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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

The displacement of a body has two components: a rigid-body displacement and a deformation.

• A rigid-body displacement consists 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 ${\displaystyle \kappa _{0}({\mathcal {B}})}$ to a current or deformed configuration ${\displaystyle \kappa _{t}({\mathcal {B}})}$ (Figure 1).

Deformation gradient tensor

The deformation gradient tensor ${\displaystyle \mathbf {F} (\mathbf {X} ,t)=F_{jK}\mathbf {e} _{j}\otimes \mathbf {I} _{K}}$ is related to both the reference and current configuration, as seen by the unit vectors ${\displaystyle \mathbf {e} _{j}}$ and ${\displaystyle \mathbf {I} _{K}\,\!}$, therefore it is a two-point tensor. Two types of deformation gradient tensor may be defined.

Due to the assumption of continuity of ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, ${\displaystyle \mathbf {F} }$ has the inverse ${\displaystyle \mathbf {H} =\mathbf {F} ^{-1}\,\!}$, where ${\displaystyle \mathbf {H} }$ is the spatial deformation gradient tensor. Then, by the implicit function theorem, the Jacobian determinant ${\displaystyle J(\mathbf {X} ,t)}$ must be nonsingular, i.e. ${\displaystyle J(\mathbf {X} ,t)=\det \mathbf {F} (\mathbf {X} ,t)\neq 0}$

The material deformation gradient tensor ${\displaystyle \mathbf {F} (\mathbf {X} ,t)=F_{jK}\mathbf {e} _{j}\otimes \mathbf {I} _{K}}$ is a second-order tensor that represents the gradient of the mapping function or functional relation ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector ${\displaystyle \mathbf {X} \,\!}$, 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 ${\displaystyle \chi (\mathbf {X} ,t)\,\!}$, i.e. differentiable function of ${\displaystyle \mathbf {X} }$ and time ${\displaystyle t\,\!}$, which implies that cracks and voids do not open or close during the deformation. Thus we have, $${\displaystyle {\begin{aligned}d\mathbf {x} &={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}\,d\mathbf {X} \qquad &{\text{or}}&\qquad dx_{j}={\frac {\partial x_{j}}{\partial X_{K}}}\,dX_{K}\\&=\nabla \chi (\mathbf {X} ,t)\,d\mathbf {X} \qquad &{\text{or}}&\qquad dx_{j}=F_{jK}\,dX_{K}\,.\\&=\mathbf {F} (\mathbf {X} ,t)\,d\mathbf {X} \end{aligned}}}$$

Relative displacement vector

Consider a particle or material point ${\displaystyle P}$ with position vector ${\displaystyle \mathbf {X} =X_{I}\mathbf {I} _{I}}$ in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by ${\displaystyle p}$ in the new configuration is given by the vector position ${\displaystyle \mathbf {x} =x_{i}\mathbf {e} _{i}\,\!}$. The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.

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

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

Taylor approximation

For an infinitesimal element ${\displaystyle d\mathbf {X} \,\!}$, and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point ${\displaystyle P\,\!}$, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle ${\displaystyle Q}$ as $${\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {X} +d\mathbf {X} )&=\mathbf {u} (\mathbf {X} )+d\mathbf {u} \quad &{\text{or}}&\quad u_{i}^{*}=u_{i}+du_{i}\\&\approx \mathbf {u} (\mathbf {X} )+\nabla _{\mathbf {X} }\mathbf {u} \cdot d\mathbf {X} \quad &{\text{or}}&\quad u_{i}^{*}\approx u_{i}+{\frac {\partial u_{i}}{\partial X_{J}}}dX_{J}\,.\end{aligned}}}$$ Thus, the previous equation ${\displaystyle d\mathbf {x} =d\mathbf {X} +d\mathbf {u} }$ can be written as $${\displaystyle {\begin{aligned}d\mathbf {x} &=d\mathbf {X} +d\mathbf {u} \\&=d\mathbf {X} +\nabla _{\mathbf {X} }\mathbf {u} \cdot d\mathbf {X} \\&=\left(\mathbf {I} +\nabla _{\mathbf {X} }\mathbf {u} \right)d\mathbf {X} \\&=\mathbf {F} d\mathbf {X} \end{aligned}}}$$

