Analytical prediction of damage in the composite part of a type-3 hydrogen storage vessel

The damage behavior of a type-3 hydrogen storage vessel is modeled. The vessel consists of a metal envelop, called liner, coated with a filament winding. The model proposed allows simulating the mechanical response of the structure to a quasi-static loading. The model is based on a meso-macro approach and takes into account the damage behavior of the composite and the elastoplastic deformation of the liner. The results obtained are compared with experimental data. Finally, the effect of stacking sequence of filament layers on the damage level in the composite is investigated.

Appendices

Appendix A

The function of loading is taken in the form

$$ {f^d} = - Y - {Y_C} - {R^d} \leqslant 0. $$

Then we can write:

$$ \begin{array}{*{20}{c}} {{{\dot{D}}_{\text{I}}} = 0\,\,\,{\text{if}}\,\,\,{f^d} < 0\,\,\,{\text{or}}\,\,\,{f^d} = 0,\frac{{\partial {f^d}}}{{\partial Y}}\dot{Y} < 0,} \\ {{{\dot{D}}_{\text{I}}} \ne 0\,\,\,{\text{if}}\,\,\,{f^d} = 0\,\,\,{\text{and}}\,\,\,\frac{{\partial {f^d}}}{{\partial Y}}\dot{Y} > 0.} \\ \end{array} $$

When loading causes damage,

$$ {f^d} = 0\,\,\,{\text{and}}\,\,\,\frac{{\partial {f^d}}}{{\partial Y}}\dot{Y} > 0. $$

The kinetics of damage is determined by the laws of evolution

$$ {\dot{D}_{\text{I}}} = - {\dot{\lambda }^d}\frac{{\partial {f^d}}}{{\partial Y}} = {\dot{\lambda }^d},\,\,\,\,{\bar{D}_{\text{I}}} = - {\dot{\lambda }^d}\frac{{\partial {f^d}}}{{\partial Y}} = {\dot{\lambda }^d}. $$

where ${\dot{\lambda }^d}$ is the Lagrange multiplier. The expressions can be obtained by using the consistency equation

$$ {f^d} = 0. $$

We can write

$$ \begin{array}{*{20}{c}} {{R_d} = {Y_c} + Y,} \\ {Y = {Y_{\text{I}}} + {Y_{\text{II}}} \times {X_{\text{II}}} + {Y_{\text{III}}} \times {X_{\text{III}}},} \\ \end{array} $$

$$ \left\{ {\begin{array}{*{20}{c}} {{Y_{\text{I}}} = {{\left[ {\frac{{\partial \psi }}{{\partial {D_{\text{I}}}}}} \right]}_{{\sigma_4},{\sigma_6}}} = \frac{{{s_{22}}}}{{2{{\left( {1 - {D_{\text{I}}}} \right)}^2}}}\sigma_{22}^2,} \hfill \\ {{Y_{\text{II}}} = {{\left[ { - \frac{{\partial \psi }}{{\partial {D_{\text{II}}}}}} \right]}_{{\sigma_2},{\sigma_4}}} = \frac{{{s_{66}}}}{{2{{\left( {1 - {D_{\text{II}}}} \right)}^2}}}\sigma_{66}^2,} \hfill \\ {{Y_{\text{III}}} = {{\left[ { - \frac{{\partial \psi }}{{\partial {D_{\text{III}}}}}} \right]}_{{\sigma_2},{\sigma_6}}} = \frac{{{s_{44}}}}{{2{{\left( {1 - {D_{\text{III}}}} \right)}^2}}}\sigma_{44}^2,} \hfill \\ {{X_{\text{II}}} = \left[ { - \frac{{\partial {D_{\text{II}}}}}{{\partial {D_{\text{I}}}}}} \right]\frac{{{s_{66}}\sqrt {{{s_{11}}{s_{22}}}} }}{{{{\left( {{s_{66}} + \frac{{{D_{\text{I}}}\sqrt {{{s_{11}}{s_{22}}}} }}{{\sqrt {{1 - {D_{\text{I}}}}} }}} \right)}^2}}}\frac{\partial }{{\partial {D_{\text{I}}}}}\left( {\frac{{{D_{\text{I}}}}}{{\sqrt {{1 - {D_{\text{I}}}}} }}} \right) = \frac{{{s_{66}}\sqrt {{{s_{11}}{s_{22}}}} }}{{{{\left( {{s_{66}} + \frac{{{D_{\text{I}}}\sqrt {{{s_{11}}{s_{22}}}} }}{{\sqrt {{1 - {D_{\text{I}}}}} }}} \right)}^2}}}\frac{{2 - {D_{\text{I}}}}}{{2{{\left( {\sqrt {{1 - {D_{\text{I}}}}} } \right)}^3}}},} \hfill \\ {{X_{\text{III}}} = \frac{{\partial {D_{\text{III}}}}}{{\partial {D_{\text{I}}}}} = \frac{{{s_{66}}{s_{22}}}}{{{{\left( {{s_{44}} + \frac{{{D_{\text{I}}}{s_{22}}}}{{\sqrt {{1 - {D_{\text{I}}}}} }}} \right)}^2}}} - \frac{{2 - {D_{\text{I}}}}}{{2{{\left( {\sqrt {{1 - {D_{\text{I}}}}} } \right)}^3}}}.} \hfill \\ \end{array} } \right. $$

