3. Hooke matrix and flexibility matrix#
3.1. Notations#
Instead of using the indices 1, 2 and 3 to locate the axes of the Cartesian coordinate system, we will use the corresponding indices \(L\), \(T\) and \(N\):
\(L\) for longitudinal
\(T\) for transversal
\(N\) for normal
The coefficients that are involved are as follows:
Keyword |
Grading |
meaning |
E_L |
\({E}_{L}\) |
Longitudinal Young’s modulus |
E_T |
\({E}_{T}\) |
Transversal Young’s modulus |
E_N |
\({E}_{N}\) |
Normal Young’s modulus |
G_LT |
\({G}_{\text{LT}}\) |
Shear modulus in plane \((L,T)\) |
G_TN |
\({G}_{\text{TN}}\) |
Shear modulus in plane \((T,N)\) |
G_LN |
\({G}_{\text{LN}}\) |
Shear modulus in plane \((L,N)\) |
NU_LT |
\({\nu }_{\text{LT}}\) |
Poisson’s ratio in plane \((L,T)\) |
NU_TN |
\({\nu }_{\text{TN}}\) |
Poisson’s ratio in plane \((T,N)\) |
NU_LN |
\({\nu }_{\text{LN}}\) |
Poisson’s ratio in plane \((L,N)\) |
ALPHA_L
|
\({\alpha }_{L}\) |
Coefficientde dilatation thermique moyen longitudinal
|
ALPHA_T |
\({\alpha }_{T}\) |
Coefficientde dilatation thermique moyen transversal
|
ALPHA_N |
\({\alpha }_{N}\) |
Coefficientde dilatation thermique moyen normal
|
Very important note:
\({\nu }_{\text{LT}}\) is different than \({\nu }_{\mathit{TL}}\) :
If traction is applied along the axis \(L\) :
\({\epsilon }_{\text{LL}}=\frac{{\sigma }_{\text{LL}}}{{E}_{L}}\) (Hooke’s law in one direction) .
This traction is accompanied, proportionally, by a contraction \(-{\nu }_{\text{LT}}\text{.}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}\) along the axis \(T\), and by a contraction \(-{\nu }_{\text{LN}}\text{.}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}\) along the axis \(N\).
The first index indicates the axis where the effect of loading occurs and the second index indicates the direction of loading.
Then we exert a pull along the \(T\) axis, then a pull along \(N\), we obtain:
\(\begin{array}{c}{\epsilon }_{\text{LL}}=\frac{{\sigma }_{\text{LL}}}{{E}_{L}}-{\nu }_{\text{TL}}\frac{{\sigma }_{\text{TT}}}{{E}_{T}}-{\nu }_{\text{NL}}\frac{{\sigma }_{\text{NN}}}{{E}_{N}}\\ {\epsilon }_{\text{TT}}=-{\nu }_{\text{LT}}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}+\frac{{\sigma }_{\text{TT}}}{{E}_{T}}-{\nu }_{\text{NT}}\frac{{\sigma }_{\text{NN}}}{{E}_{N}}\\ {\epsilon }_{\text{NN}}=-{\nu }_{\text{LN}}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}-{\nu }_{\text{TN}}\frac{{\sigma }_{\text{TT}}}{{E}_{T}}+\frac{{\sigma }_{\text{NN}}}{{E}_{N}}\end{array}\}\) [éq3.1-1]
The flexibility matrix \({\left[\mathrm{H}\right]}^{-1}\) being symmetric; from this we deduce:
\(\frac{{\nu }_{\text{LT}}}{{E}_{L}}=\frac{{\nu }_{\text{TL}}}{{E}_{T}}\); \(\frac{{\nu }_{\text{LN}}}{{E}_{L}}=\frac{{\nu }_{\text{NL}}}{{E}_{N}}\); \(\frac{{\nu }_{\text{TN}}}{{E}_{T}}=\frac{{\nu }_{\text{NT}}}{{E}_{N}}\)
3.2. 3D case#
3.2.1. Orthotropy#
3.2.1.1. Flexibility matrix#
