A universal material model subroutine for soft matter systems

TL;DR

A universal material subroutine is designed, which automates the integration of novel constitutive models of varying complexity in non-linear finite element packages, with no additional analytical derivations and algorithmic implementations.

Abstract

Soft materials play an integral part in many aspects of modern life including autonomy, sustainability, and human health, and their accurate modeling is critical to understand their unique properties and functions. Today's finite element analysis packages come with a set of pre-programmed material models, which may exhibit restricted validity in capturing the intricate mechanical behavior of these materials. Regrettably, incorporating a modified or novel material model in a finite element analysis package requires non-trivial in-depth knowledge of tensor algebra, continuum mechanics, and computer programming, making it a complex task that is prone to human error. Here we design a universal material subroutine, which automates the integration of novel constitutive models of varying complexity in non-linear finite element packages, with no additional analytical derivations and algorithmic implementations. We demonstrate the versatility of our approach to seamlessly integrate innovative constituent models from the material point to the structural level through a variety of soft matter case studies: a frontal impact to the brain; reconstructive surgery of the scalp; diastolic loading of arteries and the human heart; and the dynamic closing of the tricuspid valve. Our universal material subroutine empowers all users, not solely experts, to conduct reliable engineering analysis of soft matter systems. We envision that this framework will become an indispensable instrument for continued innovation and discovery within the soft matter community at large.

Figures (8)

• Figure 1: Constitutive neural network architecture. Anisotropic, compressible, feed forward constitutive neural network with three hidden layers to approximate the single scalar-valued free energy $\psi(\bar{I}_1, \bar{I}_2, I_3, \bar{I}_{\rm{4\alpha\beta}},\bar{I}_{\rm{5\alpha\beta}})$, as a function of 15 invariants of the left Cauchy-Green deformation tensor $\bm b$. The zeroth layer generates identity $(\circ)$, the rectified linear unit $\langle \circ \rangle$, and the absolute value $\langle \circ \rangle$ of the 15 invariants. The first layer generates powers $(\circ)$, $(\circ)^2$, $(\circ)^3$, etc. and the second layer applies the identity $(\circ)$, the exponential $(\rm{exp}(\circ)-1)$, and the logarithm $(-\rm{ln}(1-(\circ)))$ to these powers. The network is not fully connected by design to satisfy the condition of polyconvexity a priori.
• Figure 2: Interaction between the finite element analysis solver and the universal material subroutine. Flowchart of the interaction between Abaqus and the UANISOHYPER_INV subroutine architecture which embeds our universal constitutive material model. During each Newton-Raphson iteration and at each Gauss integration point, the UANISOHYPER_INV subroutine computes the strain energy function $\psi$, its first derivatives with respect to the deformation invariants ${\partial \psi}/{\partial \bar{I}_i}$, and its second derivatives with respect to the deformation invariants ${\partial^2 \psi}/{\partial \bar{I}_i \partial \bar{I}_j}$. These quantities are used by Abaqus to compute the components of the Cauchy stress tensor and the material tangent stiffness tensor, to construct the element force vector and stiffness matrix, and to assemble the global righthand side vector and stiffness matrix. Abaqus then performs a Newton-Raphson iteration based on the residual between the internal and external forces, until it achieves convergence.
• Figure 3: Universal material model subroutine schematic. Our universal material model user subroutine computes the strain energy function $\psi$ (= UA(1)), its first derivatives ${\partial \psi}/{\partial I_i}$ (= UI1(NINV)), and its second derivatives ${\partial^2 \psi}/{\partial \bar{I}_i \partial \bar{I}_j}$ (=UI2(NINV$^{*}$(NINV+1)/2)) with respect to the scalar invariants $\bar{I}_i$, derived from the deformation gradient $\bm{F}$. These functions and derivatives are computed based on a triple set of nested activation functions $f_{0}$ (= UCANN_h0), $f_{1}$ (= UCANN_h1), and $f_{2}$ (= UCANN_h2), where each unique constitutive path forms an additive constitutive neuron towards the total free energy and its derivatives.
• Figure 4: Universal material modeling of the human brain. Deformation and stress profiles for frontal impact to the human brain. The finite element models simulate the deformation and internal tissue loading corresponding to best-fit Mooney Rivlin, Blatz Ko, and newly discovered constitutive models from left to right. All simulations leverage our universal material model subroutine and only differ in the definition of the UNIVERSAL_TAB constitutive parameter table in the finite element analysis input file.
• Figure 5: Universal material modeling of skin. Deformation and stress profiles in the human scalp following a melanoma resection reconstruction procedure. The finite element models simulate the deformation and internal tissue loading corresponding a two-stage flap rotation and suturing procedure, with the first stage shown in the top row and the second stage shown in the bottom row. The remaining wound is closed with a skin graft to avoid excessive tissue stresses and damage. Both tissue manipulations are modeled using the best-fit constitutive neural network model in the three left columns. For comparison, we also showcase the resulting stress profiles for the best-fit neo Hooke Holzapfel model in the right column. All simulations leverage our universal material model subroutine and only differ in the definition of the UNIVERSAL_TAB constitutive parameter table in the finite element analysis input file.