Finite deformation tensors

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 field

Deformation gradient tensor

The deformation gradient tensor $\mathbf F(\mathbf X,t) = F_ \mathbf e_j \otimes \mathbf I_K$ is related to both the reference and current configuration, as seen by the unit vectors $\mathbf e_j$ and $\mathbf I_K\,\!$, therefore it is a '' two-point tensor''.
Two types of deformation gradient tensor may be defined.

Due to the assumption of continuity of $\chi(\mathbf X,t)\,\!$, $\mathbf F$ has the inverse $\mathbf H = \mathbf F^\,\!$, where $\mathbf H$ is the spatial deformation gradient tensor. Then, by the implicit function theorem, the Jacobian determinant $J(\mathbf X,t)$ must be nonsingular, i.e. $J(\mathbf X,t) = \det \mathbf F(\mathbf X,t) \neq 0$

The material deformation gradient tensor $\mathbf F(\mathbf X,t) = F_ \mathbf e_j\otimes\mathbf I_K$ is a second-order tensor that represents the gradient of the mapping function or functional relation $\chi(\mathbf X,t)\,\!$, which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector $\mathbf X\,\!$, 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 $\chi(\mathbf X,t)\,\!$, i.e. differentiable function of $\mathbf$ and time $t\,\!$, which implies that cracks and voids do not open or close during the deformation. Thus we have,
$\begin d\mathbf &= \frac \,d\mathbf \qquad &\text& \qquad dx_j =\frac\,dX_K \\ &= \nabla \chi(\mathbf X,t) \,d\mathbf \qquad &\text& \qquad dx_j =F_\,dX_K \,. \\ & = \mathbf F(\mathbf X,t) \,d\mathbf \end$

Relative displacement vector

Consider a particle or material point $P$ with position vector $\mathbf X = X_I \mathbf I_I$ in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by $p$ in the new configuration is given by the vector position $\mathbf = x_i \mathbf e_i\,\!$. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.

Consider now a material point $Q$ neighboring $P\,\!$, with position vector $\mathbf+ \Delta \mathbf = (X_I+\Delta X_I) \mathbf I_I\,\!$. In the deformed configuration this particle has a new position $q$ given by the position vector $\mathbf+ \Delta \mathbf\,\!$. Assuming that the line segments $\Delta X$ and $\Delta \mathbf x$ joining the particles $P$ and $Q$ in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as $d\mathbf X$ and $d\mathbf x\,\!$. Thus from Figure 2 we have
$\begin \mathbf+ d\mathbf&= \mathbf+d\mathbf+\mathbf(\mathbf+d\mathbf) \\ d\mathbf &= \mathbf-\mathbf+d\mathbf+ \mathbf(\mathbf+d\mathbf) \\ &= d\mathbf+\mathbf(\mathbf+d\mathbf)- \mathbf(\mathbf) \\ &= d\mathbf+d\mathbf \\ \end$

where $\mathbf$ is the relative displacement vector, which represents the relative displacement of $Q$ with respect to $P$ in the deformed configuration.

Taylor approximation

For an infinitesimal element $d\mathbf X\,\!$, and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point $P\,\!$, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle $Q$ as
$\begin \mathbf(\mathbf+d\mathbf) &= \mathbf(\mathbf)+d\mathbf \quad & \text & \quad u_i^* = u_i+du_i \\ &\approx \mathbf(\mathbf)+\nabla_\mathbf u\cdot d\mathbf X \quad & \text & \quad u_i^* \approx u_i + \fracdX_J \,. \end$
Thus, the previous equation $d\mathbf x = d\mathbf + d\mathbf$ can be written as
$\begin d\mathbf x&=d\mathbf X+d\mathbf u \\ &=d\mathbf X+\nabla_\mathbf u\cdot d\mathbf X\\ &=\left(\mathbf I + \nabla_\mathbf u\right)d\mathbf X\\ &=\mathbf F d\mathbf X \end$

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 geometryA. Yavari, J.E. Marsden, and M. Ortiz
On spatial and material covariant balance laws in elasticity
Journal of Mathematical Physics, 47, 2006, 042903; pp. 1–53. but we avoid those issues in this article.

