Modelling Anisotropic, Hyperelastic Materials in ABAQUS

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1 Modelling Anisotropic, Hyperelastic Materials in ABAQUS Salvatore Federico and Walter Herzog Human Performance Laboratory, Faculty of Kinesiology, The University of Calgary 2500 University Drive NW, Calgary, Alberta, CANADA T2N 1N4 Tel: Fax: Abstract: ABAQUS offers the possibility to model non-linear isotropic materials as well as linear anisotropic materials. However, it is not possible to use these two capabilities at the same time in order to model anisotropic hyperelastic materials. Gurvich proposed to model such materials by means of the linear superposition of two fictitious materials: a linear anisotropic, and a hyperelastic isotropic material. Therefore, the modelling of a hyperelastic anisotropic material is converted into a material characterization problem, i.e., fitting the parameters in the strain potentials of the two fictitious materials to experimental stress strain curves. In this work, we propose an alternative solution, based on the same principle. Assuming that the anisotropic elastic tensor is known from experiments, we first obtain its directional average, which is isotropic. Then, we choose a form for the hyperelastic potential among the ones provided by ABAQUS, make it fit the linear elastic stress strain relationship generated by the isotropic tensor, and subtract the (quadratic) potential generated by the isotropic elastic tensor. This results in a hyperelastic potential with zero stiffness at zero strain. By summing this zero-zero potential to the (quadratic) potential generated by the anisotropic elastic tensor, we correct it without the need to calculate several parameters for two fictitious materials. As an example, we show the simulation of an unconfined compression test of articular cartilage. Keywords: Anisotropy, Hyperelasticity, Constitutive Model, Composites, Biological Tissues, Articular Cartilage 1. Introduction The efforts in investigating the nature and mechanical behaviour of biological materials, as well as the interest in studying high performance industrial composites, increase the demand for a multipurpose Finite Element software which is able to deal with anisotropy and non linear elasticity. ABAQUS has capabilities to model several isotropic hyperelastic potentials, as well as linear 2005 ABAQUS Users Conference 1

2 anisotropic materials. However, modelling of an anisotropic hyperelastic material in ABAQUS would require complex programming of user subroutines, which is not feasible for most users. Gurvich (2004) proposed a method for the implementation of hyperelastic, anisotropic materials in ABAQUS, based on the assumption that such materials can be approximated by the linear superposition of two fictitious materials: a linear elastic anisotropic material (a-material), and a hyperelastic isotropic material (h-material). In ABAQUS, the linear superposition of the h-material and the a-material is implemented by means of the definition of two superimposed meshes with identical nodes. One of the two is assigned the h-material; the other the a-material (Gurvich, 2004). While this method provides a convenient alternative to programming a complex user material subroutine, it cannot say anything on how the properties of the fictitious a-material and h-material should be chosen. Therefore, the problem is moved from programming to material characterization. This means fitting parameters to the experimental curves for the actual material. For the h-material, the number of parameters depends on the form of the elastic potential. For the a-material, it depends on the grade of anisotropy of the material: e.g., 5 for transverse isotropy, 9 for orthotropy. In order to shorten and simplify the material characterization, we propose to represent an anisotropic, hyperelastic material using the superposition of its linearization (which is no longer a fictitious material) and a suitable isotropic, hyperelastic material. In Section 2, we present basic aspects of the theory of fourth order tensors, and summarize Gurvich s (2004) method. In Section 3, we describe the proposed method. In Section 4, we provide an example: the modelling of articular cartilage. In Section 5 we discuss the proposed method. 2. Preliminary Theoretical Analysis In this Section we introduce basic concepts on fourth order tensors, and review Gurvich s (2004) method for hyperelastic, anisotropic materials in ABAQUS. 2.1 Isotropic Fourth Order Tensors and Directional Averaging 3 Fourth order tensors on R constitute an 81-dimensional space. Tensors with a given symmetry constitute subspaces of lower dimensionality. Isotropic tensors constitute a two dimensional subspace, a basis for which is (Walpole, 1981): K J = δδ 1 kl 3 kl = ( δδ + δδ δδ ) 1 2 kl 2 ik jl il jk 3 kl (1) In this basis, a fourth order isotropic tensor is written as: ABAQUS Users Conference

