Euler.E2.80.93Cauchy stress principle .E2.80.93 stress vector Cauchy stress tensor

figure 2.1a internal distribution of contact forces , couple stresses on differential



d
s


{\displaystyle ds}

of internal surface



s


{\displaystyle s}

in continuum, result of interaction between 2 portions of continuum separated surface



figure 2.1b internal distribution of contact forces , couple stresses on differential



d
s


{\displaystyle ds}

of internal surface



s


{\displaystyle s}

in continuum, result of interaction between 2 portions of continuum separated surface



figure 2.1c stress vector on internal surface s normal vector n. depending on orientation of plane under consideration, stress vector may not perpendicular plane, i.e. parallel




n



{\displaystyle \mathbf {n} }

, , can resolved 2 components: 1 component normal plane, called normal stress




σ


n





{\displaystyle \sigma _{\mathrm {n} }}

, , component parallel plane, called shearing stress



τ


{\displaystyle \tau }

.

the euler–cauchy stress principle states upon surface (real or imaginary) divides body, action of 1 part of body on other equivalent (equipollent) system of distributed forces , couples on surface dividing body, , represented field





t


(

n

)




{\displaystyle \mathbf {t} ^{(\mathbf {n} )}}

, called stress vector, defined on surface



s


{\displaystyle s}

, assumed depend continuously on surface s unit vector




n



{\displaystyle \mathbf {n} }

.

to formulate euler–cauchy stress principle, consider imaginary surface



s


{\displaystyle s}

passing through internal material point



p


{\displaystyle p}

dividing continuous body 2 segments, seen in figure 2.1a or 2.1b (one may use either cutting plane diagram or diagram arbitrary volume inside continuum enclosed surface



s


{\displaystyle s}

).

following classical dynamics of newton , euler, motion of material body produced action of externally applied forces assumed of 2 kinds: surface forces




f



{\displaystyle \mathbf {f} }

, body forces




b



{\displaystyle \mathbf {b} }

. thus, total force





f




{\displaystyle {\mathcal {f}}}

applied body or portion of body can expressed as:

f


=

b

+

f



{\displaystyle {\mathcal {f}}=\mathbf {b} +\mathbf {f} }

only surface forces discussed in article relevant cauchy stress tensor.

when body subjected external surface forces or contact forces




f



{\displaystyle \mathbf {f} }

, following euler s equations of motion, internal contact forces , moments transmitted point point in body, , 1 segment other through dividing surface



s


{\displaystyle s}

, due mechanical contact of 1 portion of continuum onto other (figure 2.1a , 2.1b). on element of area



Δ
s


{\displaystyle \delta s}

containing



p


{\displaystyle p}

, normal vector




n



{\displaystyle \mathbf {n} }

, force distribution equipollent contact force



Δ

f



{\displaystyle \delta \mathbf {f} }

exerted @ point p , surface moment



Δ

m



{\displaystyle \delta \mathbf {m} }

. in particular, contact force given by

Δ

f

=


t


(

n

)



Δ
s


{\displaystyle \delta \mathbf {f} =\mathbf {t} ^{(\mathbf {n} )}\,\delta s}

where





t


(

n

)




{\displaystyle \mathbf {t} ^{(\mathbf {n} )}}

mean surface traction.

cauchy’s stress principle asserts



Δ
s


{\displaystyle \delta s}

becomes small , tends 0 ratio



Δ

f


/

Δ
s


{\displaystyle \delta \mathbf {f} /\delta s}

becomes



d

f


/

d
s


{\displaystyle d\mathbf {f} /ds}

, couple stress vector



Δ

m



{\displaystyle \delta \mathbf {m} }

vanishes. in specific fields of continuum mechanics couple stress assumed not vanish; however, classical branches of continuum mechanics address non-polar materials not consider couple stresses , body moments.

the resultant vector



d

f


/

d
s


{\displaystyle d\mathbf {f} /ds}

defined surface traction, called stress vector, traction, or traction vector. given





t


(

n

)


=

t

i


(

n

)




e


i




{\displaystyle \mathbf {t} ^{(\mathbf {n} )}=t_{i}^{(\mathbf {n} )}\mathbf {e} _{i}}

@ point



p


{\displaystyle p}

associated plane normal vector




n



{\displaystyle \mathbf {n} }

:

t

i


(

n

)


=

lim

Δ
s
→
0





Δ

f

i




Δ
s



=



d

f

i




d
s



.


{\displaystyle t_{i}^{(\mathbf {n} )}=\lim _{\delta s\to 0}{\frac {\delta f_{i}}{\delta s}}={df_{i} \over ds}.}

this equation means stress vector depends on location in body , orientation of plane on acting.

this implies balancing action of internal contact forces generates contact force density or cauchy traction field




t

(

n

,

x

,
t
)


{\displaystyle \mathbf {t} (\mathbf {n} ,\mathbf {x} ,t)}

represents distribution of internal contact forces throughout volume of body in particular configuration of body @ given time



t


{\displaystyle t}

. not vector field because depends not on position




x



{\displaystyle \mathbf {x} }

of particular material point, on local orientation of surface element defined normal vector




n



{\displaystyle \mathbf {n} }

.

depending on orientation of plane under consideration, stress vector may not perpendicular plane, i.e. parallel




n



{\displaystyle \mathbf {n} }

, , can resolved 2 components (figure 2.1c):

one normal plane, called normal stress








σ


n




=

lim

Δ
s
→
0





Δ

f


n





Δ
s



=



d

f


n





d
s



,


{\displaystyle \mathbf {\sigma _{\mathrm {n} }} =\lim _{\delta s\to 0}{\frac {\delta f_{\mathrm {n} }}{\delta s}}={\frac {df_{\mathrm {n} }}{ds}},}




where



d

f


n





{\displaystyle df_{\mathrm {n} }}

normal component of force



d

f



{\displaystyle d\mathbf {f} }

differential area



d
s


{\displaystyle ds}




and other parallel plane, called shear stress







τ

=

lim

Δ
s
→
0





Δ

f


s





Δ
s



=



d

f


s





d
s



,


{\displaystyle \mathbf {\tau } =\lim _{\delta s\to 0}{\frac {\delta f_{\mathrm {s} }}{\delta s}}={\frac {df_{\mathrm {s} }}{ds}},}




where



d

f


s





{\displaystyle df_{\mathrm {s} }}

tangential component of force



d

f



{\displaystyle d\mathbf {f} }

differential surface area



d
s


{\displaystyle ds}

. shear stress can further decomposed 2 mutually perpendicular vectors.

cauchy’s postulate

according cauchy postulate, stress vector





t


(

n

)




{\displaystyle \mathbf {t} ^{(\mathbf {n} )}}

remains unchanged surfaces passing through point



p


{\displaystyle p}

, having same normal vector




n



{\displaystyle \mathbf {n} }

@



p


{\displaystyle p}

, i.e., having common tangent @



p


{\displaystyle p}

. means stress vector function of normal vector




n



{\displaystyle \mathbf {n} }

only, , not influenced curvature of internal surfaces.

cauchy’s fundamental lemma

a consequence of cauchy’s postulate cauchy’s fundamental lemma, called cauchy reciprocal theorem, states stress vectors acting on opposite sides of same surface equal in magnitude , opposite in direction. cauchy’s fundamental lemma equivalent newton s third law of motion of action , reaction, , expressed as

−


t


(

n

)


=


t


(
−

n

)


.


{\displaystyle -\mathbf {t} ^{(\mathbf {n} )}=\mathbf {t} ^{(-\mathbf {n} )}.}






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