Handling tensors using tensorial Kelvin bases: application to olivine polycrystal deformation modeling using elastically anistropic CPFEM

Abstract

In this work we present a simple and convenient method for handling tensors within computational mechanics frameworks based on the Kelvin decomposition. This methodology was set up within a crystal plasticity framework which permits, using the Kelvin base related to the crystal symmetries, to account for elastic anisotropy. The classical mixed velocity pressure finite element formulation has been modified in order to account for the elastic anisotropic behavior introduced into the crystal plasticity model. Moreover this modification of the mixed formulations allows to account for volume/pressure variations that can stream from constitutive models that could allow present compressible plasticity. Using this numerical framework, we explore the influence of elastic anisotropy onto the mechanical behavior of olivine. Our results suggest that at the polycrystal scale, the elastic anisotropy is not of first order importance. However the local changes on the stress state can be important for some physical phenomena such as recrystallization and damage.

Appendices

Appendices

A Orthorhombic Kelvin base

In the following, $C_{ij}$ denotes the elastic stiffness matrix written in the Voigt notation such as, for an orthorhombic material:

$$\begin{aligned} C_{ij}= \begin{pmatrix} C_{11} &{} C_{12} &{} C_{13} &{} 0 &{} 0 &{} 0 \\ C_{12} &{} C_{22} &{} C_{23} &{} 0 &{} 0 &{} 0 \\ C_{13} &{} C_{23} &{} C_{33} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} C_{44} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} C_{55} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} C_{66} \end{pmatrix}. \end{aligned}$$

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The first three eigenvalues $\lambda _{k=1,2,3}$ of the Kelvin base for this type of symmetry are obtained by searching the values for which the following matrix has a null determinant [41]:

$$\begin{aligned} \begin{pmatrix} C_{11} - \lambda &{} C_{12} &{} C_{13} \\ C_{12} &{} C_{22} - \lambda &{} C_{23} \\ C_{13} &{} C_{23} &{} C_{33} - \lambda \end{pmatrix}\text {.} \end{aligned}$$

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The associated eigentensors can be written as:

$$\begin{aligned} \begin{matrix} T_{ij}^{k}= L(\lambda _{k}, \lambda _{i}, \lambda _{j}) \times \\ \tiny \begin{pmatrix} C_{12}C_{23} - C_{13}(C_{22}-\lambda _{k}) &{} 0 &{} 0 \\ 0 &{} C_{12}C_{13}-C_{23}(C_{11}-\lambda _{k}) &{} 0 \\ 0 &{} 0 &{} (C_{11}-\lambda _{k})(C_{22}-\lambda _{k}) - C_{12}^2 \end{pmatrix} \end{matrix}\nonumber \\ \end{aligned}$$

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where:

$$\begin{aligned} \begin{aligned} L(\lambda _{k},\lambda _{i},\lambda _{j}) = \frac{1}{(\lambda _{k}-\lambda _{i})(\lambda _{k}-\lambda _{j})[C_{12}(C_{13}^2 - C_{23}^2)]} \\ \times [ [ C_{12} C_{13} - C_{23}(C_{11} - \lambda _{i}) ] [ (C_{11} - \lambda _{j}) \epsilon _{11} + C_{12} \epsilon _{22} + C_{13} \epsilon _{33} ] \\ - [ C_{12}C_{23} - C_{13} (C_{22} - \lambda _{i} ) ] [ C_{12} \epsilon _{11} + (C_{22} - \lambda _{j}) \epsilon _{22} + C_{23} \epsilon _{33} ] ] \end{aligned}\nonumber \\ \end{aligned}$$

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The last three eigentensors associated to the eigenvalues $\lambda _4 = 2C_{44}, \lambda _5 = 2C_{55}, \lambda _6 = 2C_{66},$ are:

$$\begin{aligned} \begin{matrix} T_{ij}^4= \begin{pmatrix} 0 &{} \frac{1}{\sqrt{2}} &{} 0 \\ \frac{1}{\sqrt{2}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix} &{} T_{ij}^5= \begin{pmatrix} 0 &{} 0 &{} \frac{1}{\sqrt{2}} \\ 0 &{} 0 &{} 0 \\ \frac{1}{\sqrt{2}} &{} 0 &{} 0 \end{pmatrix} &{} T_{ij}^6= \begin{pmatrix} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{\sqrt{2}} \\ 0 &{} \frac{1}{\sqrt{2}} &{} 0 \end{pmatrix}\text {.} \end{matrix}\nonumber \\ \end{aligned}$$

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