3. Principle of virtual work#

The principle of virtual work is the weak formulation of the static balance of internal forces and external forces:

\(\delta {\pi }_{\text{int}}\mathrm{-}\delta {\pi }_{\text{ext}}\mathrm{=}0\)

The non-linearity of the equilibrium equations leads us to solve the above system iteratively by a Newton method. We thus proceed to the exact linearization of the principle of virtual work at each iteration, which leads to equality:

\({\Delta \delta \pi }_{\text{int}}^{}\mathrm{-}{\Delta \delta \pi }_{\text{ext}}^{}\mathrm{=}{\delta \pi }_{\text{ext}}^{}\mathrm{-}{\delta \pi }_{\text{int}}^{}\)

3.1. Internal virtual work#

The virtual work of the internal forces can be written on the initial configuration in the form:

\(\delta {\pi }_{\text{int}}\mathrm{=}{\mathrm{\int }}_{\Omega }(\delta \tilde{\mathrm{E}}\text{.}\tilde{\mathrm{S}})d\Omega\)

where \(\tilde{E}\) and \(\tilde{S}\) are the Green-Lagrange deformation and Piola-Kirchhoff stress vectors of the second kind respectively, expressed in the local coordinate system. Indeed, as the stress state is flat for Piola-Kirchhoff of the second kind, we use the formulation of the principle of virtual work in the local coordinate system. However, to limit the passages from the local coordinate system to the global coordinate system and vice versa, the local deformation and stress vectors are not calculated explicitly in the local coordinate system but they are obtained by rotating their representation in the global coordinate system.

3.1.1. Incremental form of internal virtual work#

The iterative variation of the work of the internal virtual work is written as:

\(\Delta \delta {\pi }_{\text{int}}\mathrm{=}{\mathrm{\int }}_{\Omega }(\delta \tilde{E}\text{.}\Delta \tilde{S}+\Delta \delta \tilde{E}\text{.}\tilde{S})d\Omega\)

In this equality, the iterative variation of the vector of local Piola-Kirchhoff constraints of the second kind is calculated by the discrete iterative form of the behavioral relationship:

\(\Delta \tilde{S}\mathrm{=}D\Delta \tilde{E}\)

3.1.2. Transition from global to local coordinate system#

In tensor form we go from the global stress tensor to the local stress tensor \(3\mathrm{\times }3\) (see [bib4] p. 111 for Cauchy constraints, the same relationships applying to Piola-Kirchhoff constraints of the second kind) using:

\(\mathrm{[}\tilde{S}\mathrm{]}\mathrm{=}P\mathrm{[}S\mathrm{]}{P}^{T}\)

and from the local stress tensor to the global stress tensor by inverting the previous relationship:

\(\mathrm{[}S\mathrm{]}\mathrm{=}{P}^{T}\mathrm{[}\tilde{S}\mathrm{]}P\)

In the two previous expressions, the transition matrix from the local coordinate system to the global coordinate system is an orthogonal matrix \({P}^{\mathrm{-}1}\mathrm{=}{P}^{T}\), and its explicit expression as a function of the unit vectors of the local orthonormal coordinate system is:

\(\mathrm{P}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{=}\left[\begin{array}{c}{\mathrm{t}}_{1}^{\mathrm{T}}({\xi }_{1},{\xi }_{2},{\xi }_{3})\\ {\mathrm{t}}_{2}^{\mathrm{T}}({\xi }_{1},{\xi }_{2},{\xi }_{3})\\ {\mathrm{n}}^{\mathrm{T}}({\xi }_{1},{\xi }_{2},{\xi }_{3})\end{array}\right]\)

In the context of conventional notation, we can note:

\(\begin{array}{c}{t}_{1}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{=}{\Lambda }_{0}{e}_{1}\\ {t}_{2}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{=}{\Lambda }_{0}{e}_{2}\\ {t}_{3}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{=}n({\xi }_{1},{\xi }_{2})\mathrm{=}{\Lambda }_{0}{e}_{3}\end{array}\)

with the orthogonal transition matrix (initial rotation):

\({\Lambda }_{0}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{=}\left[{t}_{1}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{:}{t}_{2}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{:}{t}_{3}({\xi }_{1},{\xi }_{2},{\xi }_{3})\right]\)

It will be noted that:

\({\Lambda }_{0}\mathrm{=}{P}^{T}\)