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 ${\displaystyle \mathbf {F} }$ is $${\displaystyle {\dot {\mathbf {F} }}={\frac {\partial \mathbf {F} }{\partial t}}={\frac {\partial }{\partial t}}\left={\frac {\partial }{\partial \mathbf {X} }}\left={\frac {\partial }{\partial \mathbf {X} }}\left}$$ where ${\displaystyle \mathbf {V} }$ 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., $${\displaystyle {\dot {\mathbf {F} }}={\frac {\partial }{\partial \mathbf {X} }}\left={\frac {\partial }{\partial \mathbf {X} }}\left=\left.{\frac {\partial }{\partial \mathbf {x} }}\left\right|_{\mathbf {x} =\mathbf {x} (\mathbf {X} ,t)}\cdot {\frac {\partial \mathbf {x} (\mathbf {X} ,t)}{\partial \mathbf {X} }}={\boldsymbol {l}}\cdot \mathbf {F} }$$ where ${\displaystyle {\boldsymbol {l}}=(\nabla _{\mathbf {x} }\mathbf {v} )^{T}}$ is the spatial velocity gradient and where ${\displaystyle \mathbf {v} (\mathbf {x} ,t)=\mathbf {V} (\mathbf {X} ,t)}$ is the spatial (Eulerian) velocity at ${\displaystyle \mathbf {x} =\mathbf {x} (\mathbf {X} ,t)}$. If the spatial velocity gradient is constant in time, the above equation can be solved exactly to give $${\displaystyle \mathbf {F} =e^{{\boldsymbol {l}}\,t}}$$ assuming ${\displaystyle \mathbf {F} =\mathbf {1} }$ at ${\displaystyle t=0}$. 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: $${\displaystyle {\boldsymbol {d}}={\tfrac {1}{2}}\left({\boldsymbol {l}}+{\boldsymbol {l}}^{T}\right)\,,~~{\boldsymbol {w}}={\tfrac {1}{2}}\left({\boldsymbol {l}}-{\boldsymbol {l}}^{T}\right)\,.}$$ 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 $${\displaystyle {\frac {\partial }{\partial t}}\left(\mathbf {F} ^{-1}\right)=-\mathbf {F} ^{-1}\cdot {\dot {\mathbf {F} }}\cdot \mathbf {F} ^{-1}\,.}$$ The above relation can be verified by taking the material time derivative of ${\displaystyle \mathbf {F} ^{-1}\cdot d\mathbf {x} =d\mathbf {X} }$ and noting that ${\displaystyle {\dot {\mathbf {X} }}=0}$.

Polar decomposition of the deformation gradient tensor

The deformation gradient ${\displaystyle \mathbf {F} \,\!}$, 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., $${\displaystyle \mathbf {F} =\mathbf {R} \mathbf {U} =\mathbf {V} \mathbf {R} }$$ where the tensor ${\displaystyle \mathbf {R} }$ is a proper orthogonal tensor, i.e., ${\displaystyle \mathbf {R} ^{-1}=\mathbf {R} ^{T}}$ and ${\displaystyle \det \mathbf {R} =+1\,\!}$, representing a rotation; the tensor ${\displaystyle \mathbf {U} }$ is the right stretch tensor; and ${\displaystyle \mathbf {V} }$ the left stretch tensor. The terms right and left means that they are to the right and left of the rotation tensor ${\displaystyle \mathbf {R} \,\!}$, respectively. ${\displaystyle \mathbf {U} }$ and ${\displaystyle \mathbf {V} }$ are both positive definite, i.e. ${\displaystyle \mathbf {x} \cdot \mathbf {U} \cdot \mathbf {x} >0}$ and ${\displaystyle \mathbf {x} \cdot \mathbf {V} \cdot \mathbf {x} >0}$ for all non-zero ${\displaystyle \mathbf {x} \in \mathbb {R} ^{3}}$, and symmetric tensors, i.e. ${\displaystyle \mathbf {U} =\mathbf {U} ^{T}}$ and ${\displaystyle \mathbf {V} =\mathbf {V} ^{T}\,\!}$, of second order.

