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Variational formulations of non-linear and non-smooth structural problems
Inr.J. Non-Linear Mechanics, Vol.28,NO.2. PP.195-208, 1993 Prmted in Great Rriiain.
Q
VARIATIONAL FORMULATIONS OF NON-LINEAR NON-SMOOTH STRUCTURAL PROBLEMS G. ROMANO, L. ROSATI and
0020- 7462193 $6.00 + 00 1993 Pergamon Press Lfd
AND
F. MAROTTI DE SCIARRA
Dipartimento di Scienza delle Costruzioni, Facolt$ di Ingegneria, Universitti di Napoli Federico II, Italy (Received 20 May 1992)
Abstract-The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibjfity, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space ofail the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibiiity, a family of non-smooth furictionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem.
1. INTRODUCTION
The variational formulation of boundary-value problems governed by linear or non-linear operators is a well-established branch of mathematical physics. An exhaustive and detailed presentation of the classical theory can be found in the books by Mikhlin [l] and Vainberg [2]. A more recent monograph on the subject is due to Oden and Reddy [33. The possibility of associating a weak formulation with problems stated in operator terms is significant both from the theoretical and the computational point of view since it allows us to investigate upon the existence and the properties of the solutions and to devise suitable numerical procedures for their computation. Once a variational formulation has been set up, the corresponding differential or algebraic boundary-value problem can be recovered by means of the Eulero-Lagrange conditions for the stationarity of the functional. Actually, of greater interest is the inverse problem, which amounts to finding a realvalued functional whose gradient yields back the assigned operator and to stating the. conditions ensuring that such a functional does, in fact, exist. The classical condition for the existence of the potential is the conservativity of the operator, i.e. the vanishing of the related circuital integral along every closed curve in the connected domain of the operator. For a differentiable operator the property of conservativity is ensured by the symmetry of its first derivative [2]. However, a large number of advanced problems in structural mechanics are characterized by non-differentiable constitutive potentials [4, 5-J. The corresponding constitutive relations are, in fact, expressed as subdifferentials of convex functionals so that the dual pairs of state variables are related by monotone multi-valued operators. Hence, for such operators, the concept of conservativity must be suitably extended. The result has been achieved in the general framework provided by the potential theory of monotone multi-valued operators recently cont~buted in [6]. For convenience, the essential features of the theory are briefly summarized at the beginning of the paper. Contributed by J. N. Reddy. A preliminary draft of this paper was presented at the Conference “Problemi di Meccanica dei Materiali e delle Strutture”, June 3-5, 1991, Amalfi, Italy. 195
196
G. KOMANO
t’l rd.
Among other results. it has been proved in [6] that the integral of monotone multi-valued operators along lines and polylines can be unambiguously defined. The concept of conservativity of monotone multi-valued operators has been introduced, as an extension of the classical one, by requiring the vanishing of the related integral along any closed path in the domain of the operator. In addition, the integral of monotone conservative multi-v.alued operators yields the expression of the potential associated with the operator, thus extending the analogous result of classical dih’erential calculus. A direct constructive method can. thus, be exploited to obtain the expression of the potential associated with such operators. The outlined methodology is applied. in particular, to the class of structural models previously addressed which covers a broad range of applications: no-tension materials, problems of unilateral frictionless contact, structures made of elements incapable of sustaining tensile or compression stresses such as cables or membranes. The formulation of the elastostatic problem for these last structural models often requires one to impose the equilibrium conditions in the current configuration so that a non-linear strain measure must be adopted. Referring to the results provided in [7], it is remarked that a consistency condition must be fulfilled by the non-linear strain operator in order that a Lagrangian description of equilibrium can be performed. Essentially, the consistency property expresses the requirement that the finite strain rate vanishes if and only if the structure undergoes a rigid act of motion. In the Lagrangian description the equilibrium equation is expressed by a non-linear operator so that the overall behaviour of such structural models is governed by a combination of non-linear field equations and monotone multi-valued constitutive operators. In order to address the inverse problem of variational calculus, the structural problem is re-formulated in terms of a single operator, which will be called the structural operator, defined in the product space of all the state variables. Basically, the structural operator encompasses in a unique expression the field and constitutive equations by means of two sub-operators. The first one is non-linear and differentiable and is termed the static-kinematic operator; it embodies jointly the equilibrium and compatibility equations of the structural model. The second one accounts simultaneously for the internal and external constitutive assumptions. Actually, a variational formulation can be derived for this class of structural problems if and only if the structural operator is conservative. Thi,s result is shown separately for the two sub-operators. First it is shown that the operator expressing jointly the equilibrium and compatibility conditions in geometrically non-linear problems has a symmetric derivative and, hence, according to a well-known result of potential theory [2], it admits a potential, On the contrary, the internal and the external constraint operators are directly assumed to be conservative since they are, respectively, assigned as subdifferential of a convex functional and superdifferential of a concave functional. In the particular case of external rigid frictionless bilateral constraints, the domain and the co-domain of the multi-valued operator expressing the relation between external forces and displacement fields are affine spaces. It is shown, however, that the original problem can be re-formulated in terms of a single-valued reduced structural operator. On the other hand, when the internal and external constraint relations are expressed by arbitrary monotone multi-valued operators, the global structural operator must be dealt with. In both cases the structural operator turns out to be conservative and it can be integrated along a ray in the product space of the state variables to obtain the corresponding potential. A family of potentials are further derived from the potential of the structural operator by reiterated substitutions of the constraint relations and of the kinematic compatibility. Three of the potentials obtained for the global structural operator turn out to be the generalization to the context of non-linear and non-smooth mechanics of the Hu-Washizu. Reissner and total potential energy functionals of linear elasticity [S]. In view of expressing the stationarity conditions of these potentials, a background of convex analysis is also reported. Particular emphasis is laid on the concept of local convexity, the associated notion of local subdifferential and related composition rules.
Non-linear
and non-smooth
structural
197
problems
The meaning of stationarity for non-smooth functionals defined in a product space is also discussed in detail. In this context the relationships among the local subdifferential of such functionals and the partial local subdifferentials evaluated separately in each space are illustrated. Local subdifferential calculus is finally applied to show that the stationarity of the potentials previously addressed yields back an operator form of the structural model.
2. LOCAL
CONVEXITY
AND
SUBDIFFERENTIABILITY
In this section we briefly report the definitions and the basic results concerning the local convexity and the subdifferentiability of a functional defined on a vector space. Let (X, X’) be an ordered pair of dual locally convex topological vector spaces [991 l] and (. ;) the related duality pairing mapping the product space X x X’ into the set of reals $3. A functional g : X H 9I is said to admit a one-sided Gateaux derivative at the point x E dom g, along the direction defined by the vector k E X, if the following limit exists: dg(x; k) = ,:;+ The functional
; [g(x + ok) - g(x)].
,f: X H \Jr defined by def f(h)
=
dg(x;h)
is positively homogeneous in k and is called the oariation of the functional g at the point x E X along the direction k. The functional g is said to be locally convex at x when f is a sublinear functional of k, that is, j’(zk) = rf(k) Vx 2 0 (positive homogeneity), ,f(k, ) +.f’(h2 1 2f(hl
Vhl, k2 E X (subadditivity).
+ h2)
The epigraph of .f‘ is then a convex cone in X x s%. Local concavity of functionals is defined in perfect analogy by reverting the inequality sign in the second expression above. If the sublinear functional fis lower-semicontinuous (I.s.c.), i.e. =f(k,)
lim inff(k)
Vk,, E X,
h - ho
its epigraph is a closed convex cone in X x ‘3 and f turns out to be the support a non-empty closed convex set K:
functional
of
f(k)=sup{(x*,k):x*EKl, with K={x*~X’:f(k)~(x*,k)Vk~X}. The local subdifSerential of the functional by
g is the multi-valued
map, 8g : X H X’, defined
def ag(x) = K.
In particular, if the functional g is differentiable at x E X, the local subdifferential is a singleton and coincides with the usual differential. A locally convex functional g is said to have a stationary point at x E dom g if the null vector 0 E X’ is included in its local subdifferential: 0 E ag(x) or, equivalently, The following
rules usually
dg(x; k) 2 0
hold for local subdifferentiability
Vk E X. [l 11:
(1) Chain rule: Given a non-linear differentiable operator A :X H Y and a functional g : Y H ‘3 which is locally subdifferentiable at y = A(x), we have a(goA)(x) where dA(x) is the derivative
=
Cd~(x)l’~g(~(x)),
of the operator
A at x E X and [dA(x)]’
is its dual operator.
G.
19x
(2) Additiuity: subdifferentiable
ROMANO et(II.
Given two functionals g i : X H $3 and at x E X, it turns out to be ?(g1 + gz)(x) = ?y,(x)
gz: X H +$l which
are locally
+ ?gz(x).
