Linear Elastic Stress (`sigma`)

The Stress (Hook's Law) calculator returns the stress matrix based on the strain tensor, shear modulus and Lame's first parameter.

INSTRUCTIONS: Enter the following:

• (ε) Strain Tensor
• (μ) Shear Modulus
• (λ) Lame's first Parameter

Stress (σ): The calculator returns the stress matrix.

The Math / Science

The Lamé parameters: λ(also called Lamé's first parameter) and μ (also called the shear modulus or Lamé's second parameter, G) are central to the representation of linear elasticity.

This equations satisfies Hooke's law in 3D for homogenous and isotropic materials, defining the stress, σ:

σ = 2⋅μ⋅ε + λ⋅tr(ε)⋅I

Inputs to this equation are:

• `epsilon` -- the strain tensor,
• I -- the identify matrix
• `mu` or G -- the shear modulus
• `lambda` -- Lame's first parameter

The first parameter λ is related to the bulk modulus and the shear modulus via K = `lambda` + (2/3) `mu` in three-dimensions and K = `lamba` + \mu in two-dimensions.

The shear modulus, μ, must be positive.

The Lamé's first parameter, λ, can be negative; for most materials it is positive.

This equation serves to simplify the stiffness matrix in Hooke's law.