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A variational justification of linear elasticity with residual stress

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J Elast (2009) 97: 189–206DOI 10.1007/s10659-009-9217-1
A Variational Justiﬁcation of Linear Elasticitywith Residual Stress
Roberto Paroni
·
Giuseppe Tomassetti
Received: 11 September 2008 / Published online: 5 August 2009© Springer Science+Business Media B.V. 2009
Abstract
We consider a residually-stressed, uniform hyperelastic body whose stored en-ergy is quadratic with respect to the Green–St. Venant strain. We show that, in the limitof vanishing loads, suitable minimizing sequences converge to the unique minimizer of theenergy functional of linear elasticity. We also deduce the standard stress-strain relations forlinear elasticity with residual stress.
Keywords
Residual stress
·
Nonlinear elasticity
·
Gamma-convergence
Mathematics Subject Classiﬁcation (2000)
74B20
·
74B10
·
49S05
1 Introduction
The theory of linear elasticity with residual stress, which according to Truesdell [14]goes back to Cauchy (1829), had apparently a cumbersome development. As Truesdellreports [14], “Cauchy’s results were not understood and were reported obscurely or evenincorrectly by nineteenth century expositors” (for detailed references see Man and Lu [10]).The formulation of the theory in its “correct form” reappeared much later in the papers of M.A. Biot (see [2] and references therein) and in a few books, for instance in Truesdell [15]
and Gurtin [8], but merely in the form of exercises. In 1986 it was rediscovered byHoger [9].
An interesting justiﬁcation of the theory of linear elasticity (without residual stress) hasbeen given recently by Dal Maso et al. in [6]. These authors denote by
u
ε
the displacementﬁeld obtained by solving the ﬁnite elasticity problem with dead loads multiplied by a small-ness parameter
ε
, and show, under suitable assumptions on the stored energy and using the
R. ParoniDipartimento di Architettura e Pianiﬁcazione, Università degli Studi di Sassari, 07041 Alghero, Italye-mail:paroni@uniss.itG. Tomassetti (
)Dipartimento di Ingegneria Civile, Università degli Studi di Roma “Tor Vergata”, 00133 Rome, Italye-mail:tomassetti@ing.uniroma2.it
190 R. Paroni, G. Tomassetti
theory of
Γ
-convergence, that the family of
scaled displacements
u
ε
/ε
converges, in a suit-able topology, as
ε
goes to 0 to the solution of an equilibrium problem that ﬁts within thetheory of linear elasticity.One main assumption made in [6] concerning the stored-energy density is that the stressin the undeformed conﬁguration, the
residual stress
, vanishes. The purpose of the presentpaper is to investigate whether this condition may be dropped.We provide a positive answer for a particular class of bodies, namely, homogeneous hy-perelastic bodies whose material elements have, when deformed from their relaxed conﬁg-urations, a quadratic stored energy with respect to the Green–St. Venant strain (a particularcase being the St. Venant–Kirchhoff materials).In Sect.2we review the theory of uniform bodies and we derive the expression of thestored energy with respect to a given reference placement for the material class we consider.These calculations are elementary, but apparently not easily found in the literature.In Sect.3we set up the problem of a hyperelastic body with residual stress. In doingso, we use the theory of material uniform bodies developed in the previous section. Webelieve that this point of view is novel and it allows us to put the theory in the context of residual-stressed hyperelastic bodies which may carry geometrically necessary dislocations.In Sect.4we formally deduce the theory of linear elasticity with residual stress. The onlypurpose of this section is to give a ﬂavor, free of technicalities, of our justiﬁcation.Section5is devoted to state the main assumptions and the main results. Our main re-sults concern the convergence of the scaled displacements to the solution of the linear limitproblem and the convergence of the ﬁrst and second Piola–Kirchhoff stresses.Finally Sect.6is devoted to the proofs of the theorems stated in Sect.5. The conver-
gence of the scaled displacements is obtained by resorting to techniques developed withinthe theory of Gamma-convergence. As a corollary of our analysis we also state a Gamma-convergence result.
