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 strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue.
Displacement.
The displacement of a body has two components: a rigid-body displacement and a deformation.
A change in the configuration of a continuum body can be described by a displacement field. 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. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.
Material coordinates (Lagrangian description).
The displacement of particles indexed by variable may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration formula_3 and deformed configuration formula_4 is called the displacement vector. Using formula_5 in place of formula_3 and formula_7 in place of formula_4, both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector:
Where formula_10 are the orthonormal unit vectors that define the basis of the spatial (lab-frame) coordinate system.
Expressed in terms of the material coordinates, the displacement field is:
Where formula_12 is the displacement vector representing rigid-body translation.
The partial derivative of the displacement vector with respect to the material coordinates yields the "material displacement gradient tensor" formula_13. Thus we have,
where formula_15 is the "deformation gradient tensor".
Spatial coordinates (Eulerian description).
In the Eulerian description, the vector extending from a particle formula_16 in the undeformed configuration to its location in the deformed configuration is called the displacement vector:
Where formula_18 are the unit vectors that define the basis of the material (body-frame) coordinate system.
Expressed in terms of spatial coordinates, the displacement field is:
The partial derivative of the displacement vector with respect to the spatial coordinates yields the "spatial displacement gradient tensor" formula_20. Thus we have,
Relationship between the material and spatial coordinate systems.
formula_22 are the direction cosines between the material and spatial coordinate systems with unit vectors formula_23 and formula_10, respectively. Thus
The relationship between formula_26 and formula_27 is then given by
Knowing that
then
Combining the coordinate systems of deformed and undeformed configurations.
It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in formula_31, and the direction cosines become Kronecker deltas, i.e.
Thus in material (undeformed) coordinates, the displacement may be expressed as:
And in spatial (deformed) coordinates, the displacement may be expressed as:
Deformation gradient tensor.
The deformation gradient tensor formula_35 is related to both the reference and current configuration, as seen by the unit vectors formula_36 and formula_37, therefore it is a "two-point tensor".
Due to the assumption of continuity of formula_38, formula_15 has the inverse formula_40, where formula_41 is the "spatial deformation gradient tensor". Then, by the implicit function theorem, the Jacobian determinant formula_42 must be nonsingular, i.e. formula_43
The "material deformation gradient tensor" formula_35 is a second-order tensor that represents the gradient of the mapping function or functional relation formula_38, which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector formula_46, 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 formula_38, i.e. differentiable function of formula_48 and time formula_49, which implies that cracks and voids do not open or close during the deformation. Thus we have,
Relative displacement vector.
Consider a particle or material point formula_16 with position vector formula_52 in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by formula_53 in the new configuration is given by the vector position formula_54. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.
Consider now a material point formula_55 neighboring formula_16, with position vector formula_57. In the deformed configuration this particle has a new position formula_58 given by the position vector formula_59. Assuming that the line segments formula_60 and formula_61 joining the particles formula_16 and formula_63 in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as formula_64 and formula_65. Thus from Figure 2 we have
where formula_67 is the relative displacement vector, which represents the relative displacement of formula_63 with respect to formula_16 in the deformed configuration.
Taylor approximation.
For an infinitesimal element formula_64, and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point formula_16, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle formula_63 as
Thus, the previous equation formula_74 can be written as
Time-derivative of the deformation gradient.
Calculations that involve the time-dependent deformation of a body often require a time derivative of the deformation gradient to be calculated. A geometrically consistent definition of such a derivative requires an excursion into differential geometry but we avoid those issues in this article.
The time derivative of formula_76 is
where formula_78 is the velocity. The derivative on the right hand side represents a material velocity gradient. It is common to convert that into a spatial gradient, i.e.,
where formula_80 is the spatial velocity gradient. If the spatial velocity gradient is constant, the above equation can be solved exactly to give
assuming formula_82 at formula_83. There are several methods of computing the exponential above.
Related quantities often used in continuum mechanics are the rate of deformation tensor and the spin tensor defined, respectively, as:
The rate of deformation tensor gives the rate of stretching of line elements while the spin tensor indicates the rate of rotation or vorticity of the motion.
The material time derivative of the inverse of the deformation gradient (keeping the reference configuration fixed) is often required in analyses that involve finite strains. This derivative is 
The above relation can be verified by taking the material time derivative of formula_86 and noting that formula_87.
Transformation of a surface and volume element.
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 Nanson's relation, expressed as
where formula_89 is an area of a region in the deformed configuration, formula_90 is the same area in the reference configuration, and formula_91 is the outward normal to the area element in the current configuration while formula_92 is the outward normal in the reference configuration, formula_93 is the deformation gradient, and formula_94.
The corresponding formula for the transformation of the volume element is 
Polar decomposition of the deformation gradient tensor.
