.612 kg MS = 1.25 kg MA = 1.37 kg h = 20×10-3 m h = 12.88×10-3 m h = 12.32×10-3 m Different h, same H S A global stiffness MH = 0.612 kg MS = 1.52 kg MA = 1.47 kg
The results for strains and stresses for low-velocity impact of these nine sandwich panels are included in Chapter 3. For the same core thickness, the FG cores weight increases 5.5 times with respect to the homogenous core but there is a significant reduction (in the range of 28.4 – 62.3 %) in maximum strains (Table 3-1) of FG cores.
112
For different core thickness and same stiffness, the FG cores weight increases around
3.15 times with respect to the homogenous core and maximum strains is reduced by 4 -
52 % (Figures 3.31 and 3.32).
Case 2: Same Mass and Flexural Rigidity
The next step was to keep the identical geometry for both sandwich structures (with
FG core and with homogeneous core) and to change the material properties (Young modulus and density) such that to obtain the same weight, same flexural rigidity, D11 and same extensional stiffness A11. For this problem only two cores are compared: homogeneous core (described by E0, ρ0) and asymmetric core (described by Et, Eb, ρb)
It was found that, for a given linear asymmetric core (as presented in Table 1), the homogeneous density and Young’s modulus given by:
ρt + ρb ρ H (z) = ρ0 = 2 (4.7) E + E E (z) = E = t b H 0 2 are required to obtain the same mass, same flexural rigidity, D11 and same extensional stiffness A11. The non-dimensional density and Young’s modulus profiles are presented in
Figure 4.3.
113
10 Homogeneous core Asymmetric core
5 mm , e t a n i d r
o 0 ess Co kn c i
h -5 T
-10 Non-dimensional core density, ρ(z)/ρ(z ) and 0 1 0 10 Non-dimensional core elastic modulus, E(z)/E(z ) 0
Figure 4.3: Through the thickness non-dimensional variations of core modulus and density for the homogeneous and linear asymmetric functionally graded sandwich beams with same mass, same flexural rigidity and same extensional stiffness.
5 x 10 8 Homogeneous core Asymmetric core 7
6 ) N ( 5 F ad,
o 4 l act
nt 3 o C 2
1
0 0 0.05 0.1 0.15 0.2 Vertical displacement, w (m) bottom
Figure 4.4: Relation between contact force and vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams.
The impact problem presented in Chapter 3 was applied to both structures described above, and the results are plotted in Figures 4.4 – 4.9. Figure 4.4 presents
114 linear relation obtained between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The slopes of these curves give the flexural rigidity and as the problem requires they are almost the same (the small difference in the slopes comes from the local indentation effects).
Figure 4.5 gives the indentation (or core compression) – contact load relation and also the approximation functions needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. For each case, ki and n are determined, such that:
n F()α = kiα (4.8)
Also from Figure 4.5, it can be noted that the contact stiffness of the asymmetric
FG core is larger than the contact stiffness of the uniform core.
5 x 10 10 Homogeneous core n Approximation function F (α)=k α H i Asymmetric core 8 n Approximation function F (α)=k α
) A i N
,F ( 6 d loa t c
a 4 ont C 2
0 0 2 4 6 8 -4 Indentation,α (m) x 10
Figure 4.5: Variation of contact force with indentation depth in functionally graded and homogeneous sandwich beams and the approximation functions.
115
5 x 10 8 Homogeneous core 7 Asymmetric core
6 ) N (
F 5 , ad
o 4 l act t 3 n
Co 2
1
0 0 1 2 3 4 5 Contact length, c (m) -3 x 10
Figure 4.6: Variation of contact force with contact length in functionally graded and homogeneous sandwich beams.
Figure 4.7: Variation of contact force with contact length in functionally graded and homogeneous sandwich beams, approximation functions and Hertz contact law.
Figures 4.6 and 4.7 present contact length – contact load relations: it can be noticed that for the same contact length, the FG core load is larger than the uniform core load.
116
Also, from Figure 4.7, it can be noted that these relations not applicable for very small contact lengths, when the contact force acts only upon the top face-sheet. Instead, in this case, the Hertz contact law (Timoshenko and Goodier, 1970) should be applied:
cE2π F = f (4.9) 4R where Ef is face-sheet Young’s modulus and R is the indenter radius. Figure 4.7 presents the approximation functions needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. For each case, F0, kc and λ are determined, such that:
λ Fc()=+F0 kcc (4.10)
0.2 Homogeneous core Asymmetric core
0.15 x ma
0.1 xx ε
0.05
0 0 1 2 3 4 5 6 7 8 Contact load, F(N) 5 x 10
Figure 4.8: Contact force vs. maximum normal strain for beams with FGM core and homogeneous core.
117
0.7 Homogeneous core 0.6 Asymmetric core
0.5
x 0.4 ma
xz
γ 0.3
0.2
0.1
0 0 2 4 6 8 5 Contact load, F(N) x 10
Figure 4.9: Contact force vs. maximum shear strain for beams with FGM core and homogeneous core.
The variation of maximum normal strain εxx and maximum shear strain γxz in the core for a given contact force are plotted in the Figures 4.8 and 4.9, respectively. Using the maximum contact force values for a given impact energy (282 J) in various panels we determined the maximum strains for that impact event. These results are summarized in
Table 4-3.
Table 4-3: Maximum normal and shear strains for a given impact energy of 282 J. FG denotes functionally graded core. The % change in strain in FG core is with respect to the maximum strain in uniform core. Core type Eh/E0 Fmax (N) εx γxz Maximum % Change Maximum % Change Uniform 1 5.33×104 0.0125 - 0.0978 - FG, 10 5.16×104 0.0135 + 7.2 0.0368 - 23.32 Asymmetric
From Table 4-3 one can notice that maximum normal strain corresponding to the maximum impact load of FG asymmetric core increases by approximately 7%. But the maximum shear strain corresponding to the maximum impact load decreases by approximately 23%.
118
Quadratic Model
Gibson and Ashby (1997) presented a very detail derivation for mechanics of foams, based on relative density and the degree to which the cells are open or closed.
Their results are compared with, and calibrated against experimental data. The linear elastic behavior of foam made of open equiaxed cell loaded in compression is described by:
2 E∗∗⎛⎞ρ = C1 ⎜⎟ (4.11) Ess⎝⎠ρ where ρ* is the foam density, ρs is the solid material density, E* is foam Young’s modulus and Es is the solid material Young’s modulus and where C1 includes all of the geometric constants of proportionality. Experimental data show C1≈1. In this chapter two optimization problems based on this model are solved.
The core density vary continuously with the core Young’s modulus:
E(z) ρ(z) = ρ(z0 ) (4.12) E(z0 ) where z0 in the thickness coordinate of a given/particular sandwich core point.
From elastic modulus point of view, as presented in Table 4-4 and Figure 4.10, three cases are investigated: uniform core, linear asymmetric and linear symmetric about the core middle line. The first step is to choose a Young’s modulus profile and a particular point z0. Then using equation (4.12) a formula for the densities is derived and then the sandwich total mass and sandwich flexural rigidities are calculated, using the equations (4.3).
