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

·σ=f

The linearized, small-displacement strain-displacement relationship is

ε=12(u+uT)

The balance of angular momentum states that stress is symmetric:

σij=σji

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

[σ11σ22σ33σ23σ13σ12]=E(1+ν)(12ν)[1ννν000ν1νν000νν1ν00000012ν00000012ν00000012ν][ε11ε22ε33ε23ε13ε12]

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

[σ11σ12σ13σ21σ22σ23σ31σ32σ33]=E(1+ν)(12ν)[1ν000ν000ν12ν000000012ν00000012ν000001ν000ν12ν00012ν0012ν01ν][ε11ε12ε13ε21ε22ε23ε31ε32ε33](1)

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

3D Linear Elasticity Problem

The toolbox form for the equation is

·(cu)=f

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

ε=12(u+uT)

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

u=[ux/xux/yux/zuy/xuy/yuy/zuz/xuz/yuz/z]

Therefore, you can write the strain-displacement equation as

ε=[100000000012012000000012000120001201200000000010000000001201200012000120000000120120000000001]uAu

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

σ=E(1+ν)(12ν)[1ν000ν000ν12ν000000012ν00000012ν000001ν000ν12ν00012ν0012ν01ν]Au=E(1+ν)(12ν)[1ν000ν000ν01/2ν01/2ν00000001/2ν0001/2ν0001/2ν01/2ν00000ν0001ν000ν000001/2ν01/2ν0001/2ν0001/2ν00000001/2ν01/2ν0ν000ν0001ν]u

Make the definitions

μ=E2(1+ν)λ=Eν(1+ν)(12ν)E(1ν)(1+ν)(12ν)=2μ+λ

and the equation becomes

σ=[2μ+λ000λ000λ0μ0μ0000000μ000μ000μ0μ00000λ0002μ+λ000λ00000μ0μ000μ000μ0000000μ0μ0λ000λ0002μ+λ]ucu

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:

[ε11ε222ε12]=1E[1ν0ν10002+2ν][σ11σ22σ12]

Inverting this equation, obtain the stress-strain relation:

(σ11σ22σ12)=E1ν2(1ν0ν10001ν2)(ε11ε222ε12)

Convert the equation for strain ε to ∇u.

ε=[10000121200121200001]uAu

Now you can rewrite the stiffness matrix as

[σ11σ12σ21σ22]=[E1ν200Eν1ν20E2(1+ν)E2(1+ν)00E2(1+ν)E2(1+ν)0Eν1ν200E1ν2]u=[2μ(μ+λ)2μ+λ002λμ2μ+λ0μμ00μμ02λμ2μ+λ002μ(μ+λ)2μ+λ]u

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:

(σ11σ22σ12)=E(1+ν)(12ν)(1νν0ν1ν00012ν2)(ε11ε222ε12)

Convert the equation for strain ε to ∇u.

ε=[10000121200121200001]uAu

Now you can rewrite the stiffness matrix as

[σ11σ12σ21σ22]=[E(1ν)(1+ν)(12ν)00Eν(1+ν)(12ν)0E2(1+ν)E2(1+ν)00E2(1+ν)E2(1+ν)0Eν(1+ν)(12ν)00E(1ν)(1+ν)(12ν)]u=[2μ+λ00λ0μμ00μμ0λ002μ+λ]u