3 L = 3κ J + 2µ K (2) If L is an elasticity tensor, the coefficients κ and µ in Equation (2) take the physical meaning of bulk modulus and shear modulus, respectively. Given an arbitrary fourth-order tensor, T, its directional average equals its projection on the isotropic subspace of fourth order tensors (Walpole, 1981, Federico et al., 2004): 2.2 Gurvich s Method T = T J + ( T T ) K (3) ppqq 5 pqpq 3 ppqq Gurvich s (2004) method of superposing a linear elastic anisotropic material (a-material), and a hyperelastic isotropic material (h-material) reads, in terms of stress: σ = σ + σ (4) ( a) ( h) ( ) where σ is the stress in the actual material to be modelled, σ a is the stress in the fictitious ( ) anisotropic linear material (a-material), and σ h is the stress in the fictitious isotropic hyperelastic material (h-material). Since stress is obtained from the elastic potential by means of a derivation with respect to the strain components, and since derivation is a linear operator, the assumption made on stresses, through Equation (4), is valid for potentials. Therefore, Equation (4) can be expressed in terms of the elastic potential, which can be written as a function of the Green strain tensor, ε: Φ ( ε) =Φ ( ε) +Φ ( ε) = ε ε +Φ ( ε) (5) ( a ) ( h ) 1 ( ) ( ) L a h 2 kl kl ( a) where the quadratic potential, Φ, of the anisotropic linear a-material is expressed by means of ( a) ( a) ( h) the fourth-order elastic tensor L. All parameters featuring in Φ and Φ must be fitted to experimental curves for the target material. 3. General Solution Based on the consideration that the linear elastic anisotropic constants (for example Young s moduli and Poisson s ratios) are usually known from experiments, and are easy to obtain, we propose to substitute the fictitious a-material in Equation (4) and Equation (5) with the actual, linearized anisotropic material. Therefore, we only need to fit the parameters of a suitable hyperelastic isotropic material, which now takes the physical meaning of a correction, to the real linearized material (c-material). With these assumptions, Equation (5) can be replaced by: Φ ( ε) =Φ ( ε) +Φ ( ε) = ε ε +Φ ( ε) (6) ( L ) ( c ) 1 ( ) L c 2 kl kl where Φ is the quadratic potential generated by the linear elastic tensor, L, of the actual anisotropic material, and Φ is the hyperelastic isotropic potential of the c-material 2005 ABAQUS Users Conference 3

4 The problem now becomes to determine the correction potential, Φ. Let us make a preliminary consideration on the stiffness of the potential, Φ, which is its second derivative with respect to the components of the Green strain tensor, ε: 2 2 Φ Φ ( ε) = Lkl + ( ε) ε ε ε ε kl kl (7) In the undeformed state ( ε = 0, ε =Ο, where O is the zero tensor), this derivative must coincide with the linear theory stiffness, i.e., the components of the elastic tensor, L. Therefore, it follows that the second derivative of the correction potential, Φ, must vanish: L kl 2 2 Φ Φ + ( Ο ) = Lkl ( Ο ) = 0 ε ε ε ε kl kl (8) We build the zero-zero correction potential starting from the anisotropic quadratic potential generated by the actual linear elasticity tensor, L, and an isotropic hyperelastic potential, Ψ, which should fit the target material. Let us first calculate the directional average of L, by projecting it onto the isotropic subspace of the space of fourth-order tensors (Equation (3)): D = L = L J + ( L L ) K (9) ppqq 5 pqpq 3 ppqq Then, we impose that D is the undeformed state stiffness of the isotropic hyperelastic potential, Ψ. Therefore. the second derivative of Ψ with respect to the components of the Green strain tensor, ε, must equal the components of D: 2 Ψ Ο = ( ) Dkl ε ε kl (10) Equation (10) provides two conditions (as many as the independent parameters characterizing an isotropic linear material) on the parameters featuring in Ψ. ( D) Now, if we subtract the quadratic potential, Φ, generated by D, from Ψ, we obtain the correction potential, with zero stiffness at zero deformation: Φ =Ψ Φ =Ψ (11) ( D) 1 ( ε) ( ε) ( ε) ( ε) D 2 kl ε ε kl In fact, by using Equation (10), the condition in Equation (8) is identically satisfied: 2 2 Φ Ψ ( Ο ) = ( Ο) Dkl = 0 ε ε ε ε kl kl (12) ABAQUS Users Conference

5 By substituting Equation (11) into Equation (6), we obtain the required general hyperelastic anisotropic potential, Φ: Φ ( ε) =Φ ( ε) Φ ( ε) +Ψ ( ε) = L εε D εε +Ψ ( ε) (13) ( D) kl kl 2 kl kl In terms of second Piola-Kirchhoff stresses, Equation (13) reads: Ψ S = L ε D ε + ( ε) = S + S kl kl kl kl ε (14) where Ψ S = Lklεkl ; S = Dklεkl + ( ε) (15) ε are the anisotropic stress and the correction stress due to the anisotropic potential, correction potential, Φ, respectively. Φ, and to the Figure 1 schematically illustrates the procedure (see circled numbers) for the derivation of the various potentials, starting from that of the actual anisotropic linear material, Φ : 1. ( D) Φ is obtained from Φ through the directional averaging on L (Equation (9)), 2. Ψ is obtained from stiffness, 3. the correction potential, ( D) Φ, imposing the condition contained in Equation (10) on the Φ, is obtained by subtraction of Ψ and ( D) Φ (Equation (11)), 4. the hyperelastic anisotropic potential, Φ, is obtained by summation of (Equations (13) and (14)). Φ and Φ Assuming that the potential Ψ is characterized by N parameters, two of which are determined by Equation (10), and that L (and, there fore, D, through Equation (9)) known from experiments, the system constituted by Equations (9), (10), and (13): D = L 3 ij J + ( L ) 5 L 3 ij K 2 Ψ ( Ο ) = Dkl, (16) ε εkl 1 1 Φ ( ε) = L ( ) 2 klεεkl D 2 klεεkl +Ψ ε provides all the parameters featuring in the potential Φ, except N 2, which have to be retrieved from fitting the potential to large-strain experimental curves ABAQUS Users Conference 5