Let us now consider the special case of convex functionals. For a convex functional g: X H $94u { + IY, ). the difference quotient in the definition of a one-sided Gateaux derivative does not increase as E decreases to zero [ 11-l 31. Hence, the limit exists at every point x E domg along any direction h E X and the following formula holds [ll]: dg(x; Iz) = (nf, ; [g(x + &) - g(x)]. A simple computation [l l] shows that the directional derivative of g is convex as a function of h and, hence, sublinear. Moreover, the definition of local subdifferential turns out to be equivalent to the usual definition of subdifferential in convex analysis [13], that is, x* E ?g(x) 0 g(y) ~ g(x) 2 (x*, J - x) The Fenchel’s
conjugate
g* : X’ H !R u ( + ~1 of g defined 8*(x*)
satisfies the Fenchel’s
= sup ((x*. 1.E.Y
y) - y(y))
2 (x*, _r>
v’!: E x, vx* 6 X’.
of pairs (x, x*) for which Fenchel’s inequality and the following relations are equivalent:
(i) g(x) + y*(x*)
by [ 131
inequality g(y) + 9*(x*)
The elements to be conjugate
v?’ E x.
holds as an equality
are said
= (x*. x).
(ii) Y* E Zg(.u), (iii) x 6 6g*(x*). Analogous results hold for concave functionals; the prefix “sub” used in the convex case has now to be replaced by “super”. In what follows, the local subdifferential (superdifferential) of a locally convex (concave) functional as well as the global subdifferential (superdifferential) of a convex (concave) functional will be denoted by the same symbol (! when no ambiguity can arise. 2.1. Stationarity property In the classical calculus of variations the stationarity condition for a differentiable functional amounts to requiring that its derivative vanishes at some point in the domain of definition. When dealing with non-smooth functionals the definition of stationarity must be suitably re-formulated. To encompass all the relevant situations we shall take into account three significant cases: locally convex, locally concave and locally saddle functionals. First we consider the stationarity condition for a locally convex functionalf‘: X x Y ++ +9I. Such a functional is said to have a stationary point at (x9 J) E domf‘if it turns out to be
where the symbol i? denotes the local subdifferential in the product space in which ./‘is defined. Denoting by 2, and by Zr the partial local subdifferentials in the spaces X and Y. respectively, it is shown in theorem Al (see Appendix) that the local subdifferential off‘ is included in the Cartesian product of the partial local subdifferentials: (7f(x, y) s aJ(4
y) x a,,/+>
4’)
199
Non-linear and non-smooth structural problems
It is then apparent that stationarity in the product space is a more stringent condition than partial stationarity in each component space. Since partial stationarity is easier to exploit, it is meaningful to find out classes of functionals for which stationarity and partial stationarity in each component space are equivalent. As is well known, any differentiable functional f meets this property; in fact, its local subdifferential turns out to be a singleton and, hence, by theorem Al, it turns out to be df(x, Y) = 4.0x,
Y) x d,f(x,
~1.
A more general class is provided by functionalsf: X x Y H 93 which are expressed as the sum of a functional g: X H 9I locally convex at x and of a functional h: Y I-+ ‘93locally convex at y. The functional f turns out to be locally convex at (x, y) and, by theorem A2, its stationarity can be equivalently enforced in each component space X and Y separately:
0 E c?,f(x, Y) *
0 E &“m Y) = &l(x), i 0 E aye,Y) = WY).
Stationarity property for locally concave functionals can be discussed in a similar way. A further significant situation in which stationarity condition has to be properly defined is given by functionalsf: X x Y H ‘3 which at a point (x, y) E X x Y turns out to be locally convex along X and locally concave along Y. A functional of this kind is said to be locally saddle at the point (x, y) and its local subdifferential is defined as def
Y) = ~x.f(x, Y) x &Jk
~fc%
YL
where i3, denotes the partial local subdifferential in the space X and ~3~denotes the partial local superdifferential in the space Y. A functional f which is locally saddle at a point (x, y) E X x Y is said to be stationary at (x, Y) if
OEaf(x7 y,-
0 E &“f(X~YL i 0
E
aYj-(x
’
y).
It has to be remarked that stationarity of a locally convex functional does not imply the existence of a local minimum as well as the stationarity of a locally saddle functional does not imply the existence of a local saddle point. However, if the functional is convex in the product space then its stationarity at a point ensures the attainment of an absolute minimum:
In the same way, the stationarity at a point of a saddle functional in the product space ensures the attainment of a saddle point:
0 E V(x, Y) -f(%
Y)
2f(x, Y) 2f(x, Y) V(K j)
E
x x y,
provided that the functional is convex along X and concave along Y.
3. POTENTIALS
OF MONOTONE
MULTI-VALUED
MAPS
To make the paper reasonably self-contained we briefly report, without proofs, some definitions and results of the potential theory of monotone multi-valued maps developed in [6], which will be referred to in the sequel. A graph G is a non-empty subset of the product space X x X’ and is said to be monotone (non-decreasing) if (XT - x:, ~2 - -WI) 2 0 V(xi, xf) E G, i = 132. Monotone non-increasing graphs are characterized by the converse inequality. A monotone graph G E X x X’ is said to be maximal if it is not properly included in any other monotone graph.
200
G. ROMANOet al
It has been proved in [4f that the integral of a monotone multi-valued map M : X t-r X’ along line segments and polylines in its domain can be unambiguously defined. In fact, given an oriented line segment with extremes a, b E dom M having a parametric representation a(t) = a + th, where h = b - u and 0 < t 5 1, the line integral of M along the segment a, b is then well-defined [6] by the formula b
1
(M(x),dx)
=
’ (M(.?(t)),h)dt
=
(ii;i(.W),
s0
I0
h)
dt,
s0
since the last integral is independent of the choice of h;i(?(t)) c M(a(t)) [6]. The integral of M along an oriented polyline 71in dom M is accordingly defined as the sum of the line integrals along each side. A monotone multi-valued map M: X H X’ is said to be ~o~ser~~ti~e if (M(x),
ds) = 0,
for every closed polyline rt E dom M. Hence, the integral of a conservative monotone multi-valued map along any oriented polyline in its domain depends only on its end points. The potential of a conservative monotone multi-valued map M having a convex domain is accordingly defined on dom M by the formula def f&l
-.f(%)
.r (M(z),
= I
dz) =
’ (M(@t)),
Ix) dr,
.rCI
x0
and it is assumed to be + m outside dom N. It has been proved in [6] that the potentials of M turns out to be the restriction to dom M of a proper 1.s.c.convex functional. If M is the subdifferential of a convex functional, continuous in the interior of its domain, its potential coincides with the restriction of the functional itself on dom M, to within an arbitrary additive constant. This result provides the generalization to non-differentiable convex functionals [6] of a classical result of integral calculus [ 11. 4. NON-LINEAR
STRUCTURAL
MODEL
Let us consider a structure undergoing a finite displacement from a reference configuration Q. to the actual one s1; O&will denote the space of the displacements that the structure can experience starting from Go, and ~2 a space of finite strain measures. With every displacement field u E ,j#, a corresponding strain measure c E 9 can be associated by means of a suitably defined non-linear di~erentiable operator D : % H 9, so that we can write E = D(U) We assume that D(0) = 0 in order to ensure that a null strain measure corresponds to a null displacement. In a Lagrangian approach the equilibrium of the structure in the current configuration Q is expressed in terms of equivalent entities pertaining to the reference configuration !&,. It has been shown in [7] that a Lagrangian description can be carried out if and only if the finite strain operator D fulfils a consistency properry. Consistency of D amounts to requiring that its rate of variation vanishes if and only if the structure undergoes a rigid act of motion. If this condition is met, the equilibrium of the external forces acting on the structure in the current configuration R can be re-formulated in terms of equivalent “reference external forces”Sdefined in the configuration Q,. It is then possible to give a consistent definition of a “reference internal force” CTwhich is in duality with the non-linear strain measure E and is related to the reference external forces j’by the reference equilibrium condition [dD(u)]‘a
=f.
Here [dD(u)] is the derivative of the strain operator operator.
D and the apex denotes the dual
Non-linear and non-smooth structural problems
201
In the sequel we shall denote by 9 the space of reference external forces which is in duality with % and by Y the space of reference internal forces d which is in duality with 9. In the next subsection we define the static-kinematic operator and prove its conservativity; in the subsequent one we deal with the multi-valued constraint operators and the related potentials. 4.1. Conservativity of the static-kinematic operator In view of the formulation of variational principles in non-linear structural mechanics, it is basic to prove that the non-linear operator governing the kinematics and the statics of the structural model does admit a potential. Let us denote by Y = 02 x ,Y the product space whose elements are the pairs y = (u, CJ) and by Y’ = 9 x 9 the corresponding dual space of the pairs y’ = (A E). The non-linear relations expressing kinematic compatibility and static equilibrium can be written in terms of a global non-linear operator A : Y H Y’ which is called the static-kinematic operator, that is,
We assume that the finite strain operator D is twice differentiable so that the operator A turns out to be differentiable. We can then appeal to a well-known result of potential theory [Z], according to which an operator admits a potential if and only if its derivative is symmetric. To establish the symmetry property of the derivative of A, we first observe that, by definition, def (CdA(~~l.h~y~)
=
lim 2-O
!jlI
~Y,),YZ) -
+
By developing the expression in the square brackets, we get
After some algebra, the bilinear form associated with the first derivative of A(y) becomes (CdA(y)]yl,
yz > = lim k {(cr,dD(u + clul )u2 - dD(u)u2 > Cr+O
which is apparently symmetric. The conservativity of A is, thus, proved. By virtue of the results provided in [6], the expression of the potential pA associated with A can be obtained by performing a direct integration in the product space Y = Q x Y along the ray to y = (u, C) to get 1
PA(Y) =
s0
dt>
and, explicitly, PAh 0) =
=
=
’ (([dD(tu)]‘trr, s0 {(to,
s
; $(trr,
u> + (IT, D(tu))jdt
[dD(tu)]u> D(tu))dt
+ ((r, D(tu)) f dt = ((T, D(u)).
