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Incontinuum mechanics, thefinite strain theory—also calledlarge strain theory, orlarge deformation theory—deals withdeformations in which strains and/or rotations are large enough to invalidate assumptions inherent ininfinitesimal 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 withelastomers,plastically deforming materials and otherfluids andbiologicalsoft tissue.

This section is an excerpt fromDisplacement field (mechanics) § Decomposition.[edit]

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

• A rigid-body displacement consists of atranslation 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).

A change in the configuration of a continuum body can be described by adisplacement field. Adisplacement field is avector 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.

Thedeformation gradient tensor is a quantity related to both the reference and current configuration, and expresses motion locally around a point.Two types of deformation gradient tensor may be defined.

Thematerial deformation gradient tensor${\displaystyle \mathbf{F}(\mathbf{X},t)=F_{jK}{\mathbf{e}}_j\otimes {\mathbf{I}}_K}$ is asecond-order tensor that represents the gradient of thesmooth and invertible mapping function${\displaystyle \chi (\mathbf{X},t)}$, which describes themotion of a continuum.In particular, the continuity of the mapping function${\displaystyle \chi (\mathbf{X},t)}$ implies thatcracks and voids do not open or close during the deformation.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.Thus we have,

Assuming that${\displaystyle \chi (\mathbf{X},t)}$ hasa smooth inverse,${\displaystyle \mathbf{F}}$ has the inverse${\displaystyle \mathbf{H}={\mathbf{F}}^{-1}=\frac{\mathrm{\partial }\mathbf{X}}{\mathrm{\partial }\mathbf{x}}}$, which is thespatial deformation gradient tensor.${\displaystyle \mathbf{F}}$ being invertible is equivalent to${\displaystyle \text{det}\mathbf{F}\ne 0}$, which corresponds to the notion that the material cannot be infinitely compressed.

Consider aparticle 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${\displaystyle \mathbf{X}+\mathrm{\Delta }\mathbf{X}=(X_I+\mathrm{\Delta }{X}_I){\mathbf{I}}_I}$. In the deformed configuration this particle has a new position${\displaystyle q}$ given by the position vector${\displaystyle \mathbf{x}+\mathrm{\Delta }\mathbf{x}}$. Assuming that the line segments${\displaystyle \mathrm{\Delta }X}$ and${\displaystyle \mathrm{\Delta }\mathbf{x}}$ 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

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

For an infinitesimal element${\displaystyle d\mathbf{X}}$, and assuming continuity on the displacement field, it is possible to use aTaylor series expansion around point${\displaystyle P}$, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle${\displaystyle Q}$ asThus, the previous equation${\displaystyle d\mathbf{x}=d\mathbf{X}+d\mathbf{u}}$ can be written as

Calculations that involve the time-dependent deformation of a body often require atime derivative of the deformation gradient to be calculated. A geometrically consistent definition of such a derivative requires an excursion intodifferential geometry^[1] but we avoid those issues in this article.

The time derivative of${\displaystyle \mathbf{F}}$ is$${\displaystyle \dot{\mathbf{F}}=\frac{\mathrm{\partial }\mathbf{F}}{\mathrm{\partial }t}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left[\frac{\mathrm{\partial }\mathbf{x}(\mathbf{X},t)}{\mathrm{\partial }\mathbf{X}}\right]=\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{X}}\left[\frac{\mathrm{\partial }\mathbf{x}(\mathbf{X},t)}{\mathrm{\partial }t}\right]=\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{X}}\left[\mathbf{V}(\mathbf{X},t)\right]}$$where${\displaystyle \mathbf{V}}$ is the (material) velocity. The derivative on the right hand side represents amaterial velocity gradient. It is common to convert that into a spatial gradient by applying the chain rule for derivatives, i.e.,$${\displaystyle \dot{\mathbf{F}}=\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{X}}\left[\mathbf{V}(\mathbf{X},t)\right]=\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{X}}\left[\mathbf{v}(\mathbf{x}(\mathbf{X},t),t)\right]={\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{x}}\left[\mathbf{v}(\mathbf{x},t)\right]|}_{\mathbf{x}=\mathbf{x}(\mathbf{X},t)}\cdot \frac{\mathrm{\partial }\mathbf{x}(\mathbf{X},t)}{\mathrm{\partial }\mathbf{X}}=\mathit{l}\cdot \mathbf{F}}$$where${\displaystyle \mathit{l}=({\mathrm{\nabla }}_{\mathbf{x}}\mathbf{v}{)}^T}$ is thespatial 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$${\displaystyle \mathbf{F}=e^{\mathit{l}t}}$$assuming${\displaystyle \mathbf{F}=\mathbf{1}}$ at${\displaystyle t=0}$. There are several methods of computing theexponential above.

