2. Cinematics
2.1. Configurations description
We consider the initial configuration \({\mathrm{\Omega }}_{0}\) and the current configuration \({\mathrm{\Omega }}_{t}\). The vector \(X\) gives the position of a particle in the initial configuration and \(x\) is its position in the current configuration. With \(u\) the movement of the particle between the two configurations:
\[\]
: label: eq-21
Two coordinate systems will be considered. The Cartesian coordinate system which has three components \((x,y,z)\) and whose base is constituted by the three vectors \({e}_{i}\) and the reference parametric coordinate system of the finite element which also has three components \(\mathrm{\xi }=(\mathrm{\xi },\mathrm{\eta },\mathrm{\zeta })\).
Ratings:
A quantity expressed in the initial configuration will be in**uppercase*;
A quantity expressed in the current configuration will be**lowercase*.
We will define the base in the initial configuration by the following three vectors:
(2.1)\[ {G} _ {i} =\ frac {\ partial X} {{\ partial\ xi} ^ {i}}\ text {for} i=\ mathrm {1,2,3}\]
The corresponding contravariant vector:
(2.2)\[ {G} _ {i}\ cdot {G} ^ {j} = {\ delta} _ {i} ^ {j}\]
Likewise, in the current configuration:
(2.3)\[ \begin{align}\begin{aligned} {g} _ {i} =\ frac {\ partial x} {{\ partial\ x} {{\ partial\ xi}} ^ {i}} = {G} _ {i} +\ frac {\ partial u} {{\ partial\ xi} {{\ partial\ xi}} ^ {i}} ^ {i}}}\ text {for} i=\ mathrm {1,2,3}}\\ \ textrm {and}\\ {g} _ {i}\ cdot {g} ^ {j} = {\ delta} _ {i} ^ {j}\end{aligned}\end{align} \]
2.2. Jacobian of the transformation
We can express the Jacobian matrix (of dimension \(3\times 3\)) of the transformation between the initial configuration and the parametric space:
(2.4)\[ J=\ frac {\ partial X} {\ partial\ mathrm {\ xi}}\]
And the inverse of the Jacobian matrix:
(2.5)\[ \ overline {J} = {\ left (\ frac {\ partial X} {\ partial\ mathrm {\ xi}}\ right)} ^ {-1}}\]
The determinant of \(J\) makes it possible to transform the integrals of the coordinate space in the initial configuration to the parametric reference space of the finite elements. It is also a measure of the real volume of the finite element:
\[\]
: label: eq-27
{J} _ {0} =mathit {det} (J)
To effectively stabilize the element and therefore to eliminate the hourglass modes arising from reduced integration, it is necessary to be able to calculate the various integral terms with sufficient precision.
As the numerical integration scheme is deliberately very poor, the best way to do this is to work with integrands that are polynomials as much as possible. For this reason, it is interesting to look for a polynomial approximation of the Jacobian \(J\) and Jacobian inverse \(\overline{J}\) matrices, which makes it possible to represent any shape of the element with sufficient precision.
The Jacobian matrix \(J\) is divided into constant, linear and bi-linear components:
\[\]
: label: eq-28
J= {J} ^ {0} +mathrm {xi} {J} {J}} ^ {mathrm {xi}} +mathrm {eta} {J} ^ {mathrm {eta}}} +mathrm {eta}}} +mathrm {eta}} ^ {mathrm {zeta}} ^ {mathrm {xi}}mathrm {eta}}} +mathrm {eta}}} ^ {mathrm {xi}}mathrm {eta}}} +mathrm {eta}} ^ {mathrm {eta}}} ^ {mathrm {eta}} ^ {mathrm {eta}}} eta} {J} ^ {mathrm {xi}mathrm {xi}}mathrm {eta}} +mathrm {zeta} {J} ^ {mathrm {eta}}mathrm {eta}}mathrm {zeta}}} +mathrm {zeta} {J} ^ {mathrm {eta}} ^ {mathrm {eta}} ^ {mathrm {eta}} ^ {mathrm {eta}} ^ {mathrm {eta}} ^ {eta} ^ {eta}} ^ {mathrm {eta}} ^ {eta} ^ {eta}} ^ {mathrm {eta}} ^ {}mathrm {zeta}}
One possible way to obtain an adequate shape of the inverse Jacobian matrix for good stabilization efficiency is to evaluate it only at the center of the element (see [Bib12] and [Bib11]). In this case, the Jacobian matrix and its inverse are constant within the element. However, this method is only robust for very fine meshes.