It is, furthermore, immediate to verify that dp, = A, i.e. the gradient of pa yields back the static-kinematic operator A.
G. ROMAF~Oet ~1.
202
4.2. Conservutivity qf’multi-valued constraint operutors We assume that the dual internal state variables (E, a) and the dual external state variables (u,f) are related by constraint conditions which are characterized, respectively, by a monotone increasing and a monotone decreasing graph. Accordingly, we consider the multi-valued operators E:9~.5//
so that the constraint
relations
G:‘Ut-+R,
and
are written
as
g E E(e).
with 0~9,
~~22,
f’~ G(u),
with u E J,, .f’~ 9.
and
To develop a variational formulation of the structural problem we must require that the multi-valued monotone constraint operators E and G are conservative. To this end they are often assigned as subdifferential operators of convex or concave constraint functionals by setting [S, 14, 151 E = c:cp, G = c’y, with (p:k++‘X u { + ;c ] convex and g:&++‘R u { + x 1 concave. As a result of the potential theory of multi-valued monotone operators developed in [6], we can state that the functional cp coincides, to within an arbitrary additive constant, with the potential of the operator E defined by the integral formula PE(E) - PE(O) =
’
When a constraint relation is not assigned as the subdifferential of a convex potential but is directly expressed by a monotone multi-valued operator, its conservativity must be checked a posteriori. To this end we have to evaluate the functional defined by the integral formula above and to verify that its subdifferential yields back the given operator.
5. VARIATIONAL
PRINCIPLES
On the basis of the results reported in the previous sections, we can now develop a general variational theory of geometrically non-linear structural problems with constraint relations governed by conservative monotone multi-valued operators. In the next section we discuss geometrically non-linear structural problems with external rigid frictionless bilateral constraints and linear elastic constitutive behaviour. 5.1. The simplest case: rigid bilaterul constraints The occurrence of multi-valued operators relating dual pairs of state variables in structural mechanics is far from being unusual even in the simplest case of external rigid frictionless bilateral constraints. In fact, the admissible displacements belong to a subspace ~2~ E ?L and the set of admissible forces is the linear variety I + %$. Here 1 is the prescribed load acting on the structure and the orthogonal complement 22 0’ is the subspace of external constraint reactions. The relation between .f and u is then expressed by the constant multi-valued operators
defined
by
The domain and the co-domain of these multi-valued operators are affine spaces; in fact, domC=/+~~anddomC’=~~‘,. For structural models with external rigid frictionless bilateral constraints, the original problem can, however, be re-formulated in terms of a single-valued operator by assuming
Non-linear
and non-smooth
structural
203
problems
the subspace of the admissible displacements as reduced ambient space; the corresponding dual space will then be the quotient space F/42; of external forces f~ F modulo the subspace of the constraint reactions 42,o”. The elements of the quotient space F/42; are the affine sets (equivalence classes) of external force systems which differ by a constraint reaction. To get the expression of the reduced static-kinematic operator, we introduce the following pair of dual linear operators: n: ?!Lo+ %
canonical injection,
If’: 9 + P/%0’
canonical projection.
The operator II maps every admissible displacement into itself considered as an element of the whole displacement space; the dual oprator TI’ maps each external force f E 9 into the corresponding equivalence class 7~ .F/%!&. The static and the kinematic compatibilities are governed by the following pair of reduced operators: ff [dD(l-Iu)]‘: Y + 9-;/C&+,
The reduced static-kinematic operator which maps each ordered pair y = (u, a) into the corresponding dual pair J’ = (c E) is, accordingly, given by An(y) =
0 on
where i= fI’I E 919’~; is the assigned reduced load. The conservativity of An(y) can be proved by the same arguments outlined in Section 4.1. The structural problem with external rigid frictionless bilateral constraints can be expressed in its reduced form by the following single-valued relations: ff’[dD(TIu)]‘o
= r” static equilibrium,
D(lTu) = E kinematic compatibility, Es = g
internal constraint,
where u E +2e, E E 9, d E Y and the elastic stiffness operator E: 9 H Y is assumed to be linear, symmetric and positive-definite. By introducing the dual product spaces X = a0 x 9’ x 9 and X’ = F;/%!,I x 9 x 9, the structural problem above can be symbolically rearranged by means of the reduced structural operator B as follows:
I” I[I[ 1
0 l-I’[dD(fIu)]’ 0 u DfI 0 -I9 a-o, E E 0 -19 [ 0 where x is the triplet (u, g, E) and I9 and Zy are the dual identity operators in the spaces ~3 and 9, respectively. The conservativity of the operator B can be inferred from the analogous property of the reduced static-kinematic operator An, from the duality existing between I9 and Zy and from the symmetry of the operator E. The potential pB of B can then be computed by a direct integration along a ray in the product space X to get OEB(X)=
1
Pet% g, 8) =
; { ([dD(Iltu)]‘ta,
(B(tx), x) dt = s0
l-h)
+ (0, D(I-Itu))
s
-(to,&)-(qte)+(E(te),~)-([u)}dt = (a, D(llu))
- (6, E) + +(EE, E) - (
The potential pB turns out to be linear in 6, non-linear and differentiable in u, convex and differentiable in E.
G.
204
ROMAN0
Yt d
Two functionals can be further obtained from the expression of the potential pB by enforcing the kinematic compatibility condition and the internal constraint relation: Pi(U,0)
= (0,D(IIu,)
- $(a,
Pi(U) = $(ED(rIu),
E-la)
D(IIu))
- (lu),
- (Cu).
These functionals assume the same value in correspondence of a solution of the structural problem x = (u, cr, e) and their stationarity yields back the corresponding operator form of the structural problem. This feature is illustrated, in particular, with reference to the potential pe. The stationarity of pe, which is locally convex at .Y = (u, (r, E), can be obtained by imposing that the null vector 0 E X’ is included in its local subdilferential at x:
The considerations provided in Section 2.1 allow us to infer that the relation equivalent to enforce the stationarity of pB separately in each space.
and that the structural
problem
0 = d,,j+#.
CT,c).
0 = d,p,(u,
(T,E),
0 = d,p,(u,
6.4,
in its reduced
operator
above
is
form is recovered.
5.2. The general non-linear model When both the external and the internal constraints are described by conservative monotone multi-valued operators, we must necessarily deal with the non-linear problem in its globality. Its operator formulation is: j‘= [dD(u)]‘a
static equilibrium,
i: = D(u)
kinematic
0E E(c) = ?C&)
internal
constraints.
,f‘~ G(u) = &J(U)
external
constraints.
compatibility,
Introducing the dual product spaces W = +Yx Y x $2 x 9 and W’ = 9 x %”x ,‘/ x J//. the problem can be rewritten in the following symbolic form by means of the structural operator
2: 0 D
0
-I:,
- I ‘,,
(7y,
I:
0
0 -1,
0
0 I[1 u
-I,-
0
o
OE Z(w) =
0
IdW41'
Gl*
.f
where w is the ordered quartet (u, CJ,~,,f‘) and the operators I#, I,F and IV, I, are the dual pairs of identity maps in the spaces %, J and 9, 8, respectively. The conservativity of the operator Z is easily inferred from the conservativity of the static-kinematic sub-operator A, the assumed conservativity of the constraint operators, and from the duality existing between the two pairs of identity maps I,&, 1.9 and I,,!, Iv. The potential pz of Z can now be computed by means of a direct integration along a ray in the product space W, 1
Pz(4
=
(Z(rw),
\v> dt,
s0
to get Pz(k
0, hf’)
=
(P(E) + s*m
+
(0,
D(u))
-
u>
-
(a,
It is apparent that the potential pz turns out to be linear in 0, non-linear in u, convex in E and concave in,f:
c>.
and differentiable
Non-linear and non-smooth structural problems
205
The following equivalent formulations can be given to the constraint relations: GE Q(E) 0 E E acp*@) 0 (P(E)+ q*(o) = (c, E), fe &I(u) * u E @I*(f) *C?(u) + s*(f)
= (f, a>.
The last terms in the previous equivalences are known in convex analysis as Fenchel’s equalities. By reiterated substitutions of the constraint relations, in the form of Fenchel’s equalities, and of the kinematic compatibility condition E = D(u), the following family of functionals can be derived from the potential pz [ 161: HI@, E, a) = cP(d - g(u) + (0, D(u)) - (a, E), ff,(k
qf,
Rl(4
= - cp*m + s*(f)
+ (a, D(u)) - CL u>,
cl = - cP”(4 - g(u) +
R2@4f) = dm))
+ s*(f)
F(u) = cpw4)
- <.A U>>
- du).
