6. Appendix: tensor ratings convention
The vectors of the deformations and the stresses in the main base \(({d}_{1},{d}_{2},{d}_{3})\) are noted:
(6.1)\[\begin{split} \ mathrm {\ epsilon} =\ left\ {\ begin {array} {c} {\ mathrm {\ epsilon}}} _ {1}\\ {\ mathrm {\ epsilon}}} _ {2}\\ {\ begin {array}}} _ {\ mathrm {\ epsilon}} _ {3}\ end {array}\ epsilon}} _ {3}\ end {array}\ epsilon}} _ {3}\ end {array}\ epsilon}} _ {3}\ end {array}\ epsilon}} _ {3}\ end {array}\ epsilon}} _ {2}\\ {epsilon}}\end{split}\]
The elasticity tensor \(C\) used to connect \(\mathrm{\epsilon }\) and \(\mathrm{\sigma }\) in the main base, such as \(\mathrm{\sigma }=C\mathrm{.}\mathrm{\epsilon }\), is written as:
(6.2)\[\begin{split} C\ mathrm {=}\ left [\ begin {array} {ccc} K+\ frac {4} {3} G& K\ mathrm {-}\ frac {2} {3} {3} G& K\ mathrm {- -}\ frac {-}\ frac {2} {3} G& K+\ frac {-}\ frac {2} {3} G& K+\ frac {3} ac {4} {3} G& K\ mathrm {-}\ frac {2} {3} G\\ K\ mathrm {-}\ frac {2} {3} G& K\ mathrm {- -}\ frac {2} {3}\ frac {2} {3} G& K+\ frac {4} {3} G\ end {array} {3} G& K\ mathrm {-} -}\ frac {-}\ frac {2} {3} G& K\ mathrm {-} -}\ frac {2} {-}\ frac {2} {3} G& K\ mathrm {-} -}\ frac {2} {-}\end{split}\]
With \(K\) the elastic compressibility module and \(G\) the elastic shear modulus. Deformations and stresses are symmetric tensors of order two. This symmetry (six independent components) is generally exploited by representing them by six-dimensional vectors resulting from the projection of these tensors into appropriate bases.
The deformations and constraints given at the input and produced at the output of the resolution of the law of behavior are expressed in the orthonormal base of symmetric tensors of order two, noted \(\stackrel{ˉ}{b}\):
(6.3)\[\begin{split} \ overline {b} =\ {\ begin {array} {c} {e} {c} {e} _ {x}\ otimes {e} _ {y}\ otimes {e} _ {y}\\ {e}\\ {e} _ {e} _ {z} _ {z}\ otimes {e} _ {y} _ {y} _ {y} + {e} _ {y}\ otimes {e} _ {x}} {\ sqrt {2}}\\ frac {{e} _ {x}\ otimes {e} _ {z} + {e} _ {e} _ {z} _ {z}}\\ otimes {e}} {z}\ otimes {e} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z} _ {z}} _ {z} + {e} _ {z}\ otimes {e} _ {y}} {\ sqrt {2}}\ end {array}\end{split}\]
Where \(\left({e}_{x},{e}_{y},{e}_{z}\right)\) represent the unit vectors of the global orthonormal Cartesian base, which is assumed to be fixed. The condensed expression for the strain and stress tensors projected in the \(\stackrel{ˉ}{b}\) database is written as:
(6.4)\[\begin{split} \ stackrel {] {\ epsilon} =\ left\ {\ begin {array} {\ begin {array} {c} {\ epsilon} _ {\ mathit {xx}}\\ {\ epsilon} _ {\ mathit {yy}}}\\ mathit {yy}}}\\ mathit {yy}}}\\ mathit {y}} _ {\ mathit {yy}}}\\ mathit {yy}}}\\ mathit {yy}}\\ mathit {yy}}}\\ mathit {yy}}}\\ mathit {xy}}\\\ sqrt {2} {\ epsilon} _ {\ mathit {yz}}\\\ sqrt {2} {\ epsilon} _ {\ mathit {xz}}}\ end {array}\ right\}\end{split}\]
This spelling makes a term appear in \(\sqrt{2}\) in front of the crossed components. It allows you to:
In fact, by noting \({\mathrm{\sigma }}_{\mathit{ij}}={C}_{\mathit{ijkl}}{\mathrm{\epsilon }}_{\mathit{kl}}\), its form projected into the \(\stackrel{̄}{b}\) base becomes \({\overline{\mathrm{\sigma }}}_{i}={\overline{C}}_{\mathit{ij}}{\overline{\mathrm{\epsilon }}}_{j}\), where we have the following expression for \(\stackrel{̄}{C}\):
