Creep¶

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

Fenomeni ya creep na additive decomposition ya strain¶

Displacement inayotegemea muda chini ya hali ya stress ya kudumu huitwa "creep".

Tabia ya viscoelastic iliyoelezwa hapo awali pia inaweza kuchukuliwa kuwa aina ya linear creep. Hapa tunaeleza baadhi ya aina za nonlinear creep. Kwa kawaida, constitutive law huundwa kwa kuongeza strain ya fenomeni hii kwenye strain inayotokea papo hapo, na strain inayotokea wakati load ya kudumu inaendelea huitwa creep strain \(\varepsilon^c\). Constitutive law inayozingatia creep kwa kawaida hutumia creep strain rate \(\dot{\varepsilon}^c\), inayofafanuliwa kama function ya stress na total creep strain.

\[ \dot{\varepsilon}^c \equiv \frac{\partial \varepsilon^c}{\partial t} = \beta(\sigma, \varepsilon^c) \]

Ikiwa strain inayotokea papo hapo ni elastic strain \(\varepsilon^e\), total strain huonyeshwa kwa kuongeza creep strain kama ifuatavyo.

\[ \varepsilon = \varepsilon^e + \varepsilon^c \]

ambapo,

\[ \varepsilon^e = c^{-1} : \sigma \]

(\(c\) ni elastic coefficient tensor).

Sheria ya Norton¶

Kama umbo mahususi la constitutive law ya creep, FrontISTR hutumia modeli ya Norton iliyo hapa chini. Katika constitutive law hii, equivalent creep strain rate \(\dot{\varepsilon}^{cr}\) huonyeshwa kama function ya Mises stress \(q\) na muda \(t\) kama ifuatavyo.

\[ \dot{\varepsilon}^{cr} = A q^n t^m \]

Hapa \(A\), \(m\), na \(n\) ni material constants.

Time integration na stress update¶

Kama ilivyo kwa nyenzo za plasticity, njia ya time integration katika numerical analysis lazima ibainishwe kwa constitutive law inayoonyesha creep. Constitutive law inapozingatia creep ni,

\[ \sigma_{n+1} = c : (\varepsilon_{n+1} - \varepsilon_{n+1}^c) \]

\[ \varepsilon_{n+1}^c = \varepsilon_n^c + \Delta t \, \beta_{n+\theta} \]

ambapo \(\beta_{n+\theta}\) ni,

\[ \beta_{n+\theta} = (1 - \theta) \beta_n + \theta \beta_{n+1} \]

Pia, creep strain increment \(\Delta \varepsilon^c\) hutumia nonlinear equation iliyorahisishwa kuwa

\[ R_{n+1} = \varepsilon_{n+1} - c^{-1} : \sigma_{n+1} - \varepsilon_n^c - \Delta t \, \beta_{n+\theta} = \mathbf{0} \]

Katika iteration ya Newton-Raphson, kwa kuchukua \(\sigma_{n+1} = \sigma_n\) kama initial value pamoja na strain increment inayopatikana kwa finite element method, iterative solution na incremental solution ni kama ifuatavyo.

\[ R_{n+1}^{(k+1)} = \mathbf{0} = R_{n+1}^{(k)} - (c^{-1} + \Delta t \, c_{n+1}^c) \, d\sigma_{n+1}^{(k)} \]

ambapo,

\[ c_{n+1}^c = \left.\frac{\partial \beta}{\partial \sigma}\right|_{n+\theta} = \theta \left.\frac{\partial \beta}{\partial \sigma}\right|_{n+1} \]

Wakati iteration inaendelea hadi residual \(R\) iwe \(\mathbf{0}\), stress \(\sigma_{n+1}\) na tangent coefficient

\[ c_{n+1}^* = (c^{-1} + \Delta t \, c_{n+1}^c)^{-1} \]

hutumiwa.

Vipengee vinavyohusiana¶

Viscoelasticity — Tabia ya viscoelastic inayolingana na linear creep
Tangent stiffness matrix — Tangent stiffness katika incremental analysis inayojumuisha creep
Data ya nyenzo (kipengele) — Jinsi ya kubainisha vigezo vya modeli ya Norton