The time derivative of $\mathbf$ is
\dot = \frac = \frac \left frac\right= \frac\left frac\right= \frac\left mathbf(\mathbf, t)\right
where $\mathbf$ is the (material) velocity. The derivative on the right hand side represents a material velocity gradient. It is common to convert that into a spatial gradient by applying the chain rule for derivatives, i.e.,

\dot =
\frac\left mathbf(\mathbf, t)\right=
\frac\left mathbf(\mathbf(\mathbf, t),t)\right=
\left.\frac\left mathbf(\mathbf,t)\right_ \cdot
\frac
= \boldsymbol\cdot\mathbf

where $\boldsymbol = (\nabla_ \mathbf)^T$ is the spatial velocity gradient and where $\mathbf(\mathbf,t) = \mathbf(\mathbf,t)$ is the spatial (Eulerian) velocity at $\mathbf = \mathbf(\mathbf, t)$. If the spatial velocity gradient is constant in time, the above equation can be solved exactly to give
$\mathbf = e^$
assuming $\mathbf = \mathbf$ at $t = 0$. 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:
$\boldsymbol = \tfrac \left(\boldsymbol + \boldsymbol^T\right) \,,~~ \boldsymbol = \tfrac \left(\boldsymbol - \boldsymbol^T\right) \,.$
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
$\frac \left(\mathbf^\right) = - \mathbf^ \cdot \dot \cdot \mathbf^ \,.$
The above relation can be verified by taking the material time derivative of $\mathbf^ \cdot d\mathbf = d\mathbf$ and noting that $\dot = 0$.

Polar decomposition of the deformation gradient tensor

The deformation gradient $\mathbf$, 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., $\mathbf = \mathbf \mathbf = \mathbf \mathbf$ where the tensor $\mathbf$ is a proper orthogonal tensor, i.e., $\mathbf R^ = \mathbf R^T$ and $\det \mathbf R = +1\,\!$, representing a rotation; the tensor $\mathbf$ is the ''right stretch tensor''; and $\mathbf$ the ''left stretch tensor''. The terms ''right'' and ''left'' means that they are to the right and left of the rotation tensor $\mathbf\,\!$, respectively. $\mathbf$ and $\mathbf$ are both positive definite, i.e. $\mathbf x \cdot \mathbf U \cdot \mathbf x > 0$ and $\mathbf x\cdot\mathbf V \cdot \mathbf x > 0$ for all non-zero $\mathbf x \in \R^3$, and symmetric tensors, i.e. $\mathbf U = \mathbf U^T$ and $\mathbf V = \mathbf V^T\,\!$, of second order.

This decomposition implies that the deformation of a line element $d\mathbf X$ in the undeformed configuration onto $d\mathbf x$ in the deformed configuration, i.e., $d\mathbf x = \mathbf F \,d\mathbf X\,\!$, may be obtained either by first stretching the element by $\mathbf U\,\!$, i.e. $d\mathbf x' = \mathbf U \,d\mathbf X\,\!$, followed by a rotation $\mathbf R\,\!$, i.e., $d\mathbf x = \mathbf R \,d\mathbf x'\,\!$; or equivalently, by applying a rigid rotation $\mathbf R$ first, i.e., $d\mathbf x' = \mathbf R \, d\mathbf X\,\!$, followed later by a stretching $\mathbf V\,\!$, i.e., $d\mathbf x = \mathbf V \, d\mathbf x'$ (See Figure 3).

Due to the orthogonality of $\mathbf R$
$\mathbf V = \mathbf R \cdot \mathbf U \cdot \mathbf R^T$
so that $\mathbf U$ and $\mathbf V$ have the same eigenvalues or ''principal stretches'', but different eigenvectors or ''principal directions'' $\mathbf_i$ and $\mathbf_i\,\!$, respectively. The principal directions are related by
$\mathbf_i = \mathbf \mathbf_i.$

This polar decomposition, which is unique as $\mathbf F$ is invertible with a positive determinant, is a corollary of the singular-value decomposition.

