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 ...
, 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
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, fro ...
s are assumed to be much smaller (indeed,
infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
and
stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a b ...
) at each point of space can be assumed to be unchanged by the deformation.
With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the
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 ...
where the opposite assumption is made.
The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the
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 ...
of structures built from relatively stiff
elastic materials like
concrete
Concrete is a composite material composed of fine and coarse aggregate bonded together with a fluid cement (cement paste) that hardens (cures) over time. Concrete is the second-most-used substance in the world after water, and is the most ...
and
steel, since a common goal in the design of such structures is to minimize their deformation under typical
loads. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.
Infinitesimal strain tensor
For ''infinitesimal deformations'' of a
continuum body, in which the
displacement gradient (2nd order tensor) is small compared to unity, i.e.
,
it is possible to perform a ''geometric linearization'' of any one of the (infinitely many possible) strain tensors used in finite strain theory, e.g. the Lagrangian strain tensor
, and the Eulerian strain tensor
. In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have
or
and
or
This linearization implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the material displacement gradient components and the spatial displacement gradient components are approximately equal. Thus we have
or
where
are the components of the ''infinitesimal strain tensor''
, also called ''Cauchy's strain tensor'', ''linear strain tensor'', or ''small strain tensor''.
or using different notation:
Furthermore, since the
deformation gradient can be expressed as
where
is the second-order identity tensor, we have
Also, from the
general expression for the Lagrangian and Eulerian finite strain tensors we have
Geometric derivation
Consider a two-dimensional deformation of an infinitesimal rectangular material element with dimensions
by
(Figure 1), which after deformation, takes the form of a rhombus. From the geometry of Figure 1 we have
For very small displacement gradients, i.e.,
, we have
The
normal strain in the
-direction of the rectangular element is defined by
and knowing that
, we have
Similarly, the normal strain in the and becomes
The
engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line
and
, is defined as
From the geometry of Figure 1 we have
For small rotations, i.e.,
and
are
we have
and, again, for small displacement gradients, we have
thus
By interchanging
and
and
and
, it can be shown that
.
Similarly, for the
-
and
-
planes, we have
It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, as
Physical interpretation
From
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 ...
we have
For infinitesimal strains then we have
Dividing by
we have
For small deformations we assume that
, thus the second term of the left hand side becomes:
.
Then we have
where
, is the unit vector in the direction of
, and the left-hand-side expression is the
normal strain in the direction of
. For the particular case of
in the
direction, i.e.,
, we have
Similarly, for
and
we can find the normal strains
and
, respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.
Strain transformation rules
If we choose an
orthonormal coordinate system (
) we can write the tensor in terms of components with respect to those base vectors as
In matrix form,
We can easily choose to use another orthonormal coordinate system (
) instead. In that case the components of the tensor are different, say
The components of the strain in the two coordinate systems are related by
where the
Einstein summation convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
for repeated indices has been used and
. In matrix form
or
Strain invariants
Certain operations on the strain tensor give the same result without regard to which orthonormal coordinate system is used to represent the components of strain. The results of these operations are called strain invariants. The most commonly used strain invariants are
In terms of components
Principal strains
It can be shown that it is possible to find a coordinate system (
) in which the components of the strain tensor are
The components of the strain tensor in the (
) coordinate system are called the principal strains and the directions
are called the directions of principal strain. Since there are no shear strain components in this coordinate system, the principal strains represent the maximum and minimum stretches of an elemental volume.
If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an
eigenvalue decomposition determined by solving the system of equations
This system of equations is equivalent to finding the vector
along which the strain tensor becomes a pure stretch with no shear component.
Volumetric strain
The ''dilatation'' (the relative variation of the volume) is the
first strain invariant or
trace of the tensor:
Actually, if we consider a cube with an edge length ''a'', it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions
and ''V''
0 = ''a''
3, thus
as we consider small deformations,
therefore the formula.
In case of pure shear, we can see that there is no change of the volume.
Strain deviator tensor
The infinitesimal strain tensor
, similarly to 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 the sum of two other tensors:
# a mean strain tensor or volumetric strain tensor or spherical strain tensor,
, related to dilation or volume change; and
# a deviatoric component called the strain deviator tensor,
, related to distortion.
where
is the mean strain given by
The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:
Octahedral strains
Let (
) be the directions of the three principal strains. An octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering
shear strain on an octahedral plane is called the octahedral shear strain and is given by
where
are the principal strains.
The
normal strain on an octahedral plane is given by
Equivalent strain
A scalar quantity called the equivalent strain, or the
von Mises equivalent strain, is often used to describe the state of strain in solids. Several definitions of equivalent strain can be found in the literature. A definition that is commonly used in the literature on
plasticity is
This quantity is work conjugate to the equivalent stress defined as
Compatibility equations
For prescribed strain components
the strain tensor equation
represents a system of six differential equations for the determination of three displacements components
, giving an over-determined system. Thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named ''compatibility equations'', are imposed upon the strain components. With the addition of the three compatibility equations the number of independent equations are reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by
Saint-Venant, and are called the "
Saint Venant compatibility equations".
The compatibility functions serve to assure a single-valued continuous displacement function
. If the elastic medium is visualised as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping.
In index notation, the compatibility equations are expressed as
In engineering notation,
*
*
*
*
*
*
Special cases
Plane strain
In real engineering components,
stress (and strain) are 3-D
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
s but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e., the normal strain
and the shear strains
and
(if the length is the 3-direction) are constrained by nearby material and are small compared to the ''cross-sectional strains''. Plane strain is then an acceptable approximation. The
strain tensor for plane strain is written as:
in which the double underline indicates a second order
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
. This strain state is called ''plane strain''. The corresponding stress tensor is:
in which the non-zero
is needed to maintain the constraint
. This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.
Antiplane strain
Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to a
screw dislocation. The
strain tensor for antiplane strain is given by
Infinitesimal rotation tensor
The infinitesimal strain tensor is defined as