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

Due to the orthogonality of ${\displaystyle \mathbf {R} }$ $${\displaystyle \mathbf {V} =\mathbf {R} \cdot \mathbf {U} \cdot \mathbf {R} ^{T}}$$ so that ${\displaystyle \mathbf {U} }$ and ${\displaystyle \mathbf {V} }$ have the same eigenvalues or principal stretches, but different eigenvectors or principal directions ${\displaystyle \mathbf {N} _{i}}$ and ${\displaystyle \mathbf {n} _{i}\,\!}$, respectively. The principal directions are related by $${\displaystyle \mathbf {n} _{i}=\mathbf {R} \mathbf {N} _{i}.}$$

This polar decomposition, which is unique as ${\displaystyle \mathbf {F} }$ 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 $${\displaystyle da~\mathbf {n} =J~dA~\mathbf {F} ^{-T}\cdot \mathbf {N} }$$ where ${\displaystyle da}$ is an area of a region in the deformed configuration, ${\displaystyle dA}$ is the same area in the reference configuration, and ${\displaystyle \mathbf {n} }$ is the outward normal to the area element in the current configuration while ${\displaystyle \mathbf {N} }$ is the outward normal in the reference configuration, ${\displaystyle \mathbf {F} }$ is the deformation gradient, and ${\displaystyle J=\det \mathbf {F} \,\!}$.

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

To see how this formula is derived, we start with the oriented area elements in the reference and current configurations: $${\displaystyle d\mathbf {A} =dA~\mathbf {N} ~;~~d\mathbf {a} =da~\mathbf {n} }$$ The reference and current volumes of an element are $${\displaystyle dV=d\mathbf {A} ^{T}\cdot d\mathbf {L} ~;~~dv=d\mathbf {a} ^{T}\cdot d\mathbf {l} }$$ where ${\displaystyle d\mathbf {l} =\mathbf {F} \cdot d\mathbf {L} \,\!}$.

Therefore, $${\displaystyle d\mathbf {a} ^{T}\cdot d\mathbf {l} =dv=J~dV=J~d\mathbf {A} ^{T}\cdot d\mathbf {L} }$$ or, $${\displaystyle d\mathbf {a} ^{T}\cdot \mathbf {F} \cdot d\mathbf {L} =dv=J~dV=J~d\mathbf {A} ^{T}\cdot d\mathbf {L} }$$ so, $${\displaystyle d\mathbf {a} ^{T}\cdot \mathbf {F} =J~d\mathbf {A} ^{T}}$$ So we get $${\displaystyle d\mathbf {a} =J~\mathbf {F} ^{-T}\cdot d\mathbf {A} }$$ or, $${\displaystyle da~\mathbf {n} =J~dA~\mathbf {F} ^{-T}\cdot \mathbf {N} }$$ Q.E.D.

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 (${\displaystyle \mathbf {R} \mathbf {R} ^{T}=\mathbf {R} ^{T}\mathbf {R} =\mathbf {I} \,\!}$) we can exclude the rotation by multiplying the deformation gradient tensor ${\displaystyle \mathbf {F} }$ 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:

$${\displaystyle \mathbf {C} =\mathbf {F} ^{T}\mathbf {F} =\mathbf {U} ^{2}\qquad {\text{or}}\qquad C_{IJ}=F_{kI}~F_{kJ}={\frac {\partial x_{k}}{\partial X_{I}}}{\frac {\partial x_{k}}{\partial X_{J}}}.}$$

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

Invariants of ${\displaystyle \mathbf {C} }$ are often used in the expressions for strain energy density functions. The most commonly used invariants are $${\displaystyle {\begin{aligned}I_{1}^{C}&:={\text{tr}}(\mathbf {C} )=C_{II}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\I_{2}^{C}&:={\tfrac {1}{2}}\left={\tfrac {1}{2}}\left=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\I_{3}^{C}&:=\det(\mathbf {C} )=J^{2}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}.\end{aligned}}}$$ where ${\displaystyle J:=\det \mathbf {F} }$ is the determinant of the deformation gradient ${\displaystyle \mathbf {F} }$ and ${\displaystyle \lambda _{i}}$ 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., ${\displaystyle \mathbf {C} ^{-1}}$, be called the Finger strain tensor. However, that nomenclature is not universally accepted in applied mechanics.

$${\displaystyle \mathbf {f} =\mathbf {C} ^{-1}=\mathbf {F} ^{-1}\mathbf {F} ^{-T}\qquad {\text{or}}\qquad f_{IJ}={\frac {\partial X_{I}}{\partial x_{k}}}{\frac {\partial X_{J}}{\partial x_{k}}}}$$

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: $${\displaystyle \mathbf {B} =\mathbf {F} \mathbf {F} ^{T}=\mathbf {V} ^{2}\qquad {\text{or}}\qquad B_{ij}={\frac {\partial x_{i}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{K}}}}$$

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 ${\displaystyle \mathbf {B} }$ are also used in the expressions for strain energy density functions. The conventional invariants are defined as $${\displaystyle {\begin{aligned}I_{1}&:={\text{tr}}(\mathbf {B} )=B_{ii}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\I_{2}&:={\tfrac {1}{2}}\left={\tfrac {1}{2}}\left(B_{ii}^{2}-B_{jk}B_{kj}\right)=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\I_{3}&:=\det \mathbf {B} =J^{2}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}\end{aligned}}}$$ where ${\displaystyle J:=\det \mathbf {F} }$ is the determinant of the deformation gradient.