119
hf hf hf
z z z h h h 0 x 0 x 0 x
(a) Homogeneous core: (b) Asymmetric core: (c) Symmetric core:
EH(z) = E0 EA(-h/2) = Eb ES(-h/2)= ES(h/2)=Eh ρ (z) = ρ E (h/2) = E E (0)=E H 0 A t S m
Figure 4.10: Young’s modulus variation for the three cases investigated: (a) sandwich beam with homogeneous core; (b) sandwich beam with linear asymmetric core and (c) sandwich beam with linear symmetric core.
Table 4-4: Variations of Young’s modulus, density, mass and flexural rigidity Homogeneous core Linear asymmetric core Linear symmetric core Core ⎛⎞EEtb−+z EtEb ⎛⎞EEhm− z Young’s EH (z) = E0 EzAb()=+E⎜⎟ EzSm()= E⎜⎟+1 modulus ⎝⎠2/EhbA22Eb ⎝⎠2/EhmS2
Core E −+EEz E EE− z ρ (z) = ρ ρρ()z =+tb tb ρρ()z = hm +1 density H 0 Ab2/Eh22E Sm bA b 2/EhmS2
⎡ 3 ⎤ ⎡ 3 ⎤ ⎛⎞E 2 ⎛⎞E 2 ⎢ t −1 ⎥ ⎢ h −1 ⎥ ⎢ ⎜⎟ ⎥ ⎢ ⎜⎟ ⎥ 2hAbρ ⎝⎠Eb 2hSmρ ⎝⎠Em 120 Total mass M =+Lb ρρh 2 h M =+Lb ⎢ 2ρ h ⎥ M =+Lb ⎢ 2ρ h ⎥ H ( 0 Hff) AfE f SfE f ⎢ 3 t −1 ⎥ ⎢ 3 h −1 ⎥ ⎢ Eb ⎥ ⎢ Em ⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥
3 3 3 Flexural 2 ⎛⎞hH EEbt+ ⎛⎞hA 3EEhm+ ⎛⎞hS DEH =+0 ⎜⎟Dfs DDA =+⎜⎟ fs DSf=+⎜⎟D s rigidity 32⎝⎠ 32⎝⎠ 62⎝⎠
In the Table 4-4 L is the beam length, b is the beam width, ρs are:
ρ (-h /2) = ρρ(h /2) = ρ ΑΑAAbt (4.13) ρρρSS(-hSS/2) == (-h /2) hρS(0) = ρm all Ei are described in Figure 4.10 and Dfg is the face-sheets flexural rigidity:
⎡ 2 ⎤ ⎛ h ⎞ ⎛ h ⎞ 2 2 3 D fs = E f ⎢2⎜ ⎟ h f + 2⎜ ⎟h f + h f ⎥ (4.14) ⎣⎢ ⎝ 2 ⎠ ⎝ 2 ⎠ 3 ⎦⎥
For a given geometry (that is for a given core thickness, h and for a given face- sheets thickness, hf) and identical face-sheets two basic optimization problems are solved:
1. Determine core Young’s modulus and density profiles such that to obtain same flexural rigidity but allow different masses;
2. Determine core Young’s modulus and density profiles such that to obtain same mass but allow different flexural rigidity;
Case 1: Same Flexural Rigidity
Based on Table 4-4, in order to fully prescribe the densities and Young’s moduli of the three sandwich structures, the following eight quantities need to be given:
• Homogeneous core: E0, ρ0 • Asymmetric core: Et, Eb, ρb • Symmetric core: Eh, Em, ρm
It is assumed that the three sandwiches have identical face-sheets and identical dimensions. It is assumed that the value of Young’s modulus at middle core in the case of a symmetric core (Em) equals the Young’s modulus value at core-bottom face-sheet interface (Eb) for the asymmetric core.
So, given Eb ,dEt , Em = Eb etermine E0 , Eh such that to obtain the same flexural rigidity:
DH = DA = DS (4.15)
122
The resulting quantities are:
E + E E = b t 0 2 (4.16) E + 2E E = b t h 3
Then using the formulas given in Table 4-4 the densities profiles are determined. In order to illustrate the above concepts the following numerical examples are considered: three sandwich beams with length L = 0.3 m, unit width b = 1 m, core thickness h =
-3 -3 20×10 m and face-sheet thickness hf = 0.3×10 m. The face-sheet Young’s modulus was chosen as 50 GPa and its density as 1400 kg/m3.
The variation of Young’s modulus for the linear asymmetric core is between Eb = 5
MPa at the core-bottom face-sheet interface and Et = 5x10 MPa at the core-top face-sheet interface. The density at the middle core for the symmetric core (ρm) equals the density at
3 the core-bottom face-sheet interface for the asymmetric core (ρb = 60 kg/m ).
The non-dimensional (with respect to Em= Eb) variations of core Young’s modulus with respect to thickness coordinate z, given in Table 4-4 and based on (4.16) are plotted in Figure 4.11. The uniform core Young’s modulus has the average value between the asymmetric top core and bottom core Young’s modulus values. In the same
trend the non-dimensional (with respect to ρm= ρb) core densities (not linear) are presented in Figure 4.12
The results for the total mass of the three sandwich structures investigated are presented in Table 4-5. In this case, the total mass of a sandwich structure with homogeneous core is larger than the total mass of a sandwich structures with FG cores.
123
10 Homogeneous
m) Linear symmetric
m Linear asymmetric ( z , 5 e t a n i d r o 0 co ess ckn i
h -5 t re Co
-10 0 2 4 6 8 10 E(z)/E b
Figure 4.11: Core Young’s modulus variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core.
Table 4-5: Total mass of sandwich structures with same geometry and different density variations (face sheets mass, the same for all cases is Mf = 0.252 kg). The % reduction in the mass in FG core is with respect to the homogenous core mass. D11=3109.2 N-m Homogeneous core Symmetric core Asymmetric core
Total mass MH = 1.0963 kg MS = 0.9528 kg MA= 1.0686 kg % Reduction - -14 -3
The impact problem, described in Chapter 3 was applied to the three structures described above and the results are plotted in Figures 4.13 – 4.20. Figure 4.13 presents linear relation obtained between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The slopes of these curves give the flexural rigidity and as the problem required they are almost the same (the small difference in the slopes comes from the local indentation effects).
124
10 Homogeneous Asymmetric
mm) Symmetric ( z
, 5 e t a n i d r o 0 ss co e kn c i -5 h t e r Co -10 1 1.5 2 2.5 3 3.5 ρ(z)/ρ b
Figure 4.12: Core densities variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core.
4 x 10 8 Homogeneous 7 Symmetric Asymmetric 6
) 5 N 4 d,F (
Loa 3
2
1
0 0 0.05 0.1 0.15 0.2 Vertical displacement, w (m)
Figure 4.13: Relation between contact force and vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams.
