In
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
, 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-
stress 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
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
as the
dynamic viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
of a fluid(not the same units); whereas in the context of
elasticity, μ is called the
shear modulus,
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 ...
E, and the notation μ is paired with the use of λ.
In homogeneous and
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
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 t ...
in 3D,
where is the
stress tensor, the
strain tensor, the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
and the
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; 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 descri ...
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. 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 coord ...
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)