Maste R's D E G Re E Thesis

Home , Orthotropic material, Transverse isotropy

Stability of End Notched Flexure Specimen THESIS

E E

Master’s (one year) Degree Thesis in Applied Mechanics Level 30 ECTS Spring term, 2010

Arun Gojuri

Supervisor: Ulf Stigh

Examiner: Thomas Carlberger MASTER’S

School of Technology & Society Skövde University BOX 408 SE-541 28 Skövde Sweden

Preface

This work has been carried out during the spring semester year 2010 at the Department of Mechanical Engineering at the University of Skövde, Sweden.

First and foremost, I would like say thanks to my supervisor Prof. Ulf Stigh not only for his help and support, but also for sharing the knowledge and for always being able to help. Moreover, for the opportunity to perform this thesis work and also for his way of making the topic exciting and interesting.

In a deep thanks to Dr. Kent Salomonsson, Dr. Tobias Andersson and Dr. Anders Biel for their great supports during my theoretical lectures works and commitment concerning various software related issues.

Also thanks to my friends Tomas Walander, Daniel Svensson, and Saidul Gundeboina for help and sharing knowledge with me during this work.

Finally yet importantly, wide thanks are dedicated to the examiner, Dr. Thomas Carlberger for giving an opportunity to present my thesis work here.

Skövde. December 2010 Arun Gojuri

Abstract

This paper deals with two-dimensional Finite Element Analysis of the End Notched Flexure (ENF) specimen. The specimen is known to be unstable if the crack length is shorter than some critical crack length acr. A geometric linear two-dimensional Finite Element (FE) analysis of the ENF specimen is performed to evaluate acr for isotropic and orthotropic elastic materials, respectively. Moreover, the Mode II Energy Release Rate (ERR) JII and the compliance of the specimen are calculated. The influence of anisotropy is studied. Comparisons are made with the results from beam theory. This work is an extension of previous work.

Keywords: Energy release rate, Finite element method, ENF-specimen, Delamination, Composite materials isotropic & orthotropic, Controlled displacement.

Table of Contents

Preface ...... 2 Abstract ...... 3 1. Introduction ...... 5 1.1 ENF specimen ...... 5 1.2 Contour integral J ...... 7 1.3 Isotropic material ...... 8 1.4 Orthotropic material ...... 9 2. Materials and method ...... 10 2.1 Geometry of specimen ...... 10 2.2 Material properties ...... 10 2.3 Finite element method simulations ...... 11 2.4 Evaluation of J ...... 12 3. Numerical studies ...... 13 4. Discussion and Conclusion ...... 16 5. References ...... 17

1. Introduction

Much work has recently been devoted to the failure mechanism of composites. Consequently, many new fracture tests have been devised for measuring the fracture properties. Most such tests and standard test procedures are limited to studies of delamination where a crack propagates between the plies. Fracture mechanics are commonly employed to accommodate crack tip singularities, and the energy release rate (ERR) is a physically well-defined quantity that is experimentally measurable using the compliance technique (Russell & Street, 1982 and Broek, 1984).

There are three types of loadings of the crack tip, called Mode I, Mode II and Mode III, respectively, cf. Fig. 1. Mode I is the opening mode. Here tensile stresses act on the plane heading the crack tip. Mode II is an in-plane sliding mode. Here the shear stresses act in the plane of the crack and perpendicularly to the crack front. Mode III is another shearing (tearing) mode. Here the shear stresses act in the plane of the crack and directed parallel to the crack front. A cracked body can be loaded in any one of these modes, or a combination of two or three modes called mixed mode.

Figure 1: Fracture mechanics failure modes: opening Mode I, shearing Mode II, and tearing Mode III.

In this work, Mode II loading is considered. In 1956, Irwin defined the strain ERR, G. This is a measure of the energy available for an increment of crack surface extension. The strain ERR can be calculated using the compliance, C, according to

(1)

Here b is the thickness, a is the crack length, and is the acting load; subscript II denotes the Mode. The compliance C is used in Eq. (1). C is defined as the ratio of the displacement , under the central loading point and the applied load P as

(2)

This is the inverse of the stiffness, K=1/C. The compliance is a function of the crack length a, i.e. C(a).

1.1. ENF specimen

The End Notched Flexure (ENF) specimen was introduced by Russell and Street (1982) as a pure mode II delamination specimen for testing of composites, see Fig. 2.