The two stress rotation relationships are also valid for Green-Lagrange strain tensors. However, writing that links the local and global deformation vectors is necessary. This relationship makes it possible to go from the vector \(6\mathrm{\times }1\) of global deformations to the vector \(6\mathrm{\times }1\) of local deformations:

\(\stackrel{6\mathrm{\times }1}{\tilde{E}}\mathrm{=}\stackrel{6\mathrm{\times }6}{\stackrel{ˉ}{H}}\stackrel{6\mathrm{\times }1}{E}\)

with the expression of the transformation matrix for deformation vectors \(6\mathrm{\times }1\) (see [bib2] p.258):

\(\stackrel{6\mathrm{\times }6}{\overline{H}}\mathrm{=}\left[\begin{array}{cccccc}{l}_{1}^{2}& {m}_{1}^{2}& {n}_{1}^{2}& {l}_{1}{m}_{1}& {m}_{1}{n}_{1}& {n}_{1}{l}_{1}\\ {l}_{2}^{2}& {m}_{2}^{2}& {n}_{2}^{2}& {l}_{2}{m}_{2}& {m}_{2}{n}_{2}& {n}_{2}{l}_{2}\\ {l}_{3}^{2}& {m}_{3}^{2}& {n}_{3}^{2}& {l}_{3}{m}_{3}& {m}_{3}{n}_{3}& {n}_{3}{l}_{3}\\ {\mathrm{2l}}_{1}{l}_{2}& {\mathrm{2m}}_{1}{m}_{2}& {\mathrm{2n}}_{1}{n}_{2}& {l}_{1}{m}_{2}+{l}_{2}{m}_{1}& {m}_{1}{n}_{2}+{m}_{2}{n}_{1}& {n}_{1}{l}_{2}+{n}_{2}{l}_{1}\\ {\mathrm{2l}}_{2}{l}_{3}& {\mathrm{2m}}_{2}{m}_{3}& {\mathrm{2n}}_{2}{n}_{3}& {l}_{2}{m}_{3}+{l}_{3}{m}_{2}& {m}_{2}{n}_{3}+{m}_{3}{n}_{2}& {n}_{2}{l}_{3}+{n}_{3}{l}_{2}\\ {\mathrm{2l}}_{3}{l}_{1}& {\mathrm{2m}}_{3}{m}_{1}& {\mathrm{2n}}_{3}{n}_{1}& {l}_{3}{m}_{1}+{l}_{1}{m}_{3}& {m}_{3}{n}_{1}+{m}_{1}{n}_{3}& {n}_{3}{l}_{1}+{n}_{1}{l}_{3}\end{array}\right]\)

and the components of the unit vectors of the local coordinate system:

\(\begin{array}{ccc}{l}_{1}\mathrm{=}{t}_{1}\text{.}{e}_{1}& {m}_{1}\mathrm{=}{t}_{1}\text{.}{e}_{2}& {n}_{1}\mathrm{=}{t}_{1}\text{.}{e}_{3}\\ {l}_{2}\mathrm{=}{t}_{2}\text{.}{e}_{1}& {m}_{2}\mathrm{=}{t}_{2}\text{.}{e}_{2}& {n}_{2}\mathrm{=}{t}_{2}\text{.}{e}_{3}\\ {l}_{3}\mathrm{=}{t}_{3}\text{.}{e}_{1}& {m}_{3}\mathrm{=}{t}_{3}\text{.}{e}_{2}& {n}_{3}\mathrm{=}{t}_{3}\text{.}{e}_{3}\end{array}\)

These expressions are general for curvilinear references. In the global Cartesian coordinate system \(\left[{e}_{1}:{e}_{2}:{e}_{3}\right]\), these components are:

\(\begin{array}{ccc}{l}_{1}\mathrm{=}{t}_{1}(1)& {m}_{1}\mathrm{=}{t}_{1}(2)& {n}_{1}\mathrm{=}{t}_{1}(3)\\ {l}_{2}\mathrm{=}{t}_{2}(1)& {m}_{2}\mathrm{=}{t}_{2}(2)& {n}_{2}\mathrm{=}{t}_{2}(3)\\ {l}_{3}\mathrm{=}{t}_{3}(1)& {m}_{3}\mathrm{=}{t}_{3}(2)& {n}_{3}\mathrm{=}{t}_{3}(3)\end{array}\)