2 Uniform Hyperelastic Bodies
We identify the three dimensional Euclidean point space with
R
3
. We consider a continuousbody
B
occupying, in its reference placement
κ
:
B
→
R
3
, an open, simply-connected re-gion
B
=
κ
(
B
)
⊂
R
3
with Lipschitz-continuous boundary. Given a material point
X
∈
B
,we denote by
X
=
κ
(X)
and
x
=
χ
(X)
the point occupied by
X
in the
reference placement
κ
and in the
current placement
χ
:
B
→
R
3
, respectively, and by
F
(
X
)
=∇
χ
κ
(
X
)
the spa-tial gradient, evaluated at
X
, of the
deformation
χ
κ
=
χ
◦
κ
−
1
that carries the body from thereference placement to the current placement. We denote by
T
(
x
)
the Cauchy stress at
x
,and by
S
(
X
)
=
det
(
F
(
X
))
T
(
x
)
F
(
X
)
−
T
(1)the
ﬁrst Piola–Kirchhoff stress relative to the placement
κ
. We identify second-order tensorswith the elements of
R
3
×
3
, the space of real 3
×
3 matrices, and we denote by
R
3
×
3Sym
,
R
3
×
3PSym
,and
R
3
×
3
+
, respectively, the spaces of symmetric, symmetric-positive deﬁnite matrices, andthe set of matrices with strictly positive determinant.Concerning the material response of the body, we make three assumptions.(i)
The body is elastic.
This means that there exists
F
κ
:
R
3
×
3
×
B
→
R
3
×
3Sym
, the
response function with respect to
κ
, such that
T
(
x
)
=
F
κ
(
F
(
X
),X).
(2)
A Variational Justiﬁcation of Linear Elasticity with Residual Stress 191
We recall for later purpose that the response function with respect to another placement
κ
must comply with
F
κ
(
A
,X)
=
F
κ
(
AP
,X)
∀
A
∈
R
3
×
3
+
,
(3)where
P
is the gradient of the deformation
π
=
κ
◦
κ
−
1
evaluated at
κ
(X)
.(ii)
The body is uniform in the sense of Noll [16 ].
For an elastic material uniformityimplies that there exists a
representative material point
X
0
∈
B
and a placement
κ
0
with thefollowing property: it is possible to associate to every
X
∈
B
a placement
κ
X
(which doesnot necessarily coincide with the reference placement
κ
) such that
F
κ
0
(
·
,X
0
)
=
F
κ
X
(
·
,X)
∀
X
∈
B
.
(4)(iii)
The representative material point
X
0
is hyperelastic
. This implies there exists asmooth function
¯
w
0
:
R
3
×
3
+
→
R
(the speciﬁc stored energy) whose gradient yields the ﬁrstPiola–Kirchhoff stress with respect to the placement
κ
0
, namely,
∂
¯
w
0
(
·
)
=
det
(
·
)
F
κ
0
(
·
,X
0
)(
·
)
−
T
.
(5)Let us denote by˚
P
X
the gradient of the mapping
κ
◦
κ
−
1
X
evaluated at
κ
X
(X)
, and let usdeﬁne the map˚
P
:
B
→
R
3
×
3
+
by˚
P
(
X
)
=
˚
P
κ
−
1
(
X
)
. We remark that˚
P
is not a gradient. From(3) and(4) we ﬁnd
F
κ
(
F
,X)
=
F
κ
X
(
F
˚
P
(
X
),X)
=
F
κ
0
(
F
˚
P
(
X
),X
0
).
(6)From (6), using (1) and(2), we obtain
S
=
det
F
F
κ
0
(
F
˚
P
(
X
),X
0
)
F
−
T
=
det˚
P
(
X
)
−
1
det
(
F
˚
P
(
X
))
F
κ
0
(
F
˚
P
(
X
),X
0
)(
F
˚
P
(
X
))
−
T
˚
P
(
X
)
T
.
(7)From (5) and (7) we arrive at
S
=
det˚
P
(
X
)
−
1
∂
¯
w
0
(
F
˚
P
(
X
))
˚
P
(
X
)
T
.
Thus, setting
¯
w
(
X
,
F
)
=
det˚
P
(
X
)
−
1
¯
w
0
(
F
˚
P
(
X
))
∀
F
∈
R
3
×
3
+
,
(8)we can write
S
(
X
)
=
∂
¯
w
(
X
,
F
(
X
)),
(9)where
¯
w
represents the
speciﬁc stored energy
in the reference placement
κ
and
∂
now de-notesthegradientwithrespecttothesecondargument.Observethatitdependson
X
through˚
P
(
X
)
only.The function
¯
w
0
must comply with the
Principle of Material Frame Indifference
, that isto say, given
F
such that det
(
F
)>
0,
¯
w
0
(
QF
)
=¯
w
0
(
F
)
∀
Q
∈
SO
(
3
),
(10)where SO
(
3
)
denotes the set of proper rotations in
R
3
. A convenient way to handle thisrestriction is to express the stored energy density in terms of an appropriate strain measure.
