Cauchy.E2.80.99s stress theorem.E2.80.94stress tensor Cauchy stress tensor

figure 2.2. stress vector acting on plane normal unit vector n.

a note on sign convention: tetrahedron formed slicing parallelepiped along arbitrary plane n. so, force acting on plane n reaction exerted other half of parallelepiped , has opposite sign.

where right-hand-side represents product of mass enclosed tetrahedron , acceleration: ρ density, acceleration, , h height of tetrahedron, considering plane n base. area of faces of tetrahedron perpendicular axes can found projecting da each face (using dot product):

d

a

1


=

(

n

⋅


e


1


)

d
a
=

n

1



d
a
,


{\displaystyle da_{1}=\left(\mathbf {n} \cdot \mathbf {e} _{1}\right)da=n_{1}\;da,}






d

a

2


=

(

n

⋅


e


2


)

d
a
=

n

2



d
a
,


{\displaystyle da_{2}=\left(\mathbf {n} \cdot \mathbf {e} _{2}\right)da=n_{2}\;da,}






d

a

3


=

(

n

⋅


e


3


)

d
a
=

n

3



d
a
,


{\displaystyle da_{3}=\left(\mathbf {n} \cdot \mathbf {e} _{3}\right)da=n_{3}\;da,}

and substituting equation cancel out da:

t


(

n

)


−


t


(


e


1


)



n

1


−


t


(


e


2


)



n

2


−


t


(


e


3


)



n

3


=
ρ

(


h
3


)


a

.


{\displaystyle \mathbf {t} ^{(\mathbf {n} )}-\mathbf {t} ^{(\mathbf {e} _{1})}n_{1}-\mathbf {t} ^{(\mathbf {e} _{2})}n_{2}-\mathbf {t} ^{(\mathbf {e} _{3})}n_{3}=\rho \left({\frac {h}{3}}\right)\mathbf {a} .}

to consider limiting case tetrahedron shrinks point, h must go 0 (intuitively, plane n translated along n toward o). result, right-hand-side of equation approaches 0, so

t


(

n

)


=


t


(


e


1


)



n

1


+


t


(


e


2


)



n

2


+


t


(


e


3


)



n

3


.


{\displaystyle \mathbf {t} ^{(\mathbf {n} )}=\mathbf {t} ^{(\mathbf {e} _{1})}n_{1}+\mathbf {t} ^{(\mathbf {e} _{2})}n_{2}+\mathbf {t} ^{(\mathbf {e} _{3})}n_{3}.}

assuming material element (figure 2.3) planes perpendicular coordinate axes of cartesian coordinate system, stress vectors associated each of element planes, i.e. t, t, , t can decomposed normal component , 2 shear components, i.e. components in direction of 3 coordinate axes. particular case of surface normal unit vector oriented in direction of x1-axis, denote normal stress σ11, , 2 shear stresses σ12 , σ13:

t


(


e


1


)


=

t

1


(


e


1


)




e


1


+

t

2


(


e


1


)




e


2


+

t

3


(


e


1


)




e


3


=

σ

11




e


1


+

σ

12




e


2


+

σ

13




e


3


,


{\displaystyle \mathbf {t} ^{(\mathbf {e} _{1})}=t_{1}^{(\mathbf {e} _{1})}\mathbf {e} _{1}+t_{2}^{(\mathbf {e} _{1})}\mathbf {e} _{2}+t_{3}^{(\mathbf {e} _{1})}\mathbf {e} _{3}=\sigma _{11}\mathbf {e} _{1}+\sigma _{12}\mathbf {e} _{2}+\sigma _{13}\mathbf {e} _{3},}








t


(


e


2


)


=

t

1


(


e


2


)




e


1


+

t

2


(


e


2


)




e


2


+

t

3


(


e


2


)




e


3


=

σ

21




e


1


+

σ

22




e


2


+

σ

23




e


3


,


{\displaystyle \mathbf {t} ^{(\mathbf {e} _{2})}=t_{1}^{(\mathbf {e} _{2})}\mathbf {e} _{1}+t_{2}^{(\mathbf {e} _{2})}\mathbf {e} _{2}+t_{3}^{(\mathbf {e} _{2})}\mathbf {e} _{3}=\sigma _{21}\mathbf {e} _{1}+\sigma _{22}\mathbf {e} _{2}+\sigma _{23}\mathbf {e} _{3},}








t


(


e


3


)