Figure 2: Geometry of End Notched Flexure (ENF) specimen

The ENF specimen gives an almost pure mode II shear loading at the crack tip. It requires a quite simple test and the specimen is easy to manufacture. Using the ENF specimen, the experimentalist is able to measure the fracture energy, . Experiments show that the specimen is sometimes stable and sometimes unstable under controlled displacement. It is noted that the specimen is always unstable under prescribed force. If the pre-crack is too short it will be unstable at prescribed displacement. That is, the crack length must be longer than some critical crack length to achieve stable crack growth. The critical crack length is the topic of the present thesis.

In experiments performed in 2002, Alfredsson et al. (2003) experienced instability with a too short crack. Carlsson, et al. (1986), and Chai & Mall (1988) developed a stability criterion. Carlsson, et al. (1986) argues that for tough resin systems, interlaminar shear effects may be significant. In turn, this requires slender specimen, i.e. large length to thickness relations. Furthermore, they argue that interlaminar shear deformation might influence the evaluation of the interlaminar Mode II fracture toughness. In an analysis based on beam theory, they show that the crack growth is stable if

. In this paper, we study the same problem using 2D elasticity using the finite element method (FEM).

The compliance of ENF-specimen is derived by Alfredsson and Stigh (2009) based on the J-integral and Euler-Bernoulli beam theory. Noting the fact that G = J for linear elasticity they derive

(3)

Here C(0) = is the compliance without a crack where E is the elastic modules and b the width. Apply C(0) in Eq. (3), the beam theory compliance of the ENF specimen is given by,

(4)

BT The compliance CBT and the Mode II ERR, GII are also given by Russell and Street (1982) for the Beam theory. Based on Eq. (1) they derive

(5)

Here h is the height of specimen, and P is the applied load. In addition, BT denotes the Beam theory

Presently, two different methods are widely used for calculating the ERR using FEM. One of these is the J -integral method, which is based on a path independent surface integral (line integral in a 2D formulation), and another one is the virtual crack extension method, which models a crack extension by a shift of node points in a Finite Element model. In this work we use the J -integral method to evaluate the ERR with the Finite Element Method (FEM).

1.2. Contour Integral - J

The J -integral is used to calculate the ERR GII in a finite element (FE) analysis. In Anderson (2005), the J integral is defined as

(7)

Here, W is the strain energy density, t is the traction vector, U the displacement vector, and x1, x2 the coordinate directions. With x1 directed in the extension of the crack.

The theoretical concept of the J- integral was developed independently in 1967 by Cherepano and in 1968 by Rice. Rice showed that the value of this integral equals the ERR in a non-linear elastic body that contains a crack.

Now J(a) informs on how J changes with a and with everything else constant. If the crack propagates, J equals the fracture energy Jc of the material. If J increases more than the fracture energy, Jc, the specimen is unstable. Thus, the condition for stability reads

There is a formula derived for this stability condition by Chai (1988) and Carlsson et al. (1986) that gives a critical crack length based on beam theory and Linear Elastic Fracture Mechanics (LEFM). In this case, J is given by

(8)

Here BT denotes the Beam theory.

If , the specimen is stable under controlled displacement based on Euler-Bernoulli beam theory.

In this work, the ENF specimen is studied with two different types of materials; Steel and Carbon Fiber Reinforced Plastic (CFRP) materials. CFRP are high performance materials for structural applications. Initially conventional test methods originally developed for determining the physical and mechanical properties of metals and other

7 homogeneous and isotropic engineering materials were used with CFRP. It was soon recognized that these new anisotropic (orthotropic) materials require special consideration for determining their physical and mechanical properties.

Carbon fiber reinforced polymer is a polymer matrix composite material reinforced by carbon fibers. Carbon fibers are very expensive and are used for reinforcing polymer matrix due to the following properties. They have very high specific modulus of elasticity, exceeding that of steel, high tensile strength, and low density, Daniel and Ishai (1994).

1.3. Isotropic material

A material is isotropic when its properties are the same in all directions or are independent of the orientation. That is, an isotropic material has an infinite number of planes of symmetry, and only two stiffness parameters 11 and 12 are required to fully define the stress strain response of the material, cf. Daniel and Ishai (1994). The stress strain relation is reduced to

(9)

For a linearly elastic material, equation 9 can be written in terms of engineering constants as

(10)

Engineers commonly use shear and Young’s modules G, E and Poisons ratio for material properties. However, these three stiffness variables are related by

(11)

- cf. Beer and Johnson, (1992).

