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Quasidifferential modelling of adhesive contact
Mathl. Comput.
Modelling Vol. 28. No. 4-8, pp. 455-467, 1998 @ 1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain 0895-7177/98 $19.00 + 0.00
PII: SO8957177(98)00135-6
Quasidifferential Modelling of Adhesive Contact Institute
Abstract-A
G. E. STAVROULAKIS of Applied Mechanics, Department of Civil Engineering Technical University of Braunschweig, D-38106 Braunschweig, Germany [email protected]
number of phenomenological models for the adhesive contact problem are presented
in this paper. The nonmonotone nature of the adhesive contact laws and the inequalities that are introduced by unilateral contact effects lead to nonsmooth and nonconvex potential energy optimization problems. For discretised problems the potential energy function is in general assumed to be qua&differentiable and/or the sets that describe the inequality subsidiary conditions are also assumed to be described by quasidifferentiable functions. The structural analysis problem is described in a systematic way by the optimality conditions of the quasidifferentiable potential energy. The arising variational problems generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a computationally efficient explication of hemivariational inequalities of nonconvex, non smooth mechanics. Nonlinear structural analysis methods are proposed for the solution of the problems by using elements of quasidifferential optimization. @ 1998 Elsevier Science Ltd. All rights reserved. Keywords-Quasidifferentiability, lit&, Hemivsziational inequalities.
Nonconvex energy, Nonsmooth mechanics, Variational inequr+
1. INTRODUCTION Adhesive contact effects and adhesive frictional effects which are possibly coupled by unilateral contact effects arise mainly in structures which include interfaces (e.g., in layered composite plates or in repaired cracked structures). A phenomenological quasistatic modelling of the mechanical behaviour of these effects leads to the adoption of a nonmonotone and possibly multivalued traction-displacement law for the adhesive interface. Nonmonotonicity copes with strength-degradation effects and multivaluedness of the law, e.g., the appearance of complete vertical branches in the graph of the law copes with locking and instantaneous cracking effects. Laws in the normal to the interface direction, the tangential direction and coupled laws can be used [1,2]. The structural analysis problem for structures which involve nonmonotone, possibly multivalued laws, can be described in certain cases by means of a nonsmooth and nonconvex superpotential energy function. Critical points, substationary points and of course local minima of the potential describe all stable and unstable equilibrium points (i.e., the solutions) of the examined structural system. They are in general solutions of hemivariational inequality problems [2-41. For the systematic study of these problems and for the construction of algorithms for their efficient solution, separate treatment of convexity and concavity information in connection with Part, of this
work has been completed in the RWTH Aachen, Germany, with the financial support of the Alexander von Humboldt Foundation, Bad Godeeberg, Bonn.
455
456
G. E. STAVROULAKIS
techniques of nonsmooth analysis has been found to be fruitful [5]. Fortunately, this information is known at the elementary level of the constitutive laws or the boundary conditions which are adopted for the mechanical modelling of the structural elements and can be traced up to the level of the structure by means of appropriate calculus rules. These steps can be performed by using the notion of the quasidifferentiability and the corresponding quasidifferential calculus [6-91. Thus, quasidifferential modelling and optimization techniques are introduced in structural analysis [lo-13). We recall here that for convex, nondifferentiable problems the adoption of the convex subdifferential give rise to the formulation and study of variational inequality problems in mechanics [2,14]. Unilateral contact and Coulomb friction effects have been tackled in a systematic way by means of convex optimization techniques and the theory of variational inequalities (see, e.g., [2,14,15] and the references therein). The convex analysis approach to contact problems will not be reviewed in details in this paper. Instead it will be used as a subproblem for the treatment of the nonconvex case. For nonconvex, nondifferentiable problems the explication of the arising optimality conditions gives rise to systems of variational inequalities [5,11] which can be solved either by quasidifferential optimization algorithms, or in some specific cases by appropriate multilevel decomposition techniques. Elements of quasidifferential modelling in mechanics of adhesive contact effects are given in this paper. The purpose of this work is to present various models which have been developed for analogous applications elsewhere, to discuss their appropriateness for the modelling of adhesive effects in a phenomenological elastostatic way and to address the reader’s attention to appropriate algorithms for their numerical solution. Details of the various presented models, the algorithms and numerical examples have not been included, since they are available in previous publications. The layout of the paper is as follows: quasidifferential energy optimization problems, both constrained and unconstrained ones and the corresponding optimality conditions are formulated in the next section. The adhesive contact problem is formulated in Section 3. In Section 4, a nonconvex qussidifferential plasticity theory is outlined which can be used for the modelling of certain adhesive and nonconvex locking effects. In Section 5, a simplified linear complementarity problem formulation with adhesive unilateral contact effects is described. It can be considered as a special case of the nonconvex plasticity approach with the advantage that it leads to more simple problems. In the last section, appropriate algorithms for the solution of the arising problems are listed and references to companion publications are given where the details of the algorithmic implementation and some representative numerical examples can be found.