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 $da~\mathbf = J~dA ~\mathbf^ \cdot \mathbf$ where $da$ is an area of a region in the deformed configuration, $dA$ is the same area in the reference configuration, and $\mathbf$ is the outward normal to the area element in the current configuration while $\mathbf$ is the outward normal in the reference configuration, $\mathbf$ is the deformation gradient, and $J = \det\mathbf\,\!$.

The corresponding formula for the transformation of the volume element is $dv = J~dV$

Fundamental strain tensors

A strain tensor is defined by the IUPAC as:

"A symmetric tensor that results when a deformation gradient tensor is factorized into a rotation tensor followed or preceded by a symmetric tensor".

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 ($\mathbf\mathbf^T=\mathbf^T\mathbf=\mathbf\,\!$) we can exclude the rotation by multiplying the deformation gradient tensor $\mathbf$ by its transpose.

Several rotation-independent deformation gradient tensors (or "deformation tensors", for short) are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy–Green deformation tensors.

Cauchy strain tensor (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'' (the IUPAC recommends that this tensor be called the Cauchy strain tensor), defined as:

$\mathbf C=\mathbf F^T\mathbf F=\mathbf U^2 \qquad \text \qquad C_=F_~F_ = \frac \frac .$

Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e. $d\mathbf x^2=d\mathbf X\cdot\mathbf C \cdot d\mathbf X$

Invariants of $\mathbf$ are often used in the expressions for strain energy density functions. The most commonly used invariants are

\begin
I_1^C & := \text(\mathbf) = C_ = \lambda_1^2 + \lambda_2^2 + \lambda_3^2 \\
I_2^C & := \tfrac\left \text~\mathbf)^2 - \text(\mathbf^2) \right = \tfrac\left C_)^2 - C_C_\right= \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 \\
I_3^C & := \det(\mathbf) = J^2 = \lambda_1^2\lambda_2^2\lambda_3^2.
\end

where $J:=\det\mathbf$ is the determinant of the deformation gradient $\mathbf$ and $\lambda_i$ 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).

Finger strain tensor

The IUPAC recommends that the inverse of the right Cauchy–Green deformation tensor (called the Cauchy strain tensor in that document), i. e., $\mathbf C^$, be called the Finger strain tensor. However, that nomenclature is not universally accepted in applied mechanics.

$\mathbf=\mathbf C^=\mathbf F^\mathbf F^ \qquad \text \qquad f_=\frac \frac$

Green strain tensor (left Cauchy–Green deformation tensor)

Reversing the order of multiplication in the formula for the right Cauchy-Green deformation tensor leads to the ''left Cauchy–Green deformation tensor'' which is defined as:
$\mathbf B = \mathbf F\mathbf F^T = \mathbf V^2 \qquad \text \qquad B_ = \frac \frac$

The left Cauchy–Green deformation tensor is often called the ''Finger deformation tensor'', named after Josef Finger (1894).

The IUPAC recommends that this tensor be called the Green strain tensor.

Invariants of $\mathbf$ are also used in the expressions for strain energy density functions. The conventional invariants are defined as

\begin
I_1 & := \text(\mathbf) = B_ = \lambda_1^2 + \lambda_2^2 + \lambda_3^2\\
I_2 & := \tfrac\left \text~\mathbf)^2 - \text(\mathbf^2)\right = \tfrac\left(B_^2 - B_B_\right) = \lambda_1^2\lambda_2^2 + \lambda_2^2\lambda_3^2 + \lambda_3^2\lambda_1^2 \\
I_3 & := \det\mathbf = J^2 = \lambda_1^2\lambda_2^2\lambda_3^2
\end

where $J:=\det\mathbf$ is the determinant of the deformation gradient.

For compressible materials, a slightly different set of invariants is used:
$(\bar_1 := J^ I_1 ~;~~ \bar_2 := J^ I_2 ~;~~ J\neq 1) ~.$

Piola strain tensor (Cauchy deformation tensor)

Earlier in 1828, Augustin-Louis Cauchy introduced a deformation tensor defined as the inverse of the left Cauchy–Green deformation tensor, $\mathbf B^\,\!$. This tensor has also been called the Piola strain tensor by the IUPAC and the Finger tensor in the rheology and fluid dynamics literature.