125
4 x 10 8
Homogeneous 7 Symmetric Asymmetric 6 ) N 5 F ( , d 4 loa t c a 3 ont C 2
1
0 0 1 2 3 4 5 -3 Contact length, c (m) x 10
Figure 4.14: Relation between contact force and contact length in functionally graded and homogeneous sandwich beams.
6
5 N) k 4 ( F , ad
o 3 l act t
n 2 Hertz law Co Homogeneous 1 Symmetric Asymmetric
0 0 1 2 3 4 5 6 -4 Contact length, c (m) x 10
Figure 4.15: Relation between contact force and contact length in functionally graded and homogeneous sandwich beams and the Hertz contact law.
Figures 4.14 and 4.15 present contact length – contact load relations: it can be noticed that for the same contact length, the asymmetric FG core contact load is larger than the uniform core contact load but the symmetric FG core load equals the uniform
126 core contact load. Also, from Figure 4.15, it can be noted that these relations are not applicable for very small contact lengths, when the contact force acts only upon the top face-sheet. Instead, in this case, the Hertz contact law (Timoshenko and Goodier, 1970) should be applied:
cE2π F = f (4.17) 4R where Ef is face-sheet Young’s modulus and R is the indenter radius.
Figure 4.16 gives the indentation (or core compression) – contact load relation needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam. For a given contact load the symmetric core produces a larger indentation than the uniform core, but the asymmetric core and the uniform core give almost the same indentation.
4 x 10 8 Homogeneous 7 Symmetric Asymmetric
) 6 N (
F 5 , ad
o 4 l act
t 3 n
Co 2
1
0 0 0.5 1 1.5 2 Indentation, α (m) -3 x 10
Figure 4.16: Relation between indentation (core compression) and contact length in functionally graded and homogeneous sandwich beams.
127
Table 4-6: Maximum normal and shear strains normal and interfacial shear stresses for a given impact energy of 282 J. FG denotes functionally graded core. The % change in strain/stress in FG core is with respect to the maximum strain/stress in uniform core. Core F Interface τ max ε γ σ (MPa) xz type (kN) x xz x (MPa) % % % % Max Max Max Max Change Change Change Change Uniform 16.91 0.0415 - 0.1000 - 2.09x102 - 0.845 - FG, 15.78 0.0448 + 7.9 0.1735 + 23.5 1.92x102 - 9.3 1.109 + 31.1 Symm FG, 16.37 0.0430 + 3.6 0.0735 - 26.5 2.0x102 - 4.7 1.254 + 48.4 Asymm
The variation of maximum normal strain εxx and maximum shear strain γxz in the core for a given contact force are plotted in the Figures 4.17 and 4.18, respectively. For small contact lengths, which mean small contact loads, the maximum strains are almost the same. Using the maximum contact force values for a given impact energy (282 J) in various panels we determined the maximum strains for that impact event. These results are summarized in Table 4-6.
From Table 4-6 one can notice that maximum normal strain corresponding to the maximum impact load slightly increases for FG cores. The maximum shear strain corresponding to the maximum impact load decreases by approximately 27% as the
Young’s modulus of the asymmetric material at top of the core is increased by a factor of
1.8. But the maximum shear strain corresponding to the maximum impact load increases by approximately 24% as the Young’s modulus of the symmetric material at top of the core is increased by a factor of 1.3. An interesting conclusion is that the maximum strains
(both normal and shear) corresponding to the maximum impact load for asymmetric core is smaller than maximum strains for symmetric core. The reason for this is that the
128 asymmetric core is stiffer in a larger region in the vicinity of contact compared to the symmetric core.
The variation of maximum normal stress σxx in the core for a given contact force is plotted in the Figures 4.19. For a given contact load, the maximum normal stress σxx in the core is the same for the three cases considered. Table 4-6 presents the maximum normal stresses contact corresponding to the maximum impact load for a given impact energy (282 J) in various panels. The FG core maximum normal stress corresponding to the maximum impact load decreases by less than 10% with respect to the uniform core maximum normal stress.
0.2 Homogeneous Symmetric Asymmetric 0.15 x ma
0.1 xx ε
0.05
0 0 2 4 6 8 Contact load, F (N) 4 x 10
Figure 4.17: Contact force vs. core maximum normal strain for beams with FGM core and homogeneous core.
Figure 4.20 gives the variation of maximum interfacial shear stress τxz for a given contact force. There is a significant increase (48% for asymmetric and 31% for symmetric) in maximum interfacial stress stresses corresponding to the maximum impact load for a given impact energy (282 J) in the panels with FG core.
129
0.5 Homogeneous Symmetric Asymmetric 0.4
0.3 x ma
xz γ 0.2
0.1
0 0 1 2 3 4 5 6 7 Contact load, F (N) 4 x 10
Figure 4.18: Contact force vs. core maximum shear strain for beams with FGM core and homogeneous core.
8 x 10 8 Homogeneous 7 Symmetric Asymmetric 6
) 5 (Pa x 4 ma
xx
σ 3
2
1
0 0 1 2 3 4 5 6 7 8 4 Contact load, F (N) x 10
Figure 4.19: Contact force vs. core maximum normal stress for beams with FGM core and homogeneous core.
130
6 x 10 3 Homogeneous Symmetric 2.5 Asymmetric ) a
(P 2 x ma
xz
τ 1.5 l ia c a f
r 1 e t n I 0.5
0 0 1 2 3 4 5 6 7 8 Contact load, F (N) 4 x 10
Figure 4.20: Contact force vs. maximum interfacial shear stress for beams with FGM core and homogeneous core.
Case 2: Same Total Mass
Based on Table 4-4, in order to fully prescribe the densities and Young’s moduli of the three sandwich structures, the following eight quantities need to be given:
• Homogeneous core: E0, ρ0 • Asymmetric core: Et, Eb, ρb • Symmetric core: Eh, Em, ρm
The above quantities were determined based on the same mass assumption:
⎧ 3 ⎛⎞E 2 ⎪ t −1 ⎪ ⎜⎟ 2 ⎝⎠Eb ⎪ρ = ρ 0 E b ⎪ 3 t −1 ⎧MMHA= ⎪ Eb ⇒ (4.18) ⎨⎨3 ⎩MMHS= ⎪ ⎛⎞E 2 ⎪ ⎜⎟h −1 ⎪ 2 E ρ = ⎝⎠m ρ ⎪ 0 3 E m ⎪ h −1 ⎩⎪ Em
From the above equations result two sufficient conditions:
131
E E t = h Eb Em (4.19)
ρb = ρm
For a given ρb, ρ0 can be determined from (4.19). On the other hand, the relation between density and Young’s modulus is:
E(z) ρ(z) = ρ(z0 ) (4.20) E(z0 )
Let z0=-h/2 and z=z* such that to obtain the homogeneous core density:
ρ(z∗) = ρ0 (4.21)
Then, from (4.20) the homogeneous core Young’s modulus is:
2 ⎛ ρ ⎞ ⎜ 0 ⎟ E(z∗) = E0 = ⎜ ⎟ Eb (4.22) ⎝ ρb ⎠
In order to illustrate the above concepts we consider the following numerical examples: three sandwich beams with length L = 0.3 m, unit width b = 1 m, core
-3 -3 thickness h = 20×10 m and face-sheet thickness hf = 0.3×10 m. The face-sheet
Young’s modulus was chosen as 50 GPa and its density as 1,400 kg/m3.
