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# Arruda-Boyce本构模型

Feb 28, 2017 • 1 min read

Mary C. Boyce is Dean of Engineering at The Fu Foundation School of Engineering and Applied Science at Columbia University in the City of New York and the Morris A. and Alma Schapiro Professor of Engineering. Prior to joining Columbia, Dean Boyce served on the faculty of the Massachusetts Institute of Technology (MIT) for over 25 years, leading the Mechanical Engineering Department from 2008 to 2013.

Her research focuses on materials and mechanics, particularly in the areas on multi-scale and nonlinear mechanics of polymers and soft composites, both those that are man-made and those formed naturally. Her leadership in the field of mechanics of materials has expanded understanding of the interplay between micro-geometry and the inherent physical behavior of a material, which has led to innovative hybrid material designs with novel properties. Her research has been documented in over 170 archival journal articles spanning materials, mechanics, and physics. She has mentored over 40 M.S. thesis students and over 25 Ph.D. students. She has been widely recognized for her scholarly contributions to the field, including election as a fellow of the American Society of Mechanical Engineers, the American Academy of Arts and Sciences, and the National Academy of Engineering.

AB 模型适用于多种高分子材料，尤其是存在应变软化的材料，可以描述无定形高分子在大变形情况下的各向异性力学行为，即在不同变形情况下，例如单轴压缩、平面应变，其力学行为不同。

## AB 本构模型

\begin{equation*} F=F^eF^p \\ F^p=F^{ps}=F^{pd} \\ L=\dot{F}F^{-1}=\dot{F}^e(F^e)^{-1}+F^e\dot{F}^p(F^p)^{-1}(F^e)^{-1}=D+W \\ L^p=\dot{F}^p(F^p)^{-1}=D^p+W^p \\ L^e=\dot{F}^e(F^e)^{-1} \\ L=L^e+F^eL^p(F^e)^{-1} \\ T=T^e=T^p \\ T^p=T^{ps}+T^{pd} \\ T^e=\frac{1}{J^{e}}\mathbf{L}^e[lnV^{e}] \\ J^{e}=det(F^{e}) \\ T^{ps}=\frac{\mu_r}{3J^{ps}}\frac{\sqrt{N}}{\lambda_{chain}}\mathscr{L}^{-1}(\frac{\lambda_{chain}}{\sqrt{N}}){\bar{B}^{ps}}^\prime \\ J^{ps}=det(F^{ps})=det(F^{p})=1 \\ \bar{F}^{ps}=(J^{ps})^{-1/3}F^{ps}=F^{ps}=F^p \\ \bar{B}^{ps}=\bar{F}^{ps}(\bar{F}^{ps})^T=F^p(F^p)^T \\ {\bar{B}^{ps}}^\prime=dev(\bar{B}^{ps}) \\ W^{p}=0 \\ D^p=D^{pd}=\frac{\dot{\gamma}^{pd}}{\sqrt{2}\tau^{pd}}{\bar{T}^{pd}}^\prime \\ \tau^{pd}=[\frac{1}{2}{\bar{T}^{pd}}^\prime{\bar{T}^{pd}}^\prime]^{1/2} \\ \bar{T}^{pd}=T^e-\frac{1}{J^e}F^eT^{ps}(F^e)^T \\ {\bar{T}^{pd}}^\prime=dev(\bar{T}^{pd}) \\ \dot{\gamma}^{pd}=\dot{\gamma}_0exp[-\frac{\Delta G}{k\Theta}(1-\frac{\tau^{pd}}{s})] \\ \dot{s}=h(1-\frac{s}{s_{ss}})\dot{\gamma}^{pd} \end{equation*}

\begin{equation*} \bar{T}^{pd}=(R^e)^T(T^e-T^{ps})R^e \end{equation*}

## VUMAT 实现方法

\begin{equation*} \dot{F}^{pd}=D^{pd}F^{pd}=f(F,F^{pd}) \\ k_1=f(F_n,F^{pd}_n) \\ k_2=f(F_{n+1/2},F^{pd}_n+k_1*\frac{dt}{2}) \\ k_3=f(F_{n+1/2},F^{pd}_n+k_2*\frac{dt}{2}) \\ k_4=f(F_{n+1},F^{pd}_n+k_3*dt) \\ F^{pd}_{n+1}=F^{pd}_{n}+(k_1+2*k_2+2*k_3+k_4)*\frac{dt}{6} \\ F_{n+1/2}=(F_{n}+F_{n+1})/2 \end{equation*}