$\mathbf=\mathbf B^=\mathbf F^\mathbf F^ \qquad \text \qquad c_=\frac \frac$

Spectral representation

If there are three distinct principal stretches $\lambda_i \,\!$, the spectral decompositions of $\mathbf$ and $\mathbf$ is given by

$\mathbf = \sum_^3 \lambda_i^2 \mathbf_i \otimes \mathbf_i \qquad \text \qquad \mathbf = \sum_^3 \lambda_i^2 \mathbf_i \otimes \mathbf_i$

Furthermore,

$\mathbf U = \sum_^3 \lambda_i \mathbf N_i \otimes \mathbf N_i ~;~~ \mathbf V = \sum_^3 \lambda_i \mathbf n_i \otimes \mathbf n_i$
$\mathbf R = \sum_^3 \mathbf n_i \otimes \mathbf N_i ~;~~ \mathbf F = \sum_^3 \lambda_i \mathbf n_i \otimes \mathbf N_i$

Observe that
$\mathbf = \mathbf~\mathbf~\mathbf^T = \sum_^3 \lambda_i~\mathbf~(\mathbf_i\otimes\mathbf_i)~\mathbf^T = \sum_^3 \lambda_i~(\mathbf~\mathbf_i)\otimes(\mathbf~\mathbf_i)$
Therefore, the uniqueness of the spectral decomposition also implies that $\mathbf_i = \mathbf~\mathbf_i \,\!$. The left stretch ($\mathbf\,\!$) is also called the ''spatial stretch tensor'' while the right stretch ($\mathbf\,\!$) is called the ''material stretch tensor''.

The effect of $\mathbf$ acting on $\mathbf_i$ is to stretch the vector by $\lambda_i$ and to rotate it to the new orientation $\mathbf_i\,\!$, i.e.,
$\mathbf~\mathbf_i = \lambda_i~(\mathbf~\mathbf_i) = \lambda_i~\mathbf_i$
In a similar vein,
$\mathbf^~\mathbf_i = \cfrac~\mathbf_i ~;~~ \mathbf^T~\mathbf_i = \lambda_i~\mathbf_i ~;~~ \mathbf^~\mathbf_i = \cfrac~\mathbf_i ~.$

Examples

; Uniaxial extension of an incompressible material
: This is the case where a specimen is stretched in 1-direction with a stretch ratio of $\mathbf\,\!$. If the volume remains constant, the contraction in the other two directions is such that $\mathbf$ or $\mathbf\,\!$. Then: $\mathbf=\begin \alpha & 0 & 0 \\ 0 & \alpha^ & 0 \\ 0 & 0 & \alpha^ \end$ $\mathbf = \mathbf = \begin \alpha^2 & 0 & 0 \\ 0 & \alpha^ & 0 \\ 0 & 0 & \alpha^ \end$
; Simple shear
:$\mathbf=\begin 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end$ $\mathbf = \begin 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end$ $\mathbf = \begin 1 & \gamma & 0 \\ \gamma & 1+\gamma^2 & 0 \\ 0 & 0 & 1 \end$
; Rigid body rotation
:$\mathbf = \begin \cos \theta & \sin \theta & 0 \\ - \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end$ $\mathbf = \mathbf = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end = \mathbf$

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
$\cfrac = \cfrac~\mathbf_i\otimes\mathbf_i = \cfrac~\mathbf^T~(\mathbf_i\otimes\mathbf_i)~\mathbf ~;~~ i=1,2,3$
and follow from the observations that
$\mathbf:(\mathbf_i\otimes\mathbf_i) = \lambda_i^2 ~;~~~~\cfrac = \mathsf^ ~;~~~~ \mathsf^:(\mathbf_i\otimes\mathbf_i)=\mathbf_i\otimes\mathbf_i.$

Physical interpretation of deformation tensors

Let $\mathbf = X^i~\boldsymbol_i$ be a Cartesian coordinate system defined on the undeformed body and let $\mathbf = x^i~\boldsymbol_i$ be another system defined on the deformed body. Let a curve $\mathbf(s)$ in the undeformed body be parametrized using s \in ,1/math>. Its image in the deformed body is $\mathbf(\mathbf(s))$.

