In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a Cauchy-elastic material is one in which the
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 ...
at each point is determined only by the current state of
deformation
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
with respect to an arbitrary reference configuration.
[R. W. Ogden, 1984, ''Non-linear Elastic Deformations'', Dover, pp. 175–204.] A Cauchy-elastic material is also called a simple elastic material.
It follows from this definition that the stress in a Cauchy-elastic material does not depend on the path of deformation or the history of deformation, or on the time taken to achieve that deformation or the rate at which the state of deformation is reached. The definition also implies that the
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
s are spatially local; that is, the stress is only affected by the state of deformation in an infinitesimal neighborhood of the point in question, without regard for the deformation or motion of the rest of the material. It also implies that body forces (such as gravity), and inertial forces cannot affect the properties of the material. Finally, a Cauchy-elastic material must satisfy the requirements of
material objectivity
Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics.
Biography ...
.
Cauchy-elastic materials are mathematical abstractions, and no real material fits this definition perfectly. However, many elastic materials of practical interest, such as steel, plastic, wood and concrete, can often be assumed to be Cauchy-elastic for the purposes of
stress analysis
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 ...
.
Mathematical definition
Formally, a material is said to be Cauchy-elastic if the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
is a function of 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 ...
(
deformation gradient
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
)
alone:
:
This definition assumes that the effect of temperature can be ignored, and the body is homogeneous. This is the
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
for a Cauchy-elastic material.
Note that the function
depends on the choice of reference configuration. Typically, the reference configuration is taken as the relaxed (zero-stress) configuration, but need not be.
Material frame-indifference requires that the constitutive relation
should not change when the location of the observer changes. Therefore the
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
for another arbitrary observer can be written
. Knowing that the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
and the
deformation gradient
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
are objective quantities, one can write:
:
where
is a proper orthogonal tensor.
The above is a condition that the
constitutive law has to respect to make sure that the response of the material will be independent of the observer. Similar conditions can be derived for
constitutive laws relating the
deformation gradient
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
to the first or second
Piola-Kirchhoff stress tensor.
Isotropic Cauchy-elastic materials
For an isotropic material the
Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
can be expressed as a function of the
left Cauchy-Green tensor . The
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
may then be written:
:
In order to find the restriction on
which will ensure the principle of material frame-indifference, one can write:
:
A
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
that respects the above condition is said to be
isotropic.
Non-conservative materials
Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses may depend on the path of deformation. Therefore a Cauchy elastic material in general has a non-conservative structure, and the stress cannot necessarily be derived from a scalar "elastic potential" function. Materials that are conservative in this sense are called
hyperelastic or "Green-elastic".
References
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Continuum mechanics
Elasticity (physics)