DOCSLIB.ORG

Numerical Simulation of Excavations Within Jointed Rock of Infinite Extent

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numerical Simulation of Excavations Within Jointed Rock of Infinite Extent NUMERICAL SIMULATION OF EXCAVATIONS WITHIN JOINTED ROCK OF INFINITE EXTENT by ALEXANDROS I . SOFIANOS (M.Sc.,D.I.C.) May 1984- - A thesis submitted for the degree of Doctor of Philosophy of the University of London Rock Mechanics Section, R.S.M.,Imperial College London SW7 2BP -2- A bstract The subject of the thesis is the development of a program to study i the behaviour of stratified and jointed rock masses around excavations. The rock mass is divided into two regions,one which is. supposed to * exhibit linear elastic behaviour,and the other which will include discontinuities that behave inelastically.The former has been simulated by a boundary integral plane strain orthotropic module,and the latter by quadratic joint,plane strain and membrane elements.The # two modules are coupled in one program.Sequences of loading include static point,pressure,body,and residual loads,construetion and excavation, and quasistatic earthquake load.The program is interactive with graphics. Problems of infinite or finite extent may be solved. Errors due to the coupling of the two numerical methods have been analysed. Through a survey of constitutive laws, idealizations of behaviour and test results for intact rock and discontinuities,appropriate models have been selected and parameter ♦ ranges identif i ed. The representation of the rock mass as an equivalent orthotropic elastic continuum has been investigated and programmed. Simplified theoretical solutions developed for the problem of a wedge on the roof of an opening have been compared with the computed results.A problem of open stoping is analysed. * ACKNOWLEDGEMENTS The author wishes to acknowledge the contribution of all members of the Rock Mechanics group at Imperial College to this work, and its full financial support by the State Scholarship Foundation of * G reece. Furthermore thanks are due to: * Dr. J. 0-Watson, for his supervision of this study,and for introducing me to the boundary element method. Dr. J.W.Bray,for discussions. Messrs S.Budd and T.Sippel,for suggestions on programming. » Mr. J.A.Samaniego,for his friendship and exchange of ideas. Last but not least I whish to express my thanks to my wife for her patience during the hard period of the study, and to my mother for dealing with all my interests during my absence from home. * * TABLE OF CONTENTS Page ABSTRACT 2 ACKNOWLEDGEMENTS 3 TABLE OF CONTENTS 4 LIST OF FIGURES 7 LIST OF TABLES 11 NOTATION AND CONVENTIONS 14 CHAPTER 1 - INTRODUCTION 20 CHAPTER 2 - NUMERICAL MODELLING OF JOINTED ROCK 23 2.0 Distribution of stresses and displacements 23 2.1 The continuum 24 2.1.1 Mechanical properties 24 2.1.2 Simulation 26 2.2 Discontinuities - A literature survey 34 2.2.1 Mechanical properties 34 2.2.2 Sim ulation 39 2.3 The joint element 47 2.3.1 The element 47 2.3.2 The constitutive law 50 2.3.3 Iterative solution 63 2.3.4 Examples 72 2.4 Change of the geometry 77 2.4.1 Excavation 77 2.4.2 Construction 79 2.5 Types of activities 80 - 5 - Page CHAPTER 3 - THE ELASTIC REGION 81 3.0 General 81 3.1 Equivalent elastic properties of a jointed rock mass 82 3.1.1 Three orthogonal sets of joints - 82 3.1.2 Two oblique sets of joints 84 3.2 Implementation