The functionals HI, RI and F are the generalization, to the present non-linear and non-smooth context, of the functionals of Hu-Washizu, Reissner and of the total potential energy in the context of linear elasticity [8]. All the functionals in the family above do assume the same value when evaluated in correspondence of a solution w = (u, 0, s,f) of the structural problem since the kinematic compatibility and the constraint relations are fulfilled. Further, the stationarity of the potential pZ as well as that of the other functionals is equivalent to the operator forms of the problem. We will show explicitly this feature with reference to the potentials pz and F. The stationarity of the potential pZ is formally expressed by 0 E @Z(W), which amounts to requiring, by virtue of the remarks of Section 2.1, the stationarity of pz separately in each space: 0 = d&u,
6, s,f),
0 = d&u,
6, s,f),
0 E &P,(4 63 Gf), OE asPz(a, 0, Gf). Here ~3,and d, denote the subdifferential and the superdifferential with respect to E andf; respectively. The first relation states that (a, dD(u)u) - (f; 21) = ([dD(u)]‘a
-f, o) = 0 VoE %,
which represents the static equilibrium condition [dD(u)]‘a = f. Likewise, the second relation yields E = D(u), and, by definition of subdifferential, we have, respectively, from the last two conditions, 0 E a&s), u E G*(f). Let us now consider the potential F(u) which is locally subdifferentiable in u since it is composed of the subdifferentiable convex functional cp and the non-linear differentiable operator D.
G. ROMANO
706
et ul
The stationarity condition states that the null vector belongs to the local subdi~erentiai of the potential F, i.e. 0 f PF(u).
This condition is equivalent to an operator formulation of the problem in terms of displacements. In fact, by the additivity property and the chain rule of the local subdifferentials, we have iiF
= c?[yl(D(u)) - &L)] = =
;i[cp(D(z*))]
-
?&4)
rd~(~)]‘~9(~(~~)) - Sg(uf.
The stationarity condition implies that there exists an internal stress CTwhich is an element of the local subdifferential of 9 at i: = D(U) and an external force j’which is an element of the superdifferential of y at II such that the equality holds between the two terms on the right-hand side of the previous relation. In formulas we have
The quartet (u, (r, s,f’) is then a solution of the given non-linear structural problem. The converse implication follows easily by reversing the steps above. Incidentally, we note that the functional F turns out to be the potential of the operator given by F(u) = [c?(9”D - ~f][U-J so that F = PF. The same feature can be shown, although in a less obvious way, for the other functionals derived from pz. Actually, each of these functionals turns out to be the potential of a structural operator associated with an operator form of the structural problem. In particular, the operator form associated with the locally convex functional Hi is given by -i?
0 E H,(u, a,,f‘) =
D - I ‘u
It is apparent that the operator Hz is conservative and then its potential turns out to be Hz, that is, Hz = 2H2. Repeating the arguments above for the functional Ri , which is concave in CIand locally convex in 14. and for the functional Rz, which is localiy convex in u and concave in,!; the following expressions of the associated operators are recovered:
and
O~R~@,f’)=
c
o’(9 ,D) _I
‘//
- I,,-
11
i)g* II
.1’ *
1
Once more, the relations RI = i?K, and R2 = r?R, can be inferred from the conservativity of the operators R, and R2. dc~~~wied~~~~n~s-- The financial gratefully acknowledged.
support
of the Italian
Ministry
for Scientific
and Technological
Research
is
Non-linear
and non-smooth
structural
problems
207
REFERENCES 1. S. G. Mikhlin, Variational Methods in Mathematical Physics. Pergamon, New York (1964). 2. M. M. Vainberg, Variational Methodsfor the Study of Nonlinear Operators. Holden Day, San Francisco, CA (1964). 3. J. T. Oden and J. N. Reddy, Variational Methods in Theoretical Mechanics, 2nd edition. Springer, Berlin (1981). 4. J. J. Moreau, On Unilateral Constraints, Friction and Plasticity. New Variational Techniques in Mathematical Physics (Edited by Cremonese), pp. 171-322. CIME, Bressanone (1974). 5. P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Birkhauser, New York (1985). 6. G. Romano, L. Rosati, F. Marotti de Sciarra and P. Bisegna, A potential theory for monotone multi-valued operators. Quart. appl. Math. (to appear). condition for finite strain measures. 7. G. Romano, L. Rosati and F. Marotti de Sciarra, On a consistency Meccanica (to appear). 8. K. Washizu, Variational Methods in Elasticity and Plasticity, 3rd edition. Pergamon, New York (1982). 9. K. Yosida, Functional Analysis, 6th edition. Springer, Berlin (1980). 10. I. Ekeland and R. Temam, Analyse Conuexe et Problbmes Variationnels. Dunod, Paris-Bruxelles-Montreal (1974). 11. A D. Ioffe and V. M. Tihomirov, The Theory of Extremal Problems. Nauka, Moscow (English translation, North-Holland, Amsterdam) (1979). Convexes, Lecture notes, siminaire: Equationes aux DBrivees Partielles. Coll&gie 12. J. J. Moreau, Fonctionelles de France (1966). 13. R. T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). 14. G. Roman0 and L. Rosati, Variational principles in convex structural analysis. Volume in onore de1 Prof. Giulio Ceradini, Roma (Luglio 1988). 15. G. Roman0 and L. Rosati, A survey on recent advances in convex structural analysis. Atti del Convegno in Ricordo di Riccardo Baldacci e Michele Capurso, Roma (27 Ottobre 1989). nella dinamica non lineare di strutture con vincoli 16. G. Romano, L. Rosati and G. Ferro. Principi variazionali convessi. Atri X Congress0 Naz. AIMETA, Pisa (2-5 Ottobre 1990).
APPENDIX Theorem Al. Given a functional
f: X x Y H ‘Jz locally convex at the point (x, y), it turns out to be Zf(x, Y) c &f(x,
where d, and 8, denote Proof By definition,
the partial
local subdifferentials
Y) x a,/(x,
a pair (x*, p*) E X’ x I” belongs to the local subdifferential dfC(x, y); (% 8)I t ((x”. y*), (x, L:))
Choosing
f=
Y),
in the spaces X and Y, respectively. offat
(x, y) if it turns out to be
v(x, j) E x x Y.
0 we have def d/[(.x, y); (% O)] 2 (x*, X> Vx E X * x* E &j-(x, y),
and, analogously,
setting X = 0, dfC(x. Y):@,
HI t (Y”, j) Vj E Y 0 y* E &f-(x, y),
which prove the theorem. If the functional to be
0
f is differentiable
at (x, y), its local subdifferential dS(x, Y) = d,f(x,
In order to prove the next theorem, we introduce A: X x YH X is the linear operator defined as A(x,y)=x and its dual operator
coincides
Y) x d,f(x,
with the differential
and it turns out
Y).
some preliminary
definitions.
The canonical projection
V(x,y)~xxY,
A’: X’ H X’ x r is the canonical injection A’x* = (r*, 0)
We can now state the following
Vx* E X’.
theorem
Theorem A2. Let 9 : X ++ ‘iRbe a locally convex functional at y E Y. The functional S: X x Y I+ R defined by
at x E X and h : Y w R be a locally convex functional
f(x> y) = g (x) + h(y) turns out to be locally convex
at the point (x, y) E X x Y and its local subdifferential WX.Y) =
%(x)xWyL
or, equivalently,
cx*.Y*) provided
that the addition
E
Nx, Y10
rule of local subdifferentials
x* E Wx), Y* E WY),
holds true.
at (x, y) is given by
G. ROMANO
208
rr ul.
Proof. Let the linear operators .4 : X x Y c 3’ and U: X x Y ++ Y be canonical projections and let A’ and B’ be the corresponding dual operators. By definition of canonical projection. the functional ,/ can be equivalently written in the form
, I\-,!‘) := (I/ .A+ II B)(u,y). so that the additivity
property
of the local subdifferentials
il(\-,J,)=i(y
A +/I
yields
B)(Y.~)-C’((/
A)(\.!‘)+;(/1
R)(IJ).
The local convexity at (1. J) of the functional $1 .A : X x 1’~ 93can be inferred from the analogous functional 9. In fact. the directional derivative of (1 4 IS given by d(c/ and, hence, by definition
A)[(u.~);(u.
r):.-l(\-. ?)I.
of .4. d(c/
From
C)] = dc/[.4(\-.
this result and the definition
.~)[(Y.T);(?.
of subdifferential.
P)] = dg(u; 7).
we Infer that the followmg
(i) (r*, J*) E r’(
2 ((.Y*.J,*).(\..
(iii) dy(.x;.C) 2 (u*, ?) + (j*,i;) Inequality
C)>
V(C P)E.Yx
I’.
V(\-. ~:JE A’ x k.