Related quantities often used in continuum mechanics are therate of deformation tensor and thespin tensor defined, respectively, as:$${\displaystyle \mathit{d}=\frac{1}{2}\left(\mathit{l}+{\mathit{l}}^T\right),\mathit{w}=\frac{1}{2}\left(\mathit{l}-{\mathit{l}}^T\right).}$$The rate of deformation tensor gives the rate of stretching of line elements while thespin tensor indicates the rate of rotation orvorticity 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$${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left({\mathbf{F}}^{-1}\right)=-{\mathbf{F}}^{-1}\cdot \dot{\mathbf{F}}\cdot {\mathbf{F}}^{-1}.}$$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 thepolar 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 aproper 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 theright stretch tensor; and${\displaystyle \mathbf{V}}$ theleft stretch tensor. The termsright andleft 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 bothpositive 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${\displaystyle \mathbf{x}\in {\mathbb{R}}^3}$, andsymmetric 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.${\displaystyle d{\mathbf{x}}^{\prime }=\mathbf{U}d\mathbf{X}}$, followed by a rotation${\displaystyle \mathbf{R}}$, i.e.,${\displaystyle d\mathbf{x}=\mathbf{R}d{\mathbf{x}}^{\prime }}$; or equivalently, by applying a rigid rotation${\displaystyle \mathbf{R}}$ first, i.e.,${\displaystyle d{\mathbf{x}}^{\prime }=\mathbf{R}d\mathbf{X}}$, followed later by a stretching${\displaystyle \mathbf{V}}$, i.e.,${\displaystyle d\mathbf{x}=\mathbf{V}d{\mathbf{x}}^{\prime }}$ (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 sameeigenvalues orprincipal stretches, but differenteigenvectors orprincipal 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 thesingular-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 useNanson's relation, expressed as$${\displaystyle da\mathbf{n}=JdA{\mathbf{F}}^{-T}\cdot \mathbf{N}}$$ 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 thedeformation gradient, and${\displaystyle J=det\mathbf{F}}$.

The corresponding formula for the transformation of thevolume element is$${\displaystyle dv=JdV}$$

A strain tensor is defined by theIUPAC as:^[3]

"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 incontinuum 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 itstranspose.

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

In 1839,George Green introduced a deformation tensor known as theright Cauchy–Green deformation tensor orGreen's deformation tensor (theIUPAC recommends that this tensor be called theCauchy strain tensor),^[3] defined as:

$${\displaystyle \mathbf{C}={\mathbf{F}}^T\mathbf{F}={\mathbf{U}}^2\text{or}{C}_{IJ}=F_{kI}{F}_{kJ}=\frac{\mathrm{\partial }{x}_k}{\mathrm{\partial }{X}_I}\frac{\mathrm{\partial }{x}_k}{\mathrm{\partial }{X}_J}.}$$

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 forstrain energy density functions. The most commonly usedinvariants arewhere${\displaystyle J:=det\mathbf{F}}$ 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).

