Elastoplasticity¶

Sura hii inaeleza mfumo wa elastoplastic constitutive law unaotumiwa na FrontISTR. Kwa maelezo ya uchaguzi wa matumizi na mipangilio ya ingizo, rejelea sehemu ya vipengele 03_material.

Fenomeni (plastic deformation)¶

TBD (itaandikwa katika awamu inayofuata).

Mfumo wa constitutive law¶

FrontISTR hutumia elastoplastic constitutive law inayofuata associated flow rule. Constitutive law hiyo inaonyesha uhusiano kati ya Jaumann rate ya Kirchhoff stress na deformation-rate tensor, na deformation analysis yake hutumia updated Lagrange method.

Tuchukulie yield condition ya elastoplastic body imetolewa kama ifuatavyo.

Initial yield condition:

\[ F(\sigma, \sigma_{y_0}) \]

Subsequent yield condition:

\[ F(\sigma, \sigma_y(\overline{e}^p)) \]

ambapo,

\(F\): yield function
\(\sigma_{y_0}\): initial yield stress
\(\sigma_y\): subsequent yield stress
\(\sigma\): stress tensor
\(e\): infinitesimal strain tensor
\(e^p\): plastic strain tensor
\(\overline{e}^p\): equivalent plastic strain

Tunachukulia kuwa uhusiano wa yield stress–equivalent plastic strain unalingana na uhusiano wa stress–plastic strain katika uniaxial state.

Uhusiano wa stress–plastic strain katika uniaxial state

\[ \sigma = H(e^p) \]

\[ \frac{d\sigma}{de^p} = H' \]

Hapa \(H'\) ni strain-hardening coefficient.

Uhusiano wa equivalent stress–equivalent plastic strain

\[ \overline{\sigma} = H(\overline{e}^p) \]

\[ \dot{\overline{\sigma}} = H' \dot{\overline{e}^p} \]

Kwa ujumla subsequent yield function ni function ya halijoto na plastic-strain work, lakini kwa urahisi hapa tunachukulia kuwa ni function ya equivalent plastic strain \(\overline{e}^p\) pekee. Kwa kuwa \(F = 0\) lazima iendelee kutimizwa wakati plastic deformation inaendelea, equation ifuatayo lazima itimizwe.

\[ \dot{F} = \frac{\partial F}{\partial \sigma} : \dot{\sigma} + \frac{\partial F}{\partial e^p} : \dot{e}^p = 0 \]

Hapa \(\dot{F}\) ni time derivative ya \(F\); kuanzia hapa, time derivative ya quantity \(A\) itaonyeshwa kwa \(\dot{A}\).

Sasa tunadhania kuwepo kwa plastic potential \(\Theta\) na kuonyesha plastic strain rate kama ifuatavyo.

\[ \dot{e}^p = \dot{\lambda} \frac{\partial \Theta}{\partial \sigma} \]

Hapa \(\dot{\lambda}\) ni coefficient.

Zaidi ya hayo, tukichukulia plastic potential \(\Theta\) kuwa sawa na yield function \(F\), tunapata associated flow rule ifuatayo.

\[ \dot{e}^p = \dot{\lambda} \frac{\partial F}{\partial \sigma} \]

Kwa kuingiza equation hii katika consistency condition, tunapata

\[ \dot{\lambda} = \frac{a^T : d_D}{A + a^T : D : a} \dot{e} \]

ambapo \(D\) ni elastic matrix, na

\[ a^T = \frac{\partial F}{\partial \sigma}, \qquad d_D = D a^T, \qquad A = -\frac{a}{\dot{\lambda}} \frac{\partial F}{\partial e^p} : \dot{e}^p \]

Uhusiano wa stress–strain wa elastoplasticity unaweza kuandikwa kama ifuatavyo.

\[ \dot{\sigma} = \left\{ D - \frac{d_D \otimes d_D^T}{A + d_D^T a} \right\} : \dot{e} \]

Ikiwa yield function \(F\) ya elastoplastic material inajulikana, constitutive law yake inaweza kupatikana kutoka equation hii.

Yield functions¶

Hapa chini tunaorodhesha elastoplastic yield functions zinazotumiwa na FrontISTR.

Von Mises yield function¶

\[ F = \sqrt{3 J_2} - \sigma_y = 0 \]

Hapa \(J_2\) ni second invariant ya deviatoric stress tensor.

Mohr-Coulomb yield function¶

\[ F = \sigma_1 - \sigma_3 + (\sigma_1 + \sigma_3) \sin\phi - 2 c \cos\phi = 0 \]

Hapa \(\sigma_1, \sigma_3\) ni maximum na minimum principal stresses, \(c\) ni cohesion, na \(\phi\) ni internal friction angle.

Drucker-Prager yield function¶

\[ F = \sqrt{J_2} - \alpha \sigma : I - \sigma_y = 0 \]

Hapa material constants \(\alpha\) na \(\sigma_y\) huhesabiwa kutoka cohesion na friction angle ya nyenzo kama ifuatavyo.

\[ \alpha = \frac{2 \sin\phi}{3 + \sin\phi}, \qquad \sigma_y = \frac{6 c \cos\phi}{3 + \sin\phi} \]

Vipengee vinavyohusiana¶

Linear elasticity — Elastic response (sehemu ya elasticity katika elastoplasticity)
Tangent stiffness matrix — Tangent stiffness ya incremental analysis inayojumuisha material nonlinearity
Data ya nyenzo (kipengele) — Chaguo na matumizi ya yield functions na hardening laws