For this reason, we approach the inverse Jacobian by means of a Taylor expansion with respect to the center of the element, following the work of [Bib10]. This method allows us to have a polynomial form of the inverse Jacobian matrix to facilitate numerical integration as required but also to allow elements that deviate from the parallelepipedic form to be taken into account in a more realistic manner.
From the decomposition (), we can approach \(\overline{J}\) by keeping only the constant term and the linear terms:
(2.6)\[ \ overline {J}\ approx {\ overline {J}}} ^ {0}} ^ {0} +\ xi {\ frac {\ partial\ overline {J}} {\ partial\ xi} |} _ {\ xi =0} +\ eta {\ frac {\ frac {\ frac {\ partial\ overline {J}}} {\ partial\ eta}} |} _ {\ xi =0} +\ zeta {\ frac {\ partial {\ partial}}\ overline {J}} {\ partial\ zeta} |} _ {\ xi =0}\]
The constant term is very easy to calculate since it is the inverse of the Jacobian constant term. For the other terms, derivatives must be evaluated. We proceed by developing the term \(\overline{J}J\) in the center of the element:
\[\]
: label: eq-30
overline {J} Japprox {overline {J}}} ^ {0}}} ^ {0} {J} +xi {frac {partial (overline {J} J)} {partialxi} |} |} _ {xi =0}} _ {xi =0}} +eta {frac {partial} J)} {partialeta} |} _ {xi =0}} +eta {frac {partial} J)} {partialeta} |} _ {xi =0}} +eta {frac {partial} J)} {partialeta} |} _ {xi =0}} +eta {frac {partial} J)} {partialeta} |} _ {=0} +zeta {frac {frac {partial (overline {J} J)} {partialzeta} |} _ {xi =0}
And like \(\overline{J}J=I\), its derivative is zero and so we have for each component \({\xi }_{i}\):
\[\]
: label: eq-31
0= {frac {partial (overline {J} J)}} {partial {xi} _ {i}} |} _ {xi =0} = {frac {partial J} {partial J}} {partial {xi}} _ {xi}} |} _ {xi}} |} _ {xi}} |} _ {xi =0} {frac {partialoverline {partial}} + {frac {partialoverline {J}} {partial {xi} _ {i}} |} _ {xi =0} {J} ^ {0}
And so:
\[\]
: label: eq-32
{frac {partialoverline {J}} {partial {xi}}} {partial {xi}}} |} _ {xi =0} =frac {- {frac {partial J}} {partial {partial J}}} {partial {xi}} {xi}} {frac {partial J}} {partial J}} {partial J}} {partial J}} {partial {xi}} {xi}} {frac {partial J}} {partial J}} {partial J}} {partial {xi}} {xi}} {frac {partial J}} {partial J}} {partial J}} {partial {xi}} {xi}} {overline {J}} ^ {0} {frac {partial J} {partial J} {partial {xi}} |} _ {xi =0} {xi =0} {overline {J}}} ^ {0}
textrm {car}
frac {1} {{J} ^ {0}} = {overline {J}}} ^ {0}
You have to evaluate the term. To do this, we start from () and we derive. For component \(\xi\) for example:
(2.7)\[ \ frac {\ partial J} {\ partial\ mathrm {\ mathrm {\ xi}}} =\ frac {\ partial {J} ^ {0}} {\ partial\ mathrm {\ xi}}} + {J}} ^ {\ mathrm {\ mathrm {\ xi}} ^ {\ mathrm {\ xi}} +\ mathrm {\ eta}}} +\ mathrm {\ eta}}} +\ mathrm {\ zeta} {J} ^ {\ mathrm {\ xi}\ mathrm {\ zeta}}\ approx {J} ^ {\ mathrm {\ xi}}}\]
