Lamé parameters
The Lamé constants (after Gabriel Lamé ) are two material constants and put all the components of the elasticity tensor of an isotropic material firmly in the framework of continuum mechanics. Your unit corresponds to a pressure ( force per unit area, in SI units).
Theory of elasticity
In the linear elasticity theory, the linear dependence of the stress tensor is described by the strain tensor by the elasticity tensor. In component form and using the Einstein summation convention is the linear relationship
The stress and strain tensors 2nd order tensors and the elasticity tensor is a tensor of fourth stage. In the case of the isotropic Hooke's law, this can be to
Simplify. The first Lame constant, and ( the shear modulus, unit) is referred to as the second Lame constant, and is the Kronecker symbol. For Poisson's ratio ( Poisson's ratio ) and elastic modulus of the context of:
See the section # Relationship between Lamé constants and elastic constants for other formulas depending on the Lamé constants.
Derivation
In the case of an isotropic, linearly elastic material, that is the stress tensor is a linear function of the components of the strain tensor from, one can define a scalar potential indicative of the energy density of the material as a function of the distortion and the relationship
A stress-strain relation is defined. This function may depend only on invariants of the strain tensor, since the choice of the coordinate system is not allowed to change the energy density of Outlined distortion condition. The strain tensor is symmetric, so it has the following invariants ( in the notation with Einstein summation convention shear )
In order to obtain a linear distortion voltage relation, the potential may only quadratically depend on the components of the strain tensor. It must have the form Thus, due to the potential Koordinateninvarianz
Have, and with arbitrary constants. Substituting this potential approach in the stress-strain relation and performs some transformations it has undergone, we obtain the relation
With the definitions
It is called now, and first and second Lamé constant. The law
Is called the generalized Hooke's law.
Fluid Mechanics
In the Navier -Stokes equations of fluid mechanics
Is the first Lamé constant and the second Lamé constant ( the dynamic viscosity, unit ).