# Biotissue Macroscale Nonlinear Analysis

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## Overview

Generally, we follow the finite element nonlinear analysis of solids presented in Bathe's Finite Element Procedures. We use the static updated Lagrangian formulation (section 6.3.2):

(KL + KNLU = PF

where KL is the linear strain portion of the stiffness matrix, KNL is the nonlinear strain portion of the stiffness matrix (for geometric nonlinearity), ΔU is the incremental displacement vector, P is external virtual work, and F is nodal point forces. The portions of these matrices and vectors corresponding to certain nodes are computed at integration points in the source file macro/src/MultiscaleIntegrator.h in the atPoint function.

The system above is solved for the incremental displacements, ΔU. The terms are given by: $K_L = \int_V B_L^T C B_L dV$ $K_{NL} = \int_V B_{NL}^T \tilde{\sigma} B_{NL} dV$ $F = \int_V B_L^T \hat{\sigma} dV$

where BL is the linear strain-displacement matrix, C is the stress-strain matrix, BNL is the nonlinear strain-displacement matrix, $\tilde{\sigma}$ is a matrix of Cauchy stresses, and $\hat{\sigma}$ is a vector of Cauchy stresses. The matrices are given by: $B_L = \begin{bmatrix} N_{1,1} & 0 & 0 & N_{2,1} & \cdots & 0 \\ 0 & N_{1,2} & 0 & 0 & \cdots & 0 \\ 0 & 0 & N_{1,3} & 0 & \cdots & N_{4,3} \\ N_{1,2} & N_{1,1} & 0 & N_{2,2} & \cdots & 0 \\ 0 & N_{1,3} & N_{1,2} & 0 & \cdots & N_{4,2} \\ N_{1,3} & 0 & N_{1,1} & N_{2,3} & \cdots & N_{4,1} \\ \end{bmatrix}$ $B_{NL} = \begin{bmatrix} \tilde{B}_{NL} & 0 & 0 \\ 0 & \tilde{B}_{NL} & 0 \\ 0 & 0 & \tilde{B}_{NL} \\ \end{bmatrix}$ $\tilde{B}_{NL} = \begin{bmatrix} N_{1,1} & 0 & 0 & N_{2,1} & \cdots & N_{4,1} \\ N_{1,2} & 0 & 0 & N_{2,2} & \cdots & N_{4,2} \\ N_{1,3} & 0 & 0 & N_{2,3} & \cdots & N_{4,3} \\ \end{bmatrix}$ $\tilde{\sigma} = \begin{bmatrix} \sigma & 0 & 0 \\ 0 & \sigma & 0 \\ 0 & 0 & \sigma \\ \end{bmatrix}$ $\sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \\ \end{bmatrix}$ $\hat{\sigma} = \begin{bmatrix} \sigma_{11} & \sigma_{22} & \sigma_{33} & \sigma_{12} & \sigma_{23} & \sigma_{31} \\ \end{bmatrix}^T$

where we've assumed 4 nodes and Ni,j denotes a shape function derivative at node i wrt spatial coordinate j.

## Incorporating Microscale Data

The microscale RVE solver returns the six unique components of the stress tensor, σ, which contributes to the nonlinear strain portion of the stiffness matrix, KNL, and the nodal point forces, F. It also returns nodal stress derivatives, $\tilde{C}$, which replaces CBL in the calculation of the linear strain portion of the stiffness matrix, KL.

## Fiber-Matrix Parallel

The fiber-matrix parallel model utilizes the fiber only RVEs and simply adds the stress found from the fiber network with stress computed from a neo-Hookean material law representing the interfibrillar matrix. This has been shown to approximate the macroscopic response of the combined fiber network and matrix.