We only kept the linear terms in \(\xi\). What’s left is:
(2.8)\[ \begin{align}\begin{aligned} {\ frac {\ partial J} {\ partial\ mathrm {\ xi}} |} |} _ {\ mathrm {\ xi} =0}\ approx {{J} ^ {\ mathrm {\ xi}} |} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |} _ {\ mathrm {\ xi}} |}\\ {\ frac {\ partial J} {\ partial\ mathrm {\ eta}} |} |} _ {\ mathrm {\ xi} =0}\ approx {{J} ^ {\ mathrm {\ eta}} |} |} |} _ {\ mathrm {\ eta}}} |} _ {\ mathrm {\ eta}} |} _ {\ mathrm {\ eta}} |} |} _ {\ mathrm {\ eta}} |} _ {\ mathrm {\ eta}} |} _ {\ mathrm {\ eta}} |} _ {\ mathrm {\ eta}} |} _ {\ mathrm {\ eta}} |}\\ {\ frac {\ partial J} {\ partial\ mathrm {\ zeta}} |} |} _ {\ mathrm {\ xi} =0}\ approx {{J} ^ {\ mathrm {\ zeta}} |} |} |} _ {\ mathrm {\ zeta}}} |} _ {\ mathrm {\ zeta}} |} _ {\ mathrm {\ zeta}} |} _ {\ mathrm {\ zeta}} |} _ {\ mathrm {\ zeta}} |} _ {\ mathrm {\ zeta}} |}\end{aligned}\end{align} \]
We recall that the linear terms for the development of \(J\) are constant on the element (no dependence on the parametric coordinate \(\xi\)). And so:
(2.9)\[ {\ frac {\ partial\ overline {J}}} {\ partial {\ xi}}} {\ partial {\ xi}}} |} _ {\ xi =0} =-\ left ({\ overline {J}}} ^ {0}\ right) {J}\ right) {J}\ right) {J}\ right) {J}}\ right) {J}}\ right) {J} ^ {0}\ right) {J}\ right) {J}}\ right) {J} ^ {0}\ right)\]
Finally, from (), we have the following estimate of the inverse Jacobian:
(2.10)\[ \ overline {J} = {\ overline {J}}} ^ {0}} -\ sum _ {i=1} ^ {3}\ left ({\ overline {J}} ^ {0}\ right) {J} _ {0} _ {0}} _ {0}}\ right) {J} _ {i}}\ left ({\ overline {J}}} ^ {0}\ right) {\ xi} _ {i}\]
Be careful to distinguish \({J}^{0}\) which is the term constant (in zero order) from the limited development of \(J\) compared to the term \({J}_{0}\) which is the exact value of \(J\) at the centre of the element (in \(\xi =0\)).
We will also need the Jacobian version of the current configuration, which is defined as follows:
(2.11)\[ j=\ frac {\ partial x} {\ partial\ mathrm {\ xi}} =\ frac {\ partial X} {\ partial\ mathrm {\ xi}}} +\ frac {\ partial u} {\ partial\ partial u} {\ partial\ mathrm {\ xi}} =J+D\]
We proceed in exactly the same way for \(D\) as for \(J\). Its decomposition in the parametric coordinate system:
(2.12)\[ D= {D} ^ {0} +\ mathrm {\ xi} {D} {D} {D} ^ {\ mathrm {\ xi}} +\ mathrm {\ eta} {D} ^ {\ mathrm {\ eta}}} +\ mathrm {\ eta}}} +\ mathrm {\ eta}} ^ {\ mathrm {\ zeta}} ^ {\ mathrm {\ xi}}\ mathrm {\ eta}}} +\ mathrm {\ xi}}\ mathrm {\ eta}}} +\ mathrm {\ eta}} ^ {\ mathrm {\ xi}}\ mathrm {\ eta}}} +\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ eta} {D} ^ {\ mathrm {\ xi}\ mathrm {\ xi}}\ mathrm {\ eta}} +\ mathrm {\ zeta} {D} ^ {\ mathrm {\ eta}\ mathrm {\ eta}}\ mathrm {\ zeta}}} +\ mathrm {\ zeta} {D} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\ mathrm {\ eta}} ^ {\}\ mathrm {\ zeta}}\]
