Civil Engineering Reference
In-Depth Information
1.10 Corollary.
(
∇
φ)
=
˜
(
∇
φ
T
∇
φ) for an isotropic and objective material,
where
˜
(C)
=
γ
0
I
+
γ
1
C
+
γ
2
C
2
,
(
1
.
28
)
and where γ
0
,γ
1
,γ
2
are functions of the invariants ı
C
.
Linear Material Laws
The stress-strain relationship can be described in terms of two parameters in the
neighborhood of a strain-free reference configuration. Setting
C
=
I
+
2
E
in
(1.28),
˜
2
E)
=
γ
0
(E) I
+
γ
1
(E) E
+
γ
2
(E) E
2
, where we have not changed
the notation for the functions.
(I
+
1.11 Theorem.
Suppose that in addition to the hypotheses of Corollary 1.10, γ
0
,γ
1
and γ
2
are differentiable functions of ı
1
(E), ı
2
(E) and ı
3
(E). Then there exist
numbers π, λ, µ with
˜
(I
+
2
E)
=−
πI
+
λ
trace
(E) I
+
2
µE
+
o(E)
as E
→
0
.
Sketch of a proof.
First note that
˜
2
E)
=
γ
0
(E) I
+
γ
1
E
+
o(E)
. In particular,
only the constant term in
γ
1
is used. By Remarks 1.9, we know that
˜
(
1
+
(I)
=−
πI
0. By (1.24), we deduce that
ı
2
=
O
(E
2
)
and
ı
3
=
O
(E
3
)
,
and only the constants and the trace remain in the terms of first order in
γ
0
(E)
.
with a suitable
π
≥
Normally, the situation
C
=
I
corresponds to an unstressed condition, and
π
=
0. The other two constants are called
Lame constants
. If we ignore the terms
of higher order, we are led to the
linear material law of Hooke
:
˜
(I
+
2
E)
=
λ
trace
(E) I
+
2
µE.
(
1
.
29
)
A material which satisfies (1.29) in general and not just for small strains is called
a
St. Venant-Kirchhoff material.
Note that in the approximation (1.6),
trace
(ε)
=
div
u,
(
1
.
30
)
and thus the Lame constant
λ
describes the stresses due to change in density. The
other Lame constant
µ
is sometimes called the
shear modulus of the material.
If we use a different set of frequently used parameters, namely
Young's mod-
ulus of elasticity
and the
Poisson ratio ν
, we have the following relationship:
E
λ
2
(λ
+
µ)
,
µ(
3
λ
+
2
µ)
λ
+
µ
ν
=
E
=
,
(
1
.
31
)
E
ν
E
λ
=
(
1
+
ν)(
1
−
2
ν)
,µ
=
2
(
1
+
ν)
.
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