من أجل تشوهات متناهية الصغر لجسم المتصل , والتي تكون فيها الازاحة وتدرج الإزاحة صغيرة مقارنة بالوحدة، أي,
‖
u
‖
≪
1
{\displaystyle \|\mathbf {u} \|\ll 1\,\!}
‖
∇
u
‖
≪
1
{\displaystyle \|\nabla \mathbf {u} \|\ll 1\,\!}
الخطية الهندسية لموتّر لاغرانج محدود الاجهاد
E
{\displaystyle \mathbf {E} \,\!}
e
{\displaystyle \mathbf {e} \,\!}
E
=
1
2
(
u
∇
X
T
+
u
∇
X
+
u
∇
X
T
u
∇
X
)
≈
1
2
(
u
∇
X
T
+
u
∇
X
)
{\displaystyle \mathbf {E} ={\frac {1}{2}}\left(\mathbf {u} \nabla _{\mathbf {X} }^{T}+\mathbf {u} \nabla _{\mathbf {X} }+\mathbf {u} \nabla _{\mathbf {X} }^{T}\mathbf {u} \nabla _{\mathbf {X} }\right)\approx {\frac {1}{2}}\left(\mathbf {u} \nabla _{\mathbf {X} }^{T}+\mathbf {u} \nabla _{\mathbf {X} }\right)\,\!}
أو
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)\,\!}
و
e
=
1
2
(
u
∇
x
T
+
u
∇
x
−
u
∇
x
T
u
∇
x
)
≈
1
2
(
u
∇
x
T
+
u
∇
x
)
{\displaystyle \mathbf {e} ={\frac {1}{2}}\left(\mathbf {u} \nabla _{\mathbf {x} }^{T}+\mathbf {u} \nabla _{\mathbf {x} }-\mathbf {u} \nabla _{\mathbf {x} }^{T}\mathbf {u} \nabla _{\mathbf {x} }\right)\approx {\frac {1}{2}}\left(\mathbf {u} \nabla _{\mathbf {x} }^{T}+\mathbf {u} \nabla _{\mathbf {x} }\right)\,\!}
أو
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)\,\!}
تقتضي هذه الخطية بأن الوصف اللاغرانجي والوصف الأويلري تكون نفسها تقريبا طالما هناك فرق بسيط في المادة والإحداثيات المكانية لنقطة مادة معطاة في المتصل. لهذا، تكون مركبات التدرج الإزاحي المادي والمكاني متساوية تقريبا وعليه
E
≈
e
≈
ε
=
1
2
(
u
∇
T
+
u
∇
)
or
E
K
L
≈
e
r
s
≈
ε
i
j
=
1
2
(
u
i
,
j
+
u
j
,
i
)
{\displaystyle \mathbf {E} \approx \mathbf {e} \approx {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left(\mathbf {u} \nabla ^{T}+\mathbf {u} \nabla \right)\qquad {\text{or}}\qquad E_{KL}\approx e_{rs}\approx \varepsilon _{ij}={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)\,\!}
حيث
ε
i
j
{\displaystyle \varepsilon _{ij}\,\!}
الاجهاد متناهي الصغر
ε
{\displaystyle {\boldsymbol {\varepsilon }}\,\!}
موتّر إجهاد كوشي , موتّر اجهاد خطي , أو موتّر إجهاد صغير .
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 \varepsilon _{ij}={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)=\left[{\begin{matrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{matrix}}\right]=\left[{\begin{matrix}{\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{matrix}}\right]}
أو باستعمال علامات أخرى:
[
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 \left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}{\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{matrix}}\right]\,\!}
أبعد من ذلك,
ε
=
1
2
(
F
+
F
T
)
−
I
{\displaystyle {\boldsymbol {\varepsilon }}={\frac {1}{2}}\left(\mathbf {F} +\mathbf {F} ^{T}\right)-\mathbf {I} \,\!}
بأخذ تعبيري لاغرانج وأويلر محدودي الاجهاد بعين الاعتبار يصبح لدينا
E
(
m
)
=
1
2
m
(
U
2
m
−
I
)
≈
ε
{\displaystyle \mathbf {E} _{(m)}={\frac {1}{2m}}(\mathbf {U} ^{2m}-I)\approx \varepsilon \,\!}
e
(
m
)
=
1
2
m
(
V
2
m
−
I
)
≈
ε
{\displaystyle \mathbf {e} _{(m)}={\frac {1}{2m}}(\mathbf {V} ^{2m}-I)\approx \varepsilon \,\!}
شكل1. تشوه هندسي ثنائي البعد لعنصر مادي متناهي الصغر.
