# Biotissue Macroscale Nonlinear Analysis

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## Contents |

## 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):

(*K*_{L} + *K*_{NL})Δ*U* = *P* − *F*

where *K*_{L} is the linear strain portion of the stiffness matrix, *K*_{NL} is the nonlinear strain portion of the stiffness matrix (for geometric nonlinearity), Δ*U* 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, Δ*U*. The terms are given by:

where *B*_{L} is the linear strain-displacement matrix, *C* is the stress-strain matrix, *B*_{NL} is the nonlinear strain-displacement matrix, is a matrix of Cauchy stresses, and is a vector of Cauchy stresses. The matrices are given by:

where we've assumed 4 nodes and *N*_{i,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, *K*_{NL}, and the nodal point forces, *F*. It also returns nodal stress derivatives, , which replaces *C**B*_{L} in the calculation of the linear strain portion of the stiffness matrix, *K*_{L}.

## 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.