Appendix B

The continuity condition for the radial displacements is

$$ \forall k \in \left[ {1,w - 1} \right],\,\,\,{U^{(k)}}\left( {r_{\text{ext}}^{(k)}} \right) = {U^{\left( {k + 1} \right)}}\left( {r_{\text{ext}}^{(k)}} \right). $$

The continuity condition for the radial stresses is

$$ \left\{ {\begin{array}{*{20}{c}} {\forall k \in \left[ {1,w - 1} \right],} \hfill & {\sigma_r^{(k)}\left( {r_{\text{ext}}^{(k)}} \right) = \sigma_{\text{ext}}^{\left( {k + 1} \right)}\left( {r_{\text{ext}}^{(k)}} \right),} \hfill \\ {\sigma_r^{(1)}\left( {{r_0}} \right) = - {p_0},} \hfill & {} \hfill \\ {\sigma_r^{(w)}\left( {{r_a}} \right) = 0.} \hfill & {} \hfill \\ \end{array} } \right. $$

The axial equilibrium condition for the solution with the closed-end effect can be expressed as

$$ 2\pi \sum\limits_{k = 1}^w {\int\limits_{{\eta_{k - 1}}}^{{r_k}} {\sigma_z^{(k)}(r)rdr = \pi r_0^2{p_0}.} } $$

The zero torsion condition is

$$ 2\pi \sum\limits_{k = 1}^w {\int\limits_{{r^{\left( {k - 1} \right)}}}^{{r^{(k)}}} {{\tau_{z\theta }}} (r){r^2}dr = 0.} $$