باعتبار تشوه ثنائي البعد لعنصر مادي مستطيل متناهي الصغر بالأبعاد
d
x
{\displaystyle dx\,\!}
d
y
{\displaystyle dy\,\!}
a
b
¯
=
(
d
x
+
∂
u
x
∂
x
d
x
)
2
+
(
∂
u
y
∂
x
d
x
)
2
=
1
+
2
∂
u
x
∂
x
+
(
∂
u
x
∂
x
)
2
+
(
∂
u
y
∂
x
)
2
d
x
{\displaystyle {\begin{aligned}{\overline {ab}}&={\sqrt {\left(dx+{\frac {\partial u_{x}}{\partial x}}dx\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}dx\right)^{2}}}\\&={\sqrt {1+2{\frac {\partial u_{x}}{\partial x}}+\left({\frac {\partial u_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial x}}\right)^{2}}}dx\\\end{aligned}}\,\!}
في كل تدرج إزاحة صغير,
‖
∇
u
‖
≪
1
{\displaystyle \|\nabla \mathbf {u} \|\ll 1\,\!}
a
b
¯
≈
d
x
+
∂
u
x
∂
x
d
x
{\displaystyle {\overline {ab}}\approx dx+{\frac {\partial u_{x}}{\partial x}}dx\,\!}
يكون التشوه باتجاه العنصر المستطيل
x
{\displaystyle x\,\!}
ε
x
=
a
b
¯
−
A
B
¯
A
B
¯
{\displaystyle \varepsilon _{x}={\frac {{\overline {ab}}-{\overline {AB}}}{\overline {AB}}}\,\!}
وإذا علم أن
A
B
¯
=
d
x
{\displaystyle {\overline {AB}}=dx\,\!}
ε
x
=
∂
u
x
∂
x
{\displaystyle \varepsilon _{x}={\frac {\partial u_{x}}{\partial x}}\,\!}
بالمثل, الاجهاد العمودي باتجاه
y
{\displaystyle y\,\!}
z
{\displaystyle z\,\!}
ε
y
=
∂
u
y
∂
y
,
ε
z
=
∂
u
z
∂
z
{\displaystyle \varepsilon _{y}={\frac {\partial u_{y}}{\partial y}}\quad ,\qquad \varepsilon _{z}={\frac {\partial u_{z}}{\partial z}}\,\!}
يعرف الاجهاد القصي الهندسي, التغير في الزاوية بين الخوط المتعامدة للمادة, في هذه الحالة الخط
A
C
¯
{\displaystyle {\overline {AC}}\,\!}
A
B
¯
{\displaystyle {\overline {AB}}\,\!}
γ
x
y
=
α
+
β
{\displaystyle \gamma _{xy}=\alpha +\beta \,\!}
من الشكل الهندسي 1 لدينا
tan
α
=
∂
u
y
∂
x
d
x
d
x
+
∂
u
x
∂
x
d
x
=
∂
u
y
∂
x
1
+
∂
u
x
∂
x
,
tan
β
=
∂
u
x
∂
y
d
y
d
y
+
∂
u
y
∂
y
d
y
=
∂
u
x
∂
y
1
+
∂
u
y
∂
y
{\displaystyle \tan \alpha ={\frac {{\dfrac {\partial u_{y}}{\partial x}}dx}{dx+{\dfrac {\partial u_{x}}{\partial x}}dx}}={\frac {\dfrac {\partial u_{y}}{\partial x}}{1+{\dfrac {\partial u_{x}}{\partial x}}}}\quad ,\qquad \tan \beta ={\frac {{\dfrac {\partial u_{x}}{\partial y}}dy}{dy+{\dfrac {\partial u_{y}}{\partial y}}dy}}={\frac {\dfrac {\partial u_{x}}{\partial y}}{1+{\dfrac {\partial u_{y}}{\partial y}}}}\,\!}
للدورانات الصغيرة, أي
α
{\displaystyle \alpha \,\!}
