Balance laws .E2.80.93 Cauchy.27s equations of motion Cauchy stress tensor

figure 4. continuum body in equilibrium


cauchy s first law of motion

according principle of conservation of linear momentum, if continuum body in static equilibrium can demonstrated components of cauchy stress tensor in every material point in body satisfy equilibrium equations.

σ

j
i
,
j


+

f

i


=
0


{\displaystyle \sigma _{ji,j}+f_{i}=0}

for example, hydrostatic fluid in equilibrium conditions, stress tensor takes on form:

σ

i
j



=
−
p


δ

i
j



,


{\displaystyle {\sigma _{ij}}=-p{\delta _{ij}},}

where



p


{\displaystyle p}

hydrostatic pressure, ,





δ

i
j



 


{\displaystyle {\delta _{ij}}\ }

kronecker delta.

cauchy s second law of motion

according principle of conservation of angular momentum, equilibrium requires summation of moments respect arbitrary point zero, leads conclusion stress tensor symmetric, having 6 independent stress components, instead of original nine:

σ

i
j


=

σ

j
i




{\displaystyle \sigma _{ij}=\sigma _{ji}}

however, in presence of couple-stresses, i.e. moments per unit volume, stress tensor non-symmetric. case when knudsen number close one,




k

n


→
1


{\displaystyle k_{n}\rightarrow 1}

, or continuum non-newtonian fluid, can lead rotationally non-invariant fluids, such polymers.