192 R. Paroni, G. Tomassetti
With a view towards the linearization process to be carried out in the forthcoming sections,we select the
Green–St. Venant strain
E
=
12
F
T
F
−
I
,
(11)where
I
denotes the identity in
R
3
×
3
. From(10) it follows that there exists a function
w
0
:
R
3
×
3Sym
→
R
such that
w
0
(
E
)
=¯
w
0
(
F
),
where
E
and
F
are related by (11). Substituting the last equation into (8) and observing that
12
((
F
˚
P
)
T
F
˚
P
−
I
)
=
12
(
˚
P
T
F
T
F
˚
P
−
˚
P
T
˚
P
+
˚
P
T
˚
P
−
I
)
=
˚
E
+
˚
P
T
E
˚
P
,
where˚
E
(
X
)
:=
12
(
˚
P
(
X
)
T
˚
P
(
X
)
−
I
),
the speciﬁc stored energy can be written in terms of
E
as
w
(
X
,
E
)
=
det˚
P
(
X
)
−
1
w
0
(
˚
E
(
X
)
+
˚
P
(
X
)
T
E
˚
P
(
X
)).
(12)We conclude this section by recalling that the
second Piola–Kirchhoff stress
is deﬁned by
=
F
−
1
S
;
(13)this is a symmetric tensor, which in view of (13) and (9) can be expressed as
(
X
)
=
∂
w
(
X
,
E
).
(14)
3 Bodies with Residual Stress
When the body is in its reference placement all the standard stress measures, namely, theCauchy stress
T
, the ﬁrst Piola–Kirchhoff stress
S
, and the second Piola–Kirchhoff stress
coincide with the
initial stress
˚
T
(
X
)
=
F
κ
(
I
,X)
=
∂
w
(
X
,
0
)
=
∂
¯
w
(
X
,
I
).
(15)By (10), we have˚
T
(
X
)
∈
R
3
×
3Sym
.
(16)The tensor ﬁeld˚
T
is called
residual stress
if it is
self-equilibrated
in the sense of
div˚
T
=
0
in
B,
˚
Tn
=
0
on
∂B,
(17)where
n
is the outward unit normal of
∂B
. Let
∂B
=
∂
D
B
∪
∂
N
B
, where
∂
D
B
and
∂
N
B
are
H
2
-measurable disjoint sets. The body is clamped on
∂
D
B
, and subject to a system of
dead
A Variational Justiﬁcation of Linear Elasticity with Residual Stress 193
loads
d
:
B
→
R
3
and
c
:
∂
N
B
:→
R
3
. Instead of using the deformation
χ
κ
as the primaryunknown, we prefer to use the
displacement ﬁeld
u
, which is related to the deformation by
u
(
X
)
=
χ
κ
(
X
)
−
X
,
X
∈
B.
In terms of
u
, the Piola stress is
S
(
X
)
=
∂
F
¯
w
(
X
,
I
+∇
u
(
X
))
for all
X
∈
B
, so that theclassical equilibrium problem consists in ﬁnding a displacement
u
:
B
→
R
3
such that
⎧⎪⎨⎪⎩
div
S
+
d
=
0
in
B,
u
=
0
on
∂
D
B,
Sn
=
c
on
∂
N
B.
(18)Let the
stored energy
and the
load potential
be deﬁned, respectively, by
W(
u
)
=
B
¯
w
(
X
,
I
+∇
u
(
X
))d
L
3
(
X
)
(19)and
U(
d
,
c
;
u
)
=
B
d
·
u
d
L
3
+
∂
N
B
c
·
u
d
H
2
.
(20)A solution of (18) is a stationary point of the
total energy
E(
u
)
=
W(
u
)
−
U(
d
,
c
;
u
).
We conclude this section by observing that in view of (12) the stored energy (19) can be
rewritten as
W(
u
)
=
B
w
(
X
,
E
(
u
)(
X
))d
L
3
(
X
),
(21)where
E
(
u
)
denotes the Green–St.Venant strain associated with the deformation
X
→
X
+
u
(
X
)
, that is
E
(
u
)
=
e
(
u
)
+
12
∇
u
T
∇
u
,
(22)where
e
(
u
)
=
12
∇
u
+∇
u
T
(23)is the strain measure of linear elasticity.
4 Formal Linearization
To capture the limit behavior of solutions for small loads, we perform the substitutions
(
d
,
c
)
→
(ε
d
,ε
c
),
u
→
ε
u
,
(24)where 0
<ε <
1 is a small parameter. Let˚
P
(
X
)
be the fourth-order tensor ﬁeld on
B
deﬁnedby˚
P
(
X
)
A
=
˚
P
(
X
)
T
A
˚
P
(
X
)
∀
A
∈
R
3
×
3Sym
.
(25)

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