1.4. Orthotropic material

Orthotropic materials are special cases of anisotropic materials. In this special case of composite material, the material has three mutually perpendicular planes of symmetry. The intersections of these planes define three mutually perpendicular axes, called the principle axes of material symmetry. The number of independent elastic constants is reduced from 36 for a general anisotropic material to 9 for an orthotropic material, cf. Daniel and Ishai (1994) and Table 1. This is because various stiffness and compliance terms are interrelated. The constitutive relation for the linear elastic orthotropic material in the material coordinate system is reduced to

(12)

Where the stress components ( ), the engineering strain components ( ), and the ( ) are elements of the compliance matrix.

Three important observations can be made with respect to the stress strain relation of Eq. (12). No interaction exists between normal stresses and shear strains, shear stresses and normal strains, and shear stresses and shear strains, cf. Daniel and Ishai (1994). Moreover, Eq. (12) can be written in terms of engineering constants as follows.

(13)

From the symmetry of the compliance matrix [ ] and Eq. (13) we conclude that

( i, j = 1, 2, 3)

For unidirectional CFRP, E2 is much smaller than E1, and 21 is much smaller than

12. For practical applications, CFRP’s are manufactured by stacking unidirectional laminas on top of each other to form a laminate. The direction of each lamina is varied.

Often a homogenized composite is achieved. In this case E1 ≈ E2 and 12 ≈ 21. That is, the laminate is isotropic in its plane. Out-of-plane, in the x3 direction, the composite is much softer.

In the present study, the specimen is deformed elastically in 2 dimensions. Material orientation is given by the fiber direction. Material data is given in Table 2.

Table 1: Independent Elastic Constants for various types of Materials

Sl. Material No. of independent No’s elastic constants 1 General anisotropic material 81 2 Anisotropic material considering symmetry of 36 stress and strain tensors( ) 3 Anisotropic material with elastic energy 21 considerations 4 General orthotropic material 9 5 Orthotropic material with transverse isotropy 5 6 Isotropic material 2

2. Materials and Method

2.1. Geometry of specimen

The geometry of the ENF specimen is shown in Fig. 2 where L is the length, b is the thickness, and 2h is total height of the specimen. Prescribed displacement , is applied at the loading point. The specimen has a crack with initial crack length, a. The end of the crack is called the crack tip. The specimen rests on two rigid supports. The loading point is symmetrically placed between the two supports.

2.2. Material properties

Here steel is considered as an isotropic material and CFRP is considered as an orthotropic material. Material data are given in Table 2.

Table 2: Material properties of isotropic and orthotropic materials

Material E G [GPa] [-] [GPa] Steel (Isotropic) 210 0.3 80.76

CFRP EF = 120, 12 = 13 = GT = G12 = G13 = (Orthotropic) ET=E22=E33=10.5 0.30 5.25, G23 = 3.48 23 = 0.51

Where E is Young’s modulus, is Poison’s ratio, G is the shear modulus and the suffix F denotes the fiber direction.

2.3. Finite Element Method simulations

In the present study, FE simulation of the ENF specimen is performed using ABAQUS solver version 6.92 to evaluate the ERR using the J -integral. The results obtained from FEM are compared with results from Beam theory.

The assembly of a specimen (Fig. 3) is constructed starting at the crack tip region. The tip region is arranged with one rectangular box and four concentric circles arranged in the box, centered at the crack tip. The upper edge of the specimen is partitioned by taking two symmetric edges to identify the centre point (node 15) where the load is applied.

Figure 3: FEM model of ENF specimen; where L= 1m, h = 0.02m, b = 0.04m, l = 0.05m is constant for all simulations and using the same meshing of the crack tip region.

The crack is assigned in the Interaction module of Abaqus. The crack extension is in the x-direction. Singularity elements are used around the crack tip. Triangular elements are formed during the meshing. These are formed by collapsing one side of the rectangular element to a single node at the crack tip. The mid nodes are moved to the quarter point towards the crack tip to achieve the correct square root singularity, Robert et al. (2002).

The applied load is simulated (Fig. 3) as a prescribed displacement at the loading point  = 1mm downwards with static load condition. Boundary conditions are arranged at 3 points; one is at node 20 where directions U1 = U2 = 0, second one and third one are at node numbers 11, 14 respectively where the direction U2 = 0 with free movement in the x- direction.

