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 m ...
, the most commonly used measure of
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 ...
is 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 complete ...
, often called simply ''the'' stress tensor or "true stress". However, several alternative measures of stress can be defined:
#The Kirchhoff stress (
).
#The Nominal stress (
).
#The first Piola–Kirchhoff stress (
). This stress tensor is the transpose of the nominal stress (
).
#The second Piola–Kirchhoff stress or PK2 stress (
).
#The Biot stress (
)
Definitions
Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.
In the reference configuration
, the outward normal to a surface element
is
and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is
leading to a force vector
. In the deformed configuration
, the surface element changes to
with outward normal
and traction vector
leading to a force
. Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity
is the
deformation gradient tensor,
is its determinant.
Cauchy stress
The Cauchy stress (or true stress) is a measure of the force acting on an element of area in the deformed configuration. This tensor is symmetric and is defined via
:
or
:
where
is the traction and
is the normal to the surface on which the traction acts.
Kirchhoff stress
The quantity,
:
is called the Kirchhoff stress tensor, with
the determinant of
. It is used widely in numerical algorithms in metal plasticity (where there
is no change in volume during plastic deformation). It can be called ''weighted Cauchy stress tensor'' as well.
Piola–Kirchhoff stress
Nominal stress/First Piola–Kirchhoff stress
The nominal stress
is the transpose of the first Piola–Kirchhoff stress (PK1 stress, also called engineering stress)
and is defined via
:
or
:
This stress is unsymmetric and is a two-point tensor like the deformation gradient.
The asymmetry derives from the fact that, as a tensor, it has one index attached to the reference configuration and one to the deformed configuration.
Second Piola–Kirchhoff stress
If we
pull back
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
to the reference configuration we obtain the traction acting on that surface before the deformation
assuming it behaves like a generic vector belonging to the deformation. In particular we have
:
or,
:
The PK2 stress (
) is symmetric and is defined via the relation
:
Therefore,
:
Biot stress
The Biot stress is useful because it is
energy conjugate to the
right stretch tensor . The Biot stress is defined as the symmetric part of the tensor
where
is the rotation tensor obtained from a
polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
of the deformation gradient. Therefore, the Biot stress tensor is defined as
:
The Biot stress is also called the Jaumann stress.
The quantity
does not have any physical interpretation. However, the unsymmetrized Biot stress has the interpretation
:
Relations
Relations between Cauchy stress and nominal stress
From
Nanson's formula relating areas in the reference and deformed configurations:
:
Now,
:
Hence,
:
or,
:
or,
:
In index notation,
:
Therefore,
:
Note that
and
are (generally) not symmetric because
is (generally) not symmetric.
Relations between nominal stress and second P–K stress
Recall that
:
and
:
Therefore,
:
or (using the symmetry of
),
:
In index notation,
:
Alternatively, we can write
:
Relations between Cauchy stress and second P–K stress
Recall that
:
In terms of the 2nd PK stress, we have
:
Therefore,
:
In index notation,
:
Since the Cauchy stress (and hence the Kirchhoff stress) is symmetric, the 2nd PK stress is also symmetric.
Alternatively, we can write
:
or,
:
Clearly, from definition of the
push-forward and
pull-back
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
operations, we have
:
and
:
Therefore,
is the pull back of
by
and
is the push forward of
.
Summary of conversion formula
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See also
*
Stress (physics)
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
*
Finite strain theory
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 ...
*
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 m ...
*
Hyperelastic material
A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density ...
*
Cauchy elastic material In physics, a Cauchy-elastic material is one in which the stress at each point is determined only by the current state of deformation with respect to an arbitrary reference configuration.R. W. Ogden, 1984, ''Non-linear Elastic Deformations'', Dover, ...
*
Critical plane analysis
Critical plane analysis refers to the analysis of stresses or strains as they are experienced by a particular plane in a material, as well as the identification of which plane is likely to experience the most extreme damage. Critical plane analys ...
References
Solid mechanics
Continuum mechanics
Gustav Kirchhoff
Tensor physical quantities