In reality, we need a script that links the local deformation vector \(5\mathrm{\times }1\) and the global deformation vector \(6\mathrm{\times }1\):

\(\stackrel{5\mathrm{\times }1}{\tilde{E}}\mathrm{=}\stackrel{5\mathrm{\times }6}{H}\stackrel{6\mathrm{\times }1}{E}\)

To do this, we forget the third line of the expression for \(\stackrel{6\mathrm{\times }6}{\overline{H}}\) (line associated with \({S}_{\text{nn}}\)):

\(\begin{array}{c}\stackrel{5\mathrm{\times }6}{H}\mathrm{=}\mathrm{[}\begin{array}{ccc}{({t}_{1}(1))}^{2}& {({t}_{1}(2))}^{2}& {({t}_{1}(3))}^{2}\\ {({t}_{2}(1))}^{2}& {({t}_{2}(2))}^{2}& {({t}_{2}(3))}^{2}\\ {\mathrm{2t}}_{1}(1){t}_{2}(1)& {\mathrm{2t}}_{1}(2){t}_{2}(2)& {\mathrm{2t}}_{1}(3){t}_{2}(3)\\ {\mathrm{2t}}_{2}(1){t}_{3}(1)& {\mathrm{2t}}_{2}(2){t}_{3}(2)& {\mathrm{2t}}_{2}(3){t}_{3}(3)\\ {\mathrm{2t}}_{3}(1){t}_{1}(1)& {\mathrm{2t}}_{3}(2){t}_{1}(2)& {\mathrm{2t}}_{3}(3){t}_{1}(3)\end{array}\\ \begin{array}{ccc}{t}_{1}(1){t}_{1}(2)& {t}_{1}(2){t}_{1}(3)& {t}_{1}(3){t}_{1}(1)\\ {t}_{2}(1){t}_{2}(2)& {t}_{2}(2){t}_{2}(3)& {t}_{2}(3){t}_{2}(1)\\ {t}_{1}(1){t}_{2}(2)+{t}_{2}(1){t}_{1}(2)& {t}_{1}(2){t}_{2}(3)+{t}_{2}(2){t}_{1}(3)& {t}_{1}(3){t}_{2}(1)+{t}_{2}(3){t}_{1}(1)\\ {t}_{2}(1){t}_{3}(2)+{t}_{3}(1){t}_{2}(2)& {t}_{2}(2){t}_{3}(3)+{t}_{3}(2){t}_{2}(3)& {t}_{2}(3){t}_{3}(1)+{t}_{3}(3){t}_{2}(1)\\ {t}_{3}(1){t}_{1}(2)+{t}_{1}(1){t}_{3}(2)& {t}_{3}(2){t}_{1}(3)+{t}_{1}(2){t}_{3}(3)& {t}_{3}(3){t}_{1}(1)+{t}_{1}(3){t}_{3}(1)\end{array}\mathrm{]}\end{array}\)

The same previous relationships can be applied for the transition from global deformation vectors to local deformation.

3.1.3. Deformation-displacement relationship#

The \(3\mathrm{\times }3\) tensor for global Green-Lagrange deformations is defined by (see for example [bib2]):

\(\mathrm{[}E\mathrm{]}\mathrm{=}\frac{1}{2}(\mathrm{\nabla }u+\mathrm{\nabla }{u}^{T}+\mathrm{\nabla }{u}^{T}\mathrm{\nabla }u)\)

with the displacement gradient tensor:

\(\mathrm{\nabla }u\mathrm{=}(\begin{array}{c}\frac{\mathrm{\partial }}{\mathrm{\partial }x}\\ \frac{\mathrm{\partial }}{\mathrm{\partial }y}\\ \frac{\mathrm{\partial }}{\mathrm{\partial }z}\end{array})\mathrm{\langle }uvw\mathrm{\rangle }\mathrm{=}\left[\begin{array}{ccc}\frac{\mathrm{\partial }u}{\mathrm{\partial }x}& \frac{\mathrm{\partial }v}{\mathrm{\partial }x}& \frac{\mathrm{\partial }w}{\mathrm{\partial }x}\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }y}& \frac{\mathrm{\partial }v}{\mathrm{\partial }y}& \frac{\mathrm{\partial }w}{\mathrm{\partial }y}\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }z}& \frac{\mathrm{\partial }v}{\mathrm{\partial }z}& \frac{\mathrm{\partial }w}{\mathrm{\partial }z}\end{array}\right]\)