Finally, the problem can be reduced to a linear system of the form

$$ X = {A^{ - 1}} \times B, $$

$$ \left[ {\begin{array}{*{20}{c}} {{D^1}} \hfill \\ {{D^2}} \hfill \\ {{D^3}} \hfill \\ {{D^4}} \hfill \\ {{E^1}} \hfill \\ {{E^2}} \hfill \\ {{E^3}} \hfill \\ {{E^4}} \hfill \\ {{\varepsilon_0}} \hfill \\ {{\gamma_0}} \hfill \\ \end{array} } \right] = {\left[ {\begin{array}{*{20}{c}} {{d_{11}}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {{e_{11}}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {{\alpha_{11}}} \hfill & {{\alpha_{12}}} \hfill \\ {{d_{21}}} \hfill & {{d_{22}}} \hfill & 0 \hfill & 0 \hfill & {{e_{21}}} \hfill & {{e_{22}}} \hfill & 0 \hfill & 0 \hfill & {{\alpha_{21}}} \hfill & {{\alpha_{22}}} \hfill \\ 0 \hfill & {{d_{32}}} \hfill & {{d_{33}}} \hfill & 0 \hfill & 0 \hfill & {{e_{32}}} \hfill & {{e_{33}}} \hfill & 0 \hfill & {{\alpha_{31}}} \hfill & {{\alpha_{32}}} \hfill \\ 0 \hfill & 0 \hfill & {{d_{43}}} \hfill & {{d_{44}}} \hfill & 0 \hfill & 0 \hfill & {{e_{43}}} \hfill & {{e_{45}}} \hfill & \alpha \hfill & \alpha \hfill \\ {{d_{15}}} \hfill & {{d_{52}}} \hfill & 0 \hfill & 0 \hfill & e \hfill & e \hfill & 0 \hfill & 0 \hfill & \alpha \hfill & \alpha \hfill \\ 0 \hfill & {{d_{62}}} \hfill & {{d_{63}}} \hfill & 0 \hfill & 0 \hfill & e \hfill & e \hfill & 0 \hfill & \alpha \hfill & \alpha \hfill \\ 0 \hfill & 0 \hfill & {{d_{73}}} \hfill & {{d_{64}}} \hfill & 0 \hfill & 0 \hfill & e \hfill & e \hfill & \alpha \hfill & \alpha \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & d \hfill & d \hfill & d \hfill & e \hfill & e \hfill & \alpha \hfill & \alpha \hfill \\ {{d_{91}}} \hfill & {{d_{92}}} \hfill & {{d_{93}}} \hfill & {{d_{94}}} \hfill & {{e_{91}}} \hfill & {{e_{92}}} \hfill & {{e_{93}}} \hfill & {{e_{94}}} \hfill & {{\alpha_{91}}} \hfill & {{\alpha_{92}}} \hfill \\ {{d_{101}}} \hfill & {{d_{102}}} \hfill & {{d_{103}}} \hfill & {{d_{104}}} \hfill & {{e_{101}}} \hfill & {{e_{102}}} \hfill & {{e_{103}}} \hfill & {{e_{104}}} \hfill & {{\alpha_{101}}} \hfill & {{\alpha_{102}}} \hfill \\ \end{array} } \right]^{ - 1}} \times \left[ {\begin{array}{*{20}{c}} { - {P_0}} \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ 0 \hfill \\ {r_0^2{{{{P_0}}} \left/ {2} \right.}} \hfill \\ 0 \hfill \\ \end{array} } \right]. $$

The nonzero entries of the quadratic matrix are calculated the boundary conditions given below for k ∈ [1, w], where w = n _L  + n _C +1.

Internal pressure condition:

$$ {A_{KK}} = C_{23}^K + {\beta^K}C_{33}^K{\left( {{r^K}} \right)^{ - {\beta^K} - 1}},{A_{Kn - 1}} = C_{23}^K + {\beta^K}C_{33}^K{\left( {{r^K}} \right)^{ - {\beta^K} - 1}}. $$

If β^K = 1, else

$$ {A_{K2n + 1}} = C_{31}^k + \frac{{N_2^K}}{2}C_{32}^K\log {r^K} + C_{33}^K\log {r^{K + 1}},{A_{K2n + 2}} = \left[ {C_{36}^K + \alpha_2^K\left( {C_{32}^K + 2C_{33}^K} \right)} \right]{r^K}; $$

If β^K ≠ 1, else

$$ {A_{K,2n - 1}} = \left( {C_{23}^K + C_{33}^k} \right){\alpha^k} + C_{13}^k,{A_{K2n + 2}} = \left[ {C_{36}^K + \alpha_2^K\left( {C_{23}^K + 2C_{33}^K} \right)} \right]{r^K}. $$

Displacement continuity condition:

$$ {A_{K,K - 1}} = {\left( {{r^K}} \right)^{{\beta^{K - 1}}}},{A_{K,K}} = {\left( {{r^K}} \right)^{{\beta^K}}},{A_{K,K + n - 1}} = {\left( {{r^K}} \right)^{ - {\beta^{K - 1}}}},{A_{K,K + n}} = {\left( {{r^K}} \right)^{ - {\beta^{k1}}}}. $$

