Processing math: 100%

Post title: Isotropic elasticity

Created on: 15 May 2018
Updated on: 25 Dec 2024

Introduction

The constitutive relation (relation between stress and strain) in linear elasticity (known as the Hook’s law) is given by the proportionality between stress and strain, σ=Dε, where σ is a second-rank stress tensor, ε is a second-rank strain tensor and D is a fourth-rank tensor of elastic constants.

The simplest but most often implemented tensor of elastic constants (mainly for polycrystalline materials) is that of isotropic elasticity.

D=[D11D12D12000D12D11D12000D12D12D11000000D44000000D44000000D44],

in which

D11=E(1ν)(1+ν)(12ν)D12=Eν(1+ν)(12ν)D44=E2(1+ν),

where E is the Young’s modulus and μ is the Poisson’s ratio. It is only these two constants sufficient for describing the isotropic behaviour.

This constitutive relation in 2D is simplified into

[σxxσyyτxy]=[C11C120C12C11000C44][εxxεyyγxy],

where the elastic constants C11, C12, and C44 are defined for

- plane strain

C11=E(1μ)(1+μ)(12μ)C12=Eμ(1+μ)(12μ)C44=E2(1+μ).

- plane stress

C11=E1μ2C12=Eμ1μ2C44=E2(1+μ).

For an elastically isotropic material the following condition holds: C44=C11C122. There is no directionality in the elastic properties. Polycrystalline materials may be considered isotropic as the sufficient amount of grains with a random (non-textured) grain orientation will average the elastic properties.