Continuum mechanics  


In continuum mechanics, the finite strain theory also called large strain theory, or large deformation theory, deals with deformations in which both rotations and strains are arbitrarily large, i.e. invalidate the assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made between them. This is commonly the case with elastomers, plasticallydeforming materials and other fluids and biological soft tissue.
A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigidbody displacement and a deformation. A rigidbody displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration to a current or deformed configuration (Figure 1).
If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the other hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero i.e. the distance between particles remains unchanged, then there is no deformation and a rigidbody displacement is said to have occurred.
The vector joining the positions of a particle in the undeformed configuration and deformed configuration is called the displacement vector , in the Lagrangian description, or , in the Eulerian description.
A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field. In general, the displacement field is expressed in terms of the material coordinates as
or in terms of the spatial coordinates as
where are the direction cosines between the material and spatial coordinate systems with unit vectors and , respectively. Thus
and the relationship between and is then given by
Knowing that
then
It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in , and the direction cosines become Kronecker deltas, i.e.
Thus, we have
or in terms of the spatial coordinates as
The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor . Thus we have,
where is the deformation gradient tensor.
Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the spatial displacement gradient tensor . Thus we have,
Consider a particle or material point with position vector in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by in the new configuration is given by the vector position . The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.
Consider now a material point neighboring , with position vector . In the deformed configuration this particle has a new position given by the position vector . Assuming that the line segments and joining the particles and in both the undeformed and deformed configuration, respectively, to be very small, then we can expressed them as and . Thus from Figure 2 we have
where is the relative displacement vector, which represents the relative displacement of with respect to in the deformed configuration.
For an infinitesimal element , and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point , neglecting higherorder terms, to approximate the components of the relative displacement vector for the neighboring particle as
Thus, the previous equation can be written as
The material deformation gradient tensor is a secondorder tensor that represents the gradient of the mapping function or functional relation , which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector , i.e. deformation at neighbouring points, by transforming (linear transformation) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function , i.e differentiable function of and time , which implies that cracks and voids do not open or close during the deformation. Thus we have,
The deformation gradient tensor is related to both the reference and current configuration, as seen by the unit vectors and , therefore it is a twopoint tensor.
Due to the assumption of continuity of , has the inverse , where is the spatial deformation gradient tensor. Then, by the implicit function theorem (Lubliner), the Jacobian determinant must be nonsingular, i.e.
To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use the Nanson's relation, expressed as
where is an area of a region in the deformed configuration, is the same area in the reference configuration, and is the outward normal to the area element in the current configuration while is the outward normal in the reference configuration, is the deformation gradient, and .
Derivation of Nanson's relation 

To see how this formula is derived, we start with the oriented
area elements
in the reference and current configurations: The reference and current volumes of an element are where . Therefore, or, or, So we get or, 
The deformation gradient , like any secondorder tensor, can be decomposed, using the polar decomposition theorem, into a product of two secondorder tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.
where the tensor is a proper orthogonal tensor, i.e. and , representing a rotation; the tensor is the right stretch tensor; and the left stretch tensor. The terms right and left means that they are to the right and left of the rotation tensor , respectively. and are both positive definite, i.e. and , and symmetric tensors, i.e. and , of second order.
This decomposition implies that the deformation of a line element in the undeformed configuration onto in the deformed configuration, i.e. , may be obtained either by first stretching the element by , i.e. , followed by a rotation , i.e. ; or equivalently, by applying a rigid rotation first, i.e. , followed later by a stretching , i.e. (See Figure 3).
It can be shown that,
so that and have the same eigenvalues or principal stretches, but different eigenvectors or principal directions and , respectively. The principal directions are related by
This polar decomposition is unique as is nonsymmetric.
Several rotationindependent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left CauchyGreen deformation tensors. The Finger deformation tensor is mainly used in describing the motion of nonlinear fluids.
Since a pure rotation should not induce any stresses in a deformable body, it is often convenient to use rotationindependent measures of deformation in continuum mechanics. As a rotation followed by its inverse rotation leads to no change () we can exclude the rotation by multiplying by its transpose.
In 1839, George Green introduced a deformation tensor known as the right CauchyGreen deformation tensor or Green's deformation tensor, defined as:
Physically, the CauchyGreen tensor gives us the square of local change in distances due to deformation, i.e.
Invariants of are often used in the expressions for strain energy density functions. The most commonly used invariants are
Reversing the order of multiplication in the formula for the right GreenCauchy deformation tensor leads to the left CauchyGreen deformation tensor which is defined as:
Invariants of are also used in the expressions for strain energy density functions. The conventional invariants are defined as
where is the determinant of the deformation gradient.
For nearly incompressible materials, a slightly different set of invariants is used:
Earlier in 1828 ^{[1]}, Augustin Louis Cauchy introduced a deformation tensor defined as the inverse of the left CauchyGreen deformation tensor, , which is often called the Cauchy deformation tensor or Finger deformation tensor, named after Josef Finger (1894).
If there are three distinct principal stretches , the spectral decompositions of and is given by
Furthermore,
Observe that
Therefore the uniqueness of the spectral decomposition also implies that . The left stretch () is also called the spatial stretch tensor while the right stretch () is called the material stretch tensor.
The effect of acting on is to stretch the vector by and to rotate it to the new orientation , i.e,
In a similar vein,
Examples 

