应变协调性(英语:strain compatibility)在连续介质力学中是指使得物体的位移单值连续的应变张量所满足的条件。应变协调是可积条件的特殊情况。1864年,法国力学家圣维南最早得到了线弹性体的协调条件。1886年,意大利数学家贝尔特拉米对此进行了严格证明。[1]
对于二维无限小应变问题,其应变-位移关系为
![{\displaystyle \varepsilon _{11}={\cfrac {\partial u_{1)){\partial x_{1))}~;~~\varepsilon _{12}={\cfrac {1}{2))\left[{\cfrac {\partial u_{1)){\partial x_{2))}+{\cfrac {\partial u_{2)){\partial x_{1))}\right]~;~~\varepsilon _{22}={\cfrac {\partial u_{2)){\partial x_{2))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/004699770b18ac4b479cf776e71702b513c64e70)
其所对应的协调条件为
![{\displaystyle {\cfrac {\partial ^{2}\varepsilon _{11)){\partial x_{2}^{2))}-2{\cfrac {\partial ^{2}\varepsilon _{12)){\partial x_{1}\partial x_{2))}+{\cfrac {\partial ^{2}\varepsilon _{22)){\partial x_{1}^{2))}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/378030f35a5614c5787dfae6ab7c051429530afd)
在三维问题中,共有六个条件需满足。除了二维问题中的一个协调条件扩展为三个条件之外,另外三个协调条件的形式为
![{\displaystyle {\cfrac {\partial ^{2}\varepsilon _{33)){\partial x_{1}\partial x_{2))}={\cfrac {\partial }{\partial x_{3))}\left[{\cfrac {\partial \varepsilon _{23)){\partial x_{1))}+{\cfrac {\partial \varepsilon _{31)){\partial x_{2))}-{\cfrac {\partial \varepsilon _{12)){\partial x_{3))}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2901bc65b63fcef94324bd1529c7b437e34a182)
使用指标记号可以将所有六个条件合写为[2]
![{\displaystyle e_{ikr}~e_{jls}~\varepsilon _{ij,kl}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a21b258db0fda79e7aa22cbbb6a253c320875fe8)
其中
为列维-奇维塔符号。使用张量符号则可以表示成
![{\displaystyle {\boldsymbol {\nabla ))\times ({\boldsymbol {\nabla ))\times {\boldsymbol {\varepsilon )))={\boldsymbol {0))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cce263a237990e6fb21b88eabbbc4abd223e714)
二阶张量
![{\displaystyle {\boldsymbol {R)):={\boldsymbol {\nabla ))\times ({\boldsymbol {\nabla ))\times {\boldsymbol {\varepsilon )))~;~~R_{rs}:=e_{ikr}~e_{jls}~\varepsilon _{ij,kl))](https://wikimedia.org/api/rest_v1/media/math/render/svg/c01657dd492f3f8c5a84796a943e6d27ba142986)
被称为不协调张量,即圣维南张量。
在有限应变理论中,协调条件为
![{\displaystyle {\boldsymbol {\nabla ))\times {\boldsymbol {F))={\boldsymbol {0))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b069c5ec539d22ca8ba17cc1896a37b2f508659c)
其中
为变形梯度张量。在笛卡尔坐标系中,该条件可表示为
![{\displaystyle e_{ABC}~{\cfrac {\partial F_{iB)){\partial X_{A))}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0bb049b1a8f6d0b765082ac0fabf057b59c481)
该条件是从映射
得到的连续变形的必要条件,同时也是保证单连通物体应变协调的充分条件。
右柯西-格林变形张量的协调条件为
![{\displaystyle R_{\alpha \beta \rho }^{\gamma }:={\frac {\partial }{\partial X^{\rho ))}[\Gamma _{\alpha \beta }^{\gamma }]-{\frac {\partial }{\partial X^{\beta ))}[\Gamma _{\alpha \rho }^{\gamma }]+\Gamma _{\mu \rho }^{\gamma }~\Gamma _{\alpha \beta }^{\mu }-\Gamma _{\mu \beta }^{\gamma }~\Gamma _{\alpha \rho }^{\mu }=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bbb50fbba94d984441ca3d47dd0ec0a8bb8159d)
其中
表示第二类克里斯托费尔符号,
则表示黎曼-克里斯托费尔曲率张量。
- ^ C Amrouche, PG Ciarlet, L Gratie, S Kesavan, On Saint Venant's compatibility conditions and Poincaré's lemma, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 887-891. doi:10.1016/j.crma.2006.03.026
- ^ Slaughter, W. S., 2003, The linearized theory of elasticity, Birkhauser