应变协调性 （英語：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]
无限小应变的协调条件
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对于二维无限小应变问题，其应变－位移关系为
ε
11
=
∂
u
1
∂
x
1
;
ε
12
=
1
2
[
∂
u
1
∂
x
2
+
∂
u
2
∂
x
1
]
;
ε
22
=
∂
u
2
∂
x
2
{\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}}}}
其所对应的协调条件为
∂
2
ε
11
∂
x
2
2
−
2
∂
2
ε
12
∂
x
1
∂
x
2
+
∂
2
ε
22
∂
x
1
2
=
0
{\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}
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
∂
2
ε
33
∂
x
1
∂
x
2
=
∂
∂
x
3
[
∂
ε
23
∂
x
1
+
∂
ε
31
∂
x
2
−
∂
ε
12
∂
x
3
]
{\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]}
使用指标记号可以将所有六个条件合写为[2]
e
i
k
r
e
j
l
s
ε
i
j
,
k
l
=
0
{\displaystyle e_{ikr}~e_{jls}~\varepsilon _{ij,kl}=0}
其中
e
i
j
k
{\displaystyle e_{ijk}}
列维-奇维塔符号 。使用张量符号则可以表示成
∇
×
(
∇
×
ε
)
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }})={\boldsymbol {0}}}
二阶张量
R
:=
∇
×
(
∇
×
ε
)
;
R
r
s
:=
e
i
k
r
e
j
l
s
ε
i
j
,
k
l
{\displaystyle {\boldsymbol {R}}:={\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }})~;~~R_{rs}:=e_{ikr}~e_{jls}~\varepsilon _{ij,kl}}
被称为不协调张量，即圣维南张量。
有限应变的协调条件
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在有限应变理论中，协调条件为
∇
×
F
=
0
{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {F}}={\boldsymbol {0}}}
其中
F
{\displaystyle {\boldsymbol {F}}}
笛卡尔坐标系 中，该条件可表示为
e
A
B
C
∂
F
i
B
∂
X
A
=
0
{\displaystyle e_{ABC}~{\cfrac {\partial F_{iB}}{\partial X_{A}}}=0}
该条件是从映射
x
=
χ
(
X
,
t
)
{\displaystyle \mathbf {x} ={\boldsymbol {\chi }}(\mathbf {X} ,t)}
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
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右柯西－格林变形张量的协调条件为
R
α
β
ρ
γ
:=
∂
∂
X
ρ
[
Γ
α
β
γ
]
−
∂
∂
X
β
[
Γ
α
ρ
γ
]
+
Γ
μ
ρ
γ
Γ
α
β
μ
−
Γ
μ
β
γ
Γ
α
ρ
μ
=
0
{\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}
其中
Γ
i
j
k
{\displaystyle \Gamma _{ij}^{k}}
克里斯托费尔符号 ，
R
i
j
k
m
{\displaystyle R_{ijk}^{m}}
黎曼－克里斯托费尔曲率张量 。
参考文献
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^ 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
![{\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)
![{\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)
![{\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)