of the direct boundary integral method 86 3.2.1 The integral equation for the complementary function 87 3.2.2 Kernels U and T 89 3.2.3 Isoparametric element 91 * 3.2.4 Nodal collocation 93 3.2.5 Numerical integration 95 3.2.6 Rotation of axes 96 3.2.7 Particular integral 97 3.2.8 Infinite domain 98 3.3 Example 103 CHAPTER 4 - COUPLING REGIONS WITH CONTINUOUS AND DISCONTINUOUS DISPLACEMENT FIELDS 107 4.0 General 107 4.1 Symmetric coupling 108 * 4.2 V alidation 110 4.3 Inherent erro rs 125 4.3.1 Causes of erro rs 125 ** 4.3.2 Examples 130 -o-/ Page CHAPTER 5 - STABILITY OF AN OVERHANGING ROCK WEDGE IN AN EXCAVATION 142 5.0 General 142 5.1 Idealised behaviour 142 5.1.1 Symmetric wedge 144 5.1.2 Asymmetric wedge 161 5.2 Numerical solution 169 5.2.1 Symmetric wedge 170 5.2.2 Asymmetric wedge 182 CHAPTER 6 - APPLICATION OF THE PROGRAM TO ORE STOPING 188 CHAPTER 7 - SUMMARY AND CONCLUSIONS 200 APPENDIX 1 - Description of input for program AJROCK 206 APPENDIX 2 - Overall structure of the program 231 APPENDIX 3 - Additional information relevant to Chapter 3 233 A3.1 O rthotropic kernels 233 A3.2 Integration of kernel - shape function products over an element containing the first argument 243 A3.3 Particular integral 248 APPENDIX 4 - Estimate of error due to the assumption of continuous tractions at nodes 252 APPENDIX 5 - Graphs for estimating the stability of a wedge in a tunnel roof. 254 REFERENCES 279 - 7 - LIST OF FIGURES Page CHAPTER 2_ % Fig. 2.1 Eight node serendipity element 27 Fig. 2.2 Axes for transverse isotropy and global cartesian system 29 * Fig. 2.3 Sign convention for internal forces 29 Fig. 2.4 Isoparametric three node membrane element 31 Fig. 2.5 Peak shear strength 36 Fig. 2.6 Peak shear strength 37 Fig. 2.7 First joint element 43 Fig. 2.8 Three dimensional joint elements 43 Fig. 2.9 Isoparametric quadratic joint element 48 * Fig. 2.10 Failure criteria and parameters 51 Fig. 2.11 Load history effect on current peak shear strength 55 Fig. 2.12 Normal stress vs normal strain law 57 Fig. 2.13 Shear strain vs shear stress 58 * Fig. 2.14 Three dimensional sketch for e s ,a , T . 60 Fig. 2.15 Dilation vs shear strain law for the two models 62 Fig. 2.16 Iterative process (for compression) - Joint 1, ♦ no dilation 65 Fig. 2.17 Iterative process - Joint 2 without dilation 66 Fig. 2.18 Iterative process - Joint 1 with dilation 67 * Fig. 2.19 Iterative process - Joint 2 with dilation 68 Fig. 2.20 Iterations for simple examples 71 Fig. 2.21 Strain softening joints (examples) 73 CHAPTER 3_ Fig. 3.1 Three orthogonal sets of joints 83 Fig. 3.2 Two oblique sets of joints 83 Fig. 3.3 Conventions for kernel arguments 90 Fig. 3.4 Isoparametric boundary element 92 Fig- 3.5 Coordinate systems H.V and 1,2 92 Fig. 3.6 Integration over remote boundary 99 Fig. 3.7 Initial meshes for the examples of Section 3.3 104 Fig. 3.8 Boundary element region subjected to gravitational fie ld 105 Fig. 3.9 Plane strain and joint elements subjected to gravitational field 106 CHAPTER 4_ Fig. 4.1 Square block in tension 111 Fig. 4.2 A circular hole under pressure 111 Fig. 4.3 Hole within infinite rock mass modelled by boundary elem ents only 113 Fig. 4.4 Hole within infinite rock mass modelled by boundary and finite elements 113 Fig. 4.5 Tension of a long plate 116 Fig. 4.6 Lined opening 119 Fig. 4.7 Excavation of a circular tunnel 122 Fig. 4.8 Excavation of a circular tunnel 122 