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Truss modular Beams with Deformation Energy
depending on Higher Displacement Gradients
Jean-Jacques Alibert, Pierre Seppecher, Francesco Dell’Isola
To cite this version:
Jean-Jacques Alibert, Pierre Seppecher, Francesco Dell’Isola. Truss modular Beams with Deformation
Energy depending on Higher Displacement Gradients. Mathematics and Mechanics of Solids, SAGE
Publications, 2003, pp.23. �hal-00497327�
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(Received 29 February 2000; Final version 7 February 2002)
$EVWUDFW Until now, no third gradient theory has been proposed to describe the homogenized ener gy associ-
ated with a microscopic structure. In this paper, we prove that this is possible using pantographic-type struc-
tures. Their deformation energies involve combinations of nodal displacements having the form of second-
order or third-order finite differences. We establish the K-convergence of these energies to second and third
gradient functionals. Some mechanical examples are provided so as to illustrate the special features of these
homogenized models.
.H\ :RU GV Second gradient theory, third gradient theory, homogenization, K-convergence, finite differences,
modular truss beam, pantograph
,1752'8&7,21
A formalized theory for constitutive equations in continuum mechanics was first developed
by Noll [1] (in particular papers 8 and 35). In the framework of the aforementioned
axiomatization it was proven by Eringen [2] and Gurtin [3] that ï if Cauchy materials are
considered ï the second principle of thermodynamics does not allow for any dependence of
stress tensor on the second gradient of placement so that ï in order to formulate a purely
mechanical model in which constitutive equations involve such a second gradient ï the
new concept of interstitial working has to be introduced [4]. Enlarging the scope of the
considered models, it has been possible to include the second gradient of placement in the set
of admissible independent variables for constitutive equations also by adding at the same time
further kinematical descriptors (e.g. directors modelling the microstructure or temperature)
for the state of material particles as done, for example, in [5], [6] and [7].
However, another equivalent formalization of continuum mechanics, based on the
principle of virtual power and stemming from the dòAlembert concept of mechanics, is
possible (for a modern description of such a point of view see, for instance, [8] and [9]).
0DWKHPDWLFV DQG 0HFKDQLFV RI 6ROLGV 51ï73, 2003 DOI: 10.1177/108128603029658
f
?2003 Sage Publications
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52 J.-J. ALIBERT ET AL.
Following the classification formalized by Germain [10], the mechanical material
behaviour of bodies can be characterized by the expression of internal (deformation) energy
in terms of the displacement gradients. Cauchy three-dimensional (3D) materials coincide
with first gradient materials; their deformation is described by, and their deformation energy
depends on, the first gradient of displacement only.
The deformation energy of
VHFRQG JUDGLHQW ' PDWHULDOV, instead, depends also on the
second gradient of displacement. Let us call
+x,
the symmetric part of the gradient ux of
the displacement
x and
+x,=@ux
+x,
its skew part. We say that a second gradient
3D material is
LQFRPSOHWH if its internal energy depends only on ux and u
+x,. These
materials are also called ñ
' PDWHULDOV ZLWK FRXSOH VWUHVVHV ò (cf [5] and [11]): PLFURURWDWLRQV
in these bodies are modelled by introducing in the constitutive equations the aforementioned
dependence on
u
+x,. Such a modelling assumption has been subsequently improved by
introducing microstructural kinematical descriptors (for more details see, for example, [6]
and [12]).
Incomplete second gradient materials have been studied for a long time. The precursor
of incomplete second gradient models is the
HODVWLFD introduced by Euler, Bernoulli and
Navier at the beginning of the eighteenth century: it is a one-dimensional (1D) model in
which: (i) the attitude of the beam sections is kinematically described by the gradient of the
vertical displacement field; (ii) the contact couple depends on the second derivative of the
vertical displacement; (iii) the deformation energy depends on the gradient of the attitude and
therefore on the second gradient of displacement.
Another example of a 1D model in which higher-order derivatives of displacement must
be introduced is given by the theory of Vlasov (see, for example, [13], [14] and [15])
describing the twist of thin-walled beams. In Vlasovòs homogenized model the phenomenon
at the micro-level to be accounted for is the
ZDUSLQJ of beam sections and the corresponding
deformation energy is shown to depend on the first and second gradients of the twist angle.
The first (incomplete) second gradient 3D model is due to E. Cosserat and F. Cosserat (at
the beginning of the nineteenth century): in [16] the deformation energy explicitly depends on
u+x,. More recently, incomplete second gradient materials have been introduced to model
granular solids; for more details and for further references, see [17]. Complete 3D models
have been introduced for describing capillary phenomena [18], [19]. These have begun to be
extensively used in the theory of damage and plasticity (see [20], [21], [22] and [23]) as they
provide a more accurate description of transition zones (e.g. of shear bands [24]) and, from a
mathematical point of view, they lead to regularized well-posed problems. The regularizing
properties of second gradient models are also exploited in the description of the mechanical
behaviour of elastic crystals (see, for instance, [25]).
Complete and incomplete second gradient materials have fundamentally different
behaviours. While, in incomplete models, boundary conditions fix the displacement and the
rotation
+x, (or their dual quantities of force and moment), in complete models one also
has to fix
+x, or its dual quantity called double force [10] to which not all mechanicians
are accustomed. Indeed, the only widely used double contact action is Vlasovòs bimoment
(see, for example, [15] and [26]) needed for describing the external action at the extremities
of thin-walled twisted tubes.
