ComputeLagrangianStressPK2

Overview

This class provides an interface to define a constitutive model in terms of the 2nd Piola-Kirchhoff stress $var element = document.getElementById("moose-equation-904079d8-123d-4016-91ec-360886b83827");katex.render("S_{IJ}", element, {displayMode:false,throwOnError:false});$ and the associated algorithmic tangent $var element = document.getElementById("moose-equation-89533543-6ef9-4eb1-86cd-3b867b0dd16b");katex.render(" T^{\\prime\\prime}_{IJKL} = \\frac{d S_{IJ}}{d S_{KL}}", element, {displayMode:true,throwOnError:false});$ It provides the Green-Lagrange strain $var element = document.getElementById("moose-equation-1a4cba7b-9272-4d9c-aa50-35867cc0f85b");katex.render(" E_{IJ} = \\frac{1}{2}\\left(F_{kI}F_{kJ} - \\delta_{IJ}\\right)", element, {displayMode:true,throwOnError:false});$ as an additional kinematic measure available for the user to use in the course of the defining the stress update. See here for a complete description of the Lagrangian kernel material system and for detailed instructions on how to implement a new material model.

Conversion

The class inherits from ComputeLagrangianStressPK1 and so must map from the 2nd Piola Kirchhoff stress to the 1st Piola Kirchhoff stress (and similarly map the tangents). The conversion formula are $var element = document.getElementById("moose-equation-37274482-731b-48b8-865f-46f0a610656d");katex.render(" P_{iJ} = F_{iK} S_{KJ}", element, {displayMode:true,throwOnError:false});$ and $var element = document.getElementById("moose-equation-86422d11-655e-4817-8b8d-a1061abe0abd");katex.render(" T_{iJkL}^{\\prime}=\\delta_{ik}S_{LJ}+F_{iT}T_{TJMN}^{\\prime\\prime}\\frac{1}{2}\\left(\\delta_{ML}F_{kN}+F_{kM}\\delta_{NL}\\right)", element, {displayMode:true,throwOnError:false});$