\(\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{L}}& \frac{-{\nu }_{\text{TL}}}{{E}_{T}}& \frac{-{\nu }_{\text{NL}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{LT}}}{{E}_{L}}& \frac{1}{{E}_{T}}& \frac{-{\nu }_{\text{NT}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{LN}}}{{E}_{L}}& \frac{-{\nu }_{\text{TN}}}{{E}_{T}}& \frac{1}{{E}_{N}}& 0& 0& 0\\ & & & \frac{1}{{G}_{\text{LT}}}& 0& 0\\ & \text{SYM}& & & \frac{1}{{G}_{\text{LN}}}& 0\\ & & & & & \frac{1}{{G}_{\text{TN}}}\phantom{\rule{2em}{0ex}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]\)
\({\left[\mathrm{H}\right]}^{-1}\) — Orthotropy
3.2.1.2. Hooke matrix#
\(\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{LL}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{\left(1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}\right)}{\mathrm{\Delta }\cdot {E}_{T}{E}_{N}}& \frac{\left({\nu }_{\text{TL}}+{\nu }_{\text{NL}}{\nu }_{\text{TN}}\right)}{\mathrm{\Delta }\cdot {E}_{T}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NL}}+{\nu }_{\text{TL}}{\nu }_{\text{NT}}\right)}{\mathrm{\Delta }\cdot {E}_{T}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{\left({\nu }_{\text{LT}}+{\nu }_{\text{LN}}{\nu }_{\text{NT}}\right)}{\mathrm{\Delta }\cdot {E}_{L}{E}_{N}}& \frac{\left(1-{\nu }_{\text{NL}}{\nu }_{\text{LN}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NT}}+{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LT}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{\left({\nu }_{\text{LN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TN}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{T}}& \frac{\left({\nu }_{\text{TN}}+{\nu }_{\text{TL}}\text{.}{\nu }_{\text{LN}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{T}}& \frac{\left(1-{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TL}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{T}}& 0& 0& 0\\ & & & \begin{array}{c}{G}_{\text{LT}}\end{array}& 0& 0\\ & \text{SYM}& & & {G}_{\text{LN}}& 0\\ & & & & & {G}_{\text{TN}}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]\)
\(\left[\mathrm{H}\right]\) — Orthotropy
with: \(\frac{{\nu }_{\text{TL}}}{{E}_{T}}=\frac{{\nu }_{\text{LT}}}{{E}_{L}}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}\frac{{\nu }_{\text{NL}}}{{E}_{N}}=\frac{{\nu }_{\text{LN}}}{{E}_{L}}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}\frac{{\nu }_{\text{NT}}}{{E}_{N}}=\frac{{\nu }_{\text{TN}}}{{E}_{T}}\)
with: \(\frac{1}{\mathrm{\Delta }}=\frac{{E}_{L}{E}_{T}{E}_{N}}{1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}-{\nu }_{\text{NL}}{\nu }_{\text{LN}}-{\nu }_{\text{LT}}{\nu }_{\text{TL}}-2{\nu }_{\text{TN}}{\nu }_{\text{NL}}{\nu }_{\text{LT}}}\)
3.2.2. Transverse isotropy#
Transverse isotropy is defined here in plane \((L,T)\), and the direction of orthotropy is therefore \(N\). The reader’s attention may be drawn to the fact that this convention differs from a usual convention which designates by « longitudinal direction » the direction of orthotropy of isotropic transverse materials.
Note that the expansion coefficients verify: \({\alpha }_{T}={\alpha }_{L}\).
3.2.2.1. Flexibility matrix#
Matrix \({\left[\mathrm{H}\right]}^{-1}\) can be deduced directly from matrix \({\left[\mathrm{H}\right]}^{-1}\) - Orthotropy using the properties of transverse isotropy.