The deformation gradient formula_93, like any invertible second-order tensor, can be decomposed, using the polar decomposition theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.
where the tensor formula_98 is a proper orthogonal tensor, i.e. formula_99 and formula_100, representing a rotation; the tensor formula_101 is the "right stretch tensor"; and formula_102 the "left stretch tensor". The terms "right" and "left" means that they are to the right and left of the rotation tensor formula_98, respectively. formula_101 and formula_102 are both positive definite, i.e. formula_106 and formula_107 for all non-zero formula_108, and symmetric tensors, i.e. formula_109 and formula_110, of second order.
This decomposition implies that the deformation of a line element formula_64 in the undeformed configuration onto formula_65 in the deformed configuration, i.e. formula_113, may be obtained either by first stretching the element by formula_114, i.e. formula_115, followed by a rotation formula_116, i.e. formula_117; or equivalently, by applying a rigid rotation formula_116 first, i.e. formula_119, followed later by a stretching formula_120, i.e. formula_121 (See Figure 3).
Due to the orthogonality of formula_122
so that formula_114 and formula_120 have the same eigenvalues or "principal stretches", but different eigenvectors or "principal directions" formula_126 and formula_127, respectively. The principal directions are related by
This polar decomposition, which is unique as formula_15 is invertible with a positive determinant, is a corrolary of the singular-value decomposition.
Deformation tensors.
Several rotation-independent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy–Green deformation tensors.
Since a pure rotation should not induce any strains in a deformable body, it is often convenient to use rotation-independent measures of deformation in continuum mechanics. As a rotation followed by its inverse rotation leads to no change (formula_130) we can exclude the rotation by multiplying formula_93 by its transpose.
The right Cauchy–Green deformation tensor.
In 1839, George Green introduced a deformation tensor known as the "right Cauchy–Green deformation tensor" or "Green's deformation tensor", defined as:
Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e. formula_133
Invariants of formula_134 are often used in the expressions for strain energy density functions. The most commonly used invariants are
where formula_136 are stretch ratios for the unit fibers that are initially oriented along the eigenvector directions of the right (reference) stretch tensor (these are not generally aligned with the three axis of the coordinate systems).
The Finger deformation tensor.
The IUPAC recommends that the inverse of the right Cauchy–Green deformation tensor (called the Cauchy tensor in that document), i. e., formula_137, be called the Finger tensor. However, that nomenclature is not universally accepted in applied mechanics.
The left Cauchy–Green or Finger deformation tensor.
Reversing the order of multiplication in the formula for the right Green–Cauchy deformation tensor leads to the "left Cauchy–Green deformation tensor" which is defined as:
The left Cauchy–Green deformation tensor is often called the "Finger deformation tensor", named after Josef Finger (1894).
Invariants of formula_140 are also used in the expressions for strain energy density functions. The conventional invariants are defined as
where formula_142 is the determinant of the deformation gradient.
For incompressible materials, a slightly different set of invariants is used:
The Cauchy deformation tensor.
Earlier in 1828, Augustin Louis Cauchy introduced a deformation tensor defined as the inverse of the left Cauchy–Green deformation tensor, formula_144. This tensor has also been called the Piola tensor and the Finger tensor in the rheology and fluid dynamics literature.
Spectral representation.
If there are three distinct principal stretches formula_146, the spectral decompositions of formula_134 and formula_140 is given by
Furthermore,
Observe that
Therefore, the uniqueness of the spectral decomposition also implies that formula_153. The left stretch (formula_102) is also called the "spatial stretch tensor" while the right stretch (formula_101) is called the "material stretch tensor".
The effect of formula_93 acting on formula_126 is to stretch the vector by formula_136 and to rotate it to the new orientation formula_127, i.e.,
In a similar vein, 
Derivatives of stretch.
Derivatives of the stretch with respect to the right Cauchy–Green deformation tensor are used to derive the stress-strain relations of many solids, particularly hyperelastic materials. These derivatives are
and follow from the observations that
Physical interpretation of deformation tensors.
Let formula_164 be a Cartesian coordinate system defined on the undeformed body and let formula_165 be another system defined on the deformed body. Let a curve formula_166 in the undeformed body be parametrized using formula_167. Its image in the deformed body is formula_168.
The undeformed length of the curve is given by
After deformation, the length becomes
Note that the right Cauchy–Green deformation tensor is defined as
Hence,
which indicates that changes in length are characterized by formula_173.
Finite strain tensors.
The concept of "strain" is used to evaluate how much a given displacement differs locally from a rigid body displacement. One of such strains for large deformations is the "Lagrangian finite strain tensor", also called the "Green-Lagrangian strain tensor" or "Green – St-Venant strain tensor", defined as
or as a function of the displacement gradient tensor
or
The Green-Lagrangian strain tensor is a measure of how much formula_177 differs from formula_178.