Fig. 4.9 Various methods to determine the limiting values of tractions at the two sides of a corner 126 Fig- 4.10 Two boundary element regions 131 Fig. 4.11 Circular disc 131 Fig. 4.12 Square block modelled by 32 boundary elements 134 Fig. 4.13 Square block modelled by boundary and plane strain elements 134 Fig. 4.14 Large problem with boundary and finite elements 138 CHAPTER _5 Fig. 5.1 Wedge id e a liz a tio n 143 Fig. 5.2 Symmetric wedge - Friction angle greater than a 145 Fig. 5.3 Symmetric wedge - Friction angle less than a 149 Fig. 5.4 Examples for very low stiffness ratio joints 151 Fig- 5.5 Behaviour of a symmetric rigid wedge 152 Fig- 5.6 Effect of intact rock flexibility 154 Fig. 5.7 Models for elastic wedge 155 Fig. 5.8 Stress redistribution 159 Fig. 5.9 Asymmetric rigid wedge 162 Fig. 5.10 Oblique wedge 167 F ig. 5.11 Wedge with ro tatio n 167 Fig. 5.12 Symmetric flexible wedge within rigid rock 171 Fig. 5.13 Symmetric elastic wedges within elastic rock ; a=20^ 175 Fig. 5.14 Example of stress redistribution in a symmetric wedge 181 Fig. 5.15 Asymmetric wedge 183 - 1 0 - Page CHAPTER 6 Fig. 6.1 Stope and drive geometry 189 % Fig. 6.2 Stope.drive,and surrounding rock discretization 189 Fig. 6.3 I n itia l mesh 193 Fig- 6.4 Gravitational loading 194 Fig. 6.5 Excavation of the drive 195 ♦ Fig. 6.6 First level ore excavation 196 Fig. 6.7 Second level ore excavation 197 Fig. 6.8 Third level ore excavation 198 APPENDICES Fig. A l.l Boundary element convention 213 * Fig. A1.2 Plane strain element convention 219 F ig . Al-3 Joint element convention 219 Fig. A2.1 Flow chart of program AJROCK 232 Fig. A3.1 Lines on which the orthotropic kernel U is undefined 240 * Fig. A3.2 Analytical integration of a logarithm - polynomial product over a straight line element 244 Fig. A3.3 Spiral method used for the determination of the diagonal terms of matrix T 244 * - 11 - LIST OF TABLES Page CHAPTER 2 Table 2.1 Shear strain regions 59 Table 2.2 Two plane strain and two joint elements 74 Table 2.3 One plane strain and three joint elements 76 CHAPTER .3 Table 3.1 Integration of kernel shape function products over an element containing the first argument 95 Table 3.2 Displacements at the nodes of a brick (example) 103 CHAPTER 4_ Table 4.1 Square block in tension 112 Table 4.2 Circular hole modelled by boundary elements only 114 Table 4.3 Circular hole modelled by boundary and finite elements - Displacements at nodes 114 Table 4.4 Circular hole modelled by boundary and finite elements - Stresses within plane strain elements 114 Table 4.5 Tension of a long plate modelled by symmetric mesh 115 Table 4.6 Tension of a long plate modelled by asymmetric mesh 117 Table 4.7 Lined circular tunnel with full adhesion on interface 123 Table 4.8 Lined circular tunnel with free slip on interface 123 Table 4.9 Excavation of a circular tunnel 124 Table 4.10 Prescribed values in finite and boundary elements 129 - 12 - Page Table 4.11 Equivalent nodal forces and displacements
Load more

Recommended publications
  • Static Analysis of Isotropic, Orthotropic and Functionally Graded Material Beams