The variation of Young’s modulus for the linear asymmetric core is between Eb = 5
MPa at the core-bottom face-sheet interface and Et = 5×10 MPa at the core-top face-sheet interface. The variation of Young’s modulus for the linear symmetric core is between Em
= 5 MPa at the middle core and Eh = 5×10 MPa at the core-top face-sheet interface. The density at the middle core for the symmetric core (ρm) equals the density at the core-
3 bottom face-sheet interface for the asymmetric core (ρb = 60 kg/m ). It is important to note that the values of Young’s modulus at the top core in the 2 FG cores are the same.
132
The non-dimensional variations of core Young’s modulus with respect to thickness coordinate z, given in Table 4 and based on (4.22) are plotted in Figure 21. For this case, the uniform core Young’s modulus based on (4.22) is E0 = 25.72 MPa. In the same style the non-dimensional core densities (not linear) are presented in Figure 4.22. For this case,
3 the uniform core density is ρ0 = 136.01 kg/m .
10 m) m ( z ,
e 5 t a n i d r o 0 Homogeneous Linear symmetric ess co Linear asymmetric ckn
i -5 h t re
Co -10 0 2 4 6 8 10 E(z)/E b
Figure 4.21: Core Young’s modulus variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core.
The results for the flexural rigidities of the three sandwich structures investigated are presented in Table 4-7. They are almost the same because the face-sheets (identical) carry the larger contribution.
Table 4-7: Flexural rigidity of sandwich structures with same geometry and different density variations (face sheets flexural rigidity, the same for all cases, is Df = 3090.9 N-m). The total mass is M = 1.0686 kg Homogeneous core Symmetric core Asymmetric core
Flexural rigidity DH = 3108.1 N-m DS = 3109.2 N-m DA=3116.7 N-m
133
10 Homogeneous
z Asymmetric , e
t Symmetric a 5 n i d r o
co 0 ess ckn i h -5 re t Co
-10 1 1.5 2 2.5 3 3.5 Non-dimensional density, ρ(z)/ρ b
Figure 4.22: Core densities variation for the three cases investigated: sandwich beam with homogeneous core; sandwich beam with linear asymmetric core and sandwich beam with linear symmetric core.
The impact problem was applied to the structures described above, and the results are plotted in Figures 4.23 – 4.28. Figure 4.23 presents linear relation obtained between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams. The slopes of these curves give the flexural rigidity and as they are almost the same because the three panels have identical face-sheets which bring the largest contribution in flexural rigidity.
Figure 4.24(a) – (c) present contact length – contact load relations: it can be noticed that for the same contact length, the FG core load is larger than the uniform core load.
Figure 4.24 presents also the Hertz law (4.9) used for very small contact lengths, when contact force acts only upon the top face-sheet. Figures 4.24(b) and (c) presents the approximation functions (4.10) needed in solving the problem of low-velocity impact of a rigid impactor and the sandwich beam.
134
x 104 8 Homogeneous 7 Symmetric Asymmetric 6 ) N ( 5 F , ad
o 4 l act t 3 n Co 2
1
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 w (m) bottom
Figure 4.23: Relation between contact force and the vertical displacement at bottom midpoint in functionally graded and homogeneous sandwich beams.
Figures 4.25-4.28 present the variation of maximum normal strain εxx, maximum
shear strain γxz, maximum normal stress σxx in the core and maximum interfacial shear
stress τxz for a given contact force. Using the maximum contact force values for a given
impact energy (282 J) in various panels the maximum strains and stresses for that impact
event were determined. These results are summarized in Table 4-8.
Table 4-8: Maximum normal and shear strains and stresses for a given impact energy of 282 J. FG denotes functionally graded core. The % change in strain/stress in FG core is with respect to the maximum strain/stress in uniform core. F Interface τ Core type max ε γ σ (MPa) xz (kN) x xz x (MPa) % % % % Max Max Max Max Change Change Change Change Uniform 16.78 0.0409 - 0.1025 - 212 - 0.827 - FG, 16.43 0.0386 - 5.6 0.1050 + 2.4 197 - 6.9 1.372 + 65.9 Symmetric FG, 16.37 0.0429 + 4.8 0.075 - 27 205 - 3.4 1.263 + 52.7 Asymmetric
135
10 Hertz law Homogeneous 8 Symmetric Asymmetric N) k (
F 6 , ad o l 4 act t n Co 2
0 0 1 2 3 4 5 6 -4 Contact length, c (m) x 10 (a)
5 5 10 10 Input data - Homogeneous Imput data -Asymmetric Aprox F(c)=F +k cλ Aprox F(c)=F +kcλ 0 i 0 ) ) N N ( F , , F ( d
ad 4 4
o 10
l 10 loa
t c a act t n ont C Co
3 103 10 -4 -3 -2 -4 -3 -2 10 10 10 10 10 10 Contact length, c (m) (b) Contact length, c (m) (c)
Figure 4.24: Relation between contact force and contact length in functionally graded and homogeneous sandwich beams, the Hertz contact law (a), and approximation functions for asymmetric core (b) and uniform core (c).
From Table 4-8 one can notice that maximum normal strain corresponding to the maximum impact load slightly increases for FG asymmetric core and slightly decreases for FG symmetric core. The maximum shear strain corresponding to the maximum impact load for symmetric core almost equals the one for uniform core. But the maximum shear strain corresponding to the maximum impact load decreases by approximately 27% as the Young’s modulus of the asymmetric material at top of the core is increased by a factor of 1.9. An interesting conclusion is that the maximum strain
136 corresponding to the maximum impact load for asymmetric core is smaller than maximum strain for symmetric core. The reason for this is that the asymmetric core is stiffer in a larger region in the vicinity of contact compared to the symmetric core.
0.2 Homogeneous Symmetric Asymmetric
0.15 x ma
0.1 xx ε
0.05
0 0 1 2 3 4 5 6 7 8 4 Contact load, F (N) x 10
Figure 4.25: Contact force vs. core maximum normal strain for beams with FGM core and homogeneous core.
0.4 Homogeneous 0.35 Symmetric Asymmetric 0.3
0.25 x ma
0.2 xz γ 0.15
0.1
0.05
0 0 1 2 3 4 5 6 4 Contact load, F (N) x 10
Figure 4.26: Contact force vs. core maximum shear strain for beams with FGM core and homogeneous core.