2. QUASIDIFFERENTIABLE POTENTIAL ENERGY OPTIMIZATION AND OPTIMALITY CONDITIONS A function f defined on an open set X E Rn is called quasidifferentiable at z E X if it is directionally differentiable [6,9] at z and its directional derivative f’(z,g) along the direction g E Rn can be written as
Here U and V are convex compact sets in Rn. The ordered pair of(z) = [V, V] is called the quasidifferential of f at z, U is called the subdifferential of f at t and is denoted by &f(z) and V is the superdifferential of f at z, denoted by at(z). Practically all convex and nonconvex, possibly nondifferentiable, finite-dimensional potential energy functions which arise in structural analysis are quasidifferentiable [12,13]. For the calculation of the quasidifferential of given structured functions, (i.e., functions which are produced from classical differentiable components and the rules of addition, subtraction, multiplication, division, finite combinations of maximum and minimum operators and of composite functions) calculus rules have been developed [7-9,131.
Quaeidifferential Modelling
457
Accordingly, a set A c Rn which is defined by a finite number of equalities and inequalities involving quasidifferential functions is called quasidifferential. For these kind of generally nonconvex sets with nonsmooth boundaries, normal and tangential vectors are exactly defined by the quasidiierentiability concept. Thus, mechanical theories which involve normals and tangentials to these sets can be constructed without the classical restriction of convexity (see, e.g., the model proposed in [16,17] for nonconvex plasticity, which will be used in the sequel for the treatment of adhesive contact problems). For quasidifferentiable optimization problems necessary and in some cases sufficient optimality conditions can be written and will be used for the description of appropriate structural analysis problems. First-order necessary (respectively, sufficient) conditions for a quasidifferentiable function f to admit a local minimum (respectively, a strict local minimum) at point t’ E Rn read (respectively, int 2 (z*)) . -df (Z*) c 2L (z*) (2) For constrained optimization, we have analogous inclusions (see [6,9,13] and the subsequent sections). For comparison with classical linearization techniques, we recall here that the notion of the quasidifferential gives rise to the following local approximation of f(z) around point x* (qussilinearization) :
with (oZ(crA)/a) t 0, as cr 1 0, VA E Rn. For an extension of the outlined theory to cover Hausdorff continuous operators (which are more tractable from the numerical point of view), the notion of the codifferentiability has been proposed (see, e.g., [9,13]). Since for the theoretical presentation of the models, no additional information is introduced by this notion, this extension will not be reviewed here.
3. STRUCTURAL ANALYSIS SUPERPOTENTIAL
PROBLEM FORM
IN Q.D.
Let us assume a structure with adhesive contact interfaces and let us assume that the phenomenological elastostatic analysis problem of this structure admits a potential characterization, i.e., the governing relations of the problem can be derived by appropriate differentiation of a potential energy function. For the modelling of interface adhesive effects, we adopt nonmonotone, possibly multivalued (i.e., with complete vertical branches) interface traction-relative displacements laws (see, e.g., [1,4], F’g 1 ure 1). A finite element discretization of the direct stiffness method with nodal displacements as the primary variables of the problem is assumed here. Let u be the n-dimensional vector of displacement degrees of freedom and e is the m-vector of element deformations. A fairly general discrete potential energy optimization problem in elastostatics reads (4) where II(e) is the elastic energy stored in the system due to deformation, O(u) is the potential that counts for various boundary, interface or skin effects and p(u) is the potential that generates the external loading vector. In general, relative interface displacements are involved in the definition of @(u) and, respectively, interface tractions appear as their work dual variables in the virtual work expression which follow. The more economical notation is preferred here. For technical details see [1,5,13,18]. The geometric compatibility transformation is written in the form of a generally nonlinear but differentiable operator d(u) : Rn + R”‘, e = d(u).
