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.
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 inσandεtakes symmetry into account.
(1) |
在前面的图中,信谊•意味着条目metric.
3D Linear Elasticity Problem
The toolbox form for the equation is
But the equations in the summary do not have ∇ualone, it appears together with its transpose:
It is a straightforward exercise to convert this equation for strainεto ∇u. In column vector form,
Therefore, you can write the strain-displacement equation as
whereAstands for the displayed matrix. So rewritingEquation 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 usingPDEModel
instead ofStructuralModel
, use theelasticityC3D(E,nu)
function (included in your software) to obtain thec
coefficient. This function uses the linearized, small-displacement assumption for an isotropic material. For examples that use this function, seeStationaryResults
.
Plane Stress
Plane stress is a condition that prevails in a flat plate in thex-yplane, loaded only in its own plane and withoutz-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 strainεto ∇u.
Now you can rewrite the stiffness matrix as
Plane Strain
Plane strain is a deformation state where there are no displacements in thez-direction, and the displacements in thex- andy-directions are functions ofxandybut notz. 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 strainεto ∇u.
Now you can rewrite the stiffness matrix as