137
x 108 8 Homogeneous 7 Symmetric Asymmetric 6
) 5 a (P x 4 ma
xx
σ 3
2
1
0 0 2 4 6 8 Contact load, F (N) 4 x 10
Figure 4.27: Contact force vs. core maximum normal stress for beams with FGM core and homogeneous core.
6 x 10 3 Homogeneous Symmetric 2.5 Asymmetric )
(Pa 2 x ma
xz τ
1.5 al aci
rf 1 e t n I 0.5
0 0 2 4 6 8 Contact load, F (N) 4 x 10
Figure 4.28: Contact force vs. maximum interfacial shear stress for beams with FGM core and homogeneous core.
From Table 4-8 one can notice that maximum normal stress corresponding to the maximum impact load slightly decreases for FG cores. But the maximum interfacial shear stress corresponding to the maximum impact load for FG cores almost equals the
138 one for uniform core. But the maximum shear stress corresponding to the maximum impact load significantly increases (by approximately 66% for symmetric core and by approximately 50% for asymmetric core) as the Young’s modulus of the FG material at top of the core is increased only by a factor of 2.
Summary and Conclusions
Based on two models found in the literature and based on different relations between relative density and elastic modulus, this chapter solves three optimization problems: the geometry of the sandwich panels is kept constant while the properties
(Young’s modulus and density) are changed such that to optimize the total mass and/or the flexural rigidity.
Choi and Sankar’s (2005) linear relation between the relative density and Young’s modulus is used to determine the FG asymmetric core and the uniform core parameters such as to obtain panels with the same mass and same flexural rigidity. The results of the impact problem prove that there is a significant reduction (approximately 24%) in the maximum shear strain corresponding to maximum impact load of FG asymmetric core compared with the uniform core maximum shear strain, while the maximum normal strains corresponding to maximum impact load are almost the same.
The second model is based on the quadratic relation between the relative density and Young’s modulus derived by Gibson and Ashby (1997). The Young’s modulus and the density were varied such as to obtain panels with same flexural rigidity or to obtain panels with the same total mass. The results of the impact analysis provided similar results for both problems: among the three sandwich panels (with homogeneous core, symmetric FG core and asymmetric FG core), the one with FG asymmetric core gives the smaller maximum shear strain corresponding to the maximum impact load
139
(approximately 27% less than the uniform core) while the difference in maximum normal strain corresponding to the maximum impact load is small. Also, when the properties are changed such as to obtain the same flexural rigidity, the sandwich panel with asymmetric
FG core gives the smaller total mass.
These conclusions based on optimization studies emphasize again the superior capability of sandwich structures with FG cores over the sandwich structures with uniform cores.
CHAPTER 5 SUMMARY AND CONCLUSIONS
New developments in manufacturing methods allow gradation in material composition, structure and properties over a broad range of scales from nanometers to meters. It has been recognized that gradients in materials compositions are more damage- resistant at surface or interface between different materials where the failure usually occurs. These type of materials can be used for a large number of applications ranging from implant teeth to rocket frames.
A sandwich structure with functionally graded core is beneficial from two aspects: as a sandwich structure (e.g. exceptionally high flexural stiffness-to-weight ratio compared to other architectures) and as a graded material (e.g. lower impact damage).
A sandwich beam with soft, transversely flexible core must be approached with a higher-order theory rather than the classical theory. Higher-order refers to the order of polynomial in thickness coordinate that describes the displacements. Few of the reasons that make higher-order theory necessary for a sandwich panel with FG core - even if this theory involves more computational effort - are:
• The core gets compressed and the thickness of the beam is changed under localized loads.
• Plane sections do not remain plane after deformation, so displacement field is not a linear function of the thickness coordinate.
• Conditions imposed on one face sheet do not necessary hold for the other.
140 141
This study presented different analytical and finite element models for sandwich structures with functionally graded core. Two analytical models for sandwich structures with two types of core Young modulus are developed: exact solutions are presented for the exponential variation of core Young modulus and a combination of Fourier series and
Galerkin method for a polynomial variation of core Young modulus. In fact, any arbitrary varaiation of properties can be solved using the Fourier-Galerkin method. Using a polynomial approximation for the exponential function it is shown that the results of the two methods coincide. Those methods are also compared with models presented in literature and modified for this type of structure: two equivalent single-layer theories based on assumed displacements, a higher-order theory and a finite-element analysis. A very good agreement among the Fourier series-Galerkin method, the higher-order theory and the finite-element analysis is found.
This research solved the problems of contact and low-velocity impact between a rigid cylindrical indenter and a sandwich structure with functionally graded core. Two numerical methods are compared with the finite element solution. For the impact problem a simple dynamic model based on quasi-static behavior of the panel is used - the sandwich beam is modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects.
Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the
142 flexural stiffness was equal to that of a beam with homogeneous core. Overall the “most graded” (with the largest difference between the largest and smallest core Young’s moduli) asymmetric core gives the smaller maximum strains. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.
The trade-off between total mass and stiffness as well as a comparison between these structures and sandwich structures with homogenous core are also presented. Based on two models found in the literature that relates Youngs’ moduli and densities, three optimization problems are solved. The geometry of the sandwich panels is kept constant while the properties (Young’s modulus and density) are changed such that to optimize the total mass and/or the flexural rigidity.
Choi and Sankar’s (2005) linear relation between the relative density and Young’s modulus is used to determine the FG asymmetric core and the uniform core parameters such as to obtain panels with same mass and same flexural rigidity. The results of the impact problem prove that there is a significant reduction (approximately 24%) in the maximum shear strain corresponding to maximum impact load of FG asymmetric core compared with the uniform core maximum shear strain, while the maximum normal strains corresponding to maximum impact load are almost the same.
The second model is based on the quadratic relation between the relative density and Young’s modulus derived by Gibson and Ashby (1997). The Young’s modulus and the density were varied such that to obtain panels with same flexural rigidity or to obtain panels with same total mass. The results of the impact analysis provided similar results for both problems: among the three sandwich panels (with homogeneous core, symmetric
143
FG core and asymmetric FG core), the one with FG asymmetric core gives the smaller maximum shear strain corresponding to the maximum impact load (approximately 27% less than the uniform core) while the difference in maximum normal strain corresponding to the maximum impact load is small. Also, when the properties are changed such as to obtain the same flexural rigidity, the sandwich panel with asymmetric FG core gives the smaller total mass. These conclusions emphasize one more time the superior capability of sandwich structures with FG cores over the sandwich structures with uniform cores.
The objective of this research was to developed analytical models of sandwich structure with functionally graded cores and to use them in order to compare these structures with traditional sandwich structure with homogenous cores. For the particular problems solved and for the particular type of materials considered, the results indicate functionally graded cores are superior to uniform core to be used in sandwich construction.
LIST OF REFERENCES
Abid Mian, M., Spencer, A. J. M., 1998. Exact Solutions for Functionally Graded and Laminated Elastic Materials, Journal of the Mechanics and Physics of Solids, 46(12), 2283-2295.