β
{\displaystyle \beta \,\!}
≪
1
{\displaystyle \ll 1\,\!}
tan
α
≈
α
,
tan
β
≈
β
{\displaystyle \tan \alpha \approx \alpha \quad ,\qquad \tan \beta \approx \beta \,\!}
مرة أخرى، لتدرجات الإزاحة الصغيرة يكون لدينا
α
=
∂
u
y
∂
x
,
β
=
∂
u
x
∂
y
{\displaystyle \alpha ={\frac {\partial u_{y}}{\partial x}}\quad ,\qquad \beta ={\frac {\partial u_{x}}{\partial y}}\,\!}
على ذلك
γ
x
y
=
α
+
β
=
∂
u
y
∂
x
+
∂
u
x
∂
y
{\displaystyle \gamma _{xy}=\alpha +\beta ={\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial y}}\,\!}
بالتبديل بين
x
{\displaystyle x\,\!}
y
{\displaystyle y\,\!}
u
x
{\displaystyle u_{x}\,\!}
u
y
{\displaystyle u_{y}\,\!}
γ
x
y
=
γ
y
x
{\displaystyle \gamma _{xy}=\gamma _{yx}\,\!}
يمكن ملاحظة أن مركبات اجهاد القص الموترية لموتر الاجهاد متناهي الصغر يمكن التعبير عنها باستخدام التعريف الهندسي للاجهاد,
γ
{\displaystyle \gamma \,\!}
[
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
]
=
[
ε
x
x
γ
x
y
/
2
γ
x
z
/
2
γ
y
x
/
2
ε
y
y
γ
y
z
/
2
γ
z
x
/
2
γ
z
y
/
2
ε
z
z
]
{\displaystyle \left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}\varepsilon _{xx}&\gamma _{xy}/2&\gamma _{xz}/2\\\gamma _{yx}/2&\varepsilon _{yy}&\gamma _{yz}/2\\\gamma _{zx}/2&\gamma _{zy}/2&\varepsilon _{zz}\\\end{matrix}}\right]\,\!}
التفسير الفيزيائي عدل من نظرية الاجهاد المحدود لدينا
d
x
2
−
d
X
2
=
d
X
⋅
2
E
⋅
d
X
or
(
d
x
)
2
−
(
d
X
)
2
=
2
E
K
L
d
X
K
d
X
L
{\displaystyle d\mathbf {x} ^{2}-d\mathbf {X} ^{2}=d\mathbf {X} \cdot 2\mathbf {E} \cdot d\mathbf {X} \quad {\text{or}}\quad (dx)^{2}-(dX)^{2}=2E_{KL}\,dX_{K}\,dX_{L}\,\!}
وعليه لدينا من الاجهاد متناهي الصغر
d
x
2
−
d
X
2
=
d
X
⋅
2
ε
⋅
d
X
or
(
d
x
)
2
−
(
d
X
)
2
=
2
ε
K
L
d
X
K
d
X
L
{\displaystyle d\mathbf {x} ^{2}-d\mathbf {X} ^{2}=d\mathbf {X} \cdot 2\mathbf {\boldsymbol {\varepsilon }} \cdot d\mathbf {X} \quad {\text{or}}\quad (dx)^{2}-(dX)^{2}=2\varepsilon _{KL}\,dX_{K}\,dX_{L}\,\!}
بالقسمة على
(
d
X
)
2
{\displaystyle (dX)^{2}\,\!}
d
x
−
d
X
d
X
d
x
+
d
X
d
X
=
2
ε
i
j
d
X
i
d
X
d
X
j
d
X
{\displaystyle {\frac {dx-dX}{dX}}{\frac {dx+dX}{dX}}=2\varepsilon _{ij}{\frac {dX_{i}}{dX}}{\frac {dX_{j}}{dX}}\,\!}
للتشوهات الصغيرة نفرض أن
d
x
≈
d
X
{\displaystyle dx\approx dX\,\!}
d
x
+
d
X
d
X
≈
2
{\displaystyle {\frac {dx+dX}{dX}}\approx 2\,\!}