TheIUPAC 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 theFinger strain tensor. However, that nomenclature is not universally accepted inapplied mechanics.

$${\displaystyle \mathbf{f}={\mathbf{C}}^{-1}={\mathbf{F}}^{-1}{\mathbf{F}}^{-T}\text{or}{f}_{IJ}=\frac{\mathrm{\partial }{X}_I}{\mathrm{\partial }{x}_k}\frac{\mathrm{\partial }{X}_J}{\mathrm{\partial }{x}_k}}$$

Reversing the order of multiplication in the formula for the right Cauchy-Green deformation tensor leads to theleft Cauchy–Green deformation tensor which is defined as:$${\displaystyle \mathbf{B}=\mathbf{F}{\mathbf{F}}^T={\mathbf{V}}^2\text{or}{B}_{ij}=\frac{\mathrm{\partial }{x}_i}{\mathrm{\partial }{X}_K}\frac{\mathrm{\partial }{x}_j}{\mathrm{\partial }{X}_K}}$$

The left Cauchy–Green deformation tensor is often called theFinger deformation tensor, named afterJosef Finger (1894).^[4]

TheIUPAC recommends that this tensor be called theGreen strain tensor.^[3]

Invariants of${\displaystyle \mathbf{B}}$ are also used in the expressions forstrain energy density functions. The conventional invariants are defined aswhere${\displaystyle J:=det\mathbf{F}}$ is the determinant of the deformation gradient.

For compressible materials, a slightly different set of invariants is used:$${\displaystyle ({\overline{I}}_1:=J^{-2/3}{I}_1;{\overline{I}}_2:=J^{-4/3}{I}_2;J\ne 1).}$$

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 thePiola strain tensor by the IUPAC^[3] and theFinger tensor^[6] in the rheology and fluid dynamics literature.

$${\displaystyle \mathbf{c}={\mathbf{B}}^{-1}={\mathbf{F}}^{-1T}{\mathbf{F}}^{-1}\text{or}{c}_{ij}=\frac{\mathrm{\partial }{X}_K}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }{X}_K}{\mathrm{\partial }{x}_j}}$$

If there are three distinct principal stretches${\displaystyle {\lambda }_i}$, thespectral decompositions of${\displaystyle \mathbf{C}}$ and${\displaystyle \mathbf{B}}$ is given by

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

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 thespatial stretch tensor while the right stretch (${\displaystyle \mathbf{U}}$) is called thematerial 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,$${\displaystyle {\mathbf{F}}^{-T}{\mathbf{N}}_i=\frac{1}{{\lambda }_i}{\mathbf{n}}_i;{\mathbf{F}}^T{\mathbf{n}}_i={\lambda }_i{\mathbf{N}}_i;{\mathbf{F}}^{-1}{\mathbf{n}}_i=\frac{1}{{\lambda }_i}{\mathbf{N}}_i.}$$

Uniaxial extension of an incompressible material

This is the case where a specimen is stretched in 1-direction with astretch ratio of${\displaystyle \alpha \mathbf{=}{\alpha }_{\mathbf{1}}}$. If the volume remains constant, the contraction in the other two directions is such that${\displaystyle {\alpha }_{\mathbf{1}}{\alpha }_{\mathbf{2}}{\alpha }_{\mathbf{3}}\mathbf{=}\mathbf{1}}$ or${\displaystyle {\alpha }_{\mathbf{2}}\mathbf{=}{\alpha }_{\mathbf{3}}\mathbf{=}{\alpha }^{\mathbf{-}\mathbf{0.5}}}$. Then:

Simple shear

Rigid body rotation

Derivatives of the stretch with respect to the right Cauchy–Green deformation tensor are used to derive the stress-strain relations of many solids, particularlyhyperelastic materials. These derivatives are$${\displaystyle \frac{\mathrm{\partial }{\lambda }_i}{\mathrm{\partial }\mathbf{C}}=\frac{1}{2{\lambda }_i}{\mathbf{N}}_i\otimes {\mathbf{N}}_i=\frac{1}{2{\lambda }_i}{\mathbf{R}}^T({\mathbf{n}}_i\otimes {\mathbf{n}}_i)\mathbf{R};i=1,2,3}$$and follow from the observations that$${\displaystyle \mathbf{C}:({\mathbf{N}}_i\otimes {\mathbf{N}}_i)={\lambda }_i^2;\frac{\mathrm{\partial }\mathbf{C}}{\mathrm{\partial }\mathbf{C}}={\mathsf{I}}^{(s)};{\mathsf{I}}^{(s)}:({\mathbf{N}}_i\otimes {\mathbf{N}}_i)={\mathbf{N}}_i\otimes {\mathbf{N}}_i.}$$

Let${\displaystyle \mathbf{X}=X^i{\mathit{E}}_i}$ be a Cartesian coordinate system defined on the undeformed body and let${\displaystyle \mathbf{x}=x^i{\mathit{E}}_i}$ 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$${\displaystyle l_X={\int }_0^1\left|\frac{d\mathbf{X}}{ds}\right|ds={\int }_0^1\sqrt{\frac{d\mathbf{X}}{ds}\cdot \frac{d\mathbf{X}}{ds}}ds={\int }_0^1\sqrt{\frac{d\mathbf{X}}{ds}\cdot \mathit{I}\cdot \frac{d\mathbf{X}}{ds}}ds}$$After deformation, the length becomesNote that the right Cauchy–Green deformation tensor is defined as$${\displaystyle \mathit{C}:={\mathit{F}}^T\cdot \mathit{F}={\left(\frac{d\mathbf{x}}{d\mathbf{X}}\right)}^T\cdot \frac{d\mathbf{x}}{d\mathbf{X}}}$$Hence,$${\displaystyle l_x={\int }_0^1\sqrt{\frac{d\mathbf{X}}{ds}\cdot \mathit{C}\cdot \frac{d\mathbf{X}}{ds}}ds}$$which indicates that changes in length are characterized by${\displaystyle \mathit{C}}$.

The concept ofstrain is used to evaluate how much a given displacement differs locally from a rigid body displacement.^[7][8][9] One of such strains for large deformations is theLagrangian finite strain tensor, also called theGreen-Lagrangian strain tensor orGreen–St-Venant strain tensor, defined as

$${\displaystyle \mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})\text{or}{E}_{KL}=\frac{1}{2}\left(\frac{\mathrm{\partial }{x}_j}{\mathrm{\partial }{X}_K}\frac{\mathrm{\partial }{x}_j}{\mathrm{\partial }{X}_L}-{\delta }_{KL}\right)}$$

or as a function of the displacement gradient tensor$${\displaystyle \mathbf{E}=\frac{1}{2}\left[({\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}{)}^T+{\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}+({\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}{)}^T\cdot {\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}\right]}$$or$${\displaystyle E_{KL}=\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_K}{\mathrm{\partial }{X}_L}+\frac{\mathrm{\partial }{u}_L}{\mathrm{\partial }{X}_K}+\frac{\mathrm{\partial }{u}_M}{\mathrm{\partial }{X}_K}\frac{\mathrm{\partial }{u}_M}{\mathrm{\partial }{X}_L}\right)}$$

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

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

$${\displaystyle \mathbf{e}=\frac{1}{2}(\mathbf{I}-\mathbf{c})=\frac{1}{2}(\mathbf{I}-{\mathbf{B}}^{-1})\text{or}{e}_{rs}=\frac{1}{2}\left({\delta }_{rs}-\frac{\mathrm{\partial }{X}_M}{\mathrm{\partial }{x}_r}\frac{\mathrm{\partial }{X}_M}{\mathrm{\partial }{x}_s}\right)}$$

or as a function of the displacement gradients we have$${\displaystyle e_{ij}=\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}-\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_i}\frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_j}\right)}$$