Abot, J.L., Daniel, I.M., Gdoutos, E., 2002. Contact Law for Composite Sandwich Beams, Journal of Sandwich Structures and Materials, 4, 157-173.
Aboudi, J., Pindera, M. J., Arnold, S.M., 1999. Higher-Order Theory for Functionally Graded Materials, Composites: Part B, 30, 777–832.
Aboudi, J., Pindera, M.J., 1995a. Thermoelastic Theory for the Response of Materials Functionally Graded in Two Directions with Applications to the Free-Edge Problem. NASA TM 106882, Lewis Research Center, Cleveland, OH.
Aboudi, J., Pindera, M.J., Arnold, S. M., 1995b. Thermo-Inelastic Response of Functionally Graded Composites, International Journal of Solids Structures 32(12), 1675-1710.
Aboudi, J., Arnold, S.M., Pindera, M.J., 1994a. Response of Functionally Graded Composites to Thermal Gradients, Composites Engineering, 4, 1-18.
Aboudi, J., Pindera, M.J., Arnold, S.M., 1994b. Elastic Response of Metal Matrix Composites with Tailored Microstructures to Thermal Gradients, International Journal of Solids Structures, 31, 1393-1428.
Abrate, S., 1998. Impact on Composite Structures, Cambridge, Cambridge University Press.
Abrate, S., 1997. Localized Impact on Sandwich Structures with Laminated Facing, Applied Mechanics Review, 50(2), 69-82.
Abrate, S., 1994. Impact on Laminated Composite: Recent Advances, Applied Mechanics Review, 47(11), 517-544.
Abrate, S., 1991. Impact on Laminated Composite Materials, Applied Mechanics Review, 44 (4), 155-190.
Alexandrov, V.M., Pozharskii, D.A., 2001. Three-Dimensional Contact Problems, Dordrecht, Kluwer Academic Publishers.
144 145
Amada, S., Untao, S., 2001. Fracture Properties of Bamboo, Composites: Part B, 32, 451-459.
Amada, S., Ichikawa, Y., Munekata, T., Nagase, V., Shimizu, H., 1997. Fiber Texture and Mechanical Graded Structure of Bamboo, Composites Part B, 28B, 13-20.
Ambur, D. R., Cruz, J., 1995. Low-Speed Impact Response Characteristics of Composite Sandwich Panels, Proceedings of 36th AIAA/ASME/ASCEAHS/ASS Structures, Structural Dynamics and Material Conference, New Orleans, LA, 2681-2689.
Anderson, T.A., 2003. A 3-D Easticity Solution for a Sandwich Composite with Functionally Graded Core Subjected to Transverse Loading by a Rigid Sphere, Composite Structures, 60, 265-274.
Anderson,T., Madenci, E., 2000. Experimental Investigation of Low-Velocity Impact Characteristics of Sandwich Composites, Composite Structures, 50, 239-247.
Apetre, N.A., Sankar, B.V., Ambur, D.R., 2003. Functionally-Graded Sandwich Core with Arbitrary Variation in Properties, AIAA Paper 2002-1683, Proceedings of the 44th AIAA Structures, Structural Dynamics and Materials Conference, Norfolk, Virginia, April 7-10.
Apetre, N.A., Sankar, B.V., Venkataraman, S., 2002. Indentation of a Sandwich Beam with Functionally Graded Core, AIAA Paper 2002-1683, Proceedings of the 43rd AIAA Structures, Structural Dynamics and Materials Conference, Denver, Colorado, April 22-25.
Ashby, M., Evans, A., Fleck, N., Gibson, L.J., Hutchinson, J.W., Wadley, H.G., 2000. Metal Foams: A Design Guide, Boston, Butterworth Heinemann.
Barut, A., Madenici, E., Anderson, T., Tessler, A., 2002. Equivalent Single-Layer Theory for a Complete Stress Field in Sandwich Panels under Arbitrary Distributed Loading, Composite Structures, 58, 483-495.
Burton,W.S., Noor, A.K., 1994. Three-Dimensional Solutions for Thermomechanical Stresses In Sandwich Panels and Shells, Journal of Engineering Mechanics, 120(10), 2044-2071.
Chen, C-F., Frederick, D., 1992. Contact of Isotropic Square Plates with Rigid Spherical Indentors, International Journal of Solids and Structures, 30(5), 637-650.
Choi, I.H., Hong, C.S., 1994. New Approach for Simple Prediction of Impact Force History on Composite Laminates, AIAA Journal, 32(10), 2067-2072.
Choi, S., Sankar, B.V., 2005. A Micromechanical Method to Predict the Fracture Toughness of Cellular Materials, International Journal of Solids and Structures, 42(5-6), 1797-1817.
146
Chomkwak, V., Avula, X.J.R., 1990. A Higher-Order Theory for Laminated Composite Plates Using Lagrange Multiplier Technique, Computers and Structures, 37(5), 845-861.
Crook, A. W., 1952, A Study of Some Impacts between Metal Bodies by a Piezoelectric Method, Proceedings of the Royal Society, London, Series A, 212, page 377.
Delale, F., Erdogan, F., 1983. The Crack Problem for a Nonhomogeneous Plane, ASME Journal of Applied Mechanics, 50, 609-614.
El-Hadek, M., Tippur, H.V., 2003. Dynamic Fracture Parameters and Constraint Effects in Functionally Graded Syntactic Epoxy Foams, International Journal of Solids Structures, 40, 1885-1906.
Essenburg, F., 1975. On the Significance of the Inclusion of the Effect of Transverse Normal Strain in Problems Involving Beams with Surface Constraints, ASME Journal of Applied Mechanics, 97, 127-132.
Fischer-Cripps, A. C., 2000. Introduction to Contact Mechanics, Berlin, Springer
Fischer-Cripps, A. C., 1999. The Hertzian Contact Surface, Journal of Materials Science, 34, 129-137.
Frostig, Y., Baruch, M., 1994. Free Vibrations of Sandwich Beams with a Transverse Flexible Core: A High Order Approach, Journal of Sound and Vibration, 176(2), 195-208.
Frostig, Y., Baruch. M., Vinay, O., Sheinman, I., 1992. High-Order Theory for Sandwich Beams Behavior with Transversely Flexible Core, Journal of Engineering Mechanics, 118(5), 1026-1043.
Frostig, Y., Baruch, M., 1990. Bending of Sandwich Beams with Transversely Flexible Core, AIAA Journal, 28(3), 523-531.
Fukui, Y., 1991. Fundamental Investigation of Functionally Graded Materials Manufacturing System Using Centrifugal Force, JSME International Journal Series III, 34(1), 144-148.
Fukui, Y., Yamanaka, N., Enokida, Y., 1997. Bending Strength of an AI-AI3Ni Functionally Graded Material, Composites Part B, 28B, 37-43.
Giannakopoulos, A.E., Suresh, S., 1997a. Indentation of Solids with Gradients in Elastic Properties: Part I. Point force, International Journal of Solids Structures, 34(19), 2357-2392.