وعليه
d
x
−
d
X
d
X
=
ε
i
j
N
i
N
j
=
N
⋅
ε
⋅
N
{\displaystyle {\frac {dx-dX}{dX}}=\varepsilon _{ij}N_{i}N_{j}=\mathbf {N} \cdot {\boldsymbol {\varepsilon }}\cdot \mathbf {N} \,\!}
حيث
N
i
=
d
X
i
d
X
{\displaystyle N_{i}={\frac {dX_{i}}{dX}}\,\!}
d
X
{\displaystyle d\mathbf {X} \,\!}
e
(
N
)
{\displaystyle e_{(\mathbf {N} )}\,\!}
N
{\displaystyle \mathbf {N} \,\!}
N
{\displaystyle \mathbf {N} \,\!}
X
1
{\displaystyle X_{1}\,\!}
N
=
I
1
{\displaystyle \mathbf {N} =\mathbf {I} _{1}\,\!}
e
(
I
1
)
=
I
1
⋅
ε
⋅
I
1
=
ε
11
{\displaystyle e_{(\mathbf {I} _{1})}=\mathbf {I} _{1}\cdot {\boldsymbol {\varepsilon }}\cdot \mathbf {I} _{1}=\varepsilon _{11}\,\!}
بالمثل,
N
=
I
2
{\displaystyle \mathbf {N} =\mathbf {I} _{2}\,\!}
N
=
I
3
{\displaystyle \mathbf {N} =\mathbf {I} _{3}\,\!}
ε
22
{\displaystyle \varepsilon _{22}\,\!}
ε
33
{\displaystyle \varepsilon _{33}\,\!}
التوتر الحجمي عدل إن التعرض (التغير النسبي في الحجم) هو أثر الموتر:
δ
=
Δ
V
V
0
=
ε
11
+
ε
22
+
ε
33
{\displaystyle \delta ={\frac {\Delta V}{V_{0}}}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\,\!}
في الواقع, إذا أخذنا مكعبا بعين الاعتبار بطول حافة a , فإنه شبه مكعب بعد التشوه (لا تتغير الزوايا في الحجم) مع الابعاد
a
⋅
(
1
+
ε
11
)
×
a
⋅
(
1
+
ε
22
)
×
a
⋅
(
1
+
ε
33
)
{\displaystyle a\cdot (1+\varepsilon _{11})\times a\cdot (1+\varepsilon _{22})\times a\cdot (1+\varepsilon _{33})\,\!}
V 0 = a 3 , وعليه
Δ
V
V
0
=
(
1
+
ε
11
+
ε
22
+
ε
33
+
ε
11
⋅
ε
22
+
ε
11
⋅
ε
33
+
ε
22
⋅
ε
33
+
ε
11
⋅
ε
22
⋅
ε
33
)
⋅
a
3
−
a
3
a
3
{\displaystyle {\frac {\Delta V}{V_{0}}}={\frac {\left(1+\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}+\varepsilon _{11}\cdot \varepsilon _{33}+\varepsilon _{22}\cdot \varepsilon _{33}+\varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\right)\cdot a^{3}-a^{3}}{a^{3}}}\,\!}
عندما نأخذ التشوهات الصغيرة بعين الاعتبار,
1
≫
ε
i
i
≫
ε
i
i
⋅
ε
j
j
≫
ε
11
⋅
ε
22
⋅
ε
33
{\displaystyle 1\gg \varepsilon _{ii}\gg \varepsilon _{ii}\cdot \varepsilon _{jj}\gg \varepsilon _{11}\cdot \varepsilon _{22}\cdot \varepsilon _{33}\,\!}
هي لذلك الصيغة.
التغيرات الحقيقية في الحجم (أعلى) والتقريب (الأسفل): يوضح الرسم الأخضر الحجم المقدر بينما الرسم البرتقالي يوضح الحجم المهمل
في حالة القص النقي, يمكن ملاحظة أنه لايوجد تغير في الحجم.
