❓What are the full steps to derive the expression for Young's modulus from the constitutive equations

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

This derivation expresses Young's Modulus ($E$) in terms of the Bulk Modulus ($K$) and Shear Modulus ($G$). The process involves combining two fundamental sets of constitutive equations: the relationship between stress ($\sigma_{ij}$) and strain ($\epsilon_{ij}$) using $K$ and $G$, and the same relationship using $E$ and Poisson's ratio ($\nu$).

Initial Constitutive Equations

We begin with the general equations for an isotropic elastic material:

1. Stress in terms of Bulk Modulus ($K$) and Shear Modulus ($G$):

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

   where:

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

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

2. Strain in terms of Young's Modulus ($E$) and Poisson's Ratio ($\nu$):

   $\epsilon_{ij} = \frac{1}{E} \left[ (1 + \nu) \sigma_{ij} - \nu \sigma_{kk} \delta_{ij} \right] \quad \text{(Eq. 2)}$

   where $\sigma_{kk} = \sigma_{11} + \sigma_{22} + \sigma_{33}$ is the hydrostatic stress (trace of the stress tensor).

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