The ENF specimen is modeled with 8 node Quad continuum elements. The mesh is divided into 3 parts (Fig. 4). Part 1 contains 148 (4 x 17, + 80) elements, part 2 contains 36 (9x2, + 9x2) and part 3 contains 92 (23x4) elements and the number of nodes is 939. The type of elements of the whole specimen is Quadratic CPS8R (Continuum element, Plane Stress eight nodes with Reduced integration) and shape of element is ‘Quad’.

Figure 4: Meshing design of ENF specimen

2.4. Evaluation of J

Based on the material properties and input data, the ERR (JFEM) is calculated by use of the contour integral J. In the present work, four contours are used around the crack tip. The results are named J1, J2, J3 & J4 respectively (in Table 4) and as shown in Fig. 5, the average values is denoted JFEM for a given specimen. The result is compared with the result from Beam theory, cf. Eq. (8). All parameters in Eq. (8) are taken from the material properties, except ‘P’ (load) which is given by the result of the FEM analysis. Here P is the reaction force at the loading point where a prescribed displacement is applied. That is at the centre of specimen at node number 15. PFEM and JBT values are given in Table 3 for a specific crack length.

Detailed results are shown in Table 4 for isotropic and orthotropic specimens with a = 0.3 m and  = 1mm. The remaining simulation results are given in chapter 3. After getting the FEM results, they are compared with the Beam theory results. Moreover, the compliances of both materials are compared with the beam theory compliance (Table 5).

The comparison shows that with a prescribed displacement JBT is always smaller than JFEM for both isotropic and orthotropic materials. This property is achieved for all simulations in this paper. Moreover, the compliance using Beam Theory is always smaller than the compliance from the FEM analysis.

Figure 5: Resulting deformation of ENF specimen after and closer view of crack tip region with 4 contours. The deformation is exaggerated.

The above procedure is the same for all remaining simulations of isotropic and orthotropic materials. Material properties are taken from Table 2.

Table 3: Numerical JBT values of crack length a = 0.3 m and resultant load from FEM

Isotropic Orthotropic

PFEM [N] 1613.55 855.41

JBT[N/m] 49.034 24.100

Table 4: Results from FEM analysis with crack length a = 0.3 m and average value of JFEM

Isotropic Orthotropic JFEM [N/m] JFEM [N/m] J1 49.994 26.258 J2 49.990 26.251 J3 49.998 26.293 J4 49.991 26.323

JFEM 49.993 26.281 average

Table 5: comparison of compliance values for isotropic & orthotropic materials of finite- element analyses and beam-theory

Material CFEM [m/N] CBT [m/N] CFEM/ CBT

Isotropic 6.197×10-7 6.156×10-7 1.006 Orthotropic 1.169×10-6 1.077×10-6 1.085

3. Numerical Studies

This work is mainly aimed at finding the critical crack length (acr). To find the critical length the simulations are performed using the procedure detailed above. By changing the crack length with constant displacement () and all other parameters constant, the critical crack length acr is given by the limiting point where J attains its maximum value, cf. Figs. 6 and 8. That is, with , J increases if the crack propagates and we risk to get unstable crack propagation. On the other hand, with , J decreases if the crack propagates and we can achieve stable crack propagation.

Here the range of crack length is varied between 0.3 m to 0.4 m. The results for both materials are shown in Table 6.

Table 6: J values for different crack lengths and  = 1mm.

Sl Crack Isotropic Orthotropic No’s length JFEM[N/m] JFEM [N/m] a [m] avg. values avg. Values

1 0.3 49.993 26.281 2 0.34 51.673 27.009 3 0.342 51.679 27.011 4 0.343 51.680 27.058 5 0.344 51.679 27.039 6 0.345 51.676 27.004 7 0.35 51.637 26.977 8 0.4 49.233 25.516

The results are shown in Figs. 6 to 9. Third order polynomials are adapted to the results for isotropic and orthotropic materials using the least square method. The maximum is achieved at the same crack length acr = 0.343m irrespective of anisotropy. That is, in this case, the critical crack length is the same for isotropic and orthotropic materials.

