Lamé parameters
Encyclopedia
In linear elasticity
, the Lamé parameters are the two parameters
which, in homogenous and isotropic materials, satisfy Hooke's law
in 3D,
where σ is the stress
, ε the strain tensor,
the identity matrix
and
the trace
function.
The first parameter λ is related to the bulk modulus and the shear modulus
, and serves to simplify the stiffness matrix in Hooke's law.
Although the shear modulus, μ, must be positive, the Lamé's first parameter, λ, can be negative, in principle; however, for most materials it is also positive. The two parameters together constitute a parametrization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli.
The parameters are named after Gabriel Lamé
.
Linear elasticity
Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...
, the Lamé parameters are the two parameters
- λ, also called Lamé's first parameter,
- μ>0, the shear modulus or Lamé's second parameter, also denoted
,
which, in homogenous and isotropic materials, satisfy Hooke's law
Hooke's law
In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law...
in 3D,
where σ is the stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...
, ε the strain tensor,
the identity matrixIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
and
the traceTrace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
function.
The first parameter λ is related to the bulk modulus and the shear modulus
, and serves to simplify the stiffness matrix in Hooke's law.Although the shear modulus, μ, must be positive, the Lamé's first parameter, λ, can be negative, in principle; however, for most materials it is also positive. The two parameters together constitute a parametrization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other elastic moduli.
The parameters are named after Gabriel Lamé
Gabriel Lamé
Gabriel Léon Jean Baptiste Lamé was a French mathematician.-Biography:Lamé was born in Tours, in today's département of Indre-et-Loire....
.