(iii) implies that 1‘* = 0 and .x* t ?y(u). so that It turns out to be iI
Repeating
property
the same arguments
for the functional
A )(Y. !‘I = ./l’i$/(Y) II 8, the theorem
is proved
expressions
are equivalent:
of the
Q
VARIATIONAL FORMULATIONS OF NON-LINEAR NON-SMOOTH STRUCTURAL PROBLEMS G. ROMANO, L. ROSATI and
0020- 7462193 $6.00 + 00 1993 Pergamon Press Lfd
AND
F. MAROTTI DE SCIARRA
Dipartimento di Scienza delle Costruzioni, Facolt$ di Ingegneria, Universitti di Napoli Federico II, Italy (Received 20 May 1992)
Abstract-The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibjfity, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space ofail the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibiiity, a family of non-smooth furictionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem.
1. INTRODUCTION
The variational formulation of boundary-value problems governed by linear or non-linear operators is a well-established branch of mathematical physics. An exhaustive and detailed presentation of the classical theory can be found in the books by Mikhlin [l] and Vainberg [2]. A more recent monograph on the subject is due to Oden and Reddy [33. The possibility of associating a weak formulation with problems stated in operator terms is significant both from the theoretical and the computational point of view since it allows us to investigate upon the existence and the properties of the solutions and to devise suitable numerical procedures for their computation. Once a variational formulation has been set up, the corresponding differential or algebraic boundary-value problem can be recovered by means of the Eulero-Lagrange conditions for the stationarity of the functional. Actually, of greater interest is the inverse problem, which amounts to finding a realvalued functional whose gradient yields back the assigned operator and to stating the. conditions ensuring that such a functional does, in fact, exist. The classical condition for the existence of the potential is the conservativity of the operator, i.e. the vanishing of the related circuital integral along every closed curve in the connected domain of the operator. For a differentiable operator the property of conservativity is ensured by the symmetry of its first derivative [2]. However, a large number of advanced problems in structural mechanics are characterized by non-differentiable constitutive potentials [4, 5-J. The corresponding constitutive relations are, in fact, expressed as subdifferentials of convex functionals so that the dual pairs of state variables are related by monotone multi-valued operators. Hence, for such operators, the concept of conservativity must be suitably extended. The result has been achieved in the general framework provided by the potential theory of monotone multi-valued operators recently cont~buted in [6]. For convenience, the essential features of the theory are briefly summarized at the beginning of the paper. Contributed by J. N. Reddy. A preliminary draft of this paper was presented at the Conference “Problemi di Meccanica dei Materiali e delle Strutture”, June 3-5, 1991, Amalfi, Italy. 195
196
G. KOMANO
t’l rd.
Among other results. it has been proved in [6] that the integral of monotone multi-valued operators along lines and polylines can be unambiguously defined. The concept of conservativity of monotone multi-valued operators has been introduced, as an extension of the classical one, by requiring the vanishing of the related integral along any closed path in the domain of the operator. In addition, the integral of monotone conservative multi-v.alued operators yields the expression of the potential associated with the operator, thus extending the analogous result of classical dih’erential calculus. A direct constructive method can. thus, be exploited to obtain the expression of the potential associated with such operators. The outlined methodology is applied. in particular, to the class of structural models previously addressed which covers a broad range of applications: no-tension materials, problems of unilateral frictionless contact, structures made of elements incapable of sustaining tensile or compression stresses such as cables or membranes. The formulation of the elastostatic problem for these last structural models often requires one to impose the equilibrium conditions in the current configuration so that a non-linear strain measure must be adopted. Referring to the results provided in [7], it is remarked that a consistency condition must be fulfilled by the non-linear strain operator in order that a Lagrangian description of equilibrium can be performed. Essentially, the consistency property expresses the requirement that the finite strain rate vanishes if and only if the structure undergoes a rigid act of motion. In the Lagrangian description the equilibrium equation is expressed by a non-linear operator so that the overall behaviour of such structural models is governed by a combination of non-linear field equations and monotone multi-valued constitutive operators. In order to address the inverse problem of variational calculus, the structural problem is re-formulated in terms of a single operator, which will be called the structural operator, defined in the product space of all the state variables. Basically, the structural operator encompasses in a unique expression the field and constitutive equations by means of two sub-operators. The first one is non-linear and differentiable and is termed the static-kinematic operator; it embodies jointly the equilibrium and compatibility equations of the structural model. The second one accounts simultaneously for the internal and external constitutive assumptions. Actually, a variational formulation can be derived for this class of structural problems if and only if the structural operator is conservative. Thi,s result is shown separately for the two sub-operators. First it is shown that the operator expressing jointly the equilibrium and compatibility conditions in geometrically non-linear problems has a symmetric derivative and, hence, according to a well-known result of potential theory [2], it admits a potential, On the contrary, the internal and the external constraint operators are directly assumed to be conservative since they are, respectively, assigned as subdifferential of a convex functional and superdifferential of a concave functional. In the particular case of external rigid frictionless bilateral constraints, the domain and the co-domain of the multi-valued operator expressing the relation between external forces and displacement fields are affine spaces. It is shown, however, that the original problem can be re-formulated in terms of a single-valued reduced structural operator. On the other hand, when the internal and external constraint relations are expressed by arbitrary monotone multi-valued operators, the global structural operator must be dealt with. In both cases the structural operator turns out to be conservative and it can be integrated along a ray in the product space of the state variables to obtain the corresponding potential. A family of potentials are further derived from the potential of the structural operator by reiterated substitutions of the constraint relations and of the kinematic compatibility. Three of the potentials obtained for the global structural operator turn out to be the generalization to the context of non-linear and non-smooth mechanics of the Hu-Washizu. Reissner and total potential energy functionals of linear elasticity [S]. In view of expressing the stationarity conditions of these potentials, a background of convex analysis is also reported. Particular emphasis is laid on the concept of local convexity, the associated notion of local subdifferential and related composition rules.
Non-linear
and non-smooth
structural
197
problems
The meaning of stationarity for non-smooth functionals defined in a product space is also discussed in detail. In this context the relationships among the local subdifferential of such functionals and the partial local subdifferentials evaluated separately in each space are illustrated. Local subdifferential calculus is finally applied to show that the stationarity of the potentials previously addressed yields back an operator form of the structural model.
2. LOCAL
CONVEXITY
AND
SUBDIFFERENTIABILITY
In this section we briefly report the definitions and the basic results concerning the local convexity and the subdifferentiability of a functional defined on a vector space. Let (X, X’) be an ordered pair of dual locally convex topological vector spaces [991 l] and (. ;) the related duality pairing mapping the product space X x X’ into the set of reals $3. A functional g : X H 9I is said to admit a one-sided Gateaux derivative at the point x E dom g, along the direction defined by the vector k E X, if the following limit exists: dg(x; k) = ,:;+ The functional
; [g(x + ok) - g(x)].
,f: X H \Jr defined by def f(h)
=
dg(x;h)
is positively homogeneous in k and is called the oariation of the functional g at the point x E X along the direction k. The functional g is said to be locally convex at x when f is a sublinear functional of k, that is, j’(zk) = rf(k) Vx 2 0 (positive homogeneity), ,f(k, ) +.f’(h2 1 2f(hl
Vhl, k2 E X (subadditivity).
+ h2)
The epigraph of .f‘ is then a convex cone in X x s%. Local concavity of functionals is defined in perfect analogy by reverting the inequality sign in the second expression above. If the sublinear functional fis lower-semicontinuous (I.s.c.), i.e. =f(k,)
lim inff(k)
Vk,, E X,
h - ho
its epigraph is a closed convex cone in X x ‘3 and f turns out to be the support a non-empty closed convex set K:
functional
of
f(k)=sup{(x*,k):x*EKl, with K={x*~X’:f(k)~(x*,k)Vk~X}. The local subdifSerential of the functional by
g is the multi-valued
map, 8g : X H X’, defined
def ag(x) = K.
In particular, if the functional g is differentiable at x E X, the local subdifferential is a singleton and coincides with the usual differential. A locally convex functional g is said to have a stationary point at x E dom g if the null vector 0 E X’ is included in its local subdifferential: 0 E ag(x) or, equivalently, The following
rules usually
dg(x; k) 2 0
hold for local subdifferentiability
Vk E X. [l 11:
(1) Chain rule: Given a non-linear differentiable operator A :X H Y and a functional g : Y H ‘3 which is locally subdifferentiable at y = A(x), we have a(goA)(x) where dA(x) is the derivative
=
Cd~(x)l’~g(~(x)),
of the operator
A at x E X and [dA(x)]’
is its dual operator.
G.
19x
(2) Additiuity: subdifferentiable
ROMANO et(II.
Given two functionals g i : X H $3 and at x E X, it turns out to be ?(g1 + gz)(x) = ?y,(x)
gz: X H +$l which
are locally
+ ?gz(x).
