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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