If β^K−1 = 1 and β^K = 1, then

$$ {A_{K,2n + 1}} = {R^K}\log {r^K}{{{\left( {N_2^{K - 1} - N_2^K} \right)}} \left/ {2} \right.},{A_{K,2n + 2}} = \left( {\alpha_2^{K - 1} - \alpha_2^K} \right){\left( {{r^K}} \right)^2}. $$

If β^K−1 = 1 and β^K ≠ 1, then

$$ {A_{K,2n + 1}} = {R^K}\log {r^K}{{{\left( {N_2^{K - 1}} \right)}} \left/ {{2 - \alpha_1^K{r^K}}} \right.},{A_{K,2n + 2}} = \left( {\alpha_2^{K - 1} - \alpha_2^K} \right){\left( {{r^K}} \right)^2}. $$

If β^K−1 ≠ 1 and β^K = 1, then

$$ {A_{K,2n + 1}} = \alpha_1^K{r^K} - {r^K}\lg {r^K}\frac{{{N^K}}}{2},{A_{K,2n + 2}} = \alpha_2^K{r^K} - \alpha_2^4{\left( {{r^K}} \right)^2}. $$

If β^K−1 ≠ 1 and β^K ≠ 1, then

$$ {A_{K,2n + 1}} = \left( {\alpha_2^{K - 1} - \alpha_1^K} \right){r^K},{A_{K,2n + 1}} = \left( {\alpha_2^{K - 1} - \alpha_1^K} \right){\left( {{r^K}} \right)^2}. $$

Strain continuity condition:

$$ \begin{array}{*{20}{c}} {{A_{n + K,K}} = \left( {C_{23}^K + {\beta^K}C_{33}^K} \right){{\left( {{r^{K + 1}}} \right)}^{{\beta^{K - 1}}}},{A_{n + K,K + 1}} = \left( {C_{23}^{K + 1} + {\beta^{K + 1}}C_{33}^{K + 1}} \right){{\left( {{r^{K + 1}}} \right)}^{{\beta^{K - 1}} - 1}},} \\ {{A_{n + K,n + K}} = \left( {C_{23}^K - {\beta^K}C_{33}^K} \right){{\left( {{r^K}} \right)}^{ - {\beta^K} - 1}},{A_{n + K,K + n + 1}} = - \left( {C_{23}^{K + 1} - {\beta^{K + 1}}C_{33}^{K + 1}} \right){{\left( {{r^{K + 1}}} \right)}^{ - {\beta^{K - 1}} - 1}}.} \\ \end{array} $$

If β^K = 1 and β^K+1 = 1, then

$$ \begin{array}{*{20}{c}} {{A_{n + K,2n + 1}} = \left[ {C_{31}^K + \frac{{N_2^K}}{2}\left( {C_{32}^K\log {r^{K + 1}} + C_{33}^{K + 1}\left( {\log {r^{K + 1}} + 1} \right)} \right)} \right] - \left[ {C_{31}^{K + 1} + \frac{{N_2^K}}{2}\left( {C_{32}^{K + 1}\log {r^{K + 1}} + C_{33}^{K + 1}\left( {\log {r^{K + 1}} + 1} \right)} \right)} \right],} \\ {{A_{n + K,K + n + 2}} = {r^{K + 1}}\left[ {\left( {C_{63}^K - C_{63}^{K + 1}} \right) + \alpha_2^K\left( {C_{32}^K - C_{32}^{K + 1}} \right) + 2\left( {C_{33}^K - C_{33}^{K + 1}} \right)} \right].} \\ \end{array} $$