موتر محرف الاجهاد عدل موتر الاجهاد متناهي الصغر
ε
i
j
{\displaystyle \varepsilon _{ij}\,\!}
موتر الاجهاد المتوسط أوموتر الاجهاد الحجمي أو موتر الاجهاد الكروي ,
ε
M
δ
i
j
{\displaystyle \varepsilon _{M}\delta _{ij}\,\!}
مركبة انحرافية موتر محرف الاجهاد ,
ε
i
j
′
{\displaystyle \varepsilon '_{ij}\,\!}
ε
i
j
=
ε
i
j
′
+
ε
M
δ
i
j
{\displaystyle \varepsilon _{ij}=\varepsilon '_{ij}+\varepsilon _{M}\delta _{ij}\,\!}
حيث
ε
M
{\displaystyle \varepsilon _{M}\,\!}
ε
M
=
ε
k
k
3
=
ε
11
+
ε
22
+
ε
33
3
=
1
3
I
1
e
{\displaystyle \varepsilon _{M}={\frac {\varepsilon _{kk}}{3}}={\frac {\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}}{3}}={\tfrac {1}{3}}I_{1}^{e}\,\!}
يمكن الحصول على موتر محرف الاجهاد بطرح موتر متوسط الاجهاد من موتر الاجهاد متناهي الصغر:
ε
i
j
′
=
ε
i
j
−
ε
k
k
3
δ
i
j
[
ε
11
′
ε
12
′
ε
13
′
ε
21
′
ε
22
′
ε
23
′
ε
31
′
ε
32
′
ε
33
′
]
=
[
ε
11
ε
12
ε
13
ε
21
ε
22
ε
23
ε
31
ε
32
ε
33
]
−
[
ε
M
0
0
0
ε
M
0
0
0
ε
M
]
=
[
ε
11
−
ε
M
ε
12
ε
13
ε
21
ε
22
−
ε
M
ε
23
ε
31
ε
32
ε
33
−
ε
M
]
{\displaystyle {\begin{aligned}\ \varepsilon '_{ij}&=\varepsilon _{ij}-{\frac {\varepsilon _{kk}}{3}}\delta _{ij}\\\left[{\begin{matrix}\varepsilon '_{11}&\varepsilon '_{12}&\varepsilon '_{13}\\\varepsilon '_{21}&\varepsilon '_{22}&\varepsilon '_{23}\\\varepsilon '_{31}&\varepsilon '_{32}&\varepsilon '_{33}\\\end{matrix}}\right]&=\left[{\begin{matrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{matrix}}\right]-\left[{\begin{matrix}\varepsilon _{M}&0&0\\0&\varepsilon _{M}&0\\0&0&\varepsilon _{M}\\\end{matrix}}\right]\\&=\left[{\begin{matrix}\varepsilon _{11}-\varepsilon _{M}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}-\varepsilon _{M}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}-\varepsilon _{M}\\\end{matrix}}\right]\\\end{aligned}}\,\!}
بالنسبة لمركبات الاجهاد الموصوفة سابقا
ε
i
j
{\displaystyle \varepsilon _{ij}\,\!}
u
i
,
j
+
u
j
,
i
=
2
ε
i
j
{\displaystyle u_{i,j}+u_{j,i}=2\varepsilon _{ij}\,\!}
u
i
{\displaystyle u_{i}\,\!}
معادلات التوافق . بإضافة معادلات التوافق الثلاث، يتم إنقاص المعادلات المستقلة إلى ثلاثة, والتي تتلائم مع مركبات الازاحة المجهولة. اكتشفت هذه القيود من قبل ساينت- فينات ، كما يطلق عليها "معادلات توافق ساينت-فينات ".
تفيد دالة التوافق في التأكد من دالة ازاحة متصلة وحيدة القيمة
u
i
{\displaystyle u_{i}\,\!}
بعلامة المعامل, يعبر عن معادلات التوافق
ε
i
j
,
k
m
+
ε
k
m
,
i
j
−
ε
i
k
,
j
m
−
ε
j
m
,
i
k
=
0
{\displaystyle \varepsilon _{ij,km}+\varepsilon _{km,ij}-\varepsilon _{ik,jm}-\varepsilon _{jm,ik}=0\,\!}
العلامة الهندسية
∂
2
ϵ
x
∂
y
2
+
∂
2
ϵ
y
∂
x
2
=
2
∂
2
ϵ
x
y
∂
x
∂
y
{\displaystyle {\frac {\partial ^{2}\epsilon _{x}}{\partial y^{2}}}+{\frac {\partial ^{2}\epsilon _{y}}{\partial x^{2}}}=2{\frac {\partial ^{2}\epsilon _{xy}}{\partial x\partial y}}\,\!}
∂
2
ϵ
y
∂
z
2
+
∂
2
ϵ
z
∂
y
2
=
2
∂
2
ϵ
y
z
∂
y
∂
z
{\displaystyle {\frac {\partial ^{2}\epsilon _{y}}{\partial z^{2}}}+{\frac {\partial ^{2}\epsilon _{z}}{\partial y^{2}}}=2{\frac {\partial ^{2}\epsilon _{yz}}{\partial y\partial z}}\,\!}
∂
2
ϵ
x
∂
z
2
+
∂
2
ϵ
z
∂
x
2
=
2
∂
2
ϵ
z
x
∂
z
∂
x