For compressible materials, a slightly different set of invariants is used: $${\displaystyle ({\bar {I}}_{1}:=J^{-2/3}I_{1}~;~~{\bar {I}}_{2}:=J^{-4/3}I_{2}~;~~J\neq 1)~.}$$

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, ${\displaystyle \mathbf {B} ^{-1}\,\!}$. This tensor has also been called the Piola strain tensor by the IUPAC and the Finger tensor in the rheology and fluid dynamics literature.

$${\displaystyle \mathbf {c} =\mathbf {B} ^{-1}=\mathbf {F} ^{-T}\mathbf {F} ^{-1}\qquad {\text{or}}\qquad c_{ij}={\frac {\partial X_{K}}{\partial x_{i}}}{\frac {\partial X_{K}}{\partial x_{j}}}}$$

Spectral representation

If there are three distinct principal stretches ${\displaystyle \lambda _{i}\,\!}$, the spectral decompositions of ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathbf {B} }$ is given by

$${\displaystyle \mathbf {C} =\sum _{i=1}^{3}\lambda _{i}^{2}\mathbf {N} _{i}\otimes \mathbf {N} _{i}\qquad {\text{and}}\qquad \mathbf {B} =\sum _{i=1}^{3}\lambda _{i}^{2}\mathbf {n} _{i}\otimes \mathbf {n} _{i}}$$

Furthermore,

$${\displaystyle \mathbf {U} =\sum _{i=1}^{3}\lambda _{i}\mathbf {N} _{i}\otimes \mathbf {N} _{i}~;~~\mathbf {V} =\sum _{i=1}^{3}\lambda _{i}\mathbf {n} _{i}\otimes \mathbf {n} _{i}}$$ $${\displaystyle \mathbf {R} =\sum _{i=1}^{3}\mathbf {n} _{i}\otimes \mathbf {N} _{i}~;~~\mathbf {F} =\sum _{i=1}^{3}\lambda _{i}\mathbf {n} _{i}\otimes \mathbf {N} _{i}}$$

Observe that $${\displaystyle \mathbf {V} =\mathbf {R} ~\mathbf {U} ~\mathbf {R} ^{T}=\sum _{i=1}^{3}\lambda _{i}~\mathbf {R} ~(\mathbf {N} _{i}\otimes \mathbf {N} _{i})~\mathbf {R} ^{T}=\sum _{i=1}^{3}\lambda _{i}~(\mathbf {R} ~\mathbf {N} _{i})\otimes (\mathbf {R} ~\mathbf {N} _{i})}$$ Therefore, the uniqueness of the spectral decomposition also implies that ${\displaystyle \mathbf {n} _{i}=\mathbf {R} ~\mathbf {N} _{i}\,\!}$. The left stretch (${\displaystyle \mathbf {V} \,\!}$) is also called the spatial stretch tensor while the right stretch (${\displaystyle \mathbf {U} \,\!}$) is called the material stretch tensor.

The effect of ${\displaystyle \mathbf {F} }$ acting on ${\displaystyle \mathbf {N} _{i}}$ is to stretch the vector by ${\displaystyle \lambda _{i}}$ and to rotate it to the new orientation ${\displaystyle \mathbf {n} _{i}\,\!}$, i.e., $${\displaystyle \mathbf {F} ~\mathbf {N} _{i}=\lambda _{i}~(\mathbf {R} ~\mathbf {N} _{i})=\lambda _{i}~\mathbf {n} _{i}}$$ In a similar vein, $${\displaystyle \mathbf {F} ^{-T}~\mathbf {N} _{i}={\cfrac {1}{\lambda _{i}}}~\mathbf {n} _{i}~;~~\mathbf {F} ^{T}~\mathbf {n} _{i}=\lambda _{i}~\mathbf {N} _{i}~;~~\mathbf {F} ^{-1}~\mathbf {n} _{i}={\cfrac {1}{\lambda _{i}}}~\mathbf {N} _{i}~.}$$

