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Uandishi wa tensor na misingi ya hisabati

Sura hii inakusanya kanuni za uandishi wa tensor, index notation na Voigt notation zinazotumiwa katika sura zinazofuata za mwongozo wa nadharia wa FrontISTR. Hapa tunaeleza tu kanuni safi za uandishi zisizotegemea ufafanuzi wa kiasi cha fizikia. Kanuni za alama zinazotegemea configuration ya continuum (reference/current configuration), kama matumizi ya herufi kubwa/ndogo na tofauti kati ya material na spatial derivatives, zinaelezwa katika Mwendo, mgeuko na strain.

Uandishi wa tensor

Uandishi wa vector na tensor kwa herufi nzito

Kama kanuni ya typeface, scalar na components za vector/tensor huandikwa kwa herufi za kawaida (kama \(\rho\), \(x_i\)), huku vector/tensor yenyewe ikiandikwa kwa herufi nzito (kama \(\boldsymbol{a}\), \(\boldsymbol{E}\), \(\boldsymbol{\sigma}\)).

Kanuni ya Einstein ya summation

Isipobainishwa vinginevyo, index ileile ikionekana mara mbili katika term moja, kanuni ya Einstein ya summation hutumika na jumla huchukuliwa juu ya index hiyo. Kwa mfano

\[ a_i b_i = \sum_i a_i b_i, \quad A_{ij} B_{ij} = \sum_{i,j} A_{ij} B_{ij}, \quad \frac{\partial a_i}{\partial x_i} = \sum_i \frac{\partial a_i}{\partial x_i} \]

kwa tensors \(\boldsymbol{A}\) na \(\boldsymbol{B}\), product \(\boldsymbol{C} = \boldsymbol{A}\boldsymbol{B}\) ina component ya \(i,j\) ambayo ni

\[ C_{ij} = A_{ik} B_{kj} \]

Kwa kanuni ya indices, isipobainishwa vinginevyo \(i, j, k, l, \ldots\) humaanisha indices za degrees of freedom (kwa 3D, \(1, 2, 3\)), na herufi ndogo za Kigiriki \(\alpha, \beta, \gamma, \ldots\) humaanisha indices za nodi zinazounda kipengele.

Inner product, transpose na tensor product

Transpose ya tensor \(\boldsymbol{A}\) huandikwa \(\boldsymbol{A}^T\). Inner product (double contraction) ya second-order tensors \(\boldsymbol{A}\) na \(\boldsymbol{B}\) huandikwa

\[ \boldsymbol{A} : \boldsymbol{B} \equiv A_{ij} B_{ij} \]

Inner product ya vectors \(\boldsymbol{a}\) na \(\boldsymbol{b}\) ni \(\boldsymbol{a} \cdot \boldsymbol{b} = a_i b_i\), na tensor product ni \((\boldsymbol{a} \otimes \boldsymbol{b})_{ij} = a_i b_j\).

Uandishi wa Voigt

Stress na strain ni symmetric second-order tensors za degrees of freedom, na coefficient inayoeleza uhusiano wa mstari kati yake ni fourth-order tensor. Kushughulikia hizi moja kwa moja kwenye programu si faida kwa gharama ya hesabu na coding (multi-dimensional arrays na nested loops nyingi), kwa hiyo programu za kawaida za FEM hutumia symmetry kuzibana: stress na strain kuwa column vectors, na fourth-order elasticity tensor kuwa 2D matrix. Hii huitwa Voigt notation.

Kuanzia hapa, matrix/vector representation ya tensor \(\boldsymbol{A}\) itaandikwa \(\hat{A}\) ili kuitofautisha na tensor ya awali.