In plan \((L,T)\), see [éq2.2-1]:
\(\begin{array}{c}{E}_{L}={E}_{T}\\ {\nu }_{\text{TL}}={\nu }_{\text{LT}}\\ {G}_{\text{LT}}=\frac{{E}_{L}}{2\left(1+{\nu }_{\text{LT}}\right)}\end{array}\)
In plans \((L,N)\) and \((T,N)\):
\(\begin{array}{c}{\nu }_{\text{NT}}={\nu }_{\text{NL}}\\ {\nu }_{\text{LN}}={\nu }_{\text{TN}}\\ {G}_{\text{TN}}={G}_{\text{LN}}\\ \frac{{\nu }_{\text{NT}}}{{E}_{N}}=\frac{{\nu }_{\text{LN}}}{{E}_{L}}\end{array}\)
\(\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{L}}& \frac{-{\nu }_{\text{LT}}}{{E}_{L}}& \frac{-{\nu }_{\text{NL}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{TL}}}{{E}_{L}}& \frac{1}{{E}_{L}}& \frac{-{\nu }_{\text{NT}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{LN}}}{{E}_{L}}& \frac{-{\nu }_{\text{TN}}}{{E}_{L}}& \frac{1}{{E}_{N}}& 0& 0& 0\\ & & & \frac{2\left(1+{\nu }_{\text{LT}}\right)}{{E}_{L}}& 0& 0\\ & \text{SYM}& & & \frac{1}{{G}_{\text{LN}}}& 0\\ & & & & & \frac{1}{{G}_{\text{TN}}}\phantom{\rule{2em}{0ex}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]\)
\({\left[\mathrm{H}\right]}^{-1}\) - Transverse isotropy
3.2.2.2. Hooke matrix#
The \(\left[\mathrm{H}\right]\) matrix has the same symmetries as \({\left[\mathrm{H}\right]}^{-1}\).
\(\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1-{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{{\nu }_{\text{LT}}+{\nu }_{\text{NL}}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{{\nu }_{\text{NL}}+{\nu }_{\text{LT}}{\nu }_{\text{NL}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{{\nu }_{\text{TL}}+{\nu }_{\text{NL}}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{1-{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{{\nu }_{\text{LN}}+{\nu }_{\text{LT}}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{{\nu }_{\text{LN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}^{2}}& \frac{{\nu }_{\text{TN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}^{2}}& \frac{1-{\nu }_{\text{LT}}^{2}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}^{2}}& 0& 0& 0\\ & & & \frac{{E}_{L}}{2\left(1+{\nu }_{\text{LT}}\right)}& & \\ & & & & {G}_{\text{LN}}& \\ & & & & & {G}_{\text{LN}\phantom{\rule{2em}{0ex}}}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]\)
\(\left[\mathrm{H}\right]\) — Transverse isotropy
with: \(\frac{1}{\mathrm{\Delta }\text{'}}=\frac{{E}_{L}^{2}\text{.}{E}_{N}}{1-2{\nu }_{\text{NL}}\cdot {\nu }_{\text{LN}}-{\nu }_{\text{LT}}^{2}-2{\nu }_{\text{NL}}\cdot {\nu }_{\text{LN}}\cdot {\nu }_{\text{LT}}}\)
3.2.3. Cubic symmetric elasticity#
Cubic-symmetric elasticity occurs when, in addition to the three Cartesian planes of symmetry, the six planes rotated at 45° are also symmetric. We then have 3 independent elastic coefficients. This corresponds to an elasticity matrix of the form:
\(\left[\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}{H}_{1111}& {H}_{1122}& {H}_{1122}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ {H}_{1122}& {H}_{1111}& {H}_{1122}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ {H}_{1122}& {H}_{1122}& {H}_{1111}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& {H}_{1212}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& {H}_{1212}& \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& {H}_{1212}\end{array}\right]\)
Given the cubic symmetry, it remains to determine 3 coefficients:
\({E}_{L}={E}_{N}={E}_{T}=E,\phantom{\rule{6em}{0ex}}{G}_{\text{LT}}={G}_{\text{LN}}={G}_{\text{TN}}=G,\phantom{\rule{6em}{0ex}}{\nu }_{\text{LN}}={\nu }_{\text{LT}}={\nu }_{\text{LN}}=\nu\)
Note that the expansion coefficients verify: \({\alpha }_{T}={\alpha }_{L}={\alpha }_{N}\).
