❓What are the full steps to derive the expression for Poisson's ratio from the constitutive equations

What are the full steps to derive the expression for Poisson's ratio from the constitutive equations and the definition of bulk modulus and shear modulus?

This derivation relies on the two fundamental constitutive equations for an isotropic, homogeneous elastic material, which relate the stress tensor ($\sigma_{ij}$) and the strain tensor ($\epsilon_{ij}$) to the material's elastic moduli.

The final expression for Poisson's ratio ($\nu$) in terms of Bulk Modulus ($K$) and Shear Modulus ($G$) is:

$\nu = \frac{3K - 2G}{6K + 2G}$

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Derivation Steps

1. The Constitutive Equations

We start with two equivalent forms of Hooke's Law for Isotropic Materials:

A. Stress in terms of $K$ and $G$ (Decomposed Strain Form)

The stress tensor ($\sigma_{ij}$) is decomposed into its hydrostatic (volumetric) and deviatoric (shear) parts:

$\sigma_{ij} = K \epsilon_{kk} \delta_{ij} + 2 G \kappa_{ij} \quad \text{(Eq. 1)}$

Where:

• $\epsilon_{kk} = \epsilon_{11} + \epsilon_{22} + \epsilon_{33}$ is the volumetric strain (trace of the strain tensor).

• $\delta_{ij}$ is the Kronecker delta (1 if $i=j$, 0 otherwise).

• $\kappa_{ij} = \epsilon_{ij} - \frac{1}{3} \epsilon_{kk} \delta_{ij}$ is the deviatoric strain (pure shear strain).

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What are the full steps to derive the expression for Poisson's ratio from the constitutive equations and the definition of bulk modulus and shear modulus? | FAQsviadean on Notion