**无限小应变理论**（infinitesimal strain theory）是连续介质力学中描述固體形變的一種數學分析法，適用在其形變量遠小於物體尺寸（無窮小量）的情形，因此若是均質材料，可以假設材料每一點的結構性質（密度及剛度）都相等，不會隨變形而不同。

在此假設下，連續介质力學的方程可以簡化。此作法也稱為是**小形變理論**、**小位移理論**或**小位移梯度理論**。无限小应变理论和有限应变理论的假設恰好相反，後者假設形變量沒有遠小於物體尺寸。

无限小应变理论常用在土木工程及機械工程中，其中會進行結構的應力分析（英语：），而材料是用強度較高的混凝土及钢製成，而結構設計的目標也是在一般結構荷重下，希望其形變量可以降到最小。不過若分析的結構物是較細較薄，較容易變形的元件（例如桿、平板及薄殼），用无限小应变理论來分析就不可靠了。

無限小應變張量
-------

在連續體的無限小變形中（其位移梯度張量遠小於1，也就是{\displaystyle \|\nabla \mathbf {u} \|\ll 1}），可以用有限應變理論中的任何一個有限應變張量（例如拉格朗日有限應變張量{\displaystyle \mathbf {E} }，或是尤拉有限應變張量{\displaystyle \mathbf {e} }）進行線性化。在線性化中，可以省略有限應變張量中的二次項或是非線性項，因此可得


E

=

1
2

(

∇

X

u

+
(

∇

X

u

)

T

+
(

∇

X

u

)

T

∇

X

u

)

≈

1
2

(

∇

X

u

+
(

∇

X

u

)

T

)

{\displaystyle \mathbf {E} ={\frac {1}{2}}\left(\nabla \_{\mathbf {X} }\mathbf {u} +(\nabla \_{\mathbf {X} }\mathbf {u} )^{T}+(\nabla \_{\mathbf {X} }\mathbf {u} )^{T}\nabla \_{\mathbf {X} }\mathbf {u} \right)\approx {\frac {1}{2}}\left(\nabla \_{\mathbf {X} }\mathbf {u} +(\nabla \_{\mathbf {X} }\mathbf {u} )^{T}\right)}

{\displaystyle \mathbf {E} ={\frac {1}{2}}\left(\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{T}+(\nabla _{\mathbf {X} }\mathbf {u} )^{T}\nabla _{\mathbf {X} }\mathbf {u} \right)\approx {\frac {1}{2}}\left(\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{T}\right)}
或


E

K
L

=

1
2

(

∂

U

K

∂

X

L


+

∂

U

L

∂

X

K


+

∂

U

M

∂

X

K



∂

U

M

∂

X

L



)

≈

1
2

(

∂

U

K

∂

X

L


+

∂

U

L

∂

X

K



)

{\displaystyle E\_{KL}={\frac {1}{2}}\left({\frac {\partial U\_{K}}{\partial X\_{L}}}+{\frac {\partial U\_{L}}{\partial X\_{K}}}+{\frac {\partial U\_{M}}{\partial X\_{K}}}{\frac {\partial U\_{M}}{\partial X\_{L}}}\right)\approx {\frac {1}{2}}\left({\frac {\partial U\_{K}}{\partial X\_{L}}}+{\frac {\partial U\_{L}}{\partial X\_{K}}}\right)}

{\displaystyle E_{KL}={\frac {1}{2}}\left({\frac {\partial U_{K}}{\partial X_{L}}}+{\frac {\partial U_{L}}{\partial X_{K}}}+{\frac {\partial U_{M}}{\partial X_{K}}}{\frac {\partial U_{M}}{\partial X_{L}}}\right)\approx {\frac {1}{2}}\left({\frac {\partial U_{K}}{\partial X_{L}}}+{\frac {\partial U_{L}}{\partial X_{K}}}\right)}
以及


e

=

1
2

(

∇

x

u

+
(

∇

x

u

)

T

−

∇

x

u

(

∇

x

u

)

T

)

≈

1
2

(

∇

x

u

+
(

∇

x

u

)

T

)

{\displaystyle \mathbf {e} ={\frac {1}{2}}\left(\nabla \_{\mathbf {x} }\mathbf {u} +(\nabla \_{\mathbf {x} }\mathbf {u} )^{T}-\nabla \_{\mathbf {x} }\mathbf {u} (\nabla \_{\mathbf {x} }\mathbf {u} )^{T}\right)\approx {\frac {1}{2}}\left(\nabla \_{\mathbf {x} }\mathbf {u} +(\nabla \_{\mathbf {x} }\mathbf {u} )^{T}\right)}

