Title

Automatic numerical evaluation of stress and tangent modulus for hyperelastic material implementation in finite element analysis Yuxiang Wang1, 2*, Gregory J. Gerling1, 3 of Systems and Information Engineering, University of Virginia 2Department of Mechanical and Aerospace Engineering, University of Virginia 3Department of Biomedical Engineering, University of Virginia 1Department

Introduction Finite element (FE) simulations of biological tissues often involve highly nonlinear hyperelastic materials. Implementation of such material, however, still remains a challenge in computational studies. While most FE toolboxes support user defined materials, the derivation and coding needs to be done by hand, which is timeconsuming for any non-trivial strain energy function, requires considerable amount of familiarity with tensor algebra, and is prone to human errors. Efforts were made to automate the implementation of hyperelastic materials based on given strain energy function. Young et al. [1] used computer algebra system to auto-implement user-defined materials for ABAQUS, but is only limited to those strain energy functions explicitly expressed in terms of Lagragian strain tensor. Sun et al. [2] took the numerical approximation approach, where the tangent modulus can be obtained by perturbing the stress tensor. This approach is very effective in reducing part of the workload, but it still requires the analytic derivation of the stress tensor given strain energy function, and therefore is not fully automatic. Here, we introduce a fully automatic implementation of hyperelastic materials, based on a double numerical differentiation algorithm. Validation with a 3D anisotropic artery inflation simulation was also performed. Methods Overall, we will first numerically differentiate strain energy function to obtain stress and then numerically differentiate stress to obtain tangent modulus. Following the notation used by Holzapfel [3], we start with a specified strain energy function Ψ(𝐅𝐅), which is related to the 2nd Piola-Kirchhoff stress in the linearized form of ∆Ψ = 𝐒𝐒: ∆𝐄𝐄. We choose the perturbation of deformation gradient to be 𝜀𝜀 ∆𝐅𝐅 (𝑖𝑖𝑖𝑖) ≈ 2𝑆𝑆 𝐅𝐅 −T �𝐞𝐞𝑖𝑖 ⊗ 𝐞𝐞𝑗𝑗 + 𝐞𝐞𝑗𝑗 ⊗ 𝐞𝐞𝑖𝑖 � (1) where 𝜀𝜀𝑆𝑆 is a small perturbation parameter. We can now eventually get components of the 2nd Piola-Kirchhoff stress from the first numerical differentiation

𝑆𝑆𝑖𝑖𝑖𝑖 ≈

Ψ(𝐅𝐅)−Ψ(𝐅𝐅� (𝑖𝑖𝑖𝑖) ) 𝜀𝜀𝑠𝑠

(2)

Then, applying the perturbation from Sun et al. [2] and plug in the stress obtained from Eqn. (2) we can get the tangent modulus from the second numerical differentiation, with another small perturbation parameter 𝜀𝜀𝐶𝐶 . Three validation experiments were performed. First, we used direct calculation to optimize perturbation parameters to minimize numerical error. Then, FE models with single element were validated. At last, we followed Zulliger et al. [4] to perform an artery inflation simulation for validation, using anisotropic Holzapfel model [3]. Results The validation tests shows that optimal values for 𝜀𝜀𝑠𝑠 and 𝜀𝜀𝐶𝐶 are 10−6 and 10−4 respectively. The average stress error with such parameter settings were 7 × 10−5 in the single-element validation at uniaxial compression/tension, biaxial tension and simple shear cases. In the artery inflation simulation, our numerical implementation yielded the exact same result compared to the analytic implementation, with a relative error of 4 × 10−6 for inflated radius at 25 kPa pressure. Conclusions We provided a fully automated implementation of arbitrary hyperelastic material. This method well matches the result yielded by the analytic method, while requiring only a strain energy function to be present. References

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