Displacement
Figure 1. Motion of a continuum body.
The displacement of a body has two components: a rigid-body displacement and a deformation.
- A rigid-body displacement consists of a simultaneous translation (physics) 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).
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 i may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration
and deformed configuration
is called the displacement vector. Using
in place of
and
in place of
, 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
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
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
. Thus we have,

where
is the deformation gradient tensor.
Spatial coordinates (Eulerian description)
In the Eulerian description, the vector extending from a particle
in the undeformed configuration to its location in the deformed configuration is called the displacement vector:

Where
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
. Thus we have,

Relationship between the material and spatial coordinate systems
are the direction cosines between the material and spatial coordinate systems with unit vectors
and
, respectively. Thus

The relationship between
and
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
, 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
Figure 2. Deformation of a continuum body.
The deformation gradient tensor
is related to both the reference and current configuration, as seen by the unit vectors
and
, therefore it is a two-point tensor.
Due to the assumption of continuity of
,
has the inverse
, where
is the spatial deformation gradient tensor. Then, by the implicit function theorem, the Jacobian determinant
must be nonsingular, i.e.
The material deformation gradient tensor
is a second-order 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,

Relative displacement vector
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 express 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.
Taylor approximation
For an infinitesimal element
, and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point
, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle
as

Thus, the previous equation
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
is
![{\dot {{\mathbf {F}}}}={\frac {\partial {\mathbf {F}}}{\partial t}}={\frac {\partial }{\partial t}}\left[{\frac {\partial {\mathbf {x}}({\mathbf {X}},t)}{\partial {\mathbf {X}}}}\right]={\frac {\partial }{\partial {\mathbf {X}}}}\left[{\frac {\partial {\mathbf {x}}({\mathbf {X}},t)}{\partial t}}\right]={\frac {\partial }{\partial {\mathbf {X}}}}\left[{\mathbf {V}}({\mathbf {X}},t)\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2666606a9db3c727de1e94fa372b590683f72c0)
where
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.,
![{\displaystyle {\dot {\mathbf {F} }}={\frac {\partial }{\partial \mathbf {X} }}\left[\mathbf {V} (\mathbf {X} ,t)\right]={\frac {\partial }{\partial \mathbf {x} }}\left[\mathbf {V} (\mathbf {X} ,t)\right]\cdot {\frac {\partial \mathbf {x} (\mathbf {X} ,t)}{\partial \mathbf {X} }}={\boldsymbol {l}}\cdot \mathbf {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eceea87251a369ad9a9d84fe6e37b32801a6241)
where
is the spatial velocity gradient. If the spatial velocity gradient is constant, the above equation can be solved exactly to give

assuming
at
. 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.
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
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
.
The corresponding formula for the transformation of the volume element is

Derivation of Nanson's relation (see also )
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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,

so,

So we get

or,

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Polar decomposition of the deformation gradient tensor
Figure 3. Representation of the polar decomposition of the deformation gradient
The deformation gradient
, 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
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
for all
, 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).
Due to the orthogonality of

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, which is unique as
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 (
) we can exclude the rotation by multiplying
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.
Invariants of
are often used in the expressions for strain energy density functions. The most commonly used invariants are
![{\begin{aligned}I_{1}^{C}&:={\text{tr}}({\mathbf {C}})=C_{{II}}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\I_{2}^{C}&:={\tfrac {1}{2}}\left[({\text{tr}}~{\mathbf {C}})^{2}-{\text{tr}}({\mathbf {C}}^{2})\right]={\tfrac {1}{2}}\left[(C_{{JJ}})^{2}-C_{{IK}}C_{{KI}}\right]=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\I_{3}^{C}&:=\det({\mathbf {C}})=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}.\end{aligned}}\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/3704e68e2c34682fc7ee876f06e3c89f4197311b)
where
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.,
, 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
are also used in the expressions for strain energy density functions. The conventional invariants are defined as
![{\begin{aligned}I_{1}&:={\text{tr}}({\mathbf {B}})=B_{{ii}}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\I_{2}&:={\tfrac {1}{2}}\left[({\text{tr}}~{\mathbf {B}})^{2}-{\text{tr}}({\mathbf {B}}^{2})\right]={\tfrac {1}{2}}\left(B_{{ii}}^{2}-B_{{jk}}B_{{kj}}\right)=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\I_{3}&:=\det {\mathbf {B}}=J^{2}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}\end{aligned}}\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3d7bf82330d0d670c9e102fc5cfa569abfbc69)
where
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,
. 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
, 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
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Uniaxial extension of an incompressible material
This is the case where a specimen is stretched in 1-direction 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
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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
be a Cartesian coordinate system defined on the undeformed body and let
be another system defined on the deformed body. Let a curve
in the undeformed body be parametrized using
. Its image in the deformed body is
.
The undeformed length of the curve is given by

