Muscle constitutive model with a tangent modulus approximation

Sophisticated muscle material models are required to perform detailed finite element simulations of soft tissue; however, state-of-the-art muscle models are not among the built-in materials in popular commercial finite element software packages. Implementing user-defined muscle material models is challenging for two reasons: deriving the tangent modulus tensor for a material with a complex strain energy function is tedious and programming the algorithm to compute it is error-prone. These challenges hinder widespread use of such models in software that employs implicit, nonlinear, Newton-type finite element methods. We implement a muscle material model in Ansys using an approximation of the tangent modulus, which simplifies its derivation and implementation. Three test models were constructed by revolving a rectangle (RR), a right trapezoid (RTR), and a generic obtuse trapezoid (RTO) around the muscle's centerline. A displacement was applied to one end of each muscle, holding the other end fixed. The results were validated against analogous simulations in FEBio, which uses the same muscle model but with the exact tangent modulus. Overall, good agreement was found between our Ansys and FEBio simulations, though some noticeable discrepancies were observed. For the elements along the muscle's centerline, the root-mean-square-percentage error in the Von Mises stress was 0.00%, 3.03%, and 6.75% for the RR, RTR, and RTO models, respectively; similar errors in longitudinal strain were observed. We provide our Ansys implementation so that others can reproduce and extend our results.Please cite the following publication:Sampaio de Oliveira ML, Uchida TK. Muscle constitutive model with a tangent modulus approximation: Ansys implementation and verification. Journal of Biomechanical Engineering 145(7): 071002, 2023. https://doi.org/10.1115/1.4056948 <br/><br/>This project includes the following software/data packages: <br/> <ul> <li> <a href="https://simtk.org/frs?group_id=2386#pack_2374">Ansys USERMAT </a> : Ansys USERMAT function implementation of a muscle material model with an approximation of the tangent modulus. Includes usermat.f, Ansys files, and example results. Please see README.txt and Sampaio de Oliveira and Uchida (2023) for more information. </li> </ul>

为执行软组织详细有限元模拟，需采用复杂的肌肉材料模型；然而，当前最先进的肌肉模型并不包含在流行的商业有限元软件包的内置材料之中。用户自定义肌肉材料模型的实现面临两大挑战：一是推导具有复杂应变能函数的材料切线模量张量繁琐且费时，二是编程计算该张量易于出错。这些挑战限制了此类模型在采用隐式、非线性、牛顿型有限元方法的软件中的广泛应用。本研究在 Ansys 中实现了一种肌肉材料模型，采用切线模量的近似计算方法，简化了其推导与实现过程。通过绕肌肉中心线旋转矩形（RR）、直角梯形（RTR）和普通钝角梯形（RTO）构建了三个测试模型。对每个肌肉的一端施加位移，另一端保持固定。结果与 FEBio 中的类似模拟进行了验证，FEBio 使用相同的肌肉模型，但具有精确的切线模量。总体而言，Ansys 与 FEBio 的模拟结果之间表现出良好的一致性，尽管存在一些显著的差异。对于沿肌肉中心线的元素，RR、RTR 和 RTO 模型的 von Mises 应力均方根百分比误差分别为 0.00%、3.03% 和 6.75%；纵向应变也观察到类似误差。我们提供 Ansys 的实现版本，以便他人能够复制和扩展我们的结果。请引用以下出版物：Sampaio de Oliveira ML, Uchida TK. 肌肉本构模型及其切线模量近似：Ansys 实现与验证。生物力学工程杂志 145(7): 071002，2023. https://doi.org/10.1115/1.4056948。本项研究涉及以下软件/数据包：

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<a href="https://simtk.org/frs?group_id=2386#pack_2374">Ansys USERMAT</a> : 包含切线模量近似肌肉材料模型的 Ansys USERMAT 函数实现。包括 usermat.f、Ansys 文件和示例结果。请参阅 README.txt 和 Sampaio de Oliveira 及 Uchida（2023）以获取更多信息。
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