Examples

Uniaxial extension of an incompressible material

This is the case where a specimen is stretched in 1-direction with a stretch ratio of ${\displaystyle \mathbf {\alpha =\alpha _{1}} \,\!}$. If the volume remains constant, the contraction in the other two directions is such that ${\displaystyle \mathbf {\alpha _{1}\alpha _{2}\alpha _{3}=1} }$ or ${\displaystyle \mathbf {\alpha _{2}=\alpha _{3}=\alpha ^{-0.5}} \,\!}$. Then: $${\displaystyle \mathbf {F} ={\begin{bmatrix}\alpha &0&0\\0&\alpha ^{-0.5}&0\\0&0&\alpha ^{-0.5}\end{bmatrix}}}$$ $${\displaystyle \mathbf {B} =\mathbf {C} ={\begin{bmatrix}\alpha ^{2}&0&0\\0&\alpha ^{-1}&0\\0&0&\alpha ^{-1}\end{bmatrix}}}$$

Simple shear

$${\displaystyle \mathbf {F} ={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}}$$ $${\displaystyle \mathbf {B} ={\begin{bmatrix}1+\gamma ^{2}&\gamma &0\\\gamma &1&0\\0&0&1\end{bmatrix}}}$$ $${\displaystyle \mathbf {C} ={\begin{bmatrix}1&\gamma &0\\\gamma &1+\gamma ^{2}&0\\0&0&1\end{bmatrix}}}$$

Rigid body rotation

$${\displaystyle \mathbf {F} ={\begin{bmatrix}\cos \theta &\sin \theta &0\\-\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}}$$ $${\displaystyle \mathbf {B} =\mathbf {C} ={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}=\mathbf {1} }$$

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 $${\displaystyle {\cfrac {\partial \lambda _{i}}{\partial \mathbf {C} }}={\cfrac {1}{2\lambda _{i}}}~\mathbf {N} _{i}\otimes \mathbf {N} _{i}={\cfrac {1}{2\lambda _{i}}}~\mathbf {R} ^{T}~(\mathbf {n} _{i}\otimes \mathbf {n} _{i})~\mathbf {R} ~;~~i=1,2,3}$$ and follow from the observations that $${\displaystyle \mathbf {C} :(\mathbf {N} _{i}\otimes \mathbf {N} _{i})=\lambda _{i}^{2}~;~~~~{\cfrac {\partial \mathbf {C} }{\partial \mathbf {C} }}={\mathsf {I}}^{(s)}~;~~~~{\mathsf {I}}^{(s)}:(\mathbf {N} _{i}\otimes \mathbf {N} _{i})=\mathbf {N} _{i}\otimes \mathbf {N} _{i}.}$$

Physical interpretation of deformation tensors

Let ${\displaystyle \mathbf {X} =X^{i}~{\boldsymbol {E}}_{i}}$ be a Cartesian coordinate system defined on the undeformed body and let ${\displaystyle \mathbf {x} =x^{i}~{\boldsymbol {E}}_{i}}$ be another system defined on the deformed body. Let a curve ${\displaystyle \mathbf {X} (s)}$ in the undeformed body be parametrized using ${\displaystyle s\in }$. Its image in the deformed body is ${\displaystyle \mathbf {x} (\mathbf {X} (s))}$.

The undeformed length of the curve is given by $${\displaystyle l_{X}=\int _{0}^{1}\left|{\cfrac {d\mathbf {X} }{ds}}\right|~ds=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {X} }{ds}}\cdot {\cfrac {d\mathbf {X} }{ds}}}}~ds=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {X} }{ds}}\cdot {\boldsymbol {I}}\cdot {\cfrac {d\mathbf {X} }{ds}}}}~ds}$$ After deformation, the length becomes $${\displaystyle {\begin{aligned}l_{x}&=\int _{0}^{1}\left|{\cfrac {d\mathbf {x} }{ds}}\right|~ds=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {x} }{ds}}\cdot {\cfrac {d\mathbf {x} }{ds}}}}~ds=\int _{0}^{1}{\sqrt {\left({\cfrac {d\mathbf {x} }{d\mathbf {X} }}\cdot {\cfrac {d\mathbf {X} }{ds}}\right)\cdot \left({\cfrac {d\mathbf {x} }{d\mathbf {X} }}\cdot {\cfrac {d\mathbf {X} }{ds}}\right)}}~ds\\&=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {X} }{ds}}\cdot \left\cdot {\cfrac {d\mathbf {X} }{ds}}}}~ds\end{aligned}}}$$ Note that the right Cauchy–Green deformation tensor is defined as $${\displaystyle {\boldsymbol {C}}:={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}=\left({\cfrac {d\mathbf {x} }{d\mathbf {X} }}\right)^{T}\cdot {\cfrac {d\mathbf {x} }{d\mathbf {X} }}}$$ Hence, $${\displaystyle l_{x}=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {X} }{ds}}\cdot {\boldsymbol {C}}\cdot {\cfrac {d\mathbf {X} }{ds}}}}~ds}$$ which indicates that changes in length are characterized by ${\displaystyle {\boldsymbol {C}}}$.