After deformation, the length becomes
![{\begin{aligned}l_{x}&=\int _{0}^{1}\left|{\cfrac {d{\mathbf {x}}}{ds}}\right|~ds=\int _{0}^{1}{\sqrt {{\cfrac {d{\mathbf {x}}}{ds}}\cdot {\cfrac {d{\mathbf {x}}}{ds}}}}~ds=\int _{0}^{1}{\sqrt {\left({\cfrac {d{\mathbf {x}}}{d{\mathbf {X}}}}\cdot {\cfrac {d{\mathbf {X}}}{ds}}\right)\cdot \left({\cfrac {d{\mathbf {x}}}{d{\mathbf {X}}}}\cdot {\cfrac {d{\mathbf {X}}}{ds}}\right)}}~ds\\&=\int _{0}^{1}{\sqrt {{\cfrac {d{\mathbf {X}}}{ds}}\cdot \left[\left({\cfrac {d{\mathbf {x}}}{d{\mathbf {X}}}}\right)^{T}\cdot {\cfrac {d{\mathbf {x}}}{d{\mathbf {X}}}}\right]\cdot {\cfrac {d{\mathbf {X}}}{ds}}}}~ds\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1b639bab0e25d96cf46894973e1d2693800d94)
Note that the right Cauchy–Green deformation tensor is defined as

Hence,

which indicates that changes in length are characterized by
.
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
![{\mathbf E}={\frac {1}{2}}\left[(\nabla _{{{\mathbf X}}}{\mathbf u})^{T}+\nabla _{{{\mathbf X}}}{\mathbf u}+(\nabla _{{{\mathbf X}}}{\mathbf u})^{T}\cdot \nabla _{{{\mathbf X}}}{\mathbf u}\right]\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb57038f53e8acd75c977dd33dc650dd27913b2c)
or

The Green-Lagrangian strain tensor is a measure of how much
differs from
.
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

Derivation of the Lagrangian and Eulerian finite strain tensors
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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 rigid-body 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 Cauchy–Green deformation tensor, . Then, replacing this equation into the first equation we have,

or

where , are the components of a second-order tensor called the Green – St-Venant 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 second-order tensor called the Eulerian-Almansi 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
![{\begin{aligned}{\mathbf E}&={\frac {1}{2}}\left({\mathbf F}^{T}{\mathbf F}-{\mathbf I}\right)\\&={\frac {1}{2}}\left[\left\{(\nabla _{{{\mathbf X}}}{\mathbf u})^{T}+{\mathbf I}\right\}\left(\nabla _{{{\mathbf X}}}{\mathbf u}+{\mathbf I}\right)-{\mathbf I}\right]\\&={\frac {1}{2}}\left[(\nabla _{{{\mathbf X}}}{\mathbf u})^{T}+\nabla _{{{\mathbf X}}}{\mathbf u}+(\nabla _{{{\mathbf X}}}{\mathbf u})^{T}\cdot \nabla _{{{\mathbf X}}}{\mathbf u}\right]\\\end{aligned}}\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/26c63bb3f2bec5cafe7a362ed4734da711ddb0fb)
or
![{\begin{aligned}E_{{KL}}&={\frac {1}{2}}\left({\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}-\delta _{{KL}}\right)\\&={\frac {1}{2}}\left[\delta _{{jM}}\left({\frac {\partial U_{M}}{\partial X_{K}}}+\delta _{{MK}}\right)\delta _{{jN}}\left({\frac {\partial U_{N}}{\partial X_{L}}}+\delta _{{NL}}\right)-\delta _{{KL}}\right]\\&={\frac {1}{2}}\left[\delta _{{MN}}\left({\frac {\partial U_{M}}{\partial X_{K}}}+\delta _{{MK}}\right)\left({\frac {\partial U_{N}}{\partial X_{L}}}+\delta _{{NL}}\right)-\delta _{{KL}}\right]\\&={\frac {1}{2}}\left[\left({\frac {\partial U_{M}}{\partial X_{K}}}+\delta _{{MK}}\right)\left({\frac {\partial U_{M}}{\partial X_{L}}}+\delta _{{ML}}\right)-\delta _{{KL}}\right]\\&={\frac {1}{2}}\left({\frac {\partial U_{K}}{\partial X_{L}}}+{\frac {\partial U_{L}}{\partial X_{K}}}+{\frac {\partial U_{M}}{\partial X_{K}}}{\frac {\partial U_{M}}{\partial X_{L}}}\right)\end{aligned}}\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2191e97fce373db79be4298a1a6769b1ceb97ce)
Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