Uniaxial extension of an incompressible
material
This is the case where a specimen is stretched in 1direction with a stretch ratio of . If the volume remains constant, the contraction in the other two directions is such that or . Then: Simple shear Rigid body rotation 
Derivatives of the stretch with respect to the right CauchyGreen deformation tensor are used to derive the stressstrain relations of many solids, particularly hyperelastic materials. These derivatives are
and follow from the observations that
The concept of strain is used to evaluate how much a given displacement differs locally from a rigid body displacement (Ref. Lubliner). One of such strains for large deformations is the Lagrangian finite strain tensor, also called the GreenLagrangian strain tensor or Green  StVenant strain tensor, defined as
or as a function of the displacement gradient tensor
or
The GreenLagrangian strain tensor is a measure of how much differs from . It can be shown that this tensor is a special case of a general formula for Lagrangian strain tensors (Hill 1968):
For different values of we have:
The EulerianAlmansi finite strain tensor, referenced to the deformed configuration, i.e. Eulerian description, is defined as
or as a function of the displacement gradients we have
Derivation of the Lagrangian and Eulerain finite strain tensors 

A measure of deformation is the difference between the squares
of the differential line element ,
in the undeformed configuration, and ,
in the deformed configuration (Figure 2). Deformation has occurred
if the difference is non zero, otherwise a rigidbody displacement
has occurred. Thus we have,
In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is Then we have, where are the components of the right CauchyGreen deformation tensor, . Then, replacing this equation into the first equation we have, or where , are the components of a secondorder tensor called the Green  StVenant strain tensor or the Lagrangian finite strain tensor, In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is where are the components of the spatial deformation gradient tensor, . Thus we have where the second order tensor is called Cauchy's deformation tensor, . Then we have, or where , are the components of a secondorder tensor called the EulerianAlmansi finite strain tensor, Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector with respect to the material coordinates to obtain the material displacement gradient tensor, Replacing this equation into the expression for the Lagrangian finite strain tensor we have or Similarly, the EulerianAlmansi finite strain tensor can be expressed as 
The stretch ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration.
The stretch ratio for the differential element (Figure) in the direction of the unit vector at the material point , in the undeformed configuration, is defined as
where is the deformed magnitude of the differential element .
Similarly, the stretch ratio for the differential element (Figure), in the direction of the unit vector at the material point , in the deformed configuration, is defined as
The normal strain in any direction can be expressed as a function of the stretch ratio,
This equation implies that the normal strain is zero, i.e. no deformation, when the stretch is equal to unity. Some materials, such as elastometers can sustain stretch ratios of 3 or 4 before they fail, whereas traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios, perhaps of the order of 1.001 (reference?)
The diagonal components of the Lagrangian finite strain tensor are related to the normal strain, e.g.
where is the normal strain or engineering strain in the direction .
The offdiagonal components of the Lagrangian finite strain tensor are related to shear strain, e.g.
where is the change in the angle between two line elements that were originally perpendicular with directions and , respectively.
Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the infinitesimal strain tensor
Derivation of the physical interpretation of the Lagrangian and Eulerian finite strain tensors 

The stretch ratio for the differential element
(Figure) in the direction of the unit vector
at the material point ,
in the undeformed configuration, is defined as
where is the deformed magnitude of the differential element . Similarly, the stretch ratio for the differential element (Figure), in the direction of the unit vector at the material point , in the deformed configuration, is defined as The square of the stretch ratio is defined as Knowing that we have where and are unit vectors. The normal strain or engineering strain in any direction can be expressed as a function of the stretch ratio, Thus, the normal strain in the direction at the material point may be expressed in terms of the stretch ratio as solving for we have The shear strain, or change in angle between two line elements and initially perpendicular, and oriented in the principal directions and , respectivelly, can also be expressed as a function of the stretch ratio. From the dot product between the deformed lines and we have where is the angle between the lines and in the deformed configuration. Defining as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have thus, then or 