The Green-Lagrange strain tensor can also be written as:

\(\left[E\right]\mathrm{=}\frac{1}{2}({F}^{T}F\mathrm{-}I)\)

with \(F\) the gradient tensor of the \(3\mathrm{\times }3\) deformations which is not symmetric:

\(F\mathrm{=}\mathrm{\nabla }{x}^{\varphi }\mathrm{=}I+\mathrm{\nabla }u\)

and \(I\) the identity tensor:

\(I\mathrm{=}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\)

The vector \(6\mathrm{\times }1\) of global Green-Lagrange deformations is ordered as follows (see [bib4] p117):

\(E\mathrm{=}(\begin{array}{c}{E}_{\text{xx}}\\ {E}_{\text{yy}}\\ {E}_{\text{zz}}\\ {\gamma }_{\text{xy}}\\ {\gamma }_{\text{xz}}\\ {\gamma }_{\text{yz}}\end{array})\mathrm{=}(\begin{array}{c}{u}_{,x}\\ {v}_{,y}\\ {w}_{,z}\\ {u}_{,y}+{v}_{,x}\\ {u}_{,z}+{w}_{,x}\\ {v}_{,z}+{w}_{,y}\end{array})+(\begin{array}{c}\frac{1}{2}({u}_{{,x}^{2}}+{v}_{{,x}^{2}}+{w}_{{,x}^{2}})\\ \frac{1}{2}({u}_{{,y}^{2}}+{v}_{{,y}^{2}}+{w}_{{,y}^{2}})\\ \frac{1}{2}({u}_{{,z}^{2}}+{v}_{{,z}^{2}}+{w}_{{,z}^{2}})\\ {u}_{,x}{u}_{,y}+{v}_{,x}{v}_{,y}+{w}_{,x}{w}_{,y}\\ {u}_{,x}{u}_{,z}+{v}_{,x}{v}_{,z}+{w}_{,x}{w}_{,z}\\ {u}_{,y}{u}_{,z}+{v}_{,y}{v}_{,z}+{w}_{,y}{w}_{,z}\end{array})\)

It is calculated as follows:

\(E\mathrm{=}\left[Q+\frac{1}{2}A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\)

with:

\(Q\mathrm{=}\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 1& 0\end{array}\right]\)

and the displacement gradient vector:

\(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\mathrm{=}(\begin{array}{c}{u}_{,x}\\ {u}_{,y}\\ {u}_{,z}\\ {v}_{,x}\\ {v}_{,y}\\ {v}_{,z}\\ {w}_{,x}\\ {w}_{,y}\\ {w}_{,z}\end{array})\)

and the \(A\) tensor depending on the displacement gradient:

\(A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\mathrm{=}\left[\begin{array}{ccccccccc}{u}_{,x}& 0& 0& {v}_{,x}& 0& 0& {w}_{,x}& 0& 0\\ 0& {u}_{,y}& 0& 0& {v}_{,y}& 0& 0& {w}_{,y}& 0\\ 0& 0& {u}_{,z}& 0& 0& {v}_{,z}& 0& 0& {w}_{,z}\\ {u}_{,y}& {u}_{,x}& 0& {v}_{,y}& {v}_{,x}& 0& {w}_{,y}& {w}_{,x}& 0\\ {u}_{,z}& 0& {u}_{,x}& {v}_{,z}& 0& {v}_{,x}& {w}_{,z}& 0& {w}_{,x}\\ 0& {u}_{,z}& {u}_{,y}& 0& {v}_{,z}& {v}_{,y}& 0& {w}_{,z}& {w}_{,y}\end{array}\right]\)

The virtual variation, noted \(\delta\), of the Green-Lagrange deformations is obtained by a differential calculation:

\(\delta E\mathrm{=}\left[Q+A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]\frac{\mathrm{\partial }\delta u}{\mathrm{\partial }x}\)

In this expression and the one that follows, we took into account the following property (see [bib4] p 141):

\(\frac{1}{2}A(\frac{\mathrm{\partial }\delta u}{\mathrm{\partial }x})\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\mathrm{=}\frac{1}{2}A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\frac{\mathrm{\partial }\delta u}{\mathrm{\partial }x}\)

The iterative variation \(\Delta\) is also obtained by a differential calculation:

\(\Delta E\mathrm{=}\left[Q+A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]\frac{\mathrm{\partial }\Delta u}{\mathrm{\partial }x}\)

