Stress na sheria za uhifadhi¶

Sura hii inapanga ufafanuzi na uhusiano wa stress tensors, pamoja na equilibrium equations na stress symmetry zinazotokana na sheria za uhifadhi wa mass, momentum na angular momentum. Aina tatu za stress tensor—Cauchy stress \(\boldsymbol{\sigma}\), first Piola-Kirchhoff stress \(\boldsymbol{P}\) na second Piola-Kirchhoff stress \(\boldsymbol{S}\)—huletwa kulingana na configuration inayotumiwa kwa reference surface na force vector. Kwa ufafanuzi wa configuration, motion na deformation gradient, rejelea Motion, deformation na strain; kwa orodha ya alama, rejelea Orodha ya alama za physical quantities.

Cauchy stress tensor¶

Tuchukulie force \(\boldsymbol{x}\) inayotenda kwenye infinitesimal surface katika point \(d\Gamma\) ya current configuration (area \(\boldsymbol{n}\), outward unit normal \(d\boldsymbol{f}\)), na tufafanue stress vector \(\boldsymbol{t}\) kama force kwa unit area, \(d\boldsymbol{f} = \boldsymbol{t}\, d\Gamma\). \(\boldsymbol{t}\) hutegemea position \(\boldsymbol{x}\) na normal \(\boldsymbol{n}\). Kwa kutumia conservation of momentum kwenye infinitesimal tetrahedron, \(\boldsymbol{t}\) huwa linear kwa \(\boldsymbol{n}\), na second-order tensor \(\boldsymbol{\sigma}(\boldsymbol{x}, t)\) ipo kiasi kwamba

\[ \boldsymbol{t} = \boldsymbol{\sigma} \boldsymbol{n}, \qquad t_i = \sigma_{ij} n_j \]

hutimizwa kwa \(\boldsymbol{n}\) yoyote (Cauchy stress theorem). Tensor hii \(\boldsymbol{\sigma}\) huitwa Cauchy stress tensor, na pia true stress kwa maana ya force kwa unit area inayotegemea geometry ya current configuration. Component \(\sigma_{ij}\) huwakilisha component ya force katika mwelekeo \(x_j\) kwa unit area ya infinitesimal surface iliyo perpendicular kwa coordinate axis \(x_i\) katika current configuration.

Kwa sheria ya conservation of angular momentum iliyoelezwa baadaye, Cauchy stress ni symmetric tensor

\[ \boldsymbol{\sigma} = \boldsymbol{\sigma}^T, \qquad \sigma_{ij} = \sigma_{ji} \]

na katika dimensionality tatu ina components 6 huru (\(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{31}\)). Uwakilishi wa stress kama vector kwa Voigt notation unategemea symmetry hii (rejelea Tensor notation na msingi wa kihisabati). Cauchy stress hutumiwa kwa stress output katika geometric nonlinear analysis ya FrontISTR kwa Updated Lagrange method na katika infinitesimal-deformation analysis.

First na second Piola-Kirchhoff stress¶

Katika finite deformation, mara nyingi ni rahisi kuonyesha stress kwa kutumia reference configuration kama reference surface na reference vector, kwa hiyo aina mbili za Piola-Kirchhoff stress tensor huletwa. Tuweke area ya infinitesimal surface katika reference configuration kuwa \(d\Gamma_0\) na outward unit normal kuwa \(\boldsymbol{N}\).

First Piola-Kirchhoff stress (first PK stress, nominal stress) \(\boldsymbol{P}\) hufafanuliwa kama stress tensor wakati force \(d\boldsymbol{f}\) ya current configuration inachukuliwa kutenda kwenye infinitesimal surface ya reference configuration:

\[ d\boldsymbol{f} = \boldsymbol{P} \boldsymbol{N}\, d\Gamma_0 \]

Kutoka Nanson formula \(\boldsymbol{n}\, d\Gamma = J \boldsymbol{F}^{-T} \boldsymbol{N}\, d\Gamma_0\) na Cauchy stress theorem, uhusiano na Cauchy stress ni

\[ \boldsymbol{P} = J\, \boldsymbol{\sigma} \boldsymbol{F}^{-T}, \qquad \boldsymbol{\sigma} = \frac{1}{J} \boldsymbol{P} \boldsymbol{F}^T \]

First PK stress kwa ujumla ni nonsymmetric tensor.

Second Piola-Kirchhoff stress (second PK stress) \(\boldsymbol{S}\) hufafanuliwa kama stress tensor wakati force \(d\boldsymbol{f}\) ya current configuration inarudishwa kwenye reference configuration kwa \(\boldsymbol{F}^{-1}\) na kisha kuchukuliwa kutenda kwenye infinitesimal surface ya reference configuration:

\[ \boldsymbol{F}^{-1}\, d\boldsymbol{f} = \boldsymbol{S} \boldsymbol{N}\, d\Gamma_0 \]

Hapa force vector na surface ya action zote zinaonyeshwa kwa quantities za reference configuration; tensor hii ni invariant kwa rigid-body rotation na ni symmetric. Uhusiano na first PK stress pamoja na transformation formula na Cauchy stress ni

\[ \boldsymbol{P} = \boldsymbol{F} \boldsymbol{S}, \qquad \boldsymbol{\sigma} = \frac{1}{J} \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T, \qquad \boldsymbol{S} = J\, \boldsymbol{F}^{-1} \boldsymbol{\sigma} \boldsymbol{F}^{-T} \]

Second PK stress hutolewa kutoka hyperelastic strain-energy function \(W(\boldsymbol{C})\) kama \(\boldsymbol{S} = 2\,\partial W / \partial \boldsymbol{C}\), na katika Total Lagrange method hutumiwa kama stress-conjugate pair \(\boldsymbol{E}\) pamoja na Green-Lagrange strain \((\boldsymbol{S}, \boldsymbol{E})\).