    Journal of Multidisciplinary Engineering Science and Technology (JMEST) ISSN: 2458-9403 Vol. 3 Issue 5, May - 2016 Static analysis of isotropic, orthotropic and functionally graded material beams Waleed M. Soliman M. Adnan Elshafei M. A. Kamel Dep. of Aeronautical Engineering Dep. of Aeronautical Engineering Dep. of Aeronautical Engineering Military Technical College Military Technical College Military Technical College Cairo, Egypt Cairo, Egypt Cairo, Egypt [email protected] [email protected] [email protected] Abstract—This paper presents static analysis of degrees of freedom for each lamina, and it can be isotropic, orthotropic and Functionally Graded used for long and short beams, this laminated finite Materials (FGMs) beams using a Finite Element element model gives good results for both stresses and deflections when compared with other solutions. Method (FEM). Ansys Workbench15 has been used to build up several models to simulate In 1993 Lidstrom [2] have used the total potential different types of beams with different boundary energy formulation to analyze equilibrium for a conditions, all beams have been subjected to both moderate deflection 3-D beam element, the condensed two-node element reduced the size of the of uniformly distributed and transversal point problem, compared with the three-node element, but loads within the experience of Timoshenko Beam increased the computing time. The condensed two- Theory and First order Shear Deformation Theory. node system was less numerically stable than the The material properties are assumed to be three-node system. Because of this fact, it was not temperature-independent, and are graded in the possible to evaluate the third and fourth-order thickness direction according to a simple power differentials of the strain energy function, and thus not law distribution of the volume fractions of the possible to determine the types of criticality constituents.
    [Show full text]
  • Constitutive Relations: Transverse Isotropy and Isotropy
    Objectives_template Module 3: 3D Constitutive Equations Lecture 11: Constitutive Relations: Transverse Isotropy and Isotropy The Lecture Contains: Transverse Isotropy Isotropic Bodies Homework References file:///D|/Web%20Course%20(Ganesh%20Rana)/Dr.%20Mohite/CompositeMaterials/lecture11/11_1.htm[8/18/2014 12:10:09 PM] Objectives_template Module 3: 3D Constitutive Equations Lecture 11: Constitutive relations: Transverse isotropy and isotropy Transverse Isotropy: Introduction: In this lecture, we are going to see some more simplifications of constitutive equation and develop the relation for isotropic materials. First we will see the development of transverse isotropy and then we will reduce from it to isotropy. First Approach: Invariance Approach This is obtained from an orthotropic material. Here, we develop the constitutive relation for a material with transverse isotropy in x2-x3 plane (this is used in lamina/laminae/laminate modeling). This is obtained with the following form of the change of axes. (3.30) Now, we have Figure 3.6: State of stress (a) in x1, x2, x3 system (b) with x1-x2 and x1-x3 planes of symmetry From this, the strains in transformed coordinate system are given as: file:///D|/Web%20Course%20(Ganesh%20Rana)/Dr.%20Mohite/CompositeMaterials/lecture11/11_2.htm[8/18/2014 12:10:09 PM] Objectives_template (3.31) Here, it is to be noted that the shear strains are the tensorial shear strain terms. For any angle α, (3.32) and therefore, W must reduce to the form (3.33) Then, for W to be invariant we must have Now, let us write the left hand side of the above equation using the matrix as given in Equation (3.26) and engineering shear strains.
