Linear Elasticity Equations

Summary of the Equations of Linear Elasticity

The stiffness matrix of linear elastic isotropic material contains two parameters:

  • E, Young's modulus (elastic modulus)

  • ν, Poisson’s ratio

Define the following quantities.

σ=stressf=body forceε=strainu=displacement

The equilibrium equation is


The linearized, small-displacement strain-displacement relationship is


The balance of angular momentum states that stress is symmetric:


The Voigt notation for the constitutive equation of the linear isotropic model is


The expanded form uses all the entries in σ and ε takes symmetry into account.


In the preceding diagram, • means the entry is symmetric.

3D Linear Elasticity Problem

The toolbox form for the equation is


But the equations in the summary do not have ∇u alone, it appears together with its transpose:


It is a straightforward exercise to convert this equation for strain ε to ∇u. In column vector form,


Therefore, you can write the strain-displacement equation as


where A stands for the displayed matrix. So rewriting Equation 1, and recalling that • means an entry is symmetric, you can write the stiffness tensor as


Make the definitions


and the equation becomes


If you are solving a 3-D linear elasticity problem by using PDEModel instead of StructuralModel, use the elasticityC3D(E,nu) function (included in your software) to obtain the c coefficient. This function uses the linearized, small-displacement assumption for an isotropic material. For examples that use this function, see Vibration of a Square Plate.

Plane Stress

Plane stress is a condition that prevails in a flat plate in the x-y plane, loaded only in its own plane and without z-direction restraint. For plane stress, σ13 = σ23 = σ31 = σ32 = σ33 = 0. Assuming isotropic conditions, the Hooke's law for plane stress gives the following strain-stress relation:


Inverting this equation, obtain the stress-strain relation:


Convert the equation for strain ε to ∇u.


Now you can rewrite the stiffness matrix as


Plane Strain

Plane strain is a deformation state where there are no displacements in the z-direction, and the displacements in the x- and y-directions are functions of x and y but not z. The stress-strain relation is only slightly different from the plane stress case, and the same set of material parameters is used.

For plane strain, ε13 = ε23 = ε31 = ε32 = ε33 = 0. Assuming isotropic conditions, the stress-strain relation can be written as follows:


Convert the equation for strain ε to ∇u.


Now you can rewrite the stiffness matrix as