Giannakopoulos, A.E., Suresh, S., 1997b. Indentation of Solids with Gradients in Elastic Properties: Part II. Axisymmetric Indentors, International Journal of Solids Structures, 34(19), 2393-242.
147
Gibson, L.J., Ashby, F., 1997. Cellular Solids: Structure and Properties, Cambridge, Cambridge University Press
Gladwell, G. M. L., 1980. Contact Problems in the Classical Theory of Elasticity, Germantown, Sijthoff & Noordhoff
Han, W., Sofonea, M., 2002. Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Providence, American Mathematical Society and International press.
Hertz, H., 1881. On the Contact of Elastic Solids, J. Reine Angew. Math., 92, 156-171. Translated and reprinted in English in Hertz’s Miscellaneous Papers, Macmillian and Co., London.
Hohe, J., Librescu, L., 2003. A Higher-Order Geometrically Nonlinear Sandwich Shell Theory Accounting for Transverse Compressibility of the Core, Proceedings at 6th International Conference on Sandwich Structures, Ft. Lauderdale, Florida, CRC Press, 796-805.
Hsueh, C.H., Lee, S., 2003. Modeling of Elastic Thermal Stresses in Two Materials Joined by a Graded Layer, Composites Part B: Engineering, 34, 747-752.
Johnson, K. L., 1985. Contact Mechanics, Cambridge, Cambridge University Press.
Khludnev, A.M., Sokolowski, J., 1997. Modeling and Control in Solid Mechanics, Basel, Birkhauser Verlag
Koissin, V. Skvortsov, V, Krahmalev, S., Shilpsha, A, 2004. The Elastic Response of Sandwich Structures to Local Loading, Composite Structures, 63, 375-385.
Koizumi M., 1997. FGM Activities in Japan, Composites Part B, 28B, 1-4.
Kollar, L.P., Springer, G.S., 2003. Mechanics of Composite Structures, Cambridge, Cambridge University Press.
Lambros, J., Santare, M.H., Li, H., Sapna, G.H., 1999. A Novel Technique for the Fabrication of Laboratory Scale Model of FGM, Experimental Mechanics, 39(3).
Lee, S. M., Tsotsis, T. K., 2000. Indentation Failure Behavior of Honeycomb Sandwich Panels, Composite Science and Technology, 60, 1147-1159.
Lee, L.J., Huang, K.Y., Fann, Y.J., 1993. Dynamic Response of Composite Sandwich Plates Impacted by a Rigid Ball, Journal of Composite Materials, 27(13), 1238- 1256.
Liao, W.C., 2001. Smooth Indentation of a Multilayer Laminated by a Cylindrical Indentor, Composite Science and Technology, 61, 1801-1809
148
Lo, K.H., Christensen, R.M., Wu, E.M., 1977. A Higher-Order Theory of Plate Deformation, Part 1: Homogeneous Plates, Part 2: Laminate Plates, ASME Journal of Applied Mechanics, 44, 663-676.
March, H.W., 1948. Effects of Shear Deformation in the Core of a Flat Rectangular Sandwich Panel – 1. Buckling under Compressive End Loads, 2. Deflection under Uniform Transverse Load, U.S. Forest Products Laboratory Report 1583.
Markworth, A.J., Ramesh, K.S., Parks, W. P., 1995. Modeling Studies Applied to Functionally Ggraded Materials, Journal of Materials Science, 30, 2183-2193.
Matthews, F.L., Davies, G.A.O., Hitchings, D., Soutis, C., 2001. Finite Element Modeling of Composite Materials and Structures, Boca Raton, CRC Press.
McCormack, T.M., Miller, R., Kesler, O., Gibson, L.J., 2001. Failure of Sandwich Beams with Metallic Cores, International Journal of Solids and Structures, 38, 4901-4920.
Meyer-Piening, H.R., 1989. Remarks on Higher Order Sandwich Stress and Deflection Analyses, Proceeding of 1st International Conference on Sandwich Construction, Stockholm Sweden, 107-127.
Midlin, R. D., 1951. Influence of Rotary Inertia and Shear on Flexural Motion of Isotropic Elastic Plates, ASME Journal of Applied Mechanics, 73, 69-77.
Mittal, R.K., 1987. A Simplified Analysis of the Effects of Transverse Shear on the Response of Elastic Plates to Impact Landing, International Journal of Solids and Structures, 23(8), 1587-1596.
Noor, A. K., Burton, W. S., Bert, C. W., 1996. Computational Models for Sandwich Panels and Shells, Applied Mechanics Review, 49, 155- 198.
Ogorkiewicz, R. M., Sayigh, A.M., 1973. Deflection of Carbon Fibre/Acrylic Foam Sandwich Beams, Composites, Nov.
Olsson, R., 1997. Theory for Small Mass Impact on Sandwich Panels, Proceedings of the EUROMECH 360 Colloquium held in Saint-ETIENNE, Fr.
Olsson, R., McManus, H. L., 1996. Improved Theory for Contact Indentation of Sandwich Panels, AIAA Journal, 34(6), 1238 – 1244.
Olsson, R., 1992. Impact Response of Orthotropic Composite Plates Predicted from a One-Parameter Differential Equation, AIAA Journal, 30(6), 1587-1596.
Pagano, N. J., 1970. Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates, Journal of Composites Materials, 4, 20-34.
149
Peled, D., Frostig, Y., 1994. High-Order Bending of Sandwich Beams with Transversely Core and Nonparallel Skins, Journal of Engineering Mechanics, 120(6), 1255- 1269.
Petras, A., Sutcliffe, M.P.F., 1999. Indentation Resistance of Sandwich Beams, Composite Structures, 46, 413-424.
Pindera, M. -J., Dunn, P., 1995. An Evaluation of Coupled Microstructural Approach for the Analysis of Functionally Graded Composites via the Finite Element Method, NASA CR 195455. Lewis Research Center, Cleveland, OH.
Reddy, J. N., 2003. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, Boca Raton, CRC Press.
Reddy, J. N., 2000. Analysis of Functionally Graded Plates, International Journal for Numerical Methods in Engineering, 47, 663-684.
Reddy, J. N., 1990. A General Non-linear Third-order Theory of Plates with Moderate Thickness, International Journal for Non-Linear Mechanics, 25, 677-686.
Reddy, J. N., 1984. A Simple Higher-order Theory for Laminated Composite Plates, Journal for Applied Mechanics, 51, 745-752.
Reddy, J.N., Cheng, Z.Q., 2001. Three-dimensional Thermomechanical Deformations of Functionally Graded Rectangular Plates, European Journal of Mechanics, A/Solids, 20, 841–855.
Reddy, J.N., Miravete, A., 1995. Practical Analysis of Composite Laminates, Boca Raton, CRC Press.
Reissner, E., 1975. On Transverse Bending of Plates, Including the Effects of Transverse Shear Deformation, International Journal of Solids and Structures, 11, 569-573.