    [Show full text]
  • Orthotropic Material Homogenization of Composite Materials Including Damping
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Lirias Orthotropic material homogenization of composite materials including damping A. Nateghia,e, A. Rezaeia,e,, E. Deckersa,d, S. Jonckheerea,b,d, C. Claeysa,d, B. Pluymersa,d, W. Van Paepegemc, W. Desmeta,d a KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B box 2420, 3001 Heverlee, Belgium. E-mail: [email protected]. b Siemens Industry Software NV, Digital Factory, Product Lifecycle Management – Simulation and Test Solutions, Interleuvenlaan 68, B-3001 Leuven, Belgium. c Ghent University, Department of Materials Science & Engineering, Technologiepark-Zwijnaarde 903, 9052 Zwijnaarde, Belgium. d Member of Flanders Make. e SIM vzw, Technologiepark Zwijnaarde 935, B-9052 Ghent, Belgium. Introduction In the multiscale analysis of composite materials obtaining the effective equivalent material properties of the composite from the properties of its constituents is an important bridge between different scales. Although, a variety of homogenization methods are already in use, there is still a need for further development of novel approaches which can deal with complexities such as frequency dependency of material properties; which is specially of importance when dealing with damped structures. Time-domain methods Frequency-domain method Material homogenization is performed in two separate steps: Wave dispersion based homogenization is performed to a static step and a dynamic one. provide homogenized damping and material
    [Show full text]
  • Analysis of Deformation
    Chapter 7: Constitutive Equations Definition: In the previous chapters we’ve learned about the definition and meaning of the concepts of stress and strain. One is an objective measure of load and the other is an objective measure of deformation. In fluids, one talks about the rate-of-deformation as opposed to simply strain (i.e. deformation alone or by itself). We all know though that deformation is caused by loads (i.e. there must be a relationship between stress and strain). A relationship between stress and strain (or rate-of-deformation tensor) is simply called a “constitutive equation”. Below we will describe how such equations are formulated. Constitutive equations between stress and strain are normally written based on phenomenological (i.e. experimental) observations and some assumption(s) on the physical behavior or response of a material to loading. Such equations can and should always be tested against experimental observations. Although there is almost an infinite amount of different materials, leading one to conclude that there is an equivalently infinite amount of constitutive equations or relations that describe such materials behavior, it turns out that there are really three major equations that cover the behavior of a wide range of materials of applied interest. One equation describes stress and small strain in solids and called “Hooke’s law”. The other two equations describe the behavior of fluidic materials. Hookean Elastic Solid: We will start explaining these equations by considering Hooke’s law first. Hooke’s law simply states that the stress tensor is assumed to be linearly related to the strain tensor.
    [Show full text]
  • Composite Laminate Modeling
    Composite Laminate Modeling White Paper for Femap and NX Nastran Users Venkata Bheemreddy, Ph.D., Senior Staff Mechanical Engineer Adrian Jensen, PE, Senior Staff Mechanical Engineer Predictive Engineering Femap 11.1.2 White Paper 2014 WHAT THIS WHITE PAPER COVERS This note is intended for new engineers interested in modeling composites and experienced engineers who would like to get acquainted with the Femap interface. This note is intended to accompany a technical seminar and will provide you a starting background on composites. The following topics are covered: o A little background on the mechanics of composites and how micromechanics can be leveraged to obtain composite material properties o 2D composite laminate modeling Defining a material model, layup, property card and material angles Symmetric vs. unsymmetric laminate and why this is important Results post processing o 3D composite laminate modeling Defining a material model, layup, property card and ply/stack orientation When is a 3D model preferred over a 2D model o Modeling a sandwich composite Methods of modeling a sandwich composite 3D vs. 2D sandwich composite models and their pros and cons o Failure modeling of a 2D composite laminate Defining a laminate failure model Post processing laminate and lamina failure indices Predictive Engineering Document, Feel Free to Share With Your Colleagues Page 2 of 66 Predictive Engineering Femap 11.1.2 White Paper 2014 TABLE OF CONTENTS 1. INTRODUCTION ...........................................................................................................................................................