Let us now consider the special case of convex functionals. For a convex functional g: X H $94u { + IY, ). the difference quotient in the definition of a one-sided Gateaux derivative does not increase as E decreases to zero [ 11-l 31. Hence, the limit exists at every point x E domg along any direction h E X and the following formula holds [ll]: dg(x; Iz) = (nf, ; [g(x + &) - g(x)]. A simple computation [l l] shows that the directional derivative of g is convex as a function of h and, hence, sublinear. Moreover, the definition of local subdifferential turns out to be equivalent to the usual definition of subdifferential in convex analysis [13], that is, x* E ?g(x) 0 g(y) ~ g(x) 2 (x*, J - x) The Fenchel’s
conjugate
g* : X’ H !R u ( + ~1 of g defined 8*(x*)
satisfies the Fenchel’s
= sup ((x*. 1.E.Y
y) - y(y))
2 (x*, _r>
v’!: E x, vx* 6 X’.
of pairs (x, x*) for which Fenchel’s inequality and the following relations are equivalent:
(i) g(x) + y*(x*)
by [ 131
inequality g(y) + 9*(x*)
The elements to be conjugate
v?’ E x.
holds as an equality
are said
= (x*. x).
(ii) Y* E Zg(.u), (iii) x 6 6g*(x*). Analogous results hold for concave functionals; the prefix “sub” used in the convex case has now to be replaced by “super”. In what follows, the local subdifferential (superdifferential) of a locally convex (concave) functional as well as the global subdifferential (superdifferential) of a convex (concave) functional will be denoted by the same symbol (! when no ambiguity can arise. 2.1. Stationarity property In the classical calculus of variations the stationarity condition for a differentiable functional amounts to requiring that its derivative vanishes at some point in the domain of definition. When dealing with non-smooth functionals the definition of stationarity must be suitably re-formulated. To encompass all the relevant situations we shall take into account three significant cases: locally convex, locally concave and locally saddle functionals. First we consider the stationarity condition for a locally convex functionalf‘: X x Y ++ +9I. Such a functional is said to have a stationary point at (x9 J) E domf‘if it turns out to be
where the symbol i? denotes the local subdifferential in the product space in which ./‘is defined. Denoting by 2, and by Zr the partial local subdifferentials in the spaces X and Y. respectively, it is shown in theorem Al (see Appendix) that the local subdifferential off‘ is included in the Cartesian product of the partial local subdifferentials: (7f(x, y) s aJ(4
y) x a,,/+>
4’)
199
Non-linear and non-smooth structural problems
It is then apparent that stationarity in the product space is a more stringent condition than partial stationarity in each component space. Since partial stationarity is easier to exploit, it is meaningful to find out classes of functionals for which stationarity and partial stationarity in each component space are equivalent. As is well known, any differentiable functional f meets this property; in fact, its local subdifferential turns out to be a singleton and, hence, by theorem Al, it turns out to be df(x, Y) = 4.0x,
Y) x d,f(x,
~1.
A more general class is provided by functionalsf: X x Y H 93 which are expressed as the sum of a functional g: X H 9I locally convex at x and of a functional h: Y I-+ ‘93locally convex at y. The functional f turns out to be locally convex at (x, y) and, by theorem A2, its stationarity can be equivalently enforced in each component space X and Y separately:
0 E c?,f(x, Y) *
0 E &“m Y) = &l(x), i 0 E aye,Y) = WY).
Stationarity property for locally concave functionals can be discussed in a similar way. A further significant situation in which stationarity condition has to be properly defined is given by functionalsf: X x Y H ‘3 which at a point (x, y) E X x Y turns out to be locally convex along X and locally concave along Y. A functional of this kind is said to be locally saddle at the point (x, y) and its local subdifferential is defined as def
Y) = ~x.f(x, Y) x &Jk
~fc%
YL
where i3, denotes the partial local subdifferential in the space X and ~3~denotes the partial local superdifferential in the space Y. A functional f which is locally saddle at a point (x, y) E X x Y is said to be stationary at (x, Y) if
OEaf(x7 y,-
0 E &“f(X~YL i 0
E
aYj-(x
’
y).
It has to be remarked that stationarity of a locally convex functional does not imply the existence of a local minimum as well as the stationarity of a locally saddle functional does not imply the existence of a local saddle point. However, if the functional is convex in the product space then its stationarity at a point ensures the attainment of an absolute minimum:
In the same way, the stationarity at a point of a saddle functional in the product space ensures the attainment of a saddle point:
0 E V(x, Y) -f(%
Y)
2f(x, Y) 2f(x, Y) V(K j)
E
x x y,
provided that the functional is convex along X and concave along Y.
3. POTENTIALS
OF MONOTONE
MULTI-VALUED
MAPS
To make the paper reasonably self-contained we briefly report, without proofs, some definitions and results of the potential theory of monotone multi-valued maps developed in [6], which will be referred to in the sequel. A graph G is a non-empty subset of the product space X x X’ and is said to be monotone (non-decreasing) if (XT - x:, ~2 - -WI) 2 0 V(xi, xf) E G, i = 132. Monotone non-increasing graphs are characterized by the converse inequality. A monotone graph G E X x X’ is said to be maximal if it is not properly included in any other monotone graph.
200
G. ROMANOet al
It has been proved in [4f that the integral of a monotone multi-valued map M : X t-r X’ along line segments and polylines in its domain can be unambiguously defined. In fact, given an oriented line segment with extremes a, b E dom M having a parametric representation a(t) = a + th, where h = b - u and 0 < t 5 1, the line integral of M along the segment a, b is then well-defined [6] by the formula b
1
(M(x),dx)
=
’ (M(.?(t)),h)dt
=
(ii;i(.W),
s0
I0
h)
dt,
s0
since the last integral is independent of the choice of h;i(?(t)) c M(a(t)) [6]. The integral of M along an oriented polyline 71in dom M is accordingly defined as the sum of the line integrals along each side. A monotone multi-valued map M: X H X’ is said to be ~o~ser~~ti~e if (M(x),
ds) = 0,
for every closed polyline rt E dom M. Hence, the integral of a conservative monotone multi-valued map along any oriented polyline in its domain depends only on its end points. The potential of a conservative monotone multi-valued map M having a convex domain is accordingly defined on dom M by the formula def f&l
-.f(%)
.r (M(z),
= I
dz) =
’ (M(@t)),
Ix) dr,
.rCI
x0
and it is assumed to be + m outside dom N. It has been proved in [6] that the potentials of M turns out to be the restriction to dom M of a proper 1.s.c.convex functional. If M is the subdifferential of a convex functional, continuous in the interior of its domain, its potential coincides with the restriction of the functional itself on dom M, to within an arbitrary additive constant. This result provides the generalization to non-differentiable convex functionals [6] of a classical result of integral calculus [ 11. 4. NON-LINEAR
STRUCTURAL
MODEL
Let us consider a structure undergoing a finite displacement from a reference configuration Q. to the actual one s1; O&will denote the space of the displacements that the structure can experience starting from Go, and ~2 a space of finite strain measures. With every displacement field u E ,j#, a corresponding strain measure c E 9 can be associated by means of a suitably defined non-linear di~erentiable operator D : % H 9, so that we can write E = D(U) We assume that D(0) = 0 in order to ensure that a null strain measure corresponds to a null displacement. In a Lagrangian approach the equilibrium of the structure in the current configuration Q is expressed in terms of equivalent entities pertaining to the reference configuration !&,. It has been shown in [7] that a Lagrangian description can be carried out if and only if the finite strain operator D fulfils a consistency properry. Consistency of D amounts to requiring that its rate of variation vanishes if and only if the structure undergoes a rigid act of motion. If this condition is met, the equilibrium of the external forces acting on the structure in the current configuration R can be re-formulated in terms of equivalent “reference external forces”Sdefined in the configuration Q,. It is then possible to give a consistent definition of a “reference internal force” CTwhich is in duality with the non-linear strain measure E and is related to the reference external forces j’by the reference equilibrium condition [dD(u)]‘a
=f.
Here [dD(u)] is the derivative of the strain operator operator.
D and the apex denotes the dual
Non-linear and non-smooth structural problems
201
In the sequel we shall denote by 9 the space of reference external forces which is in duality with % and by Y the space of reference internal forces d which is in duality with 9. In the next subsection we define the static-kinematic operator and prove its conservativity; in the subsequent one we deal with the multi-valued constraint operators and the related potentials. 4.1. Conservativity of the static-kinematic operator In view of the formulation of variational principles in non-linear structural mechanics, it is basic to prove that the non-linear operator governing the kinematics and the statics of the structural model does admit a potential. Let us denote by Y = 02 x ,Y the product space whose elements are the pairs y = (u, CJ) and by Y’ = 9 x 9 the corresponding dual space of the pairs y’ = (A E). The non-linear relations expressing kinematic compatibility and static equilibrium can be written in terms of a global non-linear operator A : Y H Y’ which is called the static-kinematic operator, that is,
We assume that the finite strain operator D is twice differentiable so that the operator A turns out to be differentiable. We can then appeal to a well-known result of potential theory [Z], according to which an operator admits a potential if and only if its derivative is symmetric. To establish the symmetry property of the derivative of A, we first observe that, by definition, def (CdA(~~l.h~y~)
=
lim 2-O
!jlI
~Y,),YZ) -
+
By developing the expression in the square brackets, we get
After some algebra, the bilinear form associated with the first derivative of A(y) becomes (CdA(y)]yl,
yz > = lim k {(cr,dD(u + clul )u2 - dD(u)u2 > Cr+O
which is apparently symmetric. The conservativity of A is, thus, proved. By virtue of the results provided in [6], the expression of the potential pA associated with A can be obtained by performing a direct integration in the product space Y = Q x Y along the ray to y = (u, C) to get 1
PA(Y) =
s0
and, explicitly, PAh 0) =
=
=
’ (([dD(tu)]‘trr, s0 {(to,
s
; $(trr,
u> + (IT, D(tu))jdt
[dD(tu)]u> D(tu))dt
+ ((r, D(tu)) f dt = ((T, D(u)).