The iterative variation of the virtual Green-Lagrange deformation is thus in the form:

\(\underset{\text{terme classique}}{\Delta \delta E\mathrm{=}A(\frac{\mathrm{\partial }\Delta u}{\mathrm{\partial }x})\frac{\mathrm{\partial }\delta u}{\mathrm{\partial }x}}+\underset{\text{terme non classique}}{\left[Q+A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]\frac{\mathrm{\partial }\Delta \delta u}{\mathrm{\partial }x}}\)

While the first term in this expression is classical for continuous 3D media, the second term, which reflects the consideration of large rotations, is less so.

3.1.4. Calculation of Cauchy stresses#

3.1.4.1. General case#

The tensor \(3\mathrm{\times }3\) of the global Piola-Kirchhoff constraints of the second kind is linked to the tensor \(3\mathrm{\times }3\) of the global Cauchy constraints by the relationship:

\(\left[S\right]\mathrm{=}\text{det}(F){F}^{\mathrm{-}1}\left[\sigma \right]{F}^{\mathrm{-}T}\)

Thus, knowing the state of Piola-Kirchhoff stresses of the second kind, it is possible to calculate the state of Cauchy stresses by the relationship:

\(\left[\sigma \right]\mathrm{=}\frac{1}{\text{det}(F)}F\left[S\right]{F}^{T}\)

It should be noted that the state of Cauchy stresses is not plane, in general, unlike the state of Piola-Kirchhoff stresses of the second kind. Moreover, the choice of a local coordinate system in which to represent this tensor is not at all obvious. However, in the following paragraph, it will be shown that in the context of small deformations, there is a local frame of reference, easily identifiable, in which the state of Cauchy stresses is also flat.

In the case of completely general laws, particular attention should be paid to numerical integration schemes that make it possible to calculate the substitution values of the \(F\) gradient at the points of normal numerical integration.

3.1.4.2. Approximation in small deformations#

We recall [bib4] that the \(F\) gradient can be written using polar decomposition in two forms:

\(F=\text{RU}=\text{VR}\)

where \(R\mathrm{=}{R}^{\mathrm{-}T}\) is an orthogonal tensor, and \(U\) and \(V\) are positive definite symmetric elongation matrices.

In the geometric nonlinear domain, we can introduce an important simplification in the polar decomposition of the deformation gradient if the deformations remain small. This simplification is not introduced in nonlinear calculation but in post-processing constraints.

The elongation at point \(Q\) being minor compared to the large rotation of the section:

\(U\mathrm{\approx }V\mathrm{\approx }I\)

We can then write:

\(F\mathrm{\approx }R\mathrm{=}\Lambda\)

where \(\Lambda\) is the high rotation tensor that turns normal \(n\) into \({n}^{\varphi }\):

\(\Lambda n\mathrm{=}{n}^{\varphi }\)

Simplification reflects the fact that on a section, the transformation is reduced to a large rotation. With this approximation of the deformation gradient, we can write:

\(F\mathrm{\approx }R\mathrm{=}\Lambda\)

and so, by exploiting the orthogonality of \(\Lambda\) we get:

\({F}^{\mathrm{-}1}\mathrm{\approx }{\Lambda }^{T}\)

and:

\(\text{det}(F)\mathrm{\approx }1\).

These simplifications lead to the final relationship:

\(\left[\sigma \right]\mathrm{\approx }\Lambda \left[S\right]{\Lambda }^{T}\)

This relationship reflects the fact that Cauchy constraints are simply obtained by the great rotation of Piola-Kirchhoff stresses of the second kind.

We can now rewrite the plane stress property of the Piola-Kirchhoff tensor of the second kind \(n\text{.}\left[S\right]n\mathrm{=}0\) in the new form:

\(n\text{.}{\Lambda }^{T}\left[s\right]\Lambda n\mathrm{=}0\)

which also leads to ownership:

\({n}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\mathrm{=}0\)

Or again:

\({\tilde{\sigma }}_{{n}^{\varphi }{n}^{\varphi }}\mathrm{=}0\)

The Cauchy constraints \(\left[\sigma ({\xi }_{1},{\xi }_{2},{\xi }_{3})\right]\) are also flat in the local coordinate system \(\left[{t}_{{1}_{}}^{\varphi }({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{:}{t}_{{2}_{}}^{\varphi }({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{:}{n}^{\varphi }({\xi }_{1},{\xi }_{2})\right]\) obtained by large rotation of the local coordinate system on the initial configuration:

\(\left[{t}_{{1}_{}}^{\varphi }\mathrm{:}{t}_{{2}_{}}^{\varphi }\mathrm{:}{n}^{\varphi }\right]\mathrm{=}\Lambda \left[{t}_{{1}_{}}\mathrm{:}{t}_{{2}_{}}\mathrm{:}n\right]\)

In this coordinate system, we can write all the components of tensor \(\mathrm{[}s\mathrm{]}\) as follows:

\(\left[\begin{array}{ccc}{\tilde{\sigma }}_{{t}_{1}^{\varphi }{t}_{1}^{\varphi }}& {\tilde{\sigma }}_{{t}_{1}^{\varphi }{t}_{2}^{\varphi }}& {\tilde{\sigma }}_{{t}_{1}^{\varphi }{n}^{\varphi }}\\ {\tilde{\sigma }}_{{t}_{2}^{\varphi }{t}_{1}^{\varphi }}& {\tilde{\sigma }}_{{t}_{2}^{\varphi }{t}_{1}^{\varphi }}& {\tilde{\sigma }}_{{t}_{2}^{\varphi }{n}^{\varphi }}\\ {\tilde{\sigma }}_{{t}_{1}^{\varphi }{n}^{\varphi }}& {\tilde{\sigma }}_{{t}_{1}^{\varphi }{n}^{\varphi }}& 0\end{array}\right]\mathrm{=}\left[\begin{array}{ccc}{t}_{1}^{\varphi }\text{.}\left[\sigma \right]{t}_{1}^{\varphi }& {t}_{1}^{\varphi }\text{.}\left[\sigma \right]{t}_{2}^{\varphi }& {t}_{1}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\\ {t}_{2}^{\varphi }\text{.}\left[\sigma \right]{t}_{1}^{\varphi }& {t}_{2}^{\varphi }\text{.}\left[\sigma \right]{t}_{2}^{\varphi }& {t}_{2}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\\ {n}^{\varphi }\text{.}\left[\sigma \right]{t}_{1}^{\varphi }& {n}^{\varphi }\text{.}\left[\sigma \right]{t}_{2}^{\varphi }& {n}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\end{array}\right]\)

Using the \(\left[\sigma \right]\mathrm{\approx }\Lambda \left[S\right]{\Lambda }^{T}\) relationship, we get:

\(\left[\begin{array}{ccc}{t}_{1}^{\varphi }\text{.}\left[\sigma \right]{t}_{1}^{\varphi }& {t}_{1}^{\varphi }\text{.}\left[\sigma \right]{t}_{2}^{\varphi }& {t}_{1}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\\ {t}_{2}^{\varphi }\text{.}\left[\sigma \right]{t}_{1}^{\varphi }& {t}_{2}^{\varphi }\text{.}\left[\sigma \right]{t}_{2}^{\varphi }& {t}_{2}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\\ {n}^{\varphi }\text{.}\left[\sigma \right]{t}_{1}^{\varphi }& {n}^{\varphi }\text{.}\left[\sigma \right]{t}_{2}^{\varphi }& {n}^{\varphi }\text{.}\left[\sigma \right]{n}^{\varphi }\end{array}\right]\mathrm{=}\left[\begin{array}{ccc}{t}_{1}\text{.}\left[S\right]{t}_{1}& {t}_{1}\text{.}\left[S\right]{t}_{2}& {t}_{1}\text{.}\left[S\right]n\\ {t}_{2}\text{.}\left[S\right]{t}_{1}& {t}_{2}\text{.}\left[S\right]{t}_{2}& {t}_{2}\text{.}\left[S\right]n\\ n\text{.}\left[S\right]{t}_{1}& n\text{.}\left[S\right]{t}_{2}& n\text{.}\left[S\right]n\end{array}\right]\)

Hence the final result:

As long as the deformation remains small, the components of the Cauchy stress tensor in the local coordinate system attached to the deformed configuration are identical to the components of the second type of Piola-Kirchhoff stress tensor in the local coordinate system attached to the initial configuration.

In the following, we decide to consider only Piola-Kirchhoff constraints of the second kind. We should note that within the framework of a more general constitutive law, we can pass from one coercive measure to another as indicated in the preceding paragraph.