The undeformed length of the curve is given by
$l_X = \int_0^1 \left, \cfrac \~ds = \int_0^1 \sqrt~ds = \int_0^1 \sqrt~ds$
After deformation, the length becomes
$\begin l_x & = \int_0^1 \left, \cfrac \~ds = \int_0^1 \sqrt~ds = \int_0^1 \sqrt~ds \\ & = \int_0^1 \sqrt~ds \end$
Note that the right Cauchy–Green deformation tensor is defined as
$\boldsymbol := \boldsymbol^T\cdot\boldsymbol = \left(\cfrac\right)^T\cdot \cfrac$
Hence,
$l_x = \int_0^1 \sqrt~ds$
which indicates that changes in length are characterized by $\boldsymbol$.

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

$\mathbf E=\frac(\mathbf C - \mathbf I)\qquad \text \qquad E_=\frac\left( \frac\frac-\delta_\right)$

or as a function of the displacement gradient tensor
\mathbf E =\frac\left (\nabla_\mathbf u)^T + \nabla_\mathbf u + (\nabla_\mathbf u)^T \cdot\nabla_\mathbf u\right/math>
or
$E_=\frac\left(\frac+\frac+\frac\frac\right)$

The Green-Lagrangian strain tensor is a measure of how much $\mathbf C$ differs from $\mathbf I\,\!$.

The Eulerian finite strain tensor, or ''Eulerian-Almansi finite strain tensor'', referenced to the deformed configuration (i.e. Eulerian description) is defined as

$\mathbf e=\frac(\mathbf I - \mathbf c)=\frac(\mathbf I - \mathbf B ^) \qquad \text \qquad e_ = \frac \left(\delta_ - \frac \frac\right)$

or as a function of the displacement gradients we have
$e_ = \frac \left(\frac + \frac - \frac \frac\right)$

d\mathbf=\mathbf F^ \, d\mathbf=\mathbf \,d\mathbf \qquad \text \qquad dX_M=\frac\, dx_n
where $\frac$ are the components of the ''spatial deformation gradient tensor'', $\mathbf\,\!$. Thus we have

$\begin d\mathbf^2 &= d\mathbf X \cdot d\mathbf X \\ &= \mathbf F^ \cdot d\mathbf x \cdot \mathbf F^ \cdot d\mathbf x \\ &= d\mathbf x \cdot \mathbf F^\mathbf F^ \cdot d\mathbf x \\ &= d\mathbf x\cdot\mathbf c\cdot d\mathbf x \end \qquad \text \qquad \begin (dX)^2&=dX_M\,dX_M \\ &= \frac\frac\,dx_r\,dx_s \\ &= c_\,dx_r\,dx_s \\ \end$
where the second order tensor $c_$ is called ''Cauchy's deformation tensor'', $\mathbf c=\mathbf F^\mathbf F^\,\!$. Then we have,

$\begin d\mathbf^2 - d\mathbf^2 &= d\mathbf x\cdot d\mathbf x-d\mathbf x\cdot\mathbf c\cdot d\mathbf x \\ &=d\mathbf x\cdot (\mathbf I - \mathbf c)\cdot d\mathbf x \\ &= d\mathbf x \cdot 2\mathbf e \cdot d\mathbf x \\ \end$
or
$\begin (dx)^2 - (dX)^2 &= dx_j\,dx_j-\frac\frac\,dx_r\,dx_s \\ &= \left(\delta_ - \frac\frac \right)\,dx_r\,dx_s \\ &=2e_\,dx_r\,dx_s \end$

where $e_\,\!$, are the components of a second-order tensor called the ''Eulerian-Almansi finite strain tensor'',
$\mathbf e=\frac(\mathbf I - \mathbf c) \qquad \text \qquad e_=\frac\left(\delta_ - \frac\frac \right)$

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 $\mathbf u(\mathbf X, t)$ with respect to the material coordinates $X_M$ to obtain the ''material displacement gradient tensor'', $\nabla_\mathbf u$