    [Show full text]
  • Dynamic Analysis of a Viscoelastic Orthotropic Cracked Body Using the Extended Finite Element Method
    Dynamic analysis of a viscoelastic orthotropic cracked body using the extended finite element method M. Toolabia, A. S. Fallah a,*, P. M. Baiz b, L.A. Louca a a Department of Civil and Environmental Engineering, Skempton Building, South Kensington Campus, Imperial College London SW7 2AZ b Department of Aeronautics, Roderic Hill Building, South Kensington Campus, Imperial College London SW7 2AZ Abstract The extended finite element method (XFEM) is found promising in approximating solutions to locally non-smooth features such as jumps, kinks, high gradients, inclusions, voids, shocks, boundary layers or cracks in solid or fluid mechanics problems. The XFEM uses the properties of the partition of unity finite element method (PUFEM) to represent the discontinuities without the corresponding finite element mesh requirements. In the present study numerical simulations of a dynamically loaded orthotropic viscoelastic cracked body are performed using XFEM and the J-integral and stress intensity factors (SIF’s) are calculated. This is achieved by fully (reproducing elements) or partially (blending elements) enriching the elements in the vicinity of the crack tip or body. The enrichment type is restricted to extrinsic mesh-based topological local enrichment in the current work. Thus two types of enrichment functions are adopted viz. the Heaviside step function replicating a jump across the crack and the asymptotic crack tip function particular to the element containing the crack tip or its immediately adjacent ones. A constitutive model for strain-rate dependent moduli and Poisson ratios (viscoelasticity) is formulated. A symmetric double cantilever beam (DCB) of a generic orthotropic material (mixed mode fracture) is studied using the developed XFEM code.
    [Show full text]
  • The Interaction Integral for Fracture of Orthotropic Functionally Graded Materials: Evaluation of Stress Intensity Factors Jeong-Ho Kim, Glaucio H
    International Journal of Solids and Structures 40 (2003) 3967–4001 www.elsevier.com/locate/ijsolstr The interaction integral for fracture of orthotropic functionally graded materials: evaluation of stress intensity factors Jeong-Ho Kim, Glaucio H. Paulino * Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205North Mathews Avenue, Urbana, IL 61801, USA Received 26 July 2002; received in revised form 11 March 2003 Abstract The interaction integral is an accurate and robust scheme for evaluating mixed-mode stress intensity factors. This paper extends the concept to orthotropic functionally graded materials and addresses fracture mechanics problems with arbitrarily oriented straight and/or curved cracks. The gradation of orthotropic material properties are smooth func- tions of spatial coordinates, which are integrated into the element stiffness matrix using the so-called ‘‘generalized isoparametric formulation’’. The types of orthotropic material gradation considered include exponential, radial, and hyperbolic-tangent functions. Stress intensity factors for mode I and mixed-mode two-dimensional problems are evaluated by means of the interaction integral and the finite element method. Extensive computational experiments have been performed to validate the proposed formulation. The accuracy of numerical results is discussed by com- parison with available analytical, semi-analytical, or numerical solutions. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Functionally graded material (FGM); Fracture mechanics; Stress intensity factor (SIF); Interaction integral; Finite element method (FEM); Generalized isoparametric formulation (GIF) 1. Introduction Functionally graded materials (FGMs) are a new class of composites in which the volume fraction of constituent materials vary smoothly, giving a nonuniform microstructure with continuously graded macro- properties (Hirai, 1993; Paulino et al., 2002).