2.4. Green-Lagrange strain tensor
The deformation gradient tensor is equal to:
(2.18)\[ F=\ frac {\ partial x} {\ partial X} {\ partial X} = {g} _ {i} =\ frac {\ partial {x} ^ {i}} {\ partial {\ xi}} {\ partial {\ xi} ^ {xi} ^ {xi} ^ {xi}} {\ partial X} ^ {i}} {e}} {e}} {e}\ otimes\ frac {\ partial {\ x} ^ {i}} {\ partial {x}} ^ {j}} {\ partial {x}} ^ {j}} {e} _ {j}\]
The « standard » Green-Lagrange tensor (unmodified):
(2.19)\[ {E} ^ {u} =\ frac {1} {2}\ left ({g} {2}\ left ({g} _ {i} - {G} _ {i}\ cdot {G} _ {j} _ {j}\ right) {J}\ right) {G}\ right) {G} _ {j}\ right) {G}}\ right) {G} ^ {i}\ right) {G} ^ {i}\ right) {G} ^ {i}} ^ {i} = {\ widehat {E}}} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j} _ {j}\ right) {G}}\ right) {G}} {u} {G} ^ {i}\ otimes {G} ^ {j} = {E} _ {\ mathit {ij}}} ^ {u} {e} _ {i}\ otimes {e} _ {j}\]
Explicitly, the coordinates in the parametric coordinate system:
(2.20)\[ {\ widehat {E}} _ {\ mathit {ij}}} ^ {j}}} ^ {u} =\ frac {1} {2}\ left ({G} _ {i}\ cdot\ frac {\ partial u}} {\ partial u}} {\ partial u}\ cdot {\ partial u}}\ cdot {G} _ {j}} {\ partial u}}\ cdot {G} _ {j}}\ cdot {G} _ {j}}} +\ frac {\ partial u} {\ partial {\ xi}} ^ {j}}\ cdot\ frac {\ partial u} {\ partial {\ xi} ^ {i}}\ right) {G}}\ right) {G} ^ {i}\ right) {G} ^ {i}\ right) {G} ^ {j}\]
In Voigt notation, this tensor has six components:
(2.21)\[ \begin{align}\begin{aligned} {E} ^ {u} =\ left\ {{E} _ {\ mathit {E}} _ {\ mathit {XX}}} ^ {u}} ^ {u}, {E} _ {\ mathit {ZZ}} _ {\ mathit {ZZ}}}} ^ {u}}} ^ {u}} ^ {U}} ^ {U}} _ {\ mathit {YZ}}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} ^ {Z}} u} ,2 {E} _ {\ mathit {XZ}}} ^ {u}\ right\}\\ {\ widehat {E}} ^ {u} =\ left\ {{\ widehat {E}}} _ {\ mathrm {\ xi}\ mathrm {\ xi}} ^ {u}} ^ {u}, {\ widehat {E}}} =\ left\ left\ {\ widehat {E}}}} = {\ widehat {E}}}} _ {\ mathrm {\ zeta}\ mathrm {\ zeta}}} ^ {zeta}}} ^ {\ zeta}}} ^ {u}} ^ {u}} ^ {u} ,2 {\ widehat {E}}, {\ widehat {E}}} _ {\ widehat {E}}, {\ widehat {E}}} _ {\ widehat {E}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}} hat {E}} _ {\ mathrm {\ xi}\ mathrm {\ zeta}} ^ {u}\ right\}\end{aligned}\end{align} \]
To go from one to the other, we will use the \(T\) matrix:
(2.22)\[ {E} ^ {u} =T {\ widehat {E}} ^ {u}\]
With pass matrix \(T\):
(2.23)\[\begin{split} T=\ left [\ begin {array} {cccccc} {\ overline {J}}} _ {11} ^ {2} & {\ overline {J}} _ {21} ^ {2} & {\ overline {J}}} & {\ overline {J}}} _ {\ overline {J}} _ {21}}} _ {21}}} _ {21}}} _ {21}}} _ {21}}} _ {\ overline {J}}} _ {21}}} _ {21}}} _ {21}}} _ {\ overline {J}}} _ {21}}} _ {21}}} _ {21}}} _ {\ overline {J}}} _ {21}}} _ {21}}} _ {21}} {J}} _ {21} {\ overline {J}}} _ {31}} _ {31} & {\ overline {J}} _ {31}\\ {overline {J}}} _ {\ overline {J}}} _ {12}}} _ {12} ^ {2}} _ {12} ^ {2}} _ {32} ^ {2} & {\ overline {J}}} _ {32} ^ {2} & {\ overline {J}}} _ {32} ^ {2}} _ {32} ^ {2}} 2} & {\ overline {J}} _ {12} {\ overline {J}}} _ {22} & {\ overline {J}} _ {22} {\ overline {J}} _ {32} _ {32} & {\ overline {J}} & {\ overline {J}} _ {\ overline {J}} _ {32}\ {\ overline {J}} _ {32}} _ {13} ^ {2} _ {13} ^ {2} & {\ overline {J}} _ {23} ^ {2} & {\ overline {J}}} _ {33} ^ {2} & {\ overline {J}}} _ {13} {\ overline {J}}} _ {\ overline {J}}} _ {23} {\ overline {J}}} _ {23} {\ overline {J}} _ {13} {\ overline {J}}} _ {13} {\ overline {J}}} _ {13} {\ overline {J}}} _ {13} {\ overline {J}}} _ {13} {\ overline {J}}} _ {13} { {33} & {\ overline {J}}} _ {13} {\ overline {J}}} _ {33}\\ 2 {\ overline {J}} _ {11} {\ overline {J}}} _ {12}} _ {12}} & 2 {\ overline {J}}} _ {J}}} _ {31} & 2 {\ overline {J}}} _ {31} {\ overline {J}} _ {32} & {\ overline {J}}} _ {12} {\ overline {J}} _ {21} + {\ overline {J}} _ {11} {\ overline {J}}} & {\ overline {J}} _ {22} {\ overline {J}} _ {31}} + {\ overline {J}} _ {31} + {\ overline {J}}} _ {21} {\ overline {J}}} _ {32} & {\ overline {J}}} _ {12} {\ overline {J}} _ {31} + {\ overline {J}}} _ {12} {12} {\ overline {J}}} _ {12} {\ overline {J}}} _ {13}} _ {13}} _ {13} & 2 {\ overline {J}} _ {22} {\ overline {J}}} _ {23}} & 2 {\ overline {J}}} _ {32} {\ overline {J}} _ {33}} _ {33}} & {\ overline {J}} & {\ overline {J}} _ {33}} & {\ overline {J}} _ {33}} & {\ overline {J}} _ {33}} & {\ overline {J}} _ {33}} & {\ overline {J}} _ {33}} & {\ overline {J}} _ {33}} & { {\ overline {J}} _ {12} {\ overline {J}}} _ {23} & {\ overline {J}} _ {23} {\ overline {J}} _ {32} + {\ overline {J}}} + {\ overline {J}}} _ {32}} + {\ overline {J}} _ {32} + {\ overline {J}} _ {32} + {\ overline {J}} + {\ overline {J}} _ {32} + {\ overline {J}} + {\ overline {J}} _ {32} + {\ overline {J}} + {\ overline {J}} _ {32} + {\ overline {J}} _ {32} + {\ overline {J}} _ {_ {32} + {\ overline {J}}} _ {12} {\ overline {J}}} _ {33}\\ 2 {\ overline {J}} _ {11} {\ overline {J}}} _ {13}} _ {13}} & 2 {\ overline {J}}} _ {31} _ {23}} & 2 {\ overline {J}}} _ {31}} {\ overline {J}} _ {33} & {\ overline {J}}} _ {13} {\ overline {J}} _ {21} + {\ overline {J}} _ {11} {\ overline {J}} {\ overline {J}}} _ {23} {\ overline {J}} _ {31}} {11} {\ overline {J}} _ {31} {11} {\ overline {J}} _ {31} {\ overline {J}} _ {31} + {\ overline {J}}}} _ {21} {\ overline {J}}} _ {33}} _ {33} & {\ overline {J}}} _ {13} {\ overline {J}} _ {31} + {\ overline {J}}}} _ {11} {11} {\ overline {J}}} _ {33}\ end {array}}\ Right]\end{split}\]
and \(\overline{J}\) the components of the inverse of the Jacobian matrix:
\[\]
: label: eq-50
overline {J} = {left (frac {partial X} {partialmathrm {xi}}right)} ^ {-1}}
We’ll use approximation () for \(\overline{J}\).