{\displaystyle \mathbf {e} ={\frac {1}{2}}\left(\nabla _{\mathbf {x} }\mathbf {u} +(\nabla _{\mathbf {x} }\mathbf {u} )^{T}-\nabla _{\mathbf {x} }\mathbf {u} (\nabla _{\mathbf {x} }\mathbf {u} )^{T}\right)\approx {\frac {1}{2}}\left(\nabla _{\mathbf {x} }\mathbf {u} +(\nabla _{\mathbf {x} }\mathbf {u} )^{T}\right)}
或


e

r
s

=

1
2

(

∂

u

r

∂

x

s


+

∂

u

s

∂

x

r


−

∂

u

k

∂

x

r



∂

u

k

∂

x

s



)

≈

1
2

(

∂

u

r

∂

x

s


+

∂

u

s

∂

x

r



)

{\displaystyle e\_{rs}={\frac {1}{2}}\left({\frac {\partial u\_{r}}{\partial x\_{s}}}+{\frac {\partial u\_{s}}{\partial x\_{r}}}-{\frac {\partial u\_{k}}{\partial x\_{r}}}{\frac {\partial u\_{k}}{\partial x\_{s}}}\right)\approx {\frac {1}{2}}\left({\frac {\partial u\_{r}}{\partial x\_{s}}}+{\frac {\partial u\_{s}}{\partial x\_{r}}}\right)}

{\displaystyle e_{rs}={\frac {1}{2}}\left({\frac {\partial u_{r}}{\partial x_{s}}}+{\frac {\partial u_{s}}{\partial x_{r}}}-{\frac {\partial u_{k}}{\partial x_{r}}}{\frac {\partial u_{k}}{\partial x_{s}}}\right)\approx {\frac {1}{2}}\left({\frac {\partial u_{r}}{\partial x_{s}}}+{\frac {\partial u_{s}}{\partial x_{r}}}\right)}
線性化意味著連續體中特定點的物質坐標（material coordinate）和空間坐標（spatial coordinate）差異很小，拉格朗日描述和尤拉描述近似相等。因此，物質位移梯度張量和空間位移梯度張量的分量也相近相等。可得


E

≈

e

≈

ε

=

1
2

(

(
∇

u

)

T

+
∇

u

)

{\displaystyle \mathbf {E} \approx \mathbf {e} \approx {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left((\nabla \mathbf {u} )^{T}+\nabla \mathbf {u} \right)}

{\displaystyle \mathbf {E} \approx \mathbf {e} \approx {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left((\nabla \mathbf {u} )^{T}+\nabla \mathbf {u} \right)}
或


E

K
L

≈

e

r
s

≈

ε

i
j

=

1
2

(

u

i
,
j

+

u

j
,
i

)

{\displaystyle E\_{KL}\approx e\_{rs}\approx \varepsilon \_{ij}={\frac {1}{2}}\left(u\_{i,j}+u\_{j,i}\right)}

{\displaystyle E_{KL}\approx e_{rs}\approx \varepsilon _{ij}={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)}
其中 \varepsilon\_{ij}是無限小應變張量（也稱為柯西應變張量、線性應變張量、小應變張量）的分量。



ε

i
j


=

1
2

(

u

i
,
j

+

u

j
,
i

)


=

[

ε

11


ε

12


ε

13


ε

21


ε

22


ε

23


ε

31


ε

32


ε

33


]



=

[



∂

u

1

∂

x

1



1
2

(

∂

u

1

∂

x

2


+

∂

u

2

∂

x

1



)


1
2

(

∂

u

1

∂

x

3


+

∂

u

3

∂

x

1



)


1
2

(

∂

u

2

∂

x

1


+

∂

u

1

∂

x

2



)



∂

u

2

∂

x

2



1
2

(

∂

u

2

∂

x

3


+

∂

u

3

∂

x

2



)


1
2

(

∂

u

3

∂

x

1


+

∂

u

1

∂

x

3



)


1
2

(

∂

u

3

∂

x

2


+

∂

u

2

∂

x

3



)



∂

u

3

∂

x

3



]