    [Show full text]
  • Plane Strain Theory for an Orthotropic Cylinder with Pure Bending
    Appendix A Plane Strain Theory for an Orthotropic Cylinder with Pure Bending Lekhnitskii (1963) derives a general theory for the state of stress in an elastic body bounded by a cylindrical surface and possessing cylindrical anisotropy where the stresses do not vary along the generator. In what follows, the analysis given in Archer (1976) for the transversely isotropic material model is extended to the orthotropic case (Archer 1985). The equations of equilibrium in cylindrical coordinates after dropping dependence on the axial coordinate, and the "torsion" equilibrium equation, and body force terms become ou, +! O't"re + u, -O"e =O or r oe r (A.1) O't"re +! OO"e + 2 't"re =O or r o(} r and the generalized Hooke's law for the orthotropic case e, [all [ Be l= a12 (A.2) Sz a13 where IX., IX9 , and IXz are prescribed strains as measured by strain relief experiments (Nicholson 1971, 1973a, Sasaki et al. 1978) and taken as independent ofz; u., ue, O"z, -r,e are stress components; e., Be, Bz, y9, are the strain components; r, (}, z the cylindrical coordinates; and all ... %6 the elastic constants. The shear stresses 't"ez and 't"rz have been uncoupled from the system (A.1-A.2) and are applicable to torsion problems which do not concern us here. The relevant strain-displacement relations are ou 1 ov u e,=or ' ee=~ oe+r (A.3) ow 1 ou ov v Bz= OZ ' l're=~ o(} +or-~ A modification of the Lekhnitskii (1963) derivation which includes the growth strains follows if we set (A.4) then since Sz and IXz are independent of z, we may write D +1Xz=a33 (Ar sin(}+ Brcos (}+C) , (A.5) 206 Plain Strain Theory for an Orthotropic Cylinder with Pure Bending where A and B correspond to "pure bending" stress terms and C to a "uniform axial" stress term.
    [Show full text]
  • Analysis of a Kinked Crack in an Anisotropic Material Under Antiplane Deformation† H
    Journal of Mechanical Science and Technology 26 (2) (2012) 411~419 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-1025-4 Analysis of a kinked crack in an anisotropic material under antiplane deformation† H. G. Beom*, J. W. Lee and C. B. Cui Department of Mechanical Engineering, Inha University, 253 Yonghyun-dong, Incheon, 402-751, Korea (Manuscript Received July 21, 2011; Revised September 23, 2011; Accepted September 23, 2011) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract This paper analyzes the asymptotic problem of a kinked crack in an anisotropic material under antiplane deformation. Using the linear transformation method proposed in this paper, a solution to the asymptotic problem of a kinked crack in an anisotropic material can be obtained from the solution of the corresponding isotropic kinked crack problem. The exact solution of the stress intensity factor for the kinked crack in the anisotropic material is obtained from the solution of the isotropic problem. The effect of the kink angle and two ani- sotropic parameters on the stress intensity factor is discussed for the inclined orthotropic material as well as the anisotropic material. In order to verify the exact solution of the stress intensity factor, numerical calculations are performed by using finite element analysis.
    [Show full text]
  • Orthotropic Materials
    Kapitel 2 Orthotropic Materials 2.1 Elastic Strain matrix Elastic strains are related to stresses by Hooke's law, as stated below. The stress- strain relationship is in each material point formulated in the local cartesian coor- dinate system. "e = Cσ (1) σ is the stress vector, which uses the same convention as the strain vector. The stress vector is given below. T σ = [σl σr σt ¿lr ¿lt ¿rt] (2) C is the material compliance matrix. The compliance matrix is given as 2 3 1 ¡ ºrl ¡ ºtl 0 0 0 El Er Et 6 ¡ ºlr 1 ¡ ºtr 0 0 0 7 6 El Er Et 7 6 º º 1 7 6 ¡ lt ¡ rt 0 0 0 7 C = 6 El Er Et 7 (3) 6 0 0 0 1 0 0 7 6 Glr 7 4 0 0 0 0 1 0 5 Glt 0 0 0 0 0 1 Grt E's are moduli of elasticity, G's are shear moduli and º's are Poissons ratios. The compliance matrix should be symmetric, which gives rise to the following restric- 4 KAPITEL 2. ORTHOTROPIC MATERIALS 2.2. COORDINATE SYSTEMS tions: Er Et Et ºrl = ºlr ; ºtl = ºlt ; ºtr = ºrt (4) El El Er The inverse of the compliance matrix is the material stiness matrix, D, which give the stresses produced by an elastic strain state. 2 3 El (1 ¡ ºrtºtr)=k El (ºrtºtl + ºrl)=k El (ºrlºtr + ºtl)=k 0 0 0 6 7 6 Er (ºltºtr + ºlr)=k Er (1 ¡ ºltºtl)=k Er (ºlrºtl + ºtr)=k 0 0 0 7 6 7 6 7 D = 6 Et (ºlrºrt + ºlt)=k Et (ºltºrl + ºrt)=k Et (1 ¡ ºlrºrl)=k 0 0 0 7 6 7 6 0 0 0 Glr 0 0 7 4 5 0 0 0 0 Glt 0 0 0 0 0 0 Grt (5) where k = 1 ¡ ºrtºtr ¡ ºlrºrl ¡ ºltºtl ¡ ºlrºrtºtl ¡ ºrlºtrºlt.
