Introduction to Linear Elasticity

Linear elasticity is valid for small elastic strains, usually less than 5%. When you specify elasticity, you can select the following aspects of the elastic material model: It can be isotropic, orthotropic, or fully anisotropic material model:

The linear elastic material model:

is valid for small elastic strains (normally less than 5%);
can be isotropic, orthotropic, or fully anisotropic;
can have properties that depend on temperature and other field variables; and
allows you to specify that compressive stress or tensile stress cannot be generated.

The total stress is defined from the total elastic strain as

σ = D^{e l} ε^{e l}

where

σ

is the total stress (“true,” or Cauchy stress in finite-strain problems),

D^{e l}

is the fourth-order elasticity tensor, and

ε^{e l}

is the total elastic strain (log strain in finite-strain problems). Do not use the linear elastic material definition when the elastic strains may become large; use a hyperelastic model instead (see Isotropic Hyperelasticity). Even in finite-strain problems the elastic strains must still be small (less than 5%).

Depending on the number of symmetry planes for the elastic properties, a material can be classified as either isotropic (an infinite number of symmetry planes passing through every point) or anisotropic (no symmetry planes). Some materials have a restricted number of symmetry planes passing through every point; for example, orthotropic materials have two orthogonal symmetry planes for the elastic properties. The number of independent components of the elasticity tensor $D^{e l}$ depends on such symmetry properties. You define the level of anisotropy and the method of defining the elastic properties, as described below. If the material is anisotropic, a local orientation must be used to define the direction of anisotropy.


Type	Description
Isotropic	A single elasticity definition for all material directions.
Orthotropic	A different elasticity definition for each primary material direction. The elasticity is defined using nine stiffness parameters, $D_{1111}$ , $D_{2222}$ , $D_{3333}$ , $D_{1122}$ , etc.
Engineering Constants	An alternate method to define orthotropic elasticity.
Lamina	Lamina elasticity is a special case of orthotropic elasticity valid only for two-dimensional structures.
Anisotropic	Provides a modeling capability for materials that exhibit highly anisotropic behavior, such as biomedical soft tissues and fiber-reinforced elastomers.
Transversely isotropic	A special subclass of orthotropy characterized by a plan of isotropy at every point in the material.
Shear	Describes the deviatoric response of materials whose volumetric response is governed by an equation of state.

Linear elastic materials must satisfy the conditions of material or Drucker stability. Stability requires that the tensor $D^{e l}$ be positive definite, which leads to certain restrictions on the values of the elastic constants. The stress-strain relations for several different classes of material symmetries are given below. The appropriate restrictions on the elastic constants stemming from the stability criterion are also given.