If β^K = 1 and β^K+1 ≠ 1

$$ \begin{array}{*{20}{c}} {{A_{n + K,2n + 1}} = C_{31}^K + \frac{{N_2^K}}{2}\left[ {C_{23}^K\log {r^{K + 1}} + C_{33}^K\log \left( {{r^{K + 1}} + 1} \right)} \right] - \left[ {C_{13}^{K + 1} + \alpha_1^K\left( {C_{23}^{K + 1} + C_{33}^{K + 1}} \right)} \right],} \\ {{A_{n + K,K + n + 2}} = \left\{ {\left[ {C_{63}^K + \alpha_2^K\left( {C_{32}^K + 2C_{33}^{K + 1}} \right)} \right] - \left[ {C_{63}^{K + 1} + \alpha_2^{K + 1}\left( {C_{32}^{K + 1} + 2C_{33}^{K + 1}} \right)} \right]} \right\}{r^{K + 1}}.} \\ \end{array} $$

If β^K ≠ 1 and β^K+1 = 1, then

$$ \begin{array}{*{20}{c}} {{A_{n + K,2n + 1}} = C_{13}^K + {\alpha^K}\left( {C_{23}^K + C_{33}^K} \right) - C_{32}^{K + 1} + \frac{{N_N^{K + 1}}}{2}C_{32}^{K + 1}\log {r^{K + 1}} + C_{33}^{K + 1}\log \left( {{r^{K + 1}} + 1} \right),} \\ {{A_{n + K,K + n + 2}} = \left\{ {\left[ {C_{63}^K + \alpha_2^K\left( {C_{32}^K + 2C_{33}^K} \right)} \right] - \left[ {C_{63}^{K + 1} + \alpha_2^{K + 1}\left( {C_{32}^{K + 1} + 2C_{33}^{K + 1}} \right)} \right]} \right\}{r^{K + 1}}.} \\ \end{array} $$

If β^K ≠ 1 and β^K+1 ≠ 1, then

$$ \begin{array}{*{20}{c}} {{A_{n + K,2n + 2}} = C_{13}^K - C_{13}^{K + 1} + \alpha_1^K\left( {C_{23}^K + C_{33}^K} \right) - \alpha_1^{K + 1}\left( {C_{23}^{K + 1} + C_{33}^{K + 1}} \right),} \\ {{A_{n + K,2n + 2}} = \left[ {\left( {C_{36}^K - C_{36}^{K + 1}} \right) + \alpha_2^K\left( {C_{23}^K - 2C_{33}^K} \right) - \alpha_2^{K + 1}\left( {C_{23}^{K + 1} + 2C_{33}^{K + 1}} \right)} \right]{r^{K + 1}}.} \\ \end{array} $$

External pressure condition:

$$ \begin{array}{*{20}{c}} {{A_{2n,K}} = \left( {C_{23}^n + {\beta^n}C_{33}^n} \right){{\left( {{r^{n + 1}}} \right)}^{{\beta^n} - 1}},{A_{2n,2n}} = \left( {C_{23}^n - {\beta^n}C_{33}^n} \right){{\left( {{r^{n + 1}}} \right)}^{{\beta^n} - 1}},} \\ {{A_{2n,2n + 1}} = C_{13}^n + \alpha_1^n\left( {C_{23}^n + C_{33}^n} \right),{A_{2n,2n + 2}} = \left[ {C_{36}^n + \alpha_1^n\left( {C_{23}^n + C_{33}^n} \right)} \right]{r^{n + 1}}.} \\ \end{array} $$

The axial equilibrium condition:

If β^K = 1, then

$$ \begin{array}{*{20}{c}} {{A_{2n + 1,K}} = \left[ {{{\left( {{r^{K + 1}}} \right)}^2} + {{\left( {{r^K}} \right)}^2}} \right]\frac{{C_{12}^K + C_{13}^K}}{2},{A_{2n + 1,K + n}} = \left( {\log {r^{K + 1}} - \log {r^{K + 1}}} \right)\left( {C_{12}^K - C_{13}^K} \right),} \\ {{A_{2n + 1,K + 2}} = \sum\limits_{K = 1}^{K = n} {C_{11}^K\frac{{{{\left( {{r^{K + 1}}} \right)}^2}}}{2} + C_{12}^K\frac{{{N_2}}}{4} \cdot \frac{1}{2}{{\left( {{r^{K + 1}}} \right)}^2}\left( {\log {r^{K + 1}} - \frac{1}{2}} \right) + C_{13}^K\frac{{{N_2}}}{2} \cdot \frac{1}{2}{{\left( {{r^{K + 1}}} \right)}^2}\left[ {\left( {\log {r^{k + 1}}} \right) + \frac{1}{2}} \right]} } \\ { - C_{11}^K{{\left( {{r^{K + 1}}} \right)}^2} + C_{12}^K\frac{{{N_2}}}{2}{{\left( {{r^K}} \right)}^2}\left( {\log {r^K} - \frac{1}{4}} \right) + C_{13}^K\frac{{{N_2}}}{2}{{\left( {{r^K}} \right)}^2}\left( {\log {r^K} - \frac{1}{4}} \right) + C_{13}^K\frac{{{N_2}}}{2} \cdot \frac{1}{2}{{\left( {{r^K}} \right)}^2}\left[ {2\log {{\left( {{r^K}} \right)}^2} + \frac{1}{2}} \right],} \\ {{A_{2n + 1,2n + K}} = \sum\limits_{k = 1}^{k = n} {\left[ {r{{\left( {{R^{K + 1}}} \right)}^3} - {{\left( {{R^K}} \right)}^3}} \right]\alpha_2^K\left( {C_{12}^K + 2C_{13}^K} \right) + C_{16}^K.} } \\ \end{array} $$

If β^K ≠ 1, then

$$ \begin{array}{*{20}{c}} {{A_{2n + 1,K}} = \left( {C_{12}^K + {\beta^K}C_{13}^K} \right)\frac{{{{\left( {{r^{K + 1}}} \right)}^{{\beta^K} + 1}} - {{\left( {{r^K}} \right)}^{{\beta^K} - 1}}}}{{1 + {\beta^K}}},} \\ {{A_{2n + 1,K + n}} = \left( {C_{12}^K - {\beta^K}C_{13}^K} \right)\frac{{{{\left( {{r^{K + 1}}} \right)}^{ - {\beta^K} + 1}} - {{\left( {{r^K}} \right)}^{ - {\beta^K} - 1}}}}{{1 - {\beta^K}}},} \\ {{A_{2n + 1,2n + 1}} = \sum\limits_{K = 1}^{K = n} {\left[ {C_{11}^K + \alpha_1^K\left( {C_{12}^K + C_{13}^K} \right)} \right]\frac{{{{\left( {{r^{K + 1}}} \right)}^2} - {{\left( {{r^K}} \right)}^2}}}{2},} } \\ {{A_{2n + 1,K + n}} = \sum\limits_{K = 1}^{k = n} {\left( {C_{16}^K + \alpha_2^KC_{12}^K + 2C_{13}^K} \right)\frac{{{{\left( {{r^{K + 1}}} \right)}^3} - {{\left( {{r^K}} \right)}^3}}}{3}.} } \\ \end{array} $$

The zero torsion condition:

If β^K = 1, then

$$ \begin{array}{*{20}{c}} {{A_{2n + 2,K}} = \left[ {{{\left( {{r^{K + 1}}} \right)}^3} - {{\left( {{r^K}} \right)}^3}} \right]\frac{{C_{62}^K + C_{63}^K}}{3},{A_{2n + 2,K + n}} = \left( {{r^{K + 1}} - {r^K}} \right)\left( {C_{62}^K - C_{63}^K} \right),} \\ {{A_{2n + 2,2n + 1}} = \sum\limits_{k = 1}^{k = n} {c_{61}^k\frac{{{{\left( {{R^{K + 1}}} \right)}^3}}}{3} + \frac{{N_2^K}}{2} \cdot \frac{1}{3}{{\left( {{r^{K + 1}}} \right)}^3}\left[ {C_{62}^K\left( {\log {r^{K + 1}} - \frac{1}{3}} \right) + C_{63}^K\frac{1}{3}\left( {\log {r^{K + 1}} + \frac{2}{3}} \right)} \right]} } \\ { - \left\{ {C_{61}^K{{\left( {{r^K}} \right)}^3} + \frac{{N_2^K}}{2} \cdot \frac{1}{3}{{\left( {{r^K}} \right)}^3}\left[ {C_{26}^K\left( {\log {r^K} - \frac{1}{3}} \right) + C_{63}^K\left( {\log {r^K} + \frac{2}{3}} \right)} \right]} \right\},} \\ {{A_{2n + 2,2n + 2}} = \sum\limits_{K = 1}^{K = n} {C_{66}^K + \alpha_2^K\left( {C_{62}^K + 2C_{63}^K} \right)\frac{{{{\left( {{R^K}} \right)}^4} - {{\left( {{r^K}} \right)}^4}}}{4}.} } \\ \end{array} $$