$\begin \mathbf u(\mathbf X,t) &= \mathbf x(\mathbf X,t) - \mathbf X \\ \nabla_\mathbf u &= \mathbf F - \mathbf I \\ \mathbf F &= \nabla_\mathbf u + \mathbf I \\ \end \qquad \text \qquad \begin u_i& = x_i-\delta_X_J \\ \delta_U_J &= x_i-\delta_X_J \\ x_i&=\delta_\left(U_J+X_J\right) \\ \frac&=\delta_\left(\frac+\delta_\right) \\ &=\frac+\delta_ \end$

Replacing this equation into the expression for the Lagrangian finite strain tensor we have
\begin
\mathbf E &= \frac\left(\mathbf F^T\mathbf F-\mathbf I\right) \\
&=\frac\left \left\\left( \nabla_\mathbf u+\mathbf I\right)-\mathbf I\right\\
&=\frac\left (\nabla_\mathbf u)^T + \nabla_\mathbf u + (\nabla_\mathbf u)^T \cdot\nabla_\mathbf u\right\\
\end
or
\begin
E_&=\frac\left( \frac \frac - \delta_\right) \\
&=\frac \left delta_\left(\frac+\delta_\right)\delta_\left(\frac+\delta_\right)-\delta_\right\\
&=\frac\left delta_\left(\frac+\delta_\right)\left(\frac+\delta_\right)-\delta_\right\\
&=\frac\left left(\frac+\delta_\right)\left(\frac+\delta_\right)-\delta_\right\\
&=\frac\left(\frac +\frac +\frac \frac\right)
\end

Similarly, the Eulerian-Almansi finite strain tensor can be expressed as

$e_ = \frac \left(\frac +\frac - \frac \frac\right)$

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)T.C. Doyle and J.L. Eriksen (1956). "Non-linear elasticity." ''Advances in Applied Mechanics'' 4, 53–115. can be expressed as

\mathbf E_=\frac(\mathbf U^- \mathbf I) = \frac\left mathbf^ - \mathbf\right/math>

For different values of $m$ we have:
* Green-Lagrangian strain tensor $\mathbf E_ = \frac (\mathbf U^- \mathbf I) = \frac (\mathbf-\mathbf)$
* Biot strain tensor $\mathbf E_ = (\mathbf U - \mathbf I) = \mathbf^-\mathbf$
* Logarithmic strain, Natural strain, True strain, or Hencky strain $\mathbf E_ = \ln \mathbf U = \frac\,\ln\mathbf$
* Almansi strain \mathbf_ = \frac\left mathbf-\mathbf^\right/math>
The second-order approximation of these tensors is
$\mathbf_ = \boldsymbol + (\nabla\mathbf)^T\cdot\nabla\mathbf - (1 - m) \boldsymbol^T\cdot\boldsymbol$
where $\boldsymbol$ is the infinitesimal strain tensor.

Many other different definitions of tensors $\mathbf$ are admissible, provided that they all satisfy the conditions that:Z.P. Bažant and L. Cedolin (1991). ''Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories.'' Oxford Univ. Press, New York (2nd ed. Dover Publ., New York 2003; 3rd ed., World Scientific 2010).
* $\mathbf$ vanishes for all rigid-body motions
* the dependence of $\mathbf$ on the displacement gradient tensor $\nabla\mathbf$ is continuous, continuously differentiable and monotonic
* it is also desired that $\mathbf$ reduces to the infinitesimal strain tensor $\boldsymbol$ as the norm $, \nabla\mathbf, \to 0$

An example is the set of tensors
$\mathbf^ = \left(^n - ^\right)/2n$
which do not belong to the Seth–Hill class, but have the same 2nd-order approximation as the Seth–Hill measures at $m=0$ for any value of $n$.Z.P. Bažant (1998).
Easy-to-compute tensors with symmetric inverse approximating Hencky finite strain and its rate
" ''Journal of Materials of Technology ASME'', 120 (April), 131–136.

Physical interpretation of the finite strain tensor

The diagonal components $E_$ of the Lagrangian finite strain tensor are related to the normal strain, e.g.

$E_=e_+\frac e_^2$

where $e_$ is the normal strain or engineering strain in the direction $\mathbf I_1\,\!$.

The off-diagonal components $E_$ of the Lagrangian finite strain tensor are related to shear strain, e.g.