(5)
458
G. E. STAVROULAKIS
Multibranch elasticity or holonomic plasticity models with ascending and descending complete vertical branches that count for crashing, cracking, and locking effects in a phenomenological way are covered by this model, by using nonsmooth and possibly nonconvex superpotential energy functions II(e) in (4). Analogously, nonmonotone relations, like stick-slip boundary or interface laws, frictional or softening frictional laws, etc., introduce nonsmooth and nonconvex potential functions Q(u) in (4) (see, [2,4,13] among others). The set of kinematically admissible displacements is in general a quasidifferentiable set, defined by Uad = {U E R” :'H(U) 5 0, G(U)=
o},
(6)
where O(u) : Rn + Rnl, ‘H(u) : R” --+ Rnz, and nr + 712 < n are general nonlinear and possibly nondifferentiable (but quasidifferentiable) equality and inequality constraints. Bilateral and unilateral contact effects and locking behaviour in a large displacement setting leads to relations of the (6) type. For simplicity only, unconstrained structural analysis problems, i.e., u ad = Rn are considered in this section.
S
de
[ul
(b)
(4
(c) Figure 1. Monotone tentials.
and nonmonotone
interface laws with corresponding
superpo-
Variational formulations for the elastostatic analysis problem described by (4) will be produced in the sequel by writing down the optimality conditions for this qussidifferentiable minimization problem and using the quasidifferential calculus for the derivation of the quasidifferential of the complete function a(u). Variational equalities for classical smooth problems, variational inequalities for nonsmooth, subdifferentiable problems (cf. [2,4,14]), and systems of variational inequalities for general quasidifferentiable problems (cf. [4,5,11,13]) are thus derived in a systematic way.
Quasidifferential
Modelling
459
We recall that for classical nonlinear elastostatic analysis problems with Ud = Rn, the structural analysis problem is written in the strong form: find u E Rn such that
Vrqu) = 0,
(7)
or in the weak (variational equality) form: find u E Rn such that l?(u,Au)
= Vl=I(~)~vu
=
1
F
T %A,
[ + [FITAu+
[F]TAu=O,
VAUER”.
Let II(e) of (4) be quasidifferentiable and let DII(e) = [aII(e),aII(e)] E Rm x R”‘. Dfi(e) E R”’ x R” can be constructed by using the rules of the quasidifferential calculus (see, [9, p. 127; 121) for the composite function II(e(u)). If moreover ‘P(u) and p(u) are differentiable, we get
Di=I(u) = [&=I(u),RI(U)] = [g-I(u) + W(u)
+ VP(U),
The optimality conditions for the quasidifferentiable unconstrained find u E Rn such that -@I(u) c El(u).
all(u)] .
optimization
(9) problem read: (IO)
They lead to the system of variational inequalities: find u E Rn such that VW E --Xl(u).
w c &I(u),
(II)
An interesting case arises if the potential energy is a difference convex (d.c.) function. For instance, let a small displacement problem be considered, i.e., (5) replaced by d(u) = GTu, with GT an (m x n) matrix and accordingly p(u) = pTu, with p the n-dimensional loading vector. Let moreover II(e) in (4) be convex and differentiable, e.g., consider for instance the linear elasticity problem with II(e) = 1/2eTKce, where Kc is the (m x m) natural stiffness matrix of the structure. Let the only cause of nonconvexity and nondifferentiability in the problem be introduced by a difference convex boundary potential energy function, i.e., a(u) is written as a difference of convex, possibly nondifferentiable terms [5,18] G(u)
In this case, by using the quasidifferential
=
@‘1(u)
-
@2(u).
(12)
calculus, we get
@I(u) = Ku + p + a@,(u),
Bl=I(u) = -z&(u).
(13)
Optimality conditions for the d.c. potentially lead to the following system of variational inequalities that describe the nonsmooth and nonconvex structural analysis problem: find u E Rn such that O~Ku+p+d@i(u)-w, (14) for each w E Rn with w E a@,(U).
(15)
Problem (14),(15) can be solved iteratively, where in each iteration cycle two convex subproblems are solved (see, [5,18]). The above decomposition is explicitly defined due to the global d.c. rep resentation of the potential energy. In the general qussidifferential case, the system of variational inequalities (11) is implicitly defined.
460
G. E. STAVROULAKIS
4. Q.D. PLASTICITY-BASED MODELLING ADHESIVE EFFECTS In this section,
we outline
a model for the adhesive
for the treatment
of nonconvex
in [11,13,16,17].
In general,
convex analysis
approach
elastoplasticity framework
to standard
yield sets are introduced.