If β^K ≠ 1, then

$$ \begin{array}{*{20}{c}} {{A_{2n + 2,K + n}} = \left( {C_{26}^K + {\beta^K}C_{36}^K} \right)\frac{{{{\left( {{r^{K + 1}}} \right)}^{{\beta^K} + 2}} + {{\left( {{r^K}} \right)}^{{\beta^K} + 2}}}}{{2 + {\beta^K}}},} \\ {{A_{2n + 2,K + n}} = \left( {C_{26}^K - {\beta^K}C_{36}^K} \right)\frac{{{{\left( {{r^{K + 1}}} \right)}^{ - {\beta^K} + 2}} + {{\left( {{r^K}} \right)}^{ - {\beta^K} + 2}}}}{{2 - {\beta^K}}},} \\ {{A_{2n + 2,2n + 1}} = \sum\limits_{K = 1}^{K = n} {\left[ {C_{16}^K + \alpha_1^K\left( {C_{26}^K + C_{36}^K} \right)} \right]\frac{{{{\left( {{r^{K + 1}}} \right)}^3} - {{\left( {{r^K}} \right)}^3}}}{3},} } \\ {{A_{2n + 2,2n + 1}} = \sum\limits_{K = 1}^{K = n} {\left[ {C_{66}^K + \alpha_1^K\left( {C_{26}^K + 2C_{36}^K} \right)} \right]\frac{{{{\left( {{r^{K + 1}}} \right)}^4} - {{\left( {{r^K}} \right)}^4}}}{4}.} } \\ \end{array} $$

The strains are given by:

If β^K ≠ 1, then

$$ \begin{array}{*{20}{c}} {\varepsilon_r^k = {D^k}\left( {{\beta^k} - 1} \right){r^{{\beta^k} - 1}} + {E^k}\left( { - {\beta^K} - 1} \right){r^{ - {\beta^k} - 1}} + \alpha_2^k{\varepsilon_0} + \frac{{{N_4}}}{2}{\gamma_0}\left[ {\frac{1}{3}{{\left( {{r^k}} \right)}^3}\ln r - \frac{1}{9}{{\left( {{r^k}} \right)}^k}} \right],} \\ {\varepsilon_\theta^k = {D^k}{r^{{\beta^k} - 1}} + {E^k}{r^{ - {\beta^k} - 1}} + \alpha_2^k{\varepsilon_0} + \frac{{{N_4}}}{2}{\gamma_0}{r^k}\ln {r^k}.} \\ \end{array} $$

If β^K =1, then

$$ \begin{array}{*{20}{c}} {\varepsilon_r^k = {D^k}{\beta^k}{r^{{\beta^r} - 1}} + {E^k}\left( { - {\beta^k}} \right){r^{ - {\beta^k} - 1}} + \alpha_2^k{\varepsilon_0} + 2\alpha_4^k{\gamma_0}{r_0},} \\ {\varepsilon_\theta^k = {D^k}{r^{{\beta^r} - 1}} + {E^k}{r^{ - {\beta^k} - 1}} + \alpha_2^k{\varepsilon_0} + \alpha_4^k{\gamma_0}{r_0}.} \\ \end{array} $$