=

t

1


(


e


3


)




e


1


+

t

2


(


e


3


)




e


2


+

t

3


(


e


3


)




e


3


=

σ

31




e


1


+

σ

32




e


2


+

σ

33




e


3


,


{\displaystyle \mathbf {t} ^{(\mathbf {e} _{3})}=t_{1}^{(\mathbf {e} _{3})}\mathbf {e} _{1}+t_{2}^{(\mathbf {e} _{3})}\mathbf {e} _{2}+t_{3}^{(\mathbf {e} _{3})}\mathbf {e} _{3}=\sigma _{31}\mathbf {e} _{1}+\sigma _{32}\mathbf {e} _{2}+\sigma _{33}\mathbf {e} _{3},}

in index notation is

t


(


e


i


)


=

t

j


(


e


i


)




e


j


=

σ

i
j




e


j


.


{\displaystyle \mathbf {t} ^{(\mathbf {e} _{i})}=t_{j}^{(\mathbf {e} _{i})}\mathbf {e} _{j}=\sigma _{ij}\mathbf {e} _{j}.}

the 9 components σij of stress vectors components of second-order cartesian tensor called cauchy stress tensor, defines state of stress @ point , given by

σ

=

σ

i
j


=

[






t


(


e


1


)








t


(


e


2


)








t


(


e


3


)






]

=

[





σ

11





σ

12





σ

13







σ

21





σ

22





σ

23







σ

31





σ

32





σ

33






]

≡

[





σ

x
x





σ

x
y





σ

x
z







σ

y
x





σ

y
y





σ

y
z







σ

z
x





σ

z
y





σ

z
z






]

≡

[





σ

x





τ

x
y





τ

x
z







τ

y
x





σ

y





τ

y
z







τ

z
x





τ

z
y





σ

z






]

,


{\displaystyle {\boldsymbol {\sigma }}=\sigma _{ij}=\left[{\begin{matrix}\mathbf {t} ^{(\mathbf {e} _{1})}\\\mathbf {t} ^{(\mathbf {e} _{2})}\\\mathbf {t} ^{(\mathbf {e} _{3})}\\\end{matrix}}\right]=\left[{\begin{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{matrix}}\right]\equiv \left[{\begin{matrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{matrix}}\right]\equiv \left[{\begin{matrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\\\end{matrix}}\right],}

where σ11, σ22, , σ33 normal stresses, , σ12, σ13, σ21, σ23, σ31, , σ32 shear stresses. first index indicates stress acts on plane normal xi -axis, , second index j denotes direction in stress acts (for example, σ12 implies stress acting on plane normal 1st axis i.e.;x1 , acts along 2nd axis i.e.;x2). stress component positive if acts in positive direction of coordinate axes, , if plane acts has outward normal vector pointing in positive coordinate direction.

thus, using components of stress tensor

t


(

n

)





=


t


(


e


1


)



n

1


+


t


(


e


2


)



n

2


+


t


(


e


3


)



n

3








=

∑

i
=
1


3




t


(


e


i


)



n

i








=

(

σ

i
j




e


j


)


n

i








=

σ

i
j



n

i




e


j








{\displaystyle {\begin{aligned}\mathbf {t} ^{(\mathbf {n} )}&=\mathbf {t} ^{(\mathbf {e} _{1})}n_{1}+\mathbf {t} ^{(\mathbf {e} _{2})}n_{2}+\mathbf {t} ^{(\mathbf {e} _{3})}n_{3}\\&=\sum _{i=1}^{3}\mathbf {t} ^{(\mathbf {e} _{i})}n_{i}\\&=\left(\sigma _{ij}\mathbf {e} _{j}\right)n_{i}\\&=\sigma _{ij}n_{i}\mathbf {e} _{j}\end{aligned}}}

or, equivalently,

t

j


(

n

)


=

σ

i
j



n

i


.


{\displaystyle t_{j}^{(\mathbf {n} )}=\sigma _{ij}n_{i}.}

alternatively, in matrix form have

[





t

1


(

n

)





t

2


(

n

)





t

3


(

n

)






]

=

[





n

1





n

2





n

3






]

⋅

[





σ

11





σ

12





σ

13







σ

21





σ

22





σ

23







σ

31





σ

32





σ

33






]

.


