應變協調性 （英語：strain compatibility ）在連續介質力學 中是指使得物體的位移 單值連續的應變 張量 所滿足的條件。應變協調是可積條件的特殊情況。1864年，法國力學家聖維南 最早得到了線彈性體的協調條件。1886年，意大利數學家貝爾特拉米 對此進行了嚴格證明。[ 1]
對於二維無限小應變問題，其應變－位移關係為
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}}}}
其所對應的協調條件為
∂
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}
在三維問題中，共有六個條件需滿足。除了二維問題中的一個協調條件擴展為三個條件之外，另外三個協調條件的形式為
∂
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]}
使用指標記號可以將所有六個條件合寫為[ 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}}
列維-奇維塔符號 。使用張量符號則可以表示成
∇
×
(
∇
×
ε
)
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times {\boldsymbol {\varepsilon }})={\boldsymbol {0}}}
二階張量
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}}
被稱為不協調張量，即聖維南張量。
在有限應變理論中，協調條件為
∇
×
F
=
0
{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {F}}={\boldsymbol {0}}}
其中
F
{\displaystyle {\boldsymbol {F}}}
笛卡爾坐標系 中，該條件可表示為
e
A
B
C
∂
F
i
B
∂
X
A
=
0
{\displaystyle e_{ABC}~{\cfrac {\partial F_{iB}}{\partial X_{A}}}=0}
該條件是從映射
x
=
χ
(
X
,
t
)
{\displaystyle \mathbf {x} ={\boldsymbol {\chi }}(\mathbf {X} ,t)}
單連通 物體應變協調的充分條件。
右柯西－格林變形張量的協調條件為
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}}
克里斯托費爾符號 ，
R
i
j
k
m
{\displaystyle R_{ijk}^{m}}
黎曼－克里斯托費爾曲率張量 。
^ 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
![{\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)