Reissner, E., 1945. The Effects of Transverse Shear Deformation on the Bending of Elastic Plates, ASME Journal of Applied Mechanics, 67, 69-77.
Rooney, F. and Ferrari, M, 1999. Tension, Bending and Flexure of Functionally Graded Cylinders, International Journal of Solids and Structures, 38, 413-421.
Sankar, B.V., 2001. An elasticity solution for functionally graded beams, Composites Science & Technology, 61, 689-696.
Sankar, B.V. 1992. Scaling of Low-velocity Impact for Symmetric Laminates. Journal of Reinforced Plastics and Composites, 11(3), 296-309.
Sankar, B.V., 1989. Smooth Indentation of Orthotropic Beams, Composite Science and Technology, 34, 94-111.
150
Sankar, B.V., Tzeng, J.T., 2002. Thermal Stresses in Functionally Graded Beams, AIAA Journal, 40(6), 1228-1232.
Sankar, B.V., Sun, C.T., 1985. Smooth Indentation of an Initially Stressed Orthotropic Beam, International Journal of Solids Structures, 21(2), 161-176.
Sankar, B.V., Sun, C.T., 1983. Indentation of a Beam by a Rigid Cylinder, International Journal of Solids and Structures, 19, 293-303.
Sarkar, P., Datta, S., Nicholson, P. S., 1997. Functionally Graded Ceramic/Ceramic and Metal/Ceramic Composites by Electrophoretic Deposition, Composites Part B, 28B, 49-56.
Sburlati, R., 2002. The Contact Behavior Between a Foam Core Sandwich Plate and a Rigid Indentor, Composites: Part B, 33, 325-332.
Shivakumar, K. N., Elber, W., Illg,W., 1985. Prediction of Impact Force and Duration During Low-Velocity Impact on Circular Composite Laminates, Trans. ASME, Journal of Applied Mechanics, 52, 674-680.
Singh, K.P., Paul, B., 1974. Numerical Solution of Non-Hertzian Elastic Contact Problems, ASME Journal of Applied Mechanics, 41, 484-490.
Sokolinsky, V. S., Nutt, S. R. 2004. Consistent Higher-Order Dynamic Equations for Soft –Core Sandwich Beams, AIAA Journal, 42(2).
Sun, C. T., Wu, C.L., 1991. Low-velocity Impact on Composite Sandwich Panels, Proceedings of 32nd AIAA/ASME/ASCEAHS/ASS Structures, Structural Dynamics and Material Conference, Baltimore, MD, 1123-1129.
Suresh, S., 2001. Graded Materials for Resistance to Contact Deformation and Damage, Science, 292, 2447-2451.
Suresh, S., Mortensen, A., 1998. Fundamentals of Functionally Graded Materials, IOM Communications Limited, London, United Kingdom.
Swanson, S. R., 2004. Core compression in Sandwich Beams under Contact Loading, Composite Structures, 64, 389-398
Swanson, S.R., 1999. An Examination of a Higher Order Theory for Sandwich Beam, Composite Structures, 44,169-177.
Tan, T.M., Sun, C. T., 1985. Use of Statical Indentation Laws in the Impact Analysis of Laminated Composite Plates. Journal of Applied Mechanics, 52, 6-12.
Tessler, A., Tsui, T., Saether, E., 1995. A {1, 2} - Order Theory for Elastodynamic Analysis of Thick Orthotropic Shells, International Journal of Solids and Structures, 32(22), 3237-3260.
151
Tessler, A., 1993. An Improved Plate Theory of {1, 2} - Order for Thick Composite Laminates, International Journal of Solids and Structures, 30(7), 981-1000.
Timoshenko,S., Goodier.J.,1970. Theory of Elasticity, New York, McGraw-Hill.
Vel, S. S., Batra, R. C., 2003. Three-dimensional Analysis of Transient Thermal Stresses in Functionally Graded Plates, International Journal of Solids and Structures, 40, 7181–7196.
Vel, S. S., Batra, R. C., 2002. Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates, AIAA Journal, 40(7).
Venkataraman, S., Sankar, B.V., 2001. Analysis of Sandwich Beams with Functionally Graded Core, AIAA Paper 2001-1281, Proceedings of the 42nd AIAA Structures, Structural Dynamics and Materials Conference, Seattle, Washington, April 16-19.
Vinson, J.R., 2001. Sandwich Structures, Applied Mechanics Review, 54(3), 201-214.
Vinson J.R., Sierakowski, R.L., 1986. The behavior of structures composed of composite materials, Dordrecht, Martinus Nijhoff Publisher.
Vinson J.R., Chou, T.W., 1975. Composite Materials and Their Use in Structures, John Wiley and Sons, New York, Applied Science Publishers LTD.
Wang, G-Y., Yew, C.H., 1989. Impact Damage in Composite Laminates, Computers and Structures, 37(6), 967-982.
Watari, F., Yokoyama, A., Saso, F., Uo, M., Kawasaki, T., 1997. Fabrication and Properties of Functionally Graded Dental Implant, Composites: Part B, 28B, 5-11.
Whitney, J. M., Pagano, N. J., 1970. Shear Deformation in Heterogeneous Anisotropic Plates, Journal of Applied Mechanics, 92, 1031-1036.
Williams D., Leggett D.M.A., Hopkins H. G., 1941. Flat Sandwich Panels under Compressive End Loads, A.R.C., R & M 1987.
Willis, J.R., 1966. Hertzian Contact of Anisotropic Bodies, Journal of Mechanics and Physics of Solids, 12, 163-176
Woo, J., Meguid, S.A., 2000. Nonlinear Analysis of Functionally Graded Plates and Shallow Shells. International Journal of Solids and Structures, 38, 7409-7421.
Wu, E., Yen, C-S., 1994. The Contact Behavior between Laminated Composite Plates and Rigid Spheres, ASME Journal of Applied Mechanics, 61, 60-66.
Yang, P.C., Norris, C.H., and Stavsky, Y., 1966. Elastic Wave Propagation in Heterogeneous Plates, International Journal of Solids and Structures, 2, 665-684.
152
Yang, S. H, Sun, C. T., 1982. Indentation Law for Composite Laminates, Composite Materials: Testing and Design, ASTM STP 787, American Society of Testing and Materials, 425-449.
Zenkert D., 1997. The Handbook of Sandwich Construction, Engineering Materials Advisory Service Ltd.
Zhu, H., Sankar, B. V., 2004. A Combined Fourier Series - Galerkin Method for the Analysis of Functionally Graded Beams. Journal of Applied Mechanics, 71(3), 421-424.
BIOGRAPHICAL SKETCH
Nicoleta A. Apetre graduated in 1998 with a Bachelor of Science in mathematics- mechanics from University of Bucharest, Romania. In 2000 she received her Master of
Science in mathematics-mechanics from the same university. Deciding to pursue a PhD in engineering, she joined the Department of Mechanical and Aerospace Engineering at
University of Florida, in January 2001.
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