    [Show full text]
  • An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials
    NASA/TM–2011-217314 An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials Michael P. Nemeth Langley Research Center, Hampton, Virginia December 2011 NASA STI Program . in Profile Since its founding, NASA has been dedicated to CONFERENCE PUBLICATION. Collected the advancement of aeronautics and space science. papers from scientific and technical The NASA scientific and technical information (STI) conferences, symposia, seminars, or other program plays a key part in helping NASA maintain meetings sponsored or co-sponsored by NASA. this important role. SPECIAL PUBLICATION. Scientific, The NASA STI program operates under the technical, or historical information from NASA auspices of the Agency Chief Information Officer. It programs, projects, and missions, often collects, organizes, provides for archiving, and concerned with subjects having substantial disseminates NASA’s STI. The NASA STI program public interest. provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical TECHNICAL TRANSLATION. English- Report Server, thus providing one of the largest language translations of foreign scientific and collections of aeronautical and space science STI in technical material pertinent to NASA’s mission. the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Specialized services also include creating custom Series, which includes the following report types: thesauri, building customized databases, and organizing and publishing research results. TECHNICAL PUBLICATION. Reports of completed research or a major significant phase For more information about the NASA STI of research that present the results of NASA program, see the following: programs and include extensive data or theoretical analysis.
    [Show full text]
  • Module 3 Constitutive Equations
    Module 3 Constitutive Equations Learning Objectives • Understand basic stress-strain response of engineering materials. • Quantify the linear elastic stress-strain response in terms of tensorial quantities and in particular the fourth-order elasticity or stiffness tensor describing Hooke's Law. • Understand the relation between internal material symmetries and macroscopic anisotropy, as well as the implications on the structure of the stiffness tensor. • Quantify the response of anisotropic materials to loadings aligned as well as rotated with respect to the material principal axes with emphasis on orthotropic and transversely- isotropic materials. • Understand the nature of temperature effects as a source of thermal expansion strains. • Quantify the linear elastic stress and strain tensors from experimental strain-gauge measurements. • Quantify the linear elastic stress and strain tensors resulting from special material loading conditions. 3.1 Linear elasticity and Hooke's Law Readings: Reddy 3.4.1 3.4.2 BC 2.6 Consider the stress strain curve σ = f() of a linear elastic material subjected to uni-axial stress loading conditions (Figure 3.1). 45 46 MODULE 3. CONSTITUTIVE EQUATIONS σ ˆ 1 2 ψ = 2 Eǫ E 1 ǫ Figure 3.1: Stress-strain curve for a linear elastic material subject to uni-axial stress σ (Note that this is not uni-axial strain due to Poisson effect) In this expression, E is Young's modulus. Strain Energy Density For a given value of the strain , the strain energy density (per unit volume) = ^(), is defined as the area under the curve. In this case, 1 () = E2 2 We note, that according to this definition, @ ^ σ = = E @ In general, for (possibly non-linear) elastic materials: @ ^ σij = σij() = (3.1) @ij Generalized Hooke's Law Defines the most general linear relation among all the components of the stress and strain tensor σij = Cijklkl (3.2) In this expression: Cijkl are the components of the fourth-order stiffness tensor of material properties or Elastic moduli.
    [Show full text]