{\displaystyle {\begin{aligned}\varepsilon \_{ij}&={\frac {1}{2}}\left(u\_{i,j}+u\_{j,i}\right)\\&={\begin{bmatrix}\varepsilon \_{11}&\varepsilon \_{12}&\varepsilon \_{13}\\\varepsilon \_{21}&\varepsilon \_{22}&\varepsilon \_{23}\\\varepsilon \_{31}&\varepsilon \_{32}&\varepsilon \_{33}\\\end{bmatrix}}\\&={\begin{bmatrix}{\frac {\partial u\_{1}}{\partial x\_{1}}}&{\frac {1}{2}}\left({\frac {\partial u\_{1}}{\partial x\_{2}}}+{\frac {\partial u\_{2}}{\partial x\_{1}}}\right)&{\frac {1}{2}}\left({\frac {\partial u\_{1}}{\partial x\_{3}}}+{\frac {\partial u\_{3}}{\partial x\_{1}}}\right)\\{\frac {1}{2}}\left({\frac {\partial u\_{2}}{\partial x\_{1}}}+{\frac {\partial u\_{1}}{\partial x\_{2}}}\right)&{\frac {\partial u\_{2}}{\partial x\_{2}}}&{\frac {1}{2}}\left({\frac {\partial u\_{2}}{\partial x\_{3}}}+{\frac {\partial u\_{3}}{\partial x\_{2}}}\right)\\{\frac {1}{2}}\left({\frac {\partial u\_{3}}{\partial x\_{1}}}+{\frac {\partial u\_{1}}{\partial x\_{3}}}\right)&{\frac {1}{2}}\left({\frac {\partial u\_{3}}{\partial x\_{2}}}+{\frac {\partial u\_{2}}{\partial x\_{3}}}\right)&{\frac {\partial u\_{3}}{\partial x\_{3}}}\\\end{bmatrix}}\end{aligned}}}

{\displaystyle {\begin{aligned}\varepsilon _{ij}&={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)\\&={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{bmatrix}}\\&={\begin{bmatrix}{\frac {\partial u_{1}}{\partial x_{1}}}&{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{1}}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{1}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{1}}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{2}}{\partial x_{1}}}+{\frac {\partial u_{1}}{\partial x_{2}}}\right)&{\frac {\partial u_{2}}{\partial x_{2}}}&{\frac {1}{2}}\left({\frac {\partial u_{2}}{\partial x_{3}}}+{\frac {\partial u_{3}}{\partial x_{2}}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{3}}{\partial x_{1}}}+{\frac {\partial u_{1}}{\partial x_{3}}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{3}}{\partial x_{2}}}+{\frac {\partial u_{2}}{\partial x_{3}}}\right)&{\frac {\partial u_{3}}{\partial x_{3}}}\\\end{bmatrix}}\end{aligned}}}
或者使用不同的表示方式：



[

ε

x
x


ε

x
y


ε

x
z


ε

y
x


ε

y
y


ε

y
z


ε

z
x


ε

z
y


ε

z
z


]

=

[



∂

u

x

∂
x


1
2

(

∂

u

x

∂
y

+

∂

u

y

∂
x

)


1
2

(

∂

u

x

∂
z

+

∂

u

z

∂
x

)


1
2

(

∂

u

y

∂
x

+

∂

u

x

∂
y

)



∂

u

y

∂
y


1
2

(

∂

u

y

∂
z

+

∂

u

z

∂
y

)


1
2

(

∂

u

z

∂
x

+

∂

u

x

∂
z

)


1
2

(

∂

u

z

∂
y

+

∂

u

y

∂
z

)



∂

u

z

∂
z



]

{\displaystyle {\begin{bmatrix}\varepsilon \_{xx}&\varepsilon \_{xy}&\varepsilon \_{xz}\\\varepsilon \_{yx}&\varepsilon \_{yy}&\varepsilon \_{yz}\\\varepsilon \_{zx}&\varepsilon \_{zy}&\varepsilon \_{zz}\\\end{bmatrix}}={\begin{bmatrix}{\frac {\partial u\_{x}}{\partial x}}&{\frac {1}{2}}\left({\frac {\partial u\_{x}}{\partial y}}+{\frac {\partial u\_{y}}{\partial x}}\right)&{\frac {1}{2}}\left({\frac {\partial u\_{x}}{\partial z}}+{\frac {\partial u\_{z}}{\partial x}}\right)\\{\frac {1}{2}}\left({\frac {\partial u\_{y}}{\partial x}}+{\frac {\partial u\_{x}}{\partial y}}\right)&{\frac {\partial u\_{y}}{\partial y}}&{\frac {1}{2}}\left({\frac {\partial u\_{y}}{\partial z}}+{\frac {\partial u\_{z}}{\partial y}}\right)\\{\frac {1}{2}}\left({\frac {\partial u\_{z}}{\partial x}}+{\frac {\partial u\_{x}}{\partial z}}\right)&{\frac {1}{2}}\left({\frac {\partial u\_{z}}{\partial y}}+{\frac {\partial u\_{y}}{\partial z}}\right)&{\frac {\partial u\_{z}}{\partial z}}\\\end{bmatrix}}}

{\displaystyle {\begin{bmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{bmatrix}}={\begin{bmatrix}{\frac {\partial u_{x}}{\partial x}}&{\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial y}}+{\frac {\partial u_{y}}{\partial x}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial z}}+{\frac {\partial u_{z}}{\partial x}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\right)&{\frac {\partial u_{y}}{\partial y}}&{\frac {1}{2}}\left({\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\right)\\{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right)&{\frac {1}{2}}\left({\frac {\partial u_{z}}{\partial y}}+{\frac {\partial u_{y}}{\partial z}}\right)&{\frac {\partial u_{z}}{\partial z}}\\\end{bmatrix}}}
進一步來說，因為形變梯度可以表示成{\displaystyle {\boldsymbol {F}}={\boldsymbol {\nabla }}\mathbf {u} +{\boldsymbol {I}}} 其中{\displaystyle {\boldsymbol {I}}}是二階單位張量，可得