{\displaystyle \left[{\begin{matrix}t_{1}^{(\mathbf {n} )}&t_{2}^{(\mathbf {n} )}&t_{3}^{(\mathbf {n} )}\end{matrix}}\right]=\left[{\begin{matrix}n_{1}&n_{2}&n_{3}\end{matrix}}\right]\cdot \left[{\begin{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{matrix}}\right].}

the voigt notation representation of cauchy stress tensor takes advantage of symmetry of stress tensor express stress six-dimensional vector of form:

σ

=



[




σ

1





σ

2





σ

3





σ

4





σ

5





σ

6





]



t


≡



[




σ

11





σ

22





σ

33





σ

23





σ

13





σ

12





]



t


.


{\displaystyle {\boldsymbol {\sigma }}={\begin{bmatrix}\sigma _{1}&\sigma _{2}&\sigma _{3}&\sigma _{4}&\sigma _{5}&\sigma _{6}\end{bmatrix}}^{t}\equiv {\begin{bmatrix}\sigma _{11}&\sigma _{22}&\sigma _{33}&\sigma _{23}&\sigma _{13}&\sigma _{12}\end{bmatrix}}^{t}.}

the voigt notation used extensively in representing stress–strain relations in solid mechanics , computational efficiency in numerical structural mechanics software.

transformation rule of stress tensor

it can shown stress tensor contravariant second order tensor, statement of how transforms under change of coordinate system. xi-system xi -system, components σij in initial system transformed components σij in new system according tensor transformation rule (figure 2.4):

σ

i
j

′

=

a

i
m



a

j
n



σ

m
n




or




σ

′

=

a


σ



a


t


,


{\displaystyle \sigma _{ij}=a_{im}a_{jn}\sigma _{mn}\quad {\text{or}}\quad {\boldsymbol {\sigma }} =\mathbf {a} {\boldsymbol {\sigma }}\mathbf {a} ^{t},}

where rotation matrix components aij. in matrix form is

[





σ

11

′




σ

12

′




σ

13

′






σ

21

′




σ

22

′




σ

23

′






σ

31

′




σ

32

′




σ

33

′





]

=

[





a

11





a

12





a

13







a

21





a

22





a

23







a

31





a

32





a

33






]


[





σ

11





σ

12





σ

13







σ

21





σ

22





σ

23







σ

31





σ

32





σ

33






]


[





a

11





a

21





a

31







a

12





a

22





a

32







a

13





a

23





a

33






]

.


{\displaystyle \left[{\begin{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{matrix}}\right]=\left[{\begin{matrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{matrix}}\right]\left[{\begin{matrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{matrix}}\right]\left[{\begin{matrix}a_{11}&a_{21}&a_{31}\\a_{12}&a_{22}&a_{32}\\a_{13}&a_{23}&a_{33}\\\end{matrix}}\right].}




figure 2.4 transformation of stress tensor

expanding matrix operation, , simplifying terms using symmetry of stress tensor, gives

σ

11

′

=

a

11


2



σ

11


+

a

12


2



σ

22


+

a

13


2



σ

33


+
2

a

11



a

12



σ

12


+
2

a

11



a

13



σ

13


+
2

a

12



a

13



σ

23


,


{\displaystyle \sigma _{11} =a_{11}^{2}\sigma _{11}+a_{12}^{2}\sigma _{22}+a_{13}^{2}\sigma _{33}+2a_{11}a_{12}\sigma _{12}+2a_{11}a_{13}\sigma _{13}+2a_{12}a_{13}\sigma _{23},}







σ

22

′

=

a

21


2



σ

11


+

a

22


2



σ

22


+

a

23


2



σ

33


+
2

a

21



a

22



σ

12


+
2

a

21



a

23



σ

13


+
2

a

22



a

23



σ

23


,


{\displaystyle \sigma _{22} =a_{21}^{2}\sigma _{11}+a_{22}^{2}\sigma _{22}+a_{23}^{2}\sigma _{33}+2a_{21}a_{22}\sigma _{12}+2a_{21}a_{23}\sigma _{13}+2a_{22}a_{23}\sigma _{23},}







σ

33

′

=

a

31


2



σ

11


+

a

32


2



σ

22


+

a

33


2



σ

33


+
2

a

31



a

32



σ

12


+
2

a

31



a

33



σ

13


+
2

a

32



a

33



σ

23


,


{\displaystyle \sigma _{33} =a_{31}^{2}\sigma _{11}+a_{32}^{2}\sigma _{22}+a_{33}^{2}\sigma _{33}+2a_{31}a_{32}\sigma _{12}+2a_{31}a_{33}\sigma _{13}+2a_{32}a_{33}\sigma _{23},}











σ

12

′

=



a

11



a

21



σ

11


+

a

12



a

22



σ

22


+

a

13



a

23



σ

33








+
(

a

11



a

22


+

a

12



a

21


)