To reproduce cubic-symmetric elasticity with ELAS_ORTH, simply calculate the orthotropy coefficients such that the elasticity matrix obtained is of the form above:
\(\begin{array}{c}{H}_{1111}=\frac{E(1-\nu )}{(1+\nu )(1-2\nu )}\\ {H}_{1122}=\frac{\nu E}{(1+\nu )(1-2\nu )}\\ {H}_{1212}={G}_{\text{LT}}={G}_{\text{LN}}={G}_{\text{TN}}\end{array}\)
so as long as \((1+\nu )(1-2\nu )\ne 0\) (i.e. \(\nu\) different from \(0.5\)).
\(\frac{{H}_{1122}}{{H}_{1111}}=\frac{\nu }{1-\nu }\) which provides \(\nu =\frac{1}{1+\frac{{H}_{1111}}{{H}_{1122}}}\) then \(E={H}_{1111}\frac{(1+\nu )(1-2\nu )}{(1-\nu )}\)
3.2.4. Isotropy#
Hooke’s law takes the following form with Lamé coefficients \(\lambda\) and \(\mu =G\):
\({\sigma }_{\text{ij}}=\lambda {\epsilon }_{\text{kk}}{\delta }_{\text{ij}}+2\mu {\epsilon }_{\text{ij}}\)
3.2.4.1. Flexibility matrix based on \(E\) and \(\nu\)#
\(\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{E}& \frac{-\nu }{E}& \frac{-\nu }{E}& 0& 0& 0\\ & \frac{1}{E}& \frac{-\nu }{E}& 0& 0& 0\\ & & \frac{1}{E}& 0& 0& 0\\ & & & \frac{1}{G}=\frac{2(1+\nu )}{E}& 0& 0\\ & \text{SYM}& & & \frac{1}{G}=\frac{2\left(1+\nu \right)}{E}& 0\\ & & & & & \frac{1}{G}=\frac{2\left(1+\nu \right)}{E}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]\)
\({\left[\mathrm{H}\right]}^{-1}\) — Complete isotropy
3.2.4.2. Hooke matrix as a function of \(E\) and \(\nu\)#
\(\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ \\ {\sigma }_{\text{LN}}\\ \\ {\sigma }_{\text{TN}}\end{array}\right]=\frac{E}{\left(1+\nu \right)\left(1-2\nu \right)}\left[\begin{array}{cccccc}1-\nu & \nu & \nu & 0& 0& 0\\ & 1-\nu & \nu & 0& 0& 0\\ & & 1-\nu & 0& 0& 0\\ & \text{SYM}& & \frac{1-2\nu }{2}& 0& 0\\ & & & & \frac{1-2\nu }{2}& 0\\ & & & & & \frac{1-2\nu }{2}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ \\ 2{\epsilon }_{\text{LN}}\\ \\ 2{\epsilon }_{\text{TN}}\end{array}\right]\)
\(\left[\mathrm{H}\right]\) — Complete isotropy
3.2.4.3. Flexibility matrix according to Lamé coefficients \(\lambda\) and \(\mu\)#
\(\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{\lambda +\mu }{\mu (3\lambda +2\mu )}& \frac{-\lambda }{2\mu (3\lambda +2\mu )}& \frac{-\lambda }{2\mu (3\lambda +2\mu )}& 0& 0& 0\\ & \frac{\lambda +\mu }{\mu (3\lambda +2\mu )}& \frac{-\lambda }{2\mu (3\lambda +2\mu )}& 0& 0& 0\\ & & \frac{\lambda +\mu }{\mu (3\lambda +2\mu )}& 0& 0& 0\\ & & & \frac{1}{\mu }& 0& 0\\ & \text{SYM}& & & \frac{1}{\mu }& 0\\ & & & & & \frac{1}{\mu }\phantom{\rule{2em}{0ex}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]\)
\({\left[\mathrm{H}\right]}^{-1}\) — Complete isotropy
3.2.4.4. Hooke matrix as a function of Lamé coefficients \(\lambda\) and \(\mu\)#
\(\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\lambda +2\mu & \lambda & \lambda & 0& 0& 0\\ & \lambda +2\mu & \lambda & 0& 0& 0\\ & & \lambda +2\mu & 0& 0& 0\\ & \text{SYM}& & \mu & 0& 0\\ & & & & \mu & 0\\ & & & & & \mu \end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]\)
\(\left[\mathrm{H}\right]\) — Complete isotropy with Lamé coefficients
3.3. 2D orthotropic case in plane and axisymmetric deformations#
3.3.1. Flexibility matrix#