Reference configurations na symmetry zimefupishwa katika jedwali lifuatalo. Katika limit ya infinitesimal deformation (\(\boldsymbol{F} \to \boldsymbol{I}\), \(J \to 1\)), aina zote tatu za stress zinafanana.

Stress tensor	Reference surface	Force vector	Symmetry	Matumizi
Cauchy stress \(\boldsymbol{\sigma}\)	Current configuration \(d\Gamma, \boldsymbol{n}\)	Current configuration \(d\boldsymbol{f}\)	Symmetric	Updated Lagrange method / infinitesimal-deformation analysis
First PK stress \(\boldsymbol{P}\)	Reference configuration \(d\Gamma_0, \boldsymbol{N}\)	Current configuration \(d\boldsymbol{f}\)	Generally nonsymmetric	Equilibrium equation in reference configuration
Second PK stress \(\boldsymbol{S}\)	Reference configuration \(d\Gamma_0, \boldsymbol{N}\)	Reference configuration \(\boldsymbol{F}^{-1} d\boldsymbol{f}\)	Symmetric	Total Lagrange method / hyperelasticity

Conservation of mass na momentum, na equilibrium equation¶

Tukiweka mass density ya current configuration kuwa \(\rho\) na ya reference configuration kuwa \(\rho_0\), conservation of mass \(\int_{\Omega} \rho\, dv = \int_{\Omega_0} \rho_0\, dV\) hupunguzwa kwa transformation ya volume element \(dv = J\, dV\) kuwa local form

\[ \rho_0 = J \rho \]

hii.

Tukiweka body force (kwa unit mass) kuwa \(\boldsymbol{g}\) na acceleration kuwa \(\boldsymbol{a}\), na kutumia Cauchy stress theorem pamoja na Gauss divergence theorem kwenye conservation of momentum (Euler first law of motion), tunapata local equilibrium equation (equation of motion) katika current configuration:

\[ \nabla_x \cdot \boldsymbol{\sigma} + \rho \boldsymbol{g} = \rho \boldsymbol{a}, \qquad \frac{\partial \sigma_{ij}}{\partial x_j} + \rho g_i = \rho a_i \]

Kwa kuandika upya integral katika reference configuration kwa Nanson formula na \(\boldsymbol{P} = J \boldsymbol{\sigma} \boldsymbol{F}^{-T}\), local form katika reference configuration ni

\[ \nabla_X \cdot \boldsymbol{P} + \rho_0 \boldsymbol{g} = \rho_0 \boldsymbol{a}, \qquad \frac{\partial P_{ij}}{\partial X_j} + \rho_0 g_i = \rho_0 a_i \]

Forms hizi mbili ni equivalent kutokana na stress transformation law na \(\rho_0 = J\rho\). Tukipuuza inertia term,

\[ \nabla_x \cdot \boldsymbol{\sigma} + \rho \boldsymbol{g} = \boldsymbol{0}, \qquad \nabla_X \cdot \boldsymbol{P} + \rho_0 \boldsymbol{g} = \boldsymbol{0} \]

hii ni static equilibrium equation na ndiyo starting equation ya static analysis ya FrontISTR (linear na nonlinear).

Conservation of angular momentum na stress symmetry¶

Kwa kuchanganya conservation of angular momentum (Euler second law of motion) na equilibrium equation kutoka conservation of momentum, tunapata symmetry ya Cauchy stress tensor:

\[ \boldsymbol{\sigma} = \boldsymbol{\sigma}^T, \qquad \sigma_{ij} = \sigma_{ji} \]

na katika dimensionality tatu ina components 6 huru. Kutoka transpose ya pande zote mbili za transformation formula \(\boldsymbol{\sigma} = J^{-1} \boldsymbol{F} \boldsymbol{S} \boldsymbol{F}^T\) pamoja na nonsingularity ya \(\boldsymbol{F}\), second PK stress pia ni symmetric:

\[ \boldsymbol{S} = \boldsymbol{S}^T \]

na hivyo ina components 6 huru. Kwa upande mwingine, first PK stress \(\boldsymbol{P} = \boldsymbol{F} \boldsymbol{S}\) kwa ujumla si symmetric; uhusiano pekee unaotimizwa ni \(\boldsymbol{P} \boldsymbol{F}^T = \boldsymbol{F} \boldsymbol{P}^T\) (yaani \(\boldsymbol{P} \boldsymbol{F}^T\) ni symmetric), na ina components 9 huru.

Symmetries hizi ndiyo msingi wa kuonyesha stress kama vector ya components 6 kwa Voigt notation. Kwa conventions za Voigt notation na uundaji wa material matrix, rejelea Tensor notation na msingi wa kihisabati na Linear elasticity.