The "Eulerian-Almansi 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
Seth–Hill family of generalized strain tensors.
B. R. Seth from the Indian Institute of Technology Kharagpur was the first to show that the Green and Almansi strain tensors are special cases of a more general strain measure. The idea was further expanded upon by Rodney Hill in 1968. The Seth–Hill family of strain measures (also called Doyle-Ericksen tensors) can be expressed as
For different values of formula_182 we have:
The second-order approximation of these tensors is
where formula_185 is the infinitesimal strain tensor.
Many other different definitions of tensors formula_186 are admissible, provided that they all satisfy the conditions that:
An example is the set of tensors 
which do not belong to the Seth–Hill class, but have the same 2nd-order approximation as the Seth–Hill measures at formula_194 for any value of formula_195.
Stretch ratio.
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 formula_196 (Figure) in the direction of the unit vector formula_197 at the material point formula_16, in the undeformed configuration, is defined as
where formula_200 is the deformed magnitude of the differential element formula_64.
Similarly, the stretch ratio for the differential element formula_202 (Figure), in the direction of the unit vector formula_203 at the material point formula_53, in the deformed configuration, is defined as
The normal strain formula_206 in any direction formula_197 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 1.1 (reference?)
Physical interpretation of the finite strain tensor.
The diagonal components formula_209 of the Lagrangian finite strain tensor are related to the normal strain, e.g.
where formula_211 is the normal strain or engineering strain in the direction formula_212.
The off-diagonal components formula_209 of the Lagrangian finite strain tensor are related to shear strain, e.g.
where formula_215 is the change in the angle between two line elements that were originally perpendicular with directions formula_212 and formula_217, 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
Deformation tensors in convected curvilinear coordinates.
A representation of deformation tensors in curvilinear coordinates is useful for many problems in continuum mechanics such as nonlinear shell theories and large plastic deformations. Let formula_218 denote the function by which a position vector in space is constructed from coordinates formula_219. The coordinates are said to be "convected" if they correspond to a one-to-one mapping to and from Lagrangian particles in a continuum body. If the coordinate grid is "painted" on the body in its initial configuration, then this grid will deform and flow with the motion of material to remain painted on the same material particles in the deformed configuration so that grid lines intersect at the same material particle in either configuration. The tangent vector to the deformed coordinate grid line curve formula_220 at formula_221 is given by
The three tangent vectors at formula_221 form a local basis. These vectors are related the reciprocal basis vectors by 
Let us define a second-order tensor field formula_225 (also called the metric tensor) with components
The Christoffel symbols of the first kind can be expressed as
To see how the Christoffel symbols are related to the Right Cauchy–Green deformation tensor let us similarly define two bases, the already mentioned one that is tangent to deformed grid lines and another that is tangent to the undeformed grid lines. Namely,
The deformation gradient in curvilinear coordinates.
Using the definition of the gradient of a vector field in curvilinear coordinates, the deformation gradient can be written as
The right Cauchy–Green tensor in curvilinear coordinates.
The right Cauchy–Green deformation tensor is given by
If we express formula_173 in terms of components with respect to the basis {formula_232} we have
Therefore, 
and the corresponding Christoffel symbol of the first kind may be written in the following form.
Some relations between deformation measures and Christoffel symbols.
Consider a one-to-one mapping from formula_236 to formula_237 and let us assume that there exist two positive-definite, symmetric second-order tensor fields formula_238 and formula_225 that satisfy
Then,
Noting that
and formula_243 we have
Define
Hence
Define
Then
Define the Christoffel symbols of the second kind as
Then
Therefore,
The invertibility of the mapping implies that
We can also formulate a similar result in terms of derivatives with respect to formula_253. Therefore,
Compatibility conditions.
The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on bodies. These allowable conditions leave the body without unphysical gaps or overlaps after a deformation. Most such conditions apply to simply-connected bodies. Additional conditions are required for the internal boundaries of multiply connected bodies.
Compatibility of the deformation gradient.
The necessary and sufficient conditions for the existence of a compatible formula_255 field over a simply connected body are
Compatibility of the right Cauchy–Green deformation tensor.
The necessary and sufficient conditions for the existence of a compatible formula_173 field over a simply connected body are
We can show these are the mixed components of the Riemann–Christoffel curvature tensor. Therefore, the necessary conditions for formula_173-compatibility are that the Riemann–Christoffel curvature of the deformation is zero.
Compatibility of the left Cauchy–Green deformation tensor.
No general sufficiency conditions are known for the left Cauchy–Green deformation tensor in three-dimensions. Compatibility conditions for two-dimensional formula_260 fields have been found by Janet Blume.