It is remarkable that the mathematically established relationships between 1D or two-
dimensional (2D) second gradient models and Cauchy materials have been investigated only
when the dependence of deformation energy on the second gradient of displacement can
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TRUSS MODULAR BEAMS WITH DEFORMATION ENERGY 53
be related, at the micro-level, to variations of attitude. Indeed, the limit analysis 3Dï1D
or 2Dï1D of plates or beams leads only to such types of second gradient model. Is there
any fundamental physical reason for this? In our opinion, this is probably due to the desire
to remain in a standard framework. For more details about these rigorous results, we cite
[27], [28], [29], [30], [31], [32] and [33]. In technical theories of beams, which supply
the mechanical grounds for the aforementioned mathematical results, the macro-models are
related to micro-models using several identification procedures; for an extensive historical
discussion we refer to Benvenuto [34], who traces back to Maxwell and de Saint-Venant
[35] the first of these analyses. That which seems to be more encompassing is based on
LGHQWLILFDWLRQ LQ H[SHQGHG SRZHU ; one postulates a macroscopic and a microscopic model,
a kinematic correspondence between the two deformations and assumes that the power
expended in corresponding motions coincides. In this way one obtains, in terms of micro
properties of the beam, the coefficients of the macro constitutive equations, the form of which
has been postulated at the beginning (see, in particular, for truss modular beams [36] and [37]).
The very nature of this procedure shows how the properties of the macro model, in general,
are not
REWDLQHG as a result of the homogenization process but are, instead, assumed D SULRUL.
Here we present a microscopic model which leads to the simplest macroscopic second
gradient model: the 1D planar beam already studied by Casal [19]. The structure we consider,
i.e. the pantographic structure, is simple and the reader may have already experimented with
it when handling a corkscrew. We assume that the considered pantograph is made of a very
large number
O of small modules and we study its limit behaviour when O tends to infinity.
In other words, we study the homogenized model for the pantograph. The computation of
the equilibrium state is straightforward and we prove rigorously, using the technique of
-
convergence, that the homogenized model is really a second gradient model (section 3).
Considering different equilibrium situations, we recall in section 4.1 the principal features
of this model and we obtain an evident and self-explanatory interpretation for its special
features, in particular for the notion of double force.
Even though the general properties of third gradient materials have been studied by
Mindlin and Tiersten [5] and Dillon and Perzyna [38], to our knowledge no homogenized
third gradient model has been recognized as necessary for describing the behaviour of a truss
structure. To find such a structure is a problem closely related to the previous problem.
Indeed, once one has obtained a complete second gradient body, it is relatively easy to
construct a third gradient body. We do this and describe a structure, based on the pantograph,
which we call the Warren-type pantographic structure. Its homogenized energy is rigorously
proven to correspond to the bending of a third gradient beam. This beam has an unusual
behaviour which we describe briefly in section 4.2.
As the theorems and proofs are quite similar in both cases, we have decided to group
them in a single theorem (section 3). This states that, for any
L3, a quadratic functional
of finite dif ferences of order
L converges to a quadratic functional of the Lth derivative. This
convergence is proven in the sense of
-conver gence with respect to the weak* topology
of measures. In this way, our result does not depend on the choice of any interpolation of
displacements which have physical meaning at nodes only. We identify the external forces
which are admissible for the considered structure; it is a class of distributions of order
L which
contains, in particular, any distribution of order
L 4. For instance, for a second gradient
beam (
L @5), the derivative of a Dirac distribution is admissible. This corresponds to the
notion of punctual double force [10].
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54 J.-J. ALIBERT ET AL.
In the mathematical literature, the problem of rigorous proof of convergence from refined
models to homogenized models is widely addressed. The results we present in this paper are
close to those found in [28], [39, [40] and [41].
75866 %($06 :,7+ 3$172*5$3+,& 68%6758&785(6
In this section, we develop the mechanical heuristic considerations leading us to formulate
the mathematical problem to be solved in the subsequent section. We introduce a modular
pantographic structure and a Warren-type pantographic structure. We describe these at a
micro-level as a truss constituted by Euler beams and find the deformation energy for it in
terms of displacements of a finite set of nodes. The obtained expression is strongly suggestive
from a mechanical point of view. Indeed, it has induced us to conjecture the following. If the
dimension of the structure module dimension tends to zero and the number of beams tends
to infinity, a macro model can be introduced in which (i) the displacements of the nodes are
characterized by a (suitably regular) field and (ii) the deformation ener gy depends on second
or third derivatives of thus introduced displacement field.
This conjecture, although mechanically well grounded, needs a mathematical proof. For
a discussion on the relationship between the discrete and the homogenized models we refer
to [27] and [28].
7KH SDQWRJUDSK
Let us consider a plane modular structure the module of which is constructed as shown in
Figure 1.
We consider inextensible but flexible beams and refer to these by their endpoints. We call
a structure made by two such beams
+#
J
%
J
.4
, and +%
J
#
J
.4
,, connected by a pivot at their
common centre
$
J
,theJth module. We consider the simplest possible geometry assuming
that
+%
J
#
J
#
J
.4
%
J
.4
, is a square.
The periodic structure (shown in Figure 2) made by
O such modules, the size of which is
O
4
, is called the pantographic structure. The Jth module is connected to the J4th module by
two pivots at
#
J
and %
J
. We assume that external forces and external constraints are applied
at nodes
$
J
only. Thus we need to express the deformation ener gy of the structure in terms of
the displacement of the nodes
$
J
. We study the behaviour of the structure in the framework of
linear elasticity. The deformation energy of an inextensible but flexible beam depends only
on its transverse displacement. This energy is proportional to the square of the curvature. The
computation of the equilibrium energy of a beam of length
A and flexural stiffness ,, subject
to given transverse displacements
B, C at the endpoints and D at the centre is standard [14]. It
reads
plq
;
?
=
A
]
3
,
5
[%+Y,gY> [+3, @ B [
A
5
@ C [+A,@D
<
@
>
@
9,
A
6
+B5D.C,
5
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