6 Figure1: Schematic representation of the construction of the correction potential, and of the anisotropic hyperelastic potential ABAQUS Users Conference

7 We remark that, while in the original method proposed by Gurvich (2004) there were two fictitious, unknown materials, our definition of the elastic potential (Equation (13)) involves three materials, of which one is the known, linearized, actual material, one is its directional average, and the third is partially determined by the first two. This method is of help in simplifying the curvefitting procedure. 4. An Application to Articular Cartilage, in ABAQUS As an example of application of the method described in Section 2, we show its implementation in an unconfined compression test of articular cartilage. Articular cartilage is known to be an anisotropic, inhomogeneous material (e.g., Cohen et al., 1998; Bursać et al., 1999, Wang et al., 2003). Often, however, Finite Element simulations of articular cartilage behaviour are performed based on a traditional linearly elastic, isotropic, homogeneous, poroelastic model. We recently developed a linear transversely isotropic, transversely homogeneous (TITH) model of articular cartilage (Federico et al., 2005), based on microstructural considerations. In the TITH model, we assumed that cartilage is a transversely isotropic, transversely homogeneous material, the symmetry axis being orthogonal to the bone-cartilage interface (tidemark). The elastic tensor, L, is thus a function of the non-dimensional tissue depth, ξ. Figure 2 shows the reference frames, and the dependence of the axial Young s modulus, E 11, the transversal Young s modulus, E 22, and the axial aggregate modulus, n (elastic modulus in uniaxial strain), on the non dimensional depth, ξ. The values are normalized to those of the proteoglycan matrix of the tissue (subscript 0). Figure 2: Reference frames and normalized elastic moduli in the TITH model 2005 ABAQUS Users Conference 7

8 Following Gurvich (2004), we assign the transversely isotropic material (described by the potential, Φ, generated by L) to one of the two superimposed meshes, and the hyperelastic correction material (described by the correction potential, Φ ) to the other. 4.1 Hyperelastic TITH Model for Articular Cartilage Once the transversely isotropic elastic tensor, L, is calculated at each depth, ξ, we can immediately obtain its isotropic projection, D, and the related bulk and shear moduli, κ and µ, through Equation (9). In order to build the correction potential, Φ, we choose, as the isotropic hyperelastic potential, Ψ, the potential of Holmes and Mow (1990), which is written as a function U of the right Cauchy stretch tensor, C, through its invariants I ( C ), I ( C ), I ( C ): UC ( ) = α exp( gc ( )) 0 g( C) = α ( I ( C) 3) + α ( I ( C) 3) βln I ( C) (17) The coefficients α 0 (with the dimension of a strain energy density), α 1, α 2 (non dimensional), are material parameters related to a material parameter β (non dimensional, and in the order of one for cartilage) and to the bulk and shear moduli, κ and µ, by the following equations: α = ( κ + µ ) 4β α = β ( κ µ ) ( κ + µ ) α = β ( κ µ ) ( κ + µ ) (18) The potential in Equation (17) can be written as a function of the Green strain tensor, ε, by use of the relationship C = 2ε + I (I being the identity tensor): Ψ ( ε) = α exp( f ( ε)) f( ε) = [(2 α + α ) I ( ε) + 4 α I ( ε) βlndet(2 ε + I)] (19) The correction potential, correction stress, S. 4.2 Input in ABAQUS Φ, is obtained with Equation (11). Equation (14) then provides the The input for the linear anisotropic material, assigned to the first mesh, is made by use of the option *ELASTIC and the parameter TYPE=ORTHOTROPIC. The input of the hyperelastic material generated by Φ, and assigned to the second mesh, is made by use of the option *HYPERELASTIC and the parameter TEST DATA INPUT, and then choosing one of the potentials in the ABAQUS libraries, and giving tabular stress-strain curves by means of the options *UNIAXIAL TEST DATA, *BIAXIAL TEST DATA, *PLANAR TEST DATA, *VOLUMETRIC TEST DATA. In our case, we chose a 6 th order Ogden potential, with uniaxial and volumetric test data ABAQUS Users Conference

