Hyperelasticity¶

Sura hii inaeleza sheria za constitutive za hyperelastic zinazotumiwa na FrontISTR. Kwa maelezo ya uchaguzi na vipimo vya ingizo, rejelea sehemu ya vipengele 03_material.

Jambo linalochanganuliwa (mwitikio unaorejea wa mgeuko mkubwa)¶

TBD (itaandikwa kikamilifu katika awamu inayofuata).

Mfumo wa sheria ya constitutive¶

Nishati potenshali ya elastiki katika nyenzo hyperelastic isotropic hutokana na mwitikio isotropic kutoka hali ya mwanzo isiyo na stress, na inaweza kuonyeshwa kama function ya right Cauchy-Green deformation tensor \(C\) kupitia principal invariants zake \((I_1, I_2, I_3)\), au kupitia principal invariants \((\overline{I}_1, \overline{I}_2, \overline{I}_3)\) za deformation tensor baada ya kuondoa volume change, yaani inaweza kuandikwa kama \(W = W(I_1, I_2, I_3)\) au \(W = W(\overline{I}_1, \overline{I}_2, \overline{I}_3)\).

Mlinganyo wa constitutive wa nyenzo hyperelastic hufafanuliwa kwa uhusiano kati ya second Piola-Kirchhoff stress na Green-Lagrange strain, na uchanganuzi wa mgeuko hutumia mbinu ya Total Lagrange.

Nishati potenshali ya elastiki \(W\) ikijulikana, second Piola-Kirchhoff stress na uhusiano wa stress-strain vinaweza kuhesabiwa kama ifuatavyo.

\[ S = 2\frac{\partial W}{\partial C} \]

\[ \mathbb{C} = 4 \frac{\partial^2 W}{\partial C \partial C} \]

Ifuatayo inaorodhesha nishati potenshali ya elastiki \(W\) kwa modeli za hyperelastic zinazotumiwa na FrontISTR.

Modeli ya hyperelastic ya Neo-Hookean¶

Modeli ya hyperelastic ya Neo-Hookean ni upanuzi wa sheria ya mstari isotropic (sheria ya Hooke) ili ishughulikie matatizo ya mgeuko mkubwa. Nishati yake potenshali ya elastiki ni kama ifuatavyo.

\[ W = C_{10} (\overline{I}_1 - 3) + \frac{1}{D} (J - 1)^2 \]

Hapa \(C_{10}\) na \(D\) ni constants za nyenzo.

Modeli ya hyperelastic ya Mooney-Rivlin¶

\[ W = C_{10} (\overline{I}_1 - 3) + C_{01} (\overline{I}_2 - 3) + \frac{1}{D} (J - 1)^2 \]

Hapa \(C_{10}\), \(C_{01}\) na \(D\) ni constants za nyenzo.

Modeli ya hyperelastic anisotropic ya Mooney-Rivlin¶

\[ W = C_{10} (\overline{I}_1 - 3) + C_{01} (\overline{I}_2 - 3) + \frac{1}{D} (J - 1)^2 + C_{42} (\overline{I}_4 - 1)^2 + C_{43} (\overline{I}_4 - 1)^3 \]

Hapa \(C_{10}\), \(C_{01}\) na \(D\) ni constants za nyenzo zinazoshirikiwa na modeli isotropic, huku \(C_{42}\) na \(C_{43}\) zikiwa constants maalum za anisotropy.

Modeli ya hyperelastic ya Arruda-Boyce¶

\[ \begin{aligned} W &= \mu \left[ \frac{1}{2}(\overline{I}_1 - 3) + \frac{1}{20\lambda_m^2}(\overline{I}_1^2 - 9) + \frac{11}{1050\lambda_m^4}(\overline{I}_1^3 - 27) \right. \\ &\quad \left. + \frac{19}{7000\lambda_m^6}(\overline{I}_1^4 - 81) + \frac{519}{673750\lambda_m^8}(\overline{I}_1^5 - 243) \right] + \frac{1}{D} \left( \frac{J^2 - 1}{2} - \ln J \right) \end{aligned} \]

\[ \mu = \frac{\mu_0}{1 + \dfrac{3}{5\lambda_m^2} + \dfrac{99}{175\lambda_m^4} + \dfrac{513}{875\lambda_m^6} + \dfrac{42039}{67375\lambda_m^8}} \]

Hapa \(\mu_0\), \(\lambda_m\) na \(D\) ni constants za nyenzo.

Vipengee vinavyohusiana¶

Elastiki ya mstari — Mwitikio wa elastiki katika mgeuko mdogo
Elastoplasticity — Sheria ya constitutive inayojumuisha mgeuko usiorejea
Data ya nyenzo (kipengele) — Chaguo za modeli na jinsi ya kuzitumia