It is, furthermore, immediate to verify that dp, = A, i.e. the gradient of pa yields back the static-kinematic operator A.
G. ROMAF~Oet ~1.
202
4.2. Conservutivity qf’multi-valued constraint operutors We assume that the dual internal state variables (E, a) and the dual external state variables (u,f) are related by constraint conditions which are characterized, respectively, by a monotone increasing and a monotone decreasing graph. Accordingly, we consider the multi-valued operators E:9~.5//
so that the constraint
relations
G:‘Ut-+R,
and
are written
as
g E E(e).
with 0~9,
~~22,
f’~ G(u),
with u E J,, .f’~ 9.
and
To develop a variational formulation of the structural problem we must require that the multi-valued monotone constraint operators E and G are conservative. To this end they are often assigned as subdifferential operators of convex or concave constraint functionals by setting [S, 14, 151 E = c:cp, G = c’y, with (p:k++‘X u { + ;c ] convex and g:&++‘R u { + x 1 concave. As a result of the potential theory of multi-valued monotone operators developed in [6], we can state that the functional cp coincides, to within an arbitrary additive constant, with the potential of the operator E defined by the integral formula PE(E) - PE(O) =
’
When a constraint relation is not assigned as the subdifferential of a convex potential but is directly expressed by a monotone multi-valued operator, its conservativity must be checked a posteriori. To this end we have to evaluate the functional defined by the integral formula above and to verify that its subdifferential yields back the given operator.
5. VARIATIONAL
PRINCIPLES
On the basis of the results reported in the previous sections, we can now develop a general variational theory of geometrically non-linear structural problems with constraint relations governed by conservative monotone multi-valued operators. In the next section we discuss geometrically non-linear structural problems with external rigid frictionless bilateral constraints and linear elastic constitutive behaviour. 5.1. The simplest case: rigid bilaterul constraints The occurrence of multi-valued operators relating dual pairs of state variables in structural mechanics is far from being unusual even in the simplest case of external rigid frictionless bilateral constraints. In fact, the admissible displacements belong to a subspace ~2~ E ?L and the set of admissible forces is the linear variety I + %$. Here 1 is the prescribed load acting on the structure and the orthogonal complement 22 0’ is the subspace of external constraint reactions. The relation between .f and u is then expressed by the constant multi-valued operators
defined
by
The domain and the co-domain of these multi-valued operators are affine spaces; in fact, domC=/+~~anddomC’=~~‘,. For structural models with external rigid frictionless bilateral constraints, the original problem can, however, be re-formulated in terms of a single-valued operator by assuming
Non-linear
and non-smooth
structural
203
problems
the subspace of the admissible displacements as reduced ambient space; the corresponding dual space will then be the quotient space F/42; of external forces f~ F modulo the subspace of the constraint reactions 42,o”. The elements of the quotient space F/42; are the affine sets (equivalence classes) of external force systems which differ by a constraint reaction. To get the expression of the reduced static-kinematic operator, we introduce the following pair of dual linear operators: n: ?!Lo+ %
canonical injection,
If’: 9 + P/%0’
canonical projection.
The operator II maps every admissible displacement into itself considered as an element of the whole displacement space; the dual oprator TI’ maps each external force f E 9 into the corresponding equivalence class 7~ .F/%!&. The static and the kinematic compatibilities are governed by the following pair of reduced operators: ff [dD(l-Iu)]‘: Y + 9-;/C&+,
The reduced static-kinematic operator which maps each ordered pair y = (u, a) into the corresponding dual pair J’ = (c E) is, accordingly, given by An(y) =
0 on
where i= fI’I E 919’~; is the assigned reduced load. The conservativity of An(y) can be proved by the same arguments outlined in Section 4.1. The structural problem with external rigid frictionless bilateral constraints can be expressed in its reduced form by the following single-valued relations: ff’[dD(TIu)]‘o
= r” static equilibrium,
D(lTu) = E kinematic compatibility, Es = g
internal constraint,
where u E +2e, E E 9, d E Y and the elastic stiffness operator E: 9 H Y is assumed to be linear, symmetric and positive-definite. By introducing the dual product spaces X = a0 x 9’ x 9 and X’ = F;/%!,I x 9 x 9, the structural problem above can be symbolically rearranged by means of the reduced structural operator B as follows:
I” I[I[ 1
0 l-I’[dD(fIu)]’ 0 u DfI 0 -I9 a-o, E E 0 -19 [ 0 where x is the triplet (u, g, E) and I9 and Zy are the dual identity operators in the spaces ~3 and 9, respectively. The conservativity of the operator B can be inferred from the analogous property of the reduced static-kinematic operator An, from the duality existing between I9 and Zy and from the symmetry of the operator E. The potential pB of B can then be computed by a direct integration along a ray in the product space X to get OEB(X)=
1
Pet% g, 8) =
; { ([dD(Iltu)]‘ta,
(B(tx), x) dt = s0
l-h)
+ (0, D(I-Itu))
s
-(to,&)-(qte)+(E(te),~)-([u)}dt = (a, D(llu))
- (6, E) + +(EE, E) - (
The potential pB turns out to be linear in 6, non-linear and differentiable in u, convex and differentiable in E.
G.
204
ROMAN0
Yt d
Two functionals can be further obtained from the expression of the potential pB by enforcing the kinematic compatibility condition and the internal constraint relation: Pi(U,0)
= (0,D(IIu,)
- $(a,
Pi(U) = $(ED(rIu),
E-la)
D(IIu))
- (lu),
- (Cu).
These functionals assume the same value in correspondence of a solution of the structural problem x = (u, cr, e) and their stationarity yields back the corresponding operator form of the structural problem. This feature is illustrated, in particular, with reference to the potential pe. The stationarity of pe, which is locally convex at .Y = (u, (r, E), can be obtained by imposing that the null vector 0 E X’ is included in its local subdilferential at x:
The considerations provided in Section 2.1 allow us to infer that the relation equivalent to enforce the stationarity of pB separately in each space.
and that the structural
problem
0 = d,,j+#.
CT,c).
0 = d,p,(u,
(T,E),
0 = d,p,(u,
6.4,
in its reduced
operator
above
is
form is recovered.
5.2. The general non-linear model When both the external and the internal constraints are described by conservative monotone multi-valued operators, we must necessarily deal with the non-linear problem in its globality. Its operator formulation is: j‘= [dD(u)]‘a
static equilibrium,
i: = D(u)
kinematic
0E E(c) = ?C&)
internal
constraints.
,f‘~ G(u) = &J(U)
external
constraints.
compatibility,
Introducing the dual product spaces W = +Yx Y x $2 x 9 and W’ = 9 x %”x ,‘/ x J//. the problem can be rewritten in the following symbolic form by means of the structural operator
2: 0 D
0
-I:,
- I ‘,,
(7y,
I:
0
0 -1,
0
0 I[1 u
-I,-
0
o
OE Z(w) =
0
IdW41'
Gl*
.f
where w is the ordered quartet (u, CJ,~,,f‘) and the operators I#, I,F and IV, I, are the dual pairs of identity maps in the spaces %, J and 9, 8, respectively. The conservativity of the operator Z is easily inferred from the conservativity of the static-kinematic sub-operator A, the assumed conservativity of the constraint operators, and from the duality existing between the two pairs of identity maps I,&, 1.9 and I,,!, Iv. The potential pz of Z can now be computed by means of a direct integration along a ray in the product space W, 1
Pz(4
=
(Z(rw),
\v> dt,
s0
to get Pz(k
0, hf’)
=
(P(E) + s*m
+
(0,
D(u))
-
u>
-
(a,
It is apparent that the potential pz turns out to be linear in 0, non-linear in u, convex in E and concave in,f:
c>.
and differentiable
Non-linear and non-smooth structural problems
205
The following equivalent formulations can be given to the constraint relations: GE Q(E) 0 E E acp*@) 0 (P(E)+ q*(o) = (c, E), fe &I(u) * u E @I*(f) *C?(u) + s*(f)
= (f, a>.
The last terms in the previous equivalences are known in convex analysis as Fenchel’s equalities. By reiterated substitutions of the constraint relations, in the form of Fenchel’s equalities, and of the kinematic compatibility condition E = D(u), the following family of functionals can be derived from the potential pz [ 161: HI@, E, a) = cP(d - g(u) + (0, D(u)) - (a, E), ff,(k
qf,
Rl(4
= - cp*m + s*(f)
+ (a, D(u)) - CL u>,
cl = - cP”(4 - g(u) +
R2@4f) = dm))
+ s*(f)
F(u) = cpw4)
- <.A U>>
- du).