The decomposition of the tensor of large deformations is of the same shape as that of small deformations ():
(2.24)\[ {\ widehat {E}} ^ {u} = {\ widehat {E}}} ^ {u\ mathrm {,0}} +\ zeta {\ widehat {E}} ^ {u,\ zeta} + {\ zeta} + {\ zeta}} ^ {2} {\ widehat {E}} ^ {u,\ widehat {E}} ^ {u,\ zeta}} ^ {u,\ zeta}} ^ {u,\ zeta}} ^ {u,\ zeta}} +\ xi {\ widehat {E}} +\ xi {\ widehat {E}} +\ xi {\ widehat {E}} +\ xi {\ widehat {E}}}} ^ {u,\ xi} +\ eta {\ widehat {E}}} ^ {u,\ eta} +\ xi\ eta {\ widehat {E}} ^ {u,\ xi\ eta} +\ xi\ zeta {\ zeta {\ zeta {E}}} ^ {u,\ xi\ zeta}} ^ {u,\ xi\ zeta}} ^ {u,\ eta\ zeta}\]
And as in the case of small deformations, using a reduced integration method, it is necessary to modify the evaluation of the Green-Lagrange deformations tensor in order to stabilize the problem (make the zero energy « hourglass » modes disappear), we will therefore write:
\[\]
: label: eq-52
{widehat {E}} ^ {u} = {widehat {E}}} ^ {u,mathit {RI}}} + {widehat {E}}} ^ {u,mathit {STAB}}
The contribution corresponding to the reduced integration in the thickness is noted \({\widehat{E}}^{u,\mathit{RI}}\) and that corresponding to the stabilization is \({\widehat{E}}^{u,\mathit{STAB}}\). By decomposing the reduced integration using a diagram of points aligned according to the thickness on the parametric components, we have:
(2.25)\[ {\ widehat {E}}} ^ {u,\ mathit {RI}}} = {\ widehat {E}} ^ {u,0} +\ zeta {\ widehat {E}} ^ {u,\ zeta} + {\ zeta}} + {\ zeta}} ^ {2} {\ widehat {E}}} ^ {u,\ zeta\ zeta} + {\ zeta} + {\ widehat {E}} ^ {u,\ zeta}} ^ {u,\ zeta}} + {\ zeta}\]
And stabilization, which corresponds to deformation modes in other directions:
(2.26)\[ {\ widehat {E}} ^ {u,\ mathit {STAB}}} =\ xi {\ widehat {E}} ^ {u,\ xi} +\ eta {\ widehat {E}}} ^ {u,\ eta}} ^ {u,\ eta}} ^ {u,\ widehat {E}} +\ xi\ zeta {\ widehat {E}} +\ xi\ zeta {\ widehat {E}} +\ xi\ zeta {\ widehat {E}}} ^ {u,\ xi\ zeta} +\ eta\ zeta {\ widehat {E}} ^ {u,\ eta\ zeta}\]
The Green-Lagrange strain tensor in the parametric coordinate system will be written as:
(2.27)\[ {\ widehat {E}} ^ {u} =\ left\ {{\ widehat {E}}} _ {\ mathrm {\ xi}\ mathrm {\ xi}} ^ {u}} ^ {u}, {\ widehat {E}}} =\ left\ left\ {\ widehat {E}}}} = {\ widehat {E}}}} _ {\ mathrm {\ zeta}\ mathrm {\ zeta}}} ^ {zeta}}} ^ {\ zeta}}} ^ {u}} ^ {u}} ^ {u} ,2 {\ widehat {E}}, {\ widehat {E}}} _ {\ widehat {E}}, {\ widehat {E}}} _ {\ widehat {E}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}}} _ {\ widehat {E}} hat {E}} _ {\ mathrm {\ xi}\ mathrm {\ zeta}} ^ {u}\ right\}\]
Be careful not to confuse the components of the deformation tensor () with those of its decomposition (-).