$E_=\frac\sqrt\sqrt\sin\phi_$

where $\phi_$ is the change in the angle between two line elements that were originally perpendicular with directions $\mathbf I_1$ and $\mathbf I_2\,\!$, 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

= \frac

The square of the stretch ratio is defined as
$\Lambda_^2=\left (\frac\right )^2$

Knowing that
$(dx)^2=C_dX_KdX_L$
we have
$\Lambda_^2 = C_ N_K N_L$
where $N_K$ and $N_L$ are unit vectors.

The normal strain or engineering strain $e_$ in any direction $\mathbf N$ can be expressed as a function of the stretch ratio,

$e_=\frac=\Lambda_-1$

Thus, the normal strain in the direction $\mathbf I_1$ at the material point $P$ may be expressed in terms of the stretch ratio as

$\begin e_=\frac&=\Lambda_-1\\ &=\sqrt -1=\sqrt-1\\ &=\sqrt-1 \end$

solving for $E_$ we have

$\begin 2E_&= \frac \\ E_&= \left(\frac\right)+ \frac \left(\frac\right)^2 \\ &=e_+\frace_^2 \end$

The ''shear strain'', or change in angle between two line elements $d\mathbf X_1$ and $d\mathbf X_2$ initially perpendicular, and oriented in the principal directions $\mathbf I_1$ and $\mathbf I_2\,\!$, respectively, can also be expressed as a function of the stretch ratio. From the dot product between the deformed lines $d\mathbf x_1$ and $d\mathbf x_2$ we have

$\begin d\mathbf x_1 \cdot d\mathbf x_2 &=dx_1 dx_2 \cos\theta_ \\ \mathbf F \cdot d\mathbf X_1\cdot \mathbf F\cdot d\mathbf X_2&= \sqrt \cdot \sqrt \cos\theta_ \\ \frac &=\frac \cos\theta_\\ \mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2&= \Lambda_\Lambda_\cos\theta_ \end$

where $\theta_$ is the angle between the lines $d\mathbf x_1$ and $d\mathbf x_2$ in the deformed configuration. Defining $\phi_$ as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have

$\phi_=\frac-\theta_$
thus,
$\cos\theta_=\sin\phi_$
then
$\mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2= \Lambda_ \Lambda_\sin\phi_$

or

$\begin C_&=\sqrt\sqrt\sin\phi_\\ 2E_+\delta_&=\sqrt\sqrt\sin\phi_\\ E_&=\frac\sqrt\sqrt\sin\phi_ \end$

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 $\boldsymbol$ field over a simply connected body are
$\boldsymbol\times\boldsymbol = \boldsymbol$

Compatibility of the right Cauchy–Green deformation tensor

The necessary and sufficient conditions for the existence of a compatible $\boldsymbol$ field over a simply connected body are

R^\gamma_ :=
\frac ,_\Gamma^\gamma_-
\frac ,_\Gamma^\gamma_+
\,_\Gamma^\gamma_\,_\Gamma^\mu_ -
\,_\Gamma^\gamma_\,_\Gamma^\mu_ = 0

We can show these are the mixed components of the Riemann–Christoffel curvature tensor. Therefore, the necessary conditions for $\boldsymbol$-compatibility are that the Riemann–Christoffel curvature of the deformation is zero.

Compatibility of the left Cauchy–Green deformation tensor

General sufficiency conditions for the left Cauchy–Green deformation tensor in three-dimensions were derived by Amit Acharya. Compatibility conditions for two-dimensional $\boldsymbol$ fields were found by Janet Blume.

See also

* Infinitesimal strain
* Compatibility (mechanics)
* Curvilinear coordinates
* Piola–Kirchhoff stress tensor, the stress tensor for finite deformations.
* Stress measures
* Strain partitioning

References

Further reading

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*{{Cite book
, last = Rees , first = David
, title = Basic Engineering Plasticity – An Introduction with Engineering and Manufacturing Applications
, publisher = Butterworth-Heinemann
, year = 2006
, url = https://books.google.com/books?id=4KWbmn_1hcYC
, isbn = 0-7506-8025-3

External links

Prof. Amit Acharya's notes on compatibility on iMechanica

Tensors
Continuum mechanics
Elasticity (physics)
Non-Newtonian fluids
Solid mechanics