ε

=

1
2

(

F

T

+

F

)

−

I

{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left({\boldsymbol {F}}^{T}+{\boldsymbol {F}}\right)-{\boldsymbol {I}}}

{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left({\boldsymbol {F}}^{T}+{\boldsymbol {F}}\right)-{\boldsymbol {I}}}
另外，根據拉格朗日有限應變張量及尤拉有限應變張量的通用表示法，可得




E

(
m
)


=

1

2
m

(

U

2
m

−

I

)
=

1

2
m

[
(

F

T

F

)

m

−

I

]
≈

1

2
m

[
{

∇

u

+
(

∇

u

)

T

+

I

}

m

−

I

]
≈

ε


e

(
m
)


=

1

2
m

(

V

2
m

−

I

)
=

1

2
m

[
(

F

F

T

)

m

−

I

]
≈

ε


{\displaystyle {\begin{aligned}\mathbf {E} \_{(m)}&={\frac {1}{2m}}(\mathbf {U} ^{2m}-{\boldsymbol {I}})={\frac {1}{2m}}[({\boldsymbol {F}}^{T}{\boldsymbol {F}})^{m}-{\boldsymbol {I}}]\approx {\frac {1}{2m}}[\{{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}+{\boldsymbol {I}}\}^{m}-{\boldsymbol {I}}]\approx {\boldsymbol {\varepsilon }}\\\mathbf {e} \_{(m)}&={\frac {1}{2m}}(\mathbf {V} ^{2m}-{\boldsymbol {I}})={\frac {1}{2m}}[({\boldsymbol {F}}{\boldsymbol {F}}^{T})^{m}-{\boldsymbol {I}}]\approx {\boldsymbol {\varepsilon }}\end{aligned}}}

{\displaystyle {\begin{aligned}\mathbf {E} _{(m)}&={\frac {1}{2m}}(\mathbf {U} ^{2m}-{\boldsymbol {I}})={\frac {1}{2m}}[({\boldsymbol {F}}^{T}{\boldsymbol {F}})^{m}-{\boldsymbol {I}}]\approx {\frac {1}{2m}}[\{{\boldsymbol {\nabla }}\mathbf {u} +({\boldsymbol {\nabla }}\mathbf {u} )^{T}+{\boldsymbol {I}}\}^{m}-{\boldsymbol {I}}]\approx {\boldsymbol {\varepsilon }}\\\mathbf {e} _{(m)}&={\frac {1}{2m}}(\mathbf {V} ^{2m}-{\boldsymbol {I}})={\frac {1}{2m}}[({\boldsymbol {F}}{\boldsymbol {F}}^{T})^{m}-{\boldsymbol {I}}]\approx {\boldsymbol {\varepsilon }}\end{aligned}}}

相關條目
----

* 形變
* 应变协调性
* 應力
* 应变片
* 应力-应变曲线
* 胡克定律
* 泊松比
* 有限应变理论
* 應變率（英语：）
* 平面應力
* 數位影像相關性（英语：）

參考資料
----

1. ↑  Boresi, Arthur P. (Arthur Peter), 1924-.. Schmidt, Richard J. (Richard Joseph), 1954- 6th. New York: John Wiley & Sons. 2003: 62. ISBN 1601199228. OCLC 430194205.

外部連結
----

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

| * 
* 论
* 编
连续介质力学 |
| --- |
|  |
| 基本定律 | 
* 质量守恒
* 动量守恒
* 能量守恒
* 熵不等式

 |  |
|  |
| 固体力学 | 
* 固体
* 形變
* 弹性
* 塑性
* 胡克定律
* 應力
* 應變
* 有限应变理论
* 无限小应变理论
* 杨氏模量
* 剪切模量
* 体积模量
* 泊松比
* 彎曲

 |
|  |
| 流体力学 | 
* 流体
* 流量
* 流体静力学
* 黏度
* 表面张力
* 流體動力學
* 牛顿流体
* 非牛頓流體
* 伯努利定律

 |
|  |
| 流变学 | 
* 黏弹性
* 流变测量
* 流变仪
* 智能流体
* 电流变液（英语：）
* 磁流变液（英语：）
* 铁磁流体

 |
|  |
| 科學家 | 
* 波义耳
* 胡克
* 牛頓
* 伯努利
* 盖-吕萨克
* 納維
* 柯西
* 斯托克斯

 |

 |