Biotissue Macroscale Nonlinear Analysis

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1 Overview
2 Incorporating Microscale Data
3 Fiber-Matrix Parallel
4 Initial linear elastic solve
5 Iterations
6 Load stepping

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):

$(K L + K N L)Δ U = P - F$

where $K L$ is the linear strain portion of the stiffness matrix, $K N L$ 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:

$K_L = \int_V B_L^T C B_L dV$

$K_{NL} = \int_V B_{NL}^T \tilde{\sigma} B_{NL} dV$

$F = \int_V B_L^T \hat{\sigma} dV$

where $B L$ is the linear strain-displacement matrix, $C$ is the stress-strain matrix, $B N L$ 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 $N i, j$ denotes a shape function derivative at node $i$ wrt spatial coordinate $j$ .

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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, $K N L$ , and the nodal point forces, $F$ . It also returns nodal stress derivatives, $\tilde{C}$ , which replaces $C B L$ in the calculation of the linear strain portion of the stiffness matrix, $K L$ .

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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.

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