{\displaystyle {\frac {\partial ^{2}\epsilon _{x}}{\partial z^{2}}}+{\frac {\partial ^{2}\epsilon _{z}}{\partial x^{2}}}=2{\frac {\partial ^{2}\epsilon _{zx}}{\partial z\partial x}}\,\!}
∂
2
ϵ
x
∂
y
∂
z
=
∂
∂
x
(
−
∂
ϵ
y
z
∂
x
+
∂
ϵ
z
x
∂
y
+
∂
ϵ
x
y
∂
z
)
{\displaystyle {\frac {\partial ^{2}\epsilon _{x}}{\partial y\partial z}}={\frac {\partial }{\partial x}}\left(-{\frac {\partial \epsilon _{yz}}{\partial x}}+{\frac {\partial \epsilon _{zx}}{\partial y}}+{\frac {\partial \epsilon _{xy}}{\partial z}}\right)\,\!}
∂
2
ϵ
y
∂
z
∂
x
=
∂
∂
y
(
∂
ϵ
y
z
∂
x
−
∂
ϵ
z
x
∂
y
+
∂
ϵ
x
y
∂
z
)
{\displaystyle {\frac {\partial ^{2}\epsilon _{y}}{\partial z\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial \epsilon _{yz}}{\partial x}}-{\frac {\partial \epsilon _{zx}}{\partial y}}+{\frac {\partial \epsilon _{xy}}{\partial z}}\right)\,\!}
∂
2
ϵ
z
∂
x
∂
y
=
∂
∂
z
(
∂
ϵ
y
z
∂
x
+
∂
ϵ
z
x
∂
y
−
∂
ϵ
x
y
∂
z
)
{\displaystyle {\frac {\partial ^{2}\epsilon _{z}}{\partial x\partial y}}={\frac {\partial }{\partial z}}\left({\frac {\partial \epsilon _{yz}}{\partial x}}+{\frac {\partial \epsilon _{zx}}{\partial y}}-{\frac {\partial \epsilon _{xy}}{\partial z}}\right)\,\!}
![{\displaystyle \varepsilon _{ij}={\frac {1}{2}}\left(u_{i,j}+u_{j,i}\right)=\left[{\begin{matrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{matrix}}\right]=\left[{\begin{matrix}{\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{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91d0f82a66f61d92a55dab6695e800e920481d5d)
![{\displaystyle \left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}{\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{matrix}}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb798dd9cff8404470ba843605146da23dbcd638)
![{\displaystyle \left[{\begin{matrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\\\end{matrix}}\right]=\left[{\begin{matrix}\varepsilon _{xx}&\gamma _{xy}/2&\gamma _{xz}/2\\\gamma _{yx}/2&\varepsilon _{yy}&\gamma _{yz}/2\\\gamma _{zx}/2&\gamma _{zy}/2&\varepsilon _{zz}\\\end{matrix}}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/700291ced7f7d577cc5a364789e9a14f4812b125)
![{\displaystyle {\begin{aligned}\ \varepsilon '_{ij}&=\varepsilon _{ij}-{\frac {\varepsilon _{kk}}{3}}\delta _{ij}\\\left[{\begin{matrix}\varepsilon '_{11}&\varepsilon '_{12}&\varepsilon '_{13}\\\varepsilon '_{21}&\varepsilon '_{22}&\varepsilon '_{23}\\\varepsilon '_{31}&\varepsilon '_{32}&\varepsilon '_{33}\\\end{matrix}}\right]&=\left[{\begin{matrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}\\\end{matrix}}\right]-\left[{\begin{matrix}\varepsilon _{M}&0&0\\0&\varepsilon _{M}&0\\0&0&\varepsilon _{M}\\\end{matrix}}\right]\\&=\left[{\begin{matrix}\varepsilon _{11}-\varepsilon _{M}&\varepsilon _{12}&\varepsilon _{13}\\\varepsilon _{21}&\varepsilon _{22}-\varepsilon _{M}&\varepsilon _{23}\\\varepsilon _{31}&\varepsilon _{32}&\varepsilon _{33}-\varepsilon _{M}\\\end{matrix}}\right]\\\end{aligned}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c16ea83935162fe6447be5762cf01547e681dc91)
مقالة مفصلة: قص ضد المستوى