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

$${\displaystyle \mathbf {E} ={\frac {1}{2}}(\mathbf {C} -\mathbf {I} )\qquad {\text{or}}\qquad E_{KL}={\frac {1}{2}}\left({\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}-\delta _{KL}\right)}$$

or as a function of the displacement gradient tensor $${\displaystyle \mathbf {E} ={\frac {1}{2}}\left}$$ or $${\displaystyle E_{KL}={\frac {1}{2}}\left({\frac {\partial u_{K}}{\partial X_{L}}}+{\frac {\partial u_{L}}{\partial X_{K}}}+{\frac {\partial u_{M}}{\partial X_{K}}}{\frac {\partial u_{M}}{\partial X_{L}}}\right)}$$

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

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

$${\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {I} -\mathbf {c} )={\frac {1}{2}}(\mathbf {I} -\mathbf {B} ^{-1})\qquad {\text{or}}\qquad e_{rs}={\frac {1}{2}}\left(\delta _{rs}-{\frac {\partial X_{M}}{\partial x_{r}}}{\frac {\partial X_{M}}{\partial x_{s}}}\right)}$$

or as a function of the displacement gradients we have $${\displaystyle e_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}-{\frac {\partial u_{k}}{\partial x_{i}}}{\frac {\partial u_{k}}{\partial x_{j}}}\right)}$$

A measure of deformation is the difference between the squares of the differential line element ${\displaystyle d\mathbf {X} \,\!}$, in the undeformed configuration, and ${\displaystyle d\mathbf {x} \,\!}$, 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,

$${\displaystyle d\mathbf {x} ^{2}-d\mathbf {X} ^{2}=d\mathbf {x} \cdot d\mathbf {x} -d\mathbf {X} \cdot d\mathbf {X} \qquad {\text{or}}\qquad (dx)^{2}-(dX)^{2}=dx_{j}dx_{j}-dX_{M}\,dX_{M}}$$

In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is

$${\displaystyle d\mathbf {x} ={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}\,d\mathbf {X} =\mathbf {F} \,d\mathbf {X} \qquad {\text{or}}\qquad dx_{j}={\frac {\partial x_{j}}{\partial X_{M}}}\,dX_{M}}$$

Then we have,

$${\displaystyle {\begin{aligned}d\mathbf {x} ^{2}&=d\mathbf {x} \cdot d\mathbf {x} \\&=\mathbf {F} \cdot d\mathbf {X} \cdot \mathbf {F} \cdot d\mathbf {X} \\&=d\mathbf {X} \cdot \mathbf {F} ^{T}\mathbf {F} \cdot d\mathbf {X} \\&=d\mathbf {X} \cdot \mathbf {C} \cdot d\mathbf {X} \end{aligned}}\qquad {\text{or}}\qquad {\begin{aligned}(dx)^{2}&=dx_{j}\,dx_{j}\\&={\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}\,dX_{K}\,dX_{L}\\&=C_{KL}\,dX_{K}\,dX_{L}\\\end{aligned}}}$$

where ${\displaystyle C_{KL}}$ are the components of the right Cauchy–Green deformation tensor, ${\displaystyle \mathbf {C} =\mathbf {F} ^{T}\mathbf {F} \,\!}$. Then, replacing this equation into the first equation we have,

