Finite strain theory

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 $${\displaystyle {\begin{aligned}\mathbf {x} &=\mathbf {X} +\mathbf {u} (\mathbf {X} ),\\\mathbf {x} +d\mathbf {x} &=\mathbf {X} +d\mathbf {X} +\mathbf {u} (\mathbf {X} +d\mathbf {X} ),\\{\text{and}}{\text{ therefore}}&\\d\mathbf {x} &=\mathbf {X} -\mathbf {x} +d\mathbf {X} +\mathbf {u} (\mathbf {X} +d\mathbf {X} )\\&=d\mathbf {X} +\mathbf {u} (\mathbf {X} +d\mathbf {X} )-\mathbf {u} (\mathbf {X} )\\&=d\mathbf {X} +d\mathbf {u} \\\end{aligned}},}$$

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.

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^[1] but we avoid those issues in this article.

The time derivative of ${\displaystyle \mathbf {F} }$ is $${\displaystyle {\dot {\mathbf {F} }}={\frac {\partial \mathbf {F} }{\partial t}}={\frac {\partial }{\partial t}}\left[{\frac {\partial \mathbf {x} (\mathbf {X} ,t)}{\partial \mathbf {X} }}\right]={\frac {\partial }{\partial \mathbf {X} }}\left[{\frac {\partial \mathbf {x} (\mathbf {X} ,t)}{\partial t}}\right]={\frac {\partial }{\partial \mathbf {X} }}\left[\mathbf {V} (\mathbf {X} ,t)\right]}$$ 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., $${\displaystyle {\dot {\mathbf {F} }}={\frac {\partial }{\partial \mathbf {X} }}\left[\mathbf {V} (\mathbf {X} ,t)\right]={\frac {\partial }{\partial \mathbf {X} }}\left[\mathbf {v} (\mathbf {x} (\mathbf {X} ,t),t)\right]=\left.{\frac {\partial }{\partial \mathbf {x} }}\left[\mathbf {v} (\mathbf {x} ,t)\right]\right|_{\mathbf {x} =\mathbf {x} (\mathbf {X} ,t)}\cdot {\frac {\partial \mathbf {x} (\mathbf {X} ,t)}{\partial \mathbf {X} }}={\boldsymbol {l}}\cdot \mathbf {F} }$$ 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 ${\displaystyle {\dot {\mathbf {X} }}=0}$.

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.

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 $${\displaystyle dv=J~dV}$$

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),^[3] 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 $${\displaystyle {\begin{aligned}I_{1}^{C}&:={\text{tr}}(\mathbf {C} )=C_{II}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\I_{2}^{C}&:={\tfrac {1}{2}}\left[({\text{tr}}~\mathbf {C} )^{2}-{\text{tr}}(\mathbf {C} ^{2})\right]={\tfrac {1}{2}}\left[(C_{JJ})^{2}-C_{IK}C_{KI}\right]=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\I_{3}^{C}&:=\det(\mathbf {C} )=J^{2}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}.\end{aligned}}}$$ 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).

The IUPAC recommends^[3] 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.

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

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).^[4]

The IUPAC recommends that this tensor be called the Green strain tensor.^[3]

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

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

Earlier in 1828,^[5]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^[3] and the Finger tensor^[6] in the rheology and fluid dynamics literature.

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

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, $${\displaystyle \mathbf {F} ^{-T}~\mathbf {N} _{i}={\cfrac {1}{\lambda _{i}}}~\mathbf {n} _{i}~;~~\mathbf {F} ^{T}~\mathbf {n} _{i}=\lambda _{i}~\mathbf {N} _{i}~;~~\mathbf {F} ^{-1}~\mathbf {n} _{i}={\cfrac {1}{\lambda _{i}}}~\mathbf {N} _{i}~.}$$

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}.}

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 ${\displaystyle s\in [0,1]}$. 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 $${\displaystyle {\begin{aligned}l_{x}&=\int _{0}^{1}\left|{\cfrac {d\mathbf {x} }{ds}}\right|~ds=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {x} }{ds}}\cdot {\cfrac {d\mathbf {x} }{ds}}}}~ds=\int _{0}^{1}{\sqrt {\left({\cfrac {d\mathbf {x} }{d\mathbf {X} }}\cdot {\cfrac {d\mathbf {X} }{ds}}\right)\cdot \left({\cfrac {d\mathbf {x} }{d\mathbf {X} }}\cdot {\cfrac {d\mathbf {X} }{ds}}\right)}}~ds\\&=\int _{0}^{1}{\sqrt {{\cfrac {d\mathbf {X} }{ds}}\cdot \left[\left({\cfrac {d\mathbf {x} }{d\mathbf {X} }}\right)^{T}\cdot {\cfrac {d\mathbf {x} }{d\mathbf {X} }}\right]\cdot {\cfrac {d\mathbf {X} }{ds}}}}~ds\end{aligned}}}$$ 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 ${\displaystyle {\boldsymbol {C}}}$.

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.^[10][11] The idea was further expanded upon by Rodney Hill in 1968.^[12] The Seth–Hill family of strain measures (also called Doyle-Ericksen tensors)^[13] can be expressed as

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