Isotropic material a vs. JFEM

51.8

51.4

51 [N/m]

50.6

FEM J 50.2

49.8 ERR ERR y = 1694.2x3 - 2588.5x2 + 1177.5x - 116.04 49.4

49 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 Crack length a [m]

Figure 6: The results of different crack lengths a vs. JFEM for Isotropic material.

51.7

51.69 [N/m]

51.68

FEM FEM J

ERR ERR 51.67

51.66 0.339 0.34 0.341 0.342 0.343 0.344 0.345 0.346 0.347 0.348 0.349 Crack length a [m]

Figure 7: Closer view of stable point for isotropic material from Fig. 6

Orthotropic material a vs. JFEM 27.5

26.5

[N/m] 26 FEM

J 25.5

25 ERR ERR y = -3249.9x3 + 2765x2 - 751.38x + 90.59 24.5

24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 Crack length a [m]

Figure 8: The results of different crack lengths a vs. JFEM for Orthotropic material

27.07

27.06

27.05

[N/m] 27.04 FEM

J 27.03

ERR ERR 27.02

27.01

27 0.339 0.34 0.341 0.342 0.343 0.344 0.345 0.346 0.347 0.348 0.349 Crack length a [m]

Figure 9: Closer view of stable point for orthotropic material from Fig. 8

4. Discussion and Conclusion

According to beam theory the critical crack length = 0.35 is independent of if the material is isotropic or orthotropic (Alfredsson and Stigh, 2009). The present study, considering 2D elasticity, shows that the critical crack length for isotropic material =

0.3428 and for orthotropic material = 0.34157 are the same considering the numerical accuracy. Both results are similar, so there appears to be no influence of if isotropic or orthotropic material is considered. Both results are smaller than the value given by beam theory. Therefore, beam theory gives us a conservative estimation of the critical crack length. The FE compliance of both materials is larger than the compliance from beam theory.

Here the 2D elasticity has improved the beam theory. The variation between in the 4 counters of JFEM are in isotropic material 0.01% and in orthotropic material 0.1%. From FEM analysis the energy release rate of the ENF specimen in both isotropic and orthotropic materials (JFEM) are nearly the same while comparing with the beam theory results of ENF specimen (JBT) for values confer Table 3 and Table 4.

5. References

1) Alfredsson K.S. and Ulf Stigh. (2009). On the stability of beam like fracture mechanics specimens. Manuscript in preparation.

2) Alfredsson K.S., Biel A. and Leffler K. (2003) An experimental method to determine the complete stress-deformation relation for a structural adhesive layer loaded in shear, In 9th International Conference on The Mechanical Behavior of Materials, Geneva, Switzerland 2002

3) Anderson T.L. (2005) Fracture Mechanics Fundamentals And Applications. (3rd ed.). CRC press, NW.

4) Beer F.P. and Johnson E.R. (1992). Journal of Mechanics of Materials. (2nd ed.). McGraw-Hill, New York.

5) Broek D.(1984). Elementary engineering fracture mechanics: (3rd ed.). The Hague: Martinus Nijhoff Publishers

6) Carlsson L.A., Gillespie J.W. and Pipes R.B. (1986). On the analysis and design of the end-notched flexure (ENF) specimens for mode II testing. Journal of Composites. Mater.20, 594–605.

7) Chai H. (1988). Shear fracture. International Journal of Fracture. 37, 137-157.

8) Chai H. and Mall S. (1988). Design aspects of the end notch adhesive joint specimen. International Journal of Fracture. 36, R3-R8.

9) Daniel I.M. and Ishai O. (1994). Engineering Mechanics of Composite Materials, Oxford University Press, New York.

10) Irwin G.R. (1956). Onset of fast crack propagation in high strength steel and aluminum alloys. Sagamore research conference proceedings. 2, 289 – 305.

11) Rice J. R. (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of applied mechanics. 35, 145 -153.

12) Robert D.C., David S.M., Michael E.P., Robert J.W. (2002) Concept And Applications Of Finite Element Analysis (4th ed.). John Wiley & Sons. Inc. USA.

13) Russell A.J., Street K.N. (1982) A factor affecting the interlaminar fracture energy of graphite/epoxy laminates. In: Hayashi T, Kawata K, Umekawa S, editors. Progress in Science and Engineering of Composites, ICCM-IV, Tokyo, Japan 279–86

14) Wikipedia.org,(2010). J–Integral. Retrieved form internet. http://en.wikipedia.org/wiki/J_integral.