$${\displaystyle {\begin{aligned}d\mathbf {x} ^{2}-d\mathbf {X} ^{2}&=d\mathbf {X} \cdot \mathbf {C} \cdot d\mathbf {X} -d\mathbf {X} \cdot d\mathbf {X} \\&=d\mathbf {X} \cdot (\mathbf {C} -\mathbf {I} )\cdot d\mathbf {X} \\&=d\mathbf {X} \cdot 2\mathbf {E} \cdot d\mathbf {X} \\\end{aligned}}}$$ or $${\displaystyle {\begin{aligned}(dx)^{2}-(dX)^{2}&={\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}\,dX_{K}\,dX_{L}-dX_{M}\,dX_{M}\\&=\left({\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}-\delta _{KL}\right)\,dX_{K}\,dX_{L}\\&=2E_{KL}\,dX_{K}\,dX_{L}\end{aligned}}}$$ where ${\displaystyle E_{KL}\,\!}$, are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor, $${\displaystyle \mathbf {E} ={\frac {1}{2}}(\mathbf {C} -\mathbf {I} )\qquad {\text{or}}\qquad E_{KL}={\frac {1}{2}}\left({\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}-\delta _{KL}\right)}$$

In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is $${\displaystyle d\mathbf {X} ={\frac {\partial \mathbf {X} }{\partial \mathbf {x} }}d\mathbf {x} =\mathbf {F} ^{-1}\,d\mathbf {x} =\mathbf {H} \,d\mathbf {x} \qquad {\text{or}}\qquad dX_{M}={\frac {\partial X_{M}}{\partial x_{n}}}\,dx_{n}}$$ where ${\displaystyle {\frac {\partial X_{M}}{\partial x_{n}}}}$ are the components of the spatial deformation gradient tensor, ${\displaystyle \mathbf {H} \,\!}$. Thus we have

$${\displaystyle {\begin{aligned}d\mathbf {X} ^{2}&=d\mathbf {X} \cdot d\mathbf {X} \\&=\mathbf {F} ^{-1}\cdot d\mathbf {x} \cdot \mathbf {F} ^{-1}\cdot d\mathbf {x} \\&=d\mathbf {x} \cdot \mathbf {F} ^{-T}\mathbf {F} ^{-1}\cdot d\mathbf {x} \\&=d\mathbf {x} \cdot \mathbf {c} \cdot d\mathbf {x} \end{aligned}}\qquad {\text{or}}\qquad {\begin{aligned}(dX)^{2}&=dX_{M}\,dX_{M}\\&={\frac {\partial X_{M}}{\partial x_{r}}}{\frac {\partial X_{M}}{\partial x_{s}}}\,dx_{r}\,dx_{s}\\&=c_{rs}\,dx_{r}\,dx_{s}\\\end{aligned}}}$$ where the second order tensor ${\displaystyle c_{rs}}$ is called Cauchy's deformation tensor, ${\displaystyle \mathbf {c} =\mathbf {F} ^{-T}\mathbf {F} ^{-1}\,\!}$. Then we have,

$${\displaystyle {\begin{aligned}d\mathbf {x} ^{2}-d\mathbf {X} ^{2}&=d\mathbf {x} \cdot d\mathbf {x} -d\mathbf {x} \cdot \mathbf {c} \cdot d\mathbf {x} \\&=d\mathbf {x} \cdot (\mathbf {I} -\mathbf {c} )\cdot d\mathbf {x} \\&=d\mathbf {x} \cdot 2\mathbf {e} \cdot d\mathbf {x} \\\end{aligned}}}$$ or $${\displaystyle {\begin{aligned}(dx)^{2}-(dX)^{2}&=dx_{j}\,dx_{j}-{\frac {\partial X_{M}}{\partial x_{r}}}{\frac {\partial X_{M}}{\partial x_{s}}}\,dx_{r}\,dx_{s}\\&=\left(\delta _{rs}-{\frac {\partial X_{M}}{\partial x_{r}}}{\frac {\partial X_{M}}{\partial x_{s}}}\right)\,dx_{r}\,dx_{s}\\&=2e_{rs}\,dx_{r}\,dx_{s}\end{aligned}}}$$

where ${\displaystyle e_{rs}\,\!}$, are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor, $${\displaystyle \mathbf {e} ={\frac {1}{2}}(\mathbf {I} -\mathbf {c} )\qquad {\text{or}}\qquad e_{rs}={\frac {1}{2}}\left(\delta _{rs}-{\frac {\partial X_{M}}{\partial x_{r}}}{\frac {\partial X_{M}}{\partial x_{s}}}\right)}$$