The functionals HI, RI and F are the generalization, to the present non-linear and non-smooth context, of the functionals of Hu-Washizu, Reissner and of the total potential energy in the context of linear elasticity [8]. All the functionals in the family above do assume the same value when evaluated in correspondence of a solution w = (u, 0, s,f) of the structural problem since the kinematic compatibility and the constraint relations are fulfilled. Further, the stationarity of the potential pZ as well as that of the other functionals is equivalent to the operator forms of the problem. We will show explicitly this feature with reference to the potentials pz and F. The stationarity of the potential pZ is formally expressed by 0 E @Z(W), which amounts to requiring, by virtue of the remarks of Section 2.1, the stationarity of pz separately in each space: 0 = d&u,
6, s,f),
0 = d&u,
6, s,f),
0 E &P,(4 63 Gf), OE asPz(a, 0, Gf). Here ~3,and d, denote the subdifferential and the superdifferential with respect to E andf; respectively. The first relation states that (a, dD(u)u) - (f; 21) = ([dD(u)]‘a
-f, o) = 0 VoE %,
which represents the static equilibrium condition [dD(u)]‘a = f. Likewise, the second relation yields E = D(u), and, by definition of subdifferential, we have, respectively, from the last two conditions, 0 E a&s), u E G*(f). Let us now consider the potential F(u) which is locally subdifferentiable in u since it is composed of the subdifferentiable convex functional cp and the non-linear differentiable operator D.
G. ROMANO
706
et ul
The stationarity condition states that the null vector belongs to the local subdi~erentiai of the potential F, i.e. 0 f PF(u).
This condition is equivalent to an operator formulation of the problem in terms of displacements. In fact, by the additivity property and the chain rule of the local subdifferentials, we have iiF
= c?[yl(D(u)) - &L)] = =
;i[cp(D(z*))]
-
?&4)
rd~(~)]‘~9(~(~~)) - Sg(uf.
The stationarity condition implies that there exists an internal stress CTwhich is an element of the local subdifferential of 9 at i: = D(U) and an external force j’which is an element of the superdifferential of y at II such that the equality holds between the two terms on the right-hand side of the previous relation. In formulas we have
The quartet (u, (r, s,f’) is then a solution of the given non-linear structural problem. The converse implication follows easily by reversing the steps above. Incidentally, we note that the functional F turns out to be the potential of the operator given by F(u) = [c?(9”D - ~f][U-J so that F = PF. The same feature can be shown, although in a less obvious way, for the other functionals derived from pz. Actually, each of these functionals turns out to be the potential of a structural operator associated with an operator form of the structural problem. In particular, the operator form associated with the locally convex functional Hi is given by -i?
0 E H,(u, a,,f‘) =
D - I ‘u
It is apparent that the operator Hz is conservative and then its potential turns out to be Hz, that is, Hz = 2H2. Repeating the arguments above for the functional Ri , which is concave in CIand locally convex in 14. and for the functional Rz, which is localiy convex in u and concave in,!; the following expressions of the associated operators are recovered:
and
O~R~@,f’)=
c
o’(9 ,D) _I
‘//
- I,,-
11
i)g* II
.1’ *
1
Once more, the relations RI = i?K, and R2 = r?R, can be inferred from the conservativity of the operators R, and R2. dc~~~wied~~~~n~s-- The financial gratefully acknowledged.
support
of the Italian
Ministry
for Scientific
and Technological
Research
is
Non-linear
and non-smooth
structural
problems
207
REFERENCES 1. S. G. Mikhlin, Variational Methods in Mathematical Physics. Pergamon, New York (1964). 2. M. M. Vainberg, Variational Methodsfor the Study of Nonlinear Operators. Holden Day, San Francisco, CA (1964). 3. J. T. Oden and J. N. Reddy, Variational Methods in Theoretical Mechanics, 2nd edition. Springer, Berlin (1981). 4. J. J. Moreau, On Unilateral Constraints, Friction and Plasticity. New Variational Techniques in Mathematical Physics (Edited by Cremonese), pp. 171-322. CIME, Bressanone (1974). 5. P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Birkhauser, New York (1985). 6. G. Romano, L. Rosati, F. Marotti de Sciarra and P. Bisegna, A potential theory for monotone multi-valued operators. Quart. appl. Math. (to appear). condition for finite strain measures. 7. G. Romano, L. Rosati and F. Marotti de Sciarra, On a consistency Meccanica (to appear). 8. K. Washizu, Variational Methods in Elasticity and Plasticity, 3rd edition. Pergamon, New York (1982). 9. K. Yosida, Functional Analysis, 6th edition. Springer, Berlin (1980). 10. I. Ekeland and R. Temam, Analyse Conuexe et Problbmes Variationnels. Dunod, Paris-Bruxelles-Montreal (1974). 11. A D. Ioffe and V. M. Tihomirov, The Theory of Extremal Problems. Nauka, Moscow (English translation, North-Holland, Amsterdam) (1979). Convexes, Lecture notes, siminaire: Equationes aux DBrivees Partielles. Coll&gie 12. J. J. Moreau, Fonctionelles de France (1966). 13. R. T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970). 14. G. Roman0 and L. Rosati, Variational principles in convex structural analysis. Volume in onore de1 Prof. Giulio Ceradini, Roma (Luglio 1988). 15. G. Roman0 and L. Rosati, A survey on recent advances in convex structural analysis. Atti del Convegno in Ricordo di Riccardo Baldacci e Michele Capurso, Roma (27 Ottobre 1989). nella dinamica non lineare di strutture con vincoli 16. G. Romano, L. Rosati and G. Ferro. Principi variazionali convessi. Atri X Congress0 Naz. AIMETA, Pisa (2-5 Ottobre 1990).
APPENDIX Theorem Al. Given a functional
f: X x Y H ‘Jz locally convex at the point (x, y), it turns out to be Zf(x, Y) c &f(x,
where d, and 8, denote Proof By definition,
the partial
local subdifferentials
Y) x a,/(x,
a pair (x*, p*) E X’ x I” belongs to the local subdifferential dfC(x, y); (% 8)I t ((x”. y*), (x, L:))
Choosing
f=
Y),
in the spaces X and Y, respectively. offat
(x, y) if it turns out to be
v(x, j) E x x Y.
0 we have def d/[(.x, y); (% O)] 2 (x*, X> Vx E X * x* E &j-(x, y),
and, analogously,
setting X = 0, dfC(x. Y):@,
HI t (Y”, j) Vj E Y 0 y* E &f-(x, y),
which prove the theorem. If the functional to be
0
f is differentiable
at (x, y), its local subdifferential dS(x, Y) = d,f(x,
In order to prove the next theorem, we introduce A: X x YH X is the linear operator defined as A(x,y)=x and its dual operator
coincides
Y) x d,f(x,
with the differential
and it turns out
Y).
some preliminary
definitions.
The canonical projection
V(x,y)~xxY,
A’: X’ H X’ x r is the canonical injection A’x* = (r*, 0)
We can now state the following
Vx* E X’.
theorem
Theorem A2. Let 9 : X ++ ‘iRbe a locally convex functional at y E Y. The functional S: X x Y I+ R defined by
at x E X and h : Y w R be a locally convex functional
f(x> y) = g (x) + h(y) turns out to be locally convex
at the point (x, y) E X x Y and its local subdifferential WX.Y) =
%(x)xWyL
or, equivalently,
cx*.Y*) provided
that the addition
E
Nx, Y10
rule of local subdifferentials
x* E Wx), Y* E WY),
holds true.
at (x, y) is given by
G. ROMANO
208
rr ul.
Proof. Let the linear operators .4 : X x Y c 3’ and U: X x Y ++ Y be canonical projections and let A’ and B’ be the corresponding dual operators. By definition of canonical projection. the functional ,/ can be equivalently written in the form
, I\-,!‘) := (I/ .A+ II B)(u,y). so that the additivity
property
of the local subdifferentials
il(\-,J,)=i(y
A +/I
yields
B)(Y.~)-C’((/
A)(\.!‘)+;(/1
R)(IJ).
The local convexity at (1. J) of the functional $1 .A : X x 1’~ 93can be inferred from the analogous functional 9. In fact. the directional derivative of (1 4 IS given by d(c/ and, hence, by definition
A)[(u.~);(u.
r):.-l(\-. ?)I.
of .4. d(c/
From
C)] = dc/[.4(\-.
this result and the definition
.~)[(Y.T);(?.
of subdifferential.
P)] = dg(u; 7).
we Infer that the followmg
(i) (r*, J*) E r’(
2 ((.Y*.J,*).(\..
(iii) dy(.x;.C) 2 (u*, ?) + (j*,i;) Inequality
C)>
V(C P)E.Yx
I’.
V(\-. ~:JE A’ x k.
(iii) implies that 1‘* = 0 and .x* t ?y(u). so that It turns out to be iI
Repeating
property
the same arguments
for the functional
A )(Y. !‘I = ./l’i$/(Y) II 8, the theorem
is proved
expressions
are equivalent:
of the