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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
![{\mathbf E}_{{(m)}}={\frac {1}{2m}}({\mathbf U}^{{2m}}-{\mathbf I})={\frac {1}{2m}}\left[{\mathbf {C}}^{{m}}-{\mathbf {I}}\right]\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/b56dbd88d4681c3732c87a09d3c96be97d04d651)
For different values of
we have:
![{\begin{aligned}{\mathbf E}_{{(1)}}&={\frac {1}{2}}({\mathbf U}^{{2}}-{\mathbf I})={\frac {1}{2}}({\mathbf {C}}-{\mathbf {I}})&\qquad {\text{Green-Lagrangian strain tensor}}\\{\mathbf E}_{{(1/2)}}&=({\mathbf U}-{\mathbf I})={\mathbf {C}}^{{1/2}}-{\mathbf {I}}&\qquad {\text{Biot strain tensor}}\\{\mathbf E}_{{(0)}}&=\ln {\mathbf U}={\frac {1}{2}}\,\ln {\mathbf {C}}&\qquad {\text{Logarithmic strain, Natural strain, True strain, or Hencky strain}}\\{\mathbf {E}}_{{(-1)}}&={\frac {1}{2}}\left[{\mathbf {I}}-{\mathbf {U}}^{{-2}}\right]&\qquad {\text{Almansi strain}}\end{aligned}}\,\!](https://wikimedia.org/api/rest_v1/media/math/render/svg/02986daeab9d6df911daf7c76fb84647fb5e9024)
The second-order approximation of these tensors is

where
is the infinitesimal strain tensor.
Many other different definitions of tensors
are admissible, provided that they all satisfy the conditions that:
vanishes for all rigid-body motions
- the dependence of
on the displacement gradient tensor
is continuous, continuously differentiable and monotonic
- it is also desired that
reduces to the infinitesimal strain tensor
as the norm 
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
for any value of
.
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
(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 1.1 (reference?)
Physical interpretation of the finite strain tensor
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 off-diagonal 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 , respectively, 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

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Deformation tensors
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
denote the function by which a position vector in space is constructed from coordinates
. 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
at
is given by

The three tangent vectors at
form a local basis. These vectors are related the reciprocal basis vectors by

Let us define a second-order tensor field
(also called the metric tensor) with components

The Christoffel symbols of the first kind can be expressed as
![\Gamma _{{ijk}}={\tfrac {1}{2}}[({\mathbf {g}}_{i}\cdot {\mathbf {g}}_{k})_{{,j}}+({\mathbf {g}}_{j}\cdot {\mathbf {g}}_{k})_{{,i}}-({\mathbf {g}}_{i}\cdot {\mathbf {g}}_{j})_{{,k}}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c84f52a8b94bf81096fd54ad218ac8c8a794593)
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
in terms of components with respect to the basis {
} we have

Therefore,

and the corresponding Christoffel symbol of the first kind may be written in the following form.
![\Gamma _{{ijk}}={\tfrac {1}{2}}[C_{{ik,j}}+C_{{jk,i}}-C_{{ij,k}}]={\tfrac {1}{2}}[({\mathbf {G}}_{i}\cdot {\boldsymbol {C}}\cdot {\mathbf {G}}_{k})_{{,j}}+({\mathbf {G}}_{j}\cdot {\boldsymbol {C}}\cdot {\mathbf {G}}_{k})_{{,i}}-({\mathbf {G}}_{i}\cdot {\boldsymbol {C}}\cdot {\mathbf {G}}_{j})_{{,k}}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/626053366fba8dffc9ee988c8728f9b476cee63a)
Some relations between deformation measures and Christoffel symbols
Consider a one-to-one mapping from
to
and let us assume that there exist two positive-definite, symmetric second-order tensor fields
and
that satisfy

Then,

Noting that

and
we have

Define

Hence

Define
![[G^{{ij}}]=[G_{{ij}}]^{{-1}}~;~~[g^{{\alpha \beta }}]=[g_{{\alpha \beta }}]^{{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6168f5195b650daf063a1dd26b314242926f774)
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
. Therefore,

Compatibility conditions
Main article: Compatibility (mechanics)
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
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
field over a simply connected body are
![R_{{\alpha \beta \rho }}^{\gamma }:={\frac {\partial }{\partial X^{\rho }}}[\,_{{(X)}}\Gamma _{{\alpha \beta }}^{\gamma }]-{\frac {\partial }{\partial X^{\beta }}}[\,_{{(X)}}\Gamma _{{\alpha \rho }}^{\gamma }]+\,_{{(X)}}\Gamma _{{\mu \rho }}^{\gamma }\,_{{(X)}}\Gamma _{{\alpha \beta }}^{\mu }-\,_{{(X)}}\Gamma _{{\mu \beta }}^{\gamma }\,_{{(X)}}\Gamma _{{\alpha \rho }}^{\mu }=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/29e1d54640569dc6ab58c0fca0c0ab634aef2180)
We can show these are the mixed components of the Riemann–Christoffel curvature tensor. Therefore, the necessary conditions for
-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
fields have been found by Janet Blume.