Vector representation ya stress na strain

Kwa symmetric stress tensor \(\boldsymbol{\sigma}\) na strain tensor \(\boldsymbol{\varepsilon}\), katika 2D tuna

\[ \hat{\sigma} = \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}, \qquad \hat{\varepsilon} = \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ 2 \varepsilon_{12} \end{bmatrix} \]

na katika 3D

\[ \hat{\sigma} = \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \\ \sigma_{23} \\ \sigma_{31} \end{bmatrix}, \qquad \hat{\varepsilon} = \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2 \varepsilon_{12} \\ 2 \varepsilon_{23} \\ 2 \varepsilon_{31} \end{bmatrix} \]

na hushughulikiwa kama column vectors. Kanuni hiyo hiyo hutumika kwa quantities zinazotokana kama variations na derivatives. Kumbuka kwamba shear components upande wa strain zina factor 2 (upande wa stress hazina). Kwa asymmetry hii, tensor inner product inaweza kuandikwa kama vector inner product

\[ \boldsymbol{\sigma} : \delta\boldsymbol{\varepsilon} = \sigma_{ij}\,\delta \varepsilon_{ij} = \sum_{i=j} \sigma_{ij}\,\delta \varepsilon_{ij} + 2\sum_{i<j} \sigma_{ij}\,\delta \varepsilon_{ij} = \hat{\sigma}^T\,\delta\hat{\varepsilon} \]

kwa umbo fupi. Katika utekelezaji ni rahisi kukosea factor 2 ya shear strain, hivyo daima zingatia kanuni "hakuna kwa stress / ipo kwa strain". Kanuni hiyo hiyo ya Voigt hutumika kwa quantities za reference configuration (second Piola-Kirchhoff stress \(\boldsymbol{S}\) na Green-Lagrange strain \(\boldsymbol{E}\)).

Matrix representation ya fourth-order tensor

Kwa uhusiano wa mstari wa stress na strain \(\boldsymbol{\sigma} = \boldsymbol{C} : \boldsymbol{\varepsilon}\) (components: \(\sigma_{ij} = C_{ijkl} \varepsilon_{kl}\)), kwa kutumia symmetry ya \(\boldsymbol{\varepsilon}\) tunaweza kuandika upya

\[ \sigma_{ij} = \tilde{C}_{ijkl} \varepsilon_{kl}, \qquad \tilde{C}_{ijkl} = \frac{1}{2}\bigl( C_{ijkl} + C_{ijlk} \bigr) \]

Katika 3D, tukipanga \(\tilde{C}\) kuwa material matrix ya 6×6 \(\hat{\tilde{C}}\), uhusiano wa awali wa tensor huwa

\[ \hat{\sigma} = \hat{\tilde{C}}\, \hat{\varepsilon} \]

Kuanzia hapa, isipobainishwa vinginevyo, material matrix itaandikwa \(D\) (au \(\hat{C}\)). Kwa components maalum za isotropic linear elasticity, rejelea Elastiki ya mstari.

Kanuni za differential operators

Symmetric gradient operator

Kwa vector field \(\boldsymbol{u}\), symmetric gradient operator \(\nabla_S\) hufafanuliwa kama

\[ \nabla_S \boldsymbol{u} = \frac{1}{2}\bigl( \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \bigr) \]

Katika components, \((\nabla_S \boldsymbol{u})_{ij} = \tfrac{1}{2}(\partial u_i / \partial x_j + \partial u_j / \partial x_i)\). Small-strain \(\boldsymbol{\varepsilon} = \nabla_S \boldsymbol{u}\) inaweza kuandikwa kwa ufupi kwa operator hii. Uchaguzi wa configuration (reference/current) ambako gradient inachukuliwa katika finite deformation unaelezwa katika Mwendo, mgeuko na strain.

Material time derivative

Material time derivative ya quantity \(A\) (time derivative inayofuata material point ileile) huonyeshwa kwa dot juu kama \(\dot{A}\):

\[ \dot{A} \equiv \frac{D A}{D t}. \]

Velocity na acceleration hufuata kanuni hii.

Vipengee vinavyohusiana

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