\(\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ \\ 0\\ {\mathrm{2\epsilon }}_{\text{LT}}\end{array}\right]=\left[\begin{array}{cccc}\frac{1}{{E}_{L}}& -\frac{{\nu }_{\text{TL}}}{{E}_{T}}& -\frac{{\nu }_{\text{NL}}}{{E}_{N}}& 0\\ -\frac{{\nu }_{\text{LT}}}{{E}_{L}}& \frac{1}{{E}_{T}}& -\frac{{\nu }_{\text{NL}}}{{E}_{N}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{\text{LT}}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ \\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\end{array}\right]\)
\({\left[\mathrm{H}\right]}^{-1}\) — Plane orthotropy in plane deformations and axisymmetry
3.3.2. Hooke matrix#
\(\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\end{array}\right]=\frac{1}{\mathrm{\Delta }}\left[\begin{array}{cccc}\frac{\left(1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}\right)}{{E}_{T}{E}_{N}}& \frac{\left({\nu }_{\text{TL}}+{\nu }_{\text{NL}}{\nu }_{\text{TN}}\right)}{{E}_{T}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NL}}+{\nu }_{\text{TL}}{\nu }_{\text{NT}}\right)}{{E}_{T}\text{.}{E}_{N}}& 0\\ \frac{\left({\nu }_{\text{LT}}+{\nu }_{\text{LN}}{\nu }_{\text{NT}}\right)}{{E}_{L}{E}_{N}}& \frac{\left(1-{\nu }_{\text{NL}}{\nu }_{\text{LN}}\right)}{{E}_{L}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NT}}+{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LT}}\right)}{{E}_{L}\text{.}{E}_{N}}& 0\\ \frac{\left({\nu }_{\text{LN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TN}}\right)}{{E}_{L}\text{.}{E}_{T}}& \frac{\left({\nu }_{\text{TN}}+{\nu }_{\text{TL}}\text{.}{\nu }_{\text{LN}}\right)}{{E}_{L}\text{.}{E}_{T}}& \frac{\left(1-{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TL}}\right)}{{E}_{L}\text{.}{E}_{T}}& 0\\ 0& 0& 0& {G}_{\text{LT}}\cdot \mathrm{\Delta }\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ 0\\ 2{\epsilon }_{\text{LT}}\end{array}\right]\)
\(\left[\mathrm{H}\right]\) — Plane orthotropy in plane deformations and axisymmetry
with: \(\frac{1}{\mathrm{\Delta }}=\frac{{E}_{L}{E}_{T}{E}_{N}}{1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}-{\nu }_{\text{NL}}{\nu }_{\text{LN}}-{\nu }_{\text{LT}}{\nu }_{\text{TL}}-2{\nu }_{\text{TN}}{\nu }_{\text{NL}}{\nu }_{\text{LT}}}\)
3.4. 2D orthotropic case in plane constraints#
3.4.1. Flexibility matrix#
\(\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ \\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\end{array}\right]=\left[\begin{array}{cccc}\frac{1}{{E}_{L}}& -\frac{{\nu }_{\text{TL}}}{{E}_{T}}& 0& 0\\ -\frac{{\nu }_{\text{LT}}}{{E}_{L}}& \frac{1}{{E}_{T}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{\text{LT}}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ \\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\end{array}\right]\)
\({\left[\mathrm{H}\right]}^{-1}\) — Plane orthotropy in plane stresses
3.4.2. Hooke matrix#
Using the system of equations [éq3.1-1], we get:
\(\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ 0\\ {\sigma }_{\text{LT}}\end{array}\right]=\frac{1}{1-{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TL}}}\left[\begin{array}{cccc}{E}_{L}& {\nu }_{\text{TL}}{E}_{T}& 0& 0\\ {\nu }_{\text{LT}}{E}_{L}& {E}_{T}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {G}_{\text{LT}}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\end{array}\right]\)
\(\left[\mathrm{H}\right]\) — Orthotropy in plane constraints