(6.5)\[\begin{split} \ stackrel {ń} {C} =\ left [\ begin {array} {\ begin {array} {cccccc} {C} _ {\ mathit {xxyy}}} & {\ mathit {xxyy}}} & {C}} & {C} _ {\ mathit {xxyy}}} & {C} _ {\ mathit {xxyy}}} &\ sqrt {2} {C}} &\ sqrt {2} {C}} _ {\ mathit {xxxz}} &\ sqrt {2} {2} {C}} _ {\ mathit {xxyz}}\\ {C} _ {\ mathit {xxyy}} & {C} _ {\ mathit {yyyy}}} & {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C} _ {\ mathit {yyyy}}} & {C}\ sqrt {2} {C} _ {\ mathit {zzxx}}} &\ sqrt {2} {C} _ {\ mathit {yyyz}}\\ {C} _ {\ mathit {zzxx}}} & {\ mathit {zzxx}}} & {C} _ {\ mathit {zzzz}} _ {\ mathit {zzxx}}} _ {\ mathit {zzxx}}} &\ sqrt {2} {zzxx}}} & {\ mathit {zzxx}}} & {\ mathit {zzxx}}} &\ sqrt {2} {zzxx}}} & {C} _ {\ mathit {zzxy}} &\ sqrt {2} {C} {C} _ {\ mathit {zzxz}}} &\ sqrt {2} {C} _ {\ mathit {zzyz}}}\\ sqrt {2} {2} {C}}}\\ sqrt {2} {C} _ {\ mathit {xyyy}}}\ sqrt {xyyy}}}\ sqrt {2} {C}} _ {\ mathit {xyyy}}} &\ sqrt {xyyy}} &\ sqrt {xyyy}} rt {2} {C} _ {\ mathit {xyzz}} & 2 {C} _ {\ mathit {xyxy}}} & 2 {C} _ {\ mathit {xyxz}} & 2 {C} _ {\ mathit {xyyz}}}\\\ sqrt {xyz}}}\\\ sqrt {xyz}}}\\ mathit {xyz}}}\\ mathit {xzyy}} &\ sqrt {2} {C}} _ {\ mathit {xyyz}}}\\ mathit {xzyy}}}\\ mathit {xzyy}}}} &\ sqrt {2} {C} _ {\ mathit {xzzz}} & 2 {C} _ {\ mathit {xzxy}} & 2 {C} _ {\ mathit {xzxz}}} & 2 {xzxz}} & 2 {C}}} & 2 {C}} _ {\ mathit {xzxz}}} & 2 {\ mathit {xzxz}}} & 2 {\ mathit {xzxz}}} & 2 {\ mathit {xzxz}}} & 2 {\ mathit {xzxz}}} & 2 {\ mathit {xzxz}}} & 2 {\ mathit {xzxz}}}\ sqrt {2} {C} _ {\ mathit {yzyy}}} &\ sqrt {2} {C} _ {\ mathit {yzzz}} & 2 {C} _ {\ mathit {yzxy}}} & 2 {\ mathit {yzxy}}} & 2 {\ mathit {yzyz}}} & 2 {C} _ {\ mathit {yzyz}}}\ end {yzyz}} & 2 {C} _ {\ mathit {yzyz}}}\ end {array}}\ right]\end{split}\]
The condensed form () is not convenient to use because of the need to manipulate terms in \(\sqrt{2}\) during matrix operations. We prefer another script, based on the projection into the so-called Voigt database, noted \(\stackrel{̃}{b}\) and having the following expression:
\[\]
: label: eq-61
widehat {b} ={begin {array} {c} {e} {e} _ {x}otimes {e} _ {y}otimes {e} _ {y} _ {y}\ {e}\ {e} _ {e} _ {z}otimes {e} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y} _ {y}}{e}{y}{e}{y}{e}{y}{e}{y}{e}{y}{e} _ {x}otimes {e} _ {z}\ {e} _ {y}otimes {e} _ {z}end {array}end {array}
The condensed expression for the strain and stress tensors projected in the Voigt \(\widehat{b}\) database is written as:
\[\]
: label: eq-62
widehat {mathrm {epsilon}} =left{epsilon}} =left{begin {array} {c} {mathrm {epsilon}} _ {mathit {xx}}}\ {mathit {xx}}}\ {mathit {xx}}}\ mathit {xx}}}\ {mathit {xx}}}\ {mathit {xx}}}\ {mathit {xx}}}\ {mathit {xx}}}\ {mathit {xx}}}\ {mathit {xx}}}\ {mathrm {epsilon}}}\ {mathit {zz}}}\ {mathrm {epsilon}}} _ {mathit {xy}}\ {mathrm {epsilon}} _ {mathit {yz}}\ {mathrm {epsilon}} _ {mathit {xz}}end {array}yz}}end {array}right}
This writing makes it possible to avoid the terms in \(\sqrt{2}\) in front of the crossed components, and is more convenient to use during the numerical resolution of the law of behavior.