In
continuum mechanics, Lamé parameters (also called the Lamé coefficients, Lamé constants or Lamé moduli) are two material-dependent quantities denoted by λ and μ that arise in
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
-
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
relationships.
In general, λ and μ are individually referred to as ''Lamé's first parameter'' and ''Lamé's second parameter'', respectively. Other names are sometimes employed for one or both parameters, depending on context. For example, the parameter μ is referred to in
fluid dynamics as the
dynamic viscosity of a fluid(not the same units); whereas in the context of
elasticity, μ is called the
shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
,
and is sometimes denoted by ''G'' instead of μ. Typically the notation G is seen paired with the use of
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...
E, and the notation μ is paired with the use of λ.
In homogeneous and
isotropic materials, these define
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
in 3D,
where is the
stress tensor, the
strain tensor
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
, the
identity matrix and the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
function. Hooke's law may be written in terms of tensor components using index notation as
where is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
.
The two parameters together constitute a parameterization of the elastic moduli for homogeneous isotropic media, popular in mathematical literature, and are thus related to the other
elastic moduli
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
; for instance, the
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describ ...
can be expressed as . Relations for other moduli are found in the (λ, G) row of the
conversions table at the end of this article.
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 parameters are named after
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
. They have the same
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
as stress and are usually given in SI unit of stress
a
Further reading
* K. Feng, Z.-C. Shi, ''Mathematical Theory of Elastic Structures'', Springer New York, , (1981)
* G. Mavko, T. Mukerji, J. Dvorkin, ''The Rock Physics Handbook'', Cambridge University Press (paperback), , (2003)
* W.S. Slaughter, ''The Linearized Theory of Elasticity'', Birkhäuser, , (2002)
References
{{DEFAULTSORT:Lame parameters
Elasticity (physics)