Biomedical Engineering Reference
In-Depth Information
σ
yy
y
0
σ
xy
σ
xx
x
0
Figure 13.2
Stress components in a plane stress state.
σ
xx
=
σ
xx
(
x
0
,
y
0
)
σ
yy
=
σ
yy
(
x
0
,
y
0
)
(13.15)
σ
xy
=
σ
yx
=
σ
xy
(
x
0
,
y
0
).
The stress components
σ
zz
,
σ
xy
=
σ
yx
and
σ
xz
=
σ
zx
are assumed to be zero
(negligible).
Based on Hooke's law (see Section
12.2
for the fully elaborated expression in
components), using
σ
zz
=
0 it is found:
ν
1
−
ν
ε
zz
=−
(
ε
xx
+
ε
yy
) ,
(13.16)
and by exploiting this equation, the description of the (linearly elastic) material
behaviour for plane stress becomes
E
σ
xx
=
2
(
ε
xx
+
νε
yy
)
(13.17)
1
−
ν
E
σ
yy
=
2
(
νε
xx
+
ε
yy
)
(13.18)
−
ν
1
E
(1
+
ν
)
ε
xy
,
σ
xy
=
(13.19)
thus coupling the relevant stresses and strains. It should be noted that in this case
for the material parameters the Young's modulus
E
and the Poisson's ratio
have
been used, instead of the compression modulus
K
and the shear modulus
G
in
previous sections.
ν