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 ${\displaystyle \mathbf {u} (\mathbf {X} ,t)}$ with respect to the material coordinates ${\displaystyle X_{M}}$ to obtain the material displacement gradient tensor, ${\displaystyle \nabla _{\mathbf {X} }\mathbf {u} }$

$${\displaystyle {\begin{aligned}\mathbf {u} (\mathbf {X} ,t)&=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \\\nabla _{\mathbf {X} }\mathbf {u} &=\mathbf {F} -\mathbf {I} \\\mathbf {F} &=\nabla _{\mathbf {X} }\mathbf {u} +\mathbf {I} \\\end{aligned}}\qquad {\text{or}}\qquad {\begin{aligned}u_{i}&=x_{i}-\delta _{iJ}X_{J}\\\delta _{iJ}U_{J}&=x_{i}-\delta _{iJ}X_{J}\\x_{i}&=\delta _{iJ}\left(U_{J}+X_{J}\right)\\{\frac {\partial x_{i}}{\partial X_{K}}}&=\delta _{iJ}\left({\frac {\partial U_{J}}{\partial X_{K}}}+\delta _{JK}\right)\\&={\frac {\partial u_{i}}{\partial X_{K}}}+\delta _{iK}\end{aligned}}}$$

Replacing this equation into the expression for the Lagrangian finite strain tensor we have $${\displaystyle {\begin{aligned}\mathbf {E} &={\frac {1}{2}}\left(\mathbf {F} ^{T}\mathbf {F} -\mathbf {I} \right)\\&={\frac {1}{2}}\left\\&={\frac {1}{2}}\left\\\end{aligned}}}$$ or $${\displaystyle {\begin{aligned}E_{KL}&={\frac {1}{2}}\left({\frac {\partial x_{j}}{\partial X_{K}}}{\frac {\partial x_{j}}{\partial X_{L}}}-\delta _{KL}\right)\\&={\frac {1}{2}}\left\\&={\frac {1}{2}}\left\\&={\frac {1}{2}}\left\\&={\frac {1}{2}}\left({\frac {\partial U_{K}}{\partial X_{L}}}+{\frac {\partial U_{L}}{\partial X_{K}}}+{\frac {\partial U_{M}}{\partial X_{K}}}{\frac {\partial U_{M}}{\partial X_{L}}}\right)\end{aligned}}}$$

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

$${\displaystyle e_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}-{\frac {\partial u_{k}}{\partial x_{i}}}{\frac {\partial u_{k}}{\partial x_{j}}}\right)}$$

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) can be expressed as

$${\displaystyle \mathbf {E} _{(m)}={\frac {1}{2m}}(\mathbf {U} ^{2m}-\mathbf {I} )={\frac {1}{2m}}\left}$$

For different values of ${\displaystyle m}$ we have:

• Green-Lagrangian strain tensor $${\displaystyle \mathbf {E} _{(1)}={\frac {1}{2}}(\mathbf {U} ^{2}-\mathbf {I} )={\frac {1}{2}}(\mathbf {C} -\mathbf {I} )}$$
• Biot strain tensor $${\displaystyle \mathbf {E} _{(1/2)}=(\mathbf {U} -\mathbf {I} )=\mathbf {C} ^{1/2}-\mathbf {I} }$$
• Logarithmic strain, Natural strain, True strain, or Hencky strain $${\displaystyle \mathbf {E} _{(0)}=\ln \mathbf {U} ={\frac {1}{2}}\,\ln \mathbf {C} }$$
• Almansi strain $${\displaystyle \mathbf {E} _{(-1)}={\frac {1}{2}}\left}$$

The second-order approximation of these tensors is $${\displaystyle \mathbf {E} _{(m)}={\boldsymbol {\varepsilon }}+{\tfrac {1}{2}}(\nabla \mathbf {u} )^{T}\cdot \nabla \mathbf {u} -(1-m){\boldsymbol {\varepsilon }}^{T}\cdot {\boldsymbol {\varepsilon }}}$$ where ${\displaystyle {\boldsymbol {\varepsilon }}}$ is the infinitesimal strain tensor.

Many other different definitions of tensors ${\displaystyle \mathbf {E} }$ are admissible, provided that they all satisfy the conditions that:

• ${\displaystyle \mathbf {E} }$ vanishes for all rigid-body motions
• the dependence of ${\displaystyle \mathbf {E} }$ on the displacement gradient tensor ${\displaystyle \nabla \mathbf {u} }$ is continuous, continuously differentiable and monotonic
• it is also desired that ${\displaystyle \mathbf {E} }$ reduces to the infinitesimal strain tensor ${\displaystyle {\boldsymbol {\varepsilon }}}$ as the norm ${\displaystyle |\nabla \mathbf {u} |\to 0}$