σ

12


+
(

a

12



a

23


+

a

13



a

22


)

σ

23


+
(

a

11



a

23


+

a

13



a

21


)

σ

13


,






{\displaystyle {\begin{aligned}\sigma _{12} =&a_{11}a_{21}\sigma _{11}+a_{12}a_{22}\sigma _{22}+a_{13}a_{23}\sigma _{33}\\&+(a_{11}a_{22}+a_{12}a_{21})\sigma _{12}+(a_{12}a_{23}+a_{13}a_{22})\sigma _{23}+(a_{11}a_{23}+a_{13}a_{21})\sigma _{13},\end{aligned}}}











σ

23

′

=



a

21



a

31



σ

11


+

a

22



a

32



σ

22


+

a

23



a

33



σ

33








+
(

a

21



a

32


+

a

22



a

31


)

σ

12


+
(

a

22



a

33


+

a

23



a

32


)

σ

23


+
(

a

21



a

33


+

a

23



a

31


)

σ

13


,






{\displaystyle {\begin{aligned}\sigma _{23} =&a_{21}a_{31}\sigma _{11}+a_{22}a_{32}\sigma _{22}+a_{23}a_{33}\sigma _{33}\\&+(a_{21}a_{32}+a_{22}a_{31})\sigma _{12}+(a_{22}a_{33}+a_{23}a_{32})\sigma _{23}+(a_{21}a_{33}+a_{23}a_{31})\sigma _{13},\end{aligned}}}











σ

13

′

=



a

11



a

31



σ

11


+

a

12



a

32



σ

22


+

a

13



a

33



σ

33








+
(

a

11



a

32


+

a

12



a

31


)

σ

12


+
(

a

12



a

33


+

a

13



a

32


)

σ

23


+
(

a

11



a

33


+

a

13



a

31


)

σ

13


.






{\displaystyle {\begin{aligned}\sigma _{13} =&a_{11}a_{31}\sigma _{11}+a_{12}a_{32}\sigma _{22}+a_{13}a_{33}\sigma _{33}\\&+(a_{11}a_{32}+a_{12}a_{31})\sigma _{12}+(a_{12}a_{33}+a_{13}a_{32})\sigma _{23}+(a_{11}a_{33}+a_{13}a_{31})\sigma _{13}.\end{aligned}}}

the mohr circle stress graphical representation of transformation of stresses.

normal , shear stresses

the magnitude of normal stress component σn of stress vector t acting on arbitrary plane normal unit vector n @ given point, in terms of components σij of stress tensor σ, dot product of stress vector , normal unit vector:

σ


n






=


t


(

n

)


⋅

n







=

t

i


(

n

)



n

i








=

σ

i
j



n

i



n

j


.






{\displaystyle {\begin{aligned}\sigma _{\mathrm {n} }&=\mathbf {t} ^{(\mathbf {n} )}\cdot \mathbf {n} \\&=t_{i}^{(\mathbf {n} )}n_{i}\\&=\sigma _{ij}n_{i}n_{j}.\end{aligned}}}

the magnitude of shear stress component τn, acting orthogonal vector n, can found using pythagorean theorem:

τ


n






=




(

t

(

n

)


)


2


−

σ


n



2










=



t

i


(

n

)



t

i


(

n

)


−

σ


n



2




,






{\displaystyle {\begin{aligned}\tau _{\mathrm {n} }&={\sqrt {\left(t^{(\mathbf {n} )}\right)^{2}-\sigma _{\mathrm {n} }^{2}}}\\&={\sqrt {t_{i}^{(\mathbf {n} )}t_{i}^{(\mathbf {n} )}-\sigma _{\mathrm {n} }^{2}}},\end{aligned}}}

where

(

t

(

n

)


)


2


=

t

i


(

n

)



t

i


(

n

)


=

(

σ

i
j



n

j


)


(

σ

i
k



n

k


)

=

σ

i
j



σ

i
k



n

j



n

k


.


{\displaystyle \left(t^{(\mathbf {n} )}\right)^{2}=t_{i}^{(\mathbf {n} )}t_{i}^{(\mathbf {n} )}=\left(\sigma _{ij}n_{j}\right)\left(\sigma _{ik}n_{k}\right)=\sigma _{ij}\sigma _{ik}n_{j}n_{k}.}






^ cite error: named reference chen invoked never defined (see page).
^ cite error: named reference atanackovic invoked never defined (see page).
^ cite error: named reference irgens invoked never defined (see page).