9 Finally, it is possible to include the poroelastic behaviour, because ABAQUS treats pore pressure as a degree of freedom, and the congruency imposed by the operation of copying nodes with no shift (Gurvich, 2004): *ELCOPY, ELEMENT SHIFT=N, SHIFT NODES=0, OLD SET=elset1, NEW SET=elset2 ensures that the two meshes will see the same pore pressure, as well as the same displacements. The poroelastic behaviour is described by means of the strain-dependent permeability of Holmes and Mow (1990), in the form of Wu and Herzog (2000), which is the most convenient for ABAQUS input, as it expresses permeability, k, as a function of the void ratio, e: 4.3 Simulation and Results e M 1+ e k = k0 expo 1 (20) e e0 κ 2 An axisymmetric sample (Figure 3a) is meshed and divided into five layers, each with different elastic properties according to the TITH model. An unconfined compression test (Figure 3b) is performed by imposing a 10 second ramp displacement to the upper surface of the sample to a final nominal axial strain of 25%, and then keeping displacement constant up to 120 seconds. The pore pressure at the end of the loading ramp ( t = 10s ) and at the end of the simulation ( t = 120s ) is shown in Figure 4. The sample shows large strain at the lateral boundary, with a nonuniform behaviour given by the combination of anisotropy and inhomogeneity. Figure 3: a) Axi-symmetric FE mesh with five layers; b) Schematic illustration of an unconfined compression test 2005 ABAQUS Users Conference 9

10 Figure 4: Pore pressure plots for a 25% nominal strain unconfined compression test, at time t = 10 s, and t = 120 s 5. Discussion As for Gurvich s (2004) method, our method can be implemented easily as a standard feature into ABAQUS. Also, as for Gurvich (2004), the procedure should: automatically create the second mesh, automatically calculate the total stresses. Furthermore, it should: automatically calculate the directional average, D, of the elastic tensor, L, of the anisotropic material, automatically adjust the 2 parameters of the potential Ψ depending on the 2 independent elastic constants of the isotropic tensor, D. The proposed method constitutes a user-friendly solution while avoiding the problems of programming complex subroutines and fitting several material parameters to experimental curves. The method is particularly convenient when the linearly elastic, anisotropic properties of the material are known from experiment or microstructural considerations, and a suitable potential can be suggested. Its major application might be in biomechanical or composite material modelling. 6. References 1. Bursać, P.M., Obitz, T.W., Eisenberg, S.R., Stamenović, D., Confined and Unconfined Stress Relaxation of Cartilage: Appropriateness of a Transversely Isotropic Analysis, Journal of Biomechanics, 32, ABAQUS Users Conference

11 2. Cohen, B., Lai, W.M., Mow, V.C., A Transversely Isotropic Biphasic Model for Unconfined Compression of Growth Plate and Chondroepiphysis, Journal of Biomechanical Engineering, 120, Federico, S., Grillo, A., Herzog, W., A Transversely Isotropic Composite with a Statistical Distribution of Spheroidal Inclusions: a Geometrical Approach to Overall Properties, Journal of the Mechanics and Physics of Solids, 52 (10), Federico, S., Grillo, A., La Rosa, G., Giaquinta, G., Herzog, W., A Transversely Isotropic, Transversely Homogeneous Microstructural-Statistical Model of Articular Cartilage, Journal of Biomechanics, in press (available on line, 29 th of December 2004). 5. Gurvich, M.R., A Constitutive Model of Hyperelastic Anisotropic Materials: Approach and Implementation in ABAQUS ABAQUS Users Conference, Holmes, M.H., Mow, V.C., The Non-linear Characteristics of Soft Gels and Hydrated Connective Tissues in Ultrafiltration, J. Biomech., 23, Walpole, L.J., Elastic Behavior of Composite Materials: Theoretical Foundations. Advances in Applied Mechanics, 21, Wang, C.C.-B., Chahine, N.O., Hung, C.T, Ateshian, G.A., Optical Determination of Anisotropic Material Properties of Bovine Articular Cartilage in Compression, Journal of Biomechanics, 36, Wu, J.Z., Herzog, W., Finite Element Simulation of Location- and Time-dependent Mechanical Behavior of Chondrocytes in Unconfined Compression Tests, Annals of Biomedical Engineering, 28, Acknowledgements The Authors would like to gratefully acknowledge Dr. Mark R. Gurvich for the useful references provided, and Azim Jinha for the hints in programming. The Canadian Institute of Health Research (CIHR); the Arthritis Society of Canada; the Canada Research Chair Programme, the Alberta Ingenuity Fund ABAQUS Users Conference 11