In turn,
the rules of the quasidifferential of plasticity. by appropriately
By means
normal calculus
of this approach,
formulated
systems
(see, [16,17] for more details). This theory will be specialized
effects which is based on a general
problems
of elastoplasticity generalized
OF
with quasidifferentiable and in particular
elastoplasticity,
and tangential
nonconvex
elastoplasticity
of convex variational
here for the treatment
sets introduced by extending
inequalities of adhesive
the convex
problem
the
quasidifferentiable
cones to the yield sets are produced
and they are used to generalize the general
approach
by
flow rules
can be described
as it will be outlined
here
effects, where the adhesive
interface (or boundary) is modelled by means of a thin layer. At this layer, relative displacement and traction vectors are denoted by [u] and S, respectively. By following the general framework of elastoplasticity, a yield locus C in the S space can be postulated and in turn adhesive effects can be described like the flow rules of plasticity. Convex models for interfaces, which include among others unilateral contact effects and Coulomb static friction effects with given normal traction, can be modelled in this way (see [19, pp. 100-103; 201). Adhesive effects require the consideration of nonconvex yield surfaces (see Figure 2). Quasidifferential modelling provides us with a tool for the treatment of this case in finite-dimensional (discretized) problems, as it will be shown here. Only associated plasticity based models will be considered since they give rise to energy optimization problems which permit a direct application of the quasidifferential optimization results of the previous section to the structural analysis problem. Nonassociated models can be constructed by following the lines of [20].
Figure 2. Nonconvex yield surface and associated
flow rule
Let for a discretized problem the only internal variables, in the standard elastoplasticity terminology, be the inelastic relative displacements [u]‘“. Let moreover the elastic region in the is used: interface traction space be a closed convex subset C in Rm. The following notation elastic strains ee, inelastic strains ein, enlarged stress vector s, and natural stiffness matrix K. The above quantities
are defined
by
Quasidifferential
Modelling
461
Here, as usual e, cr, Ko refer to the elastic substructure, i.e., all the finite elements of the structure except of the inelastic quantities of the adhesive interface. Interface quantities are denoted by [u], [u]~“,S, Kin, where Kin is the elastic stiffness matrix of the interface and [,lin counts for inelastic effects. For notational simplicity, vectors and matrices are not written boldface in this section. Moreover, as in standard elastoplasticity, we assume the existence of a convex free energy potential W (e - e’“) = f (e - ein)T KO (e - e’“) ,
(17)
and a convex, nonnegative, sublinear, and positively homogeneous of order one internal dissipation potential
v = I; ([a]‘“)f
mg
{ST[tip} .
(18)
Here I6 is the support function of the convex set C (see, e.g., [2, p. 551). The governing relations for the structural analysis problem are derived by differentiating (respectively, subdierentiating) (17) (respectively, (18)) with respect to e (respectively, [ti]‘“):
[;I=[7 Iin] [[u]‘Iu]~~]~ s = aI&
(19) (20)
([ti]‘“) .
The latter relation can be inverted to give the flow rule, in local variables
[ii]‘” E
aIc(s)
= NC(S),
(21)
= J+(s),
(22)
or in global variables P E &(a).
with 6 = R” x C. Here Ic denotes the indicator function of C and NC is the normal cone to the set C. If C is described by the convex yield locus F(S) as C = {S E R”’ 1F(S) 5 0))
(23)
relation (21) is equivalent to the convex variational inequality
[ti]‘”T (T
- S) 5 0,
VT E C, S E C, or VT s.t. F(T) 5 0.
(24
The rate elastoplastic analysis problem is discretized by appropriate schemes and by the assump tion that within the time step At = t,,+l - t, the behaviour is holonomic. Thus, a stepwise holonomic problem arises. For instance, by an implicit Euler scheme (21) yields in
%+1
-
At
ek E
dIc(sn+l)
* eF+,;, - ek E X&s,+l).
(25)
Relations (25) and (19) written for time step (n + 1) (with F = K-l) relation 0 E Fs,,+l - e,+l + ek + a+ (sn+l) , or equivalently to the minimization min &t+rEC
Jn+i(sn+l)
lead to the incremental (26)
(stepwise holonomic) problem = ;s;+iFs,+i
-
e,T+lsn+l
in
+ e,
T
(27)
G. E. STAVROULAKIS
462
Solutions of problem (27) are also solutions of the differential inclusion (multivalued equation)
or %+J(%+1
) fl {-q+?I+1)}
#
0.
(29)
An equivalent convex variational inequality problem reads: find s,+r such that Vf(Sn+l)(T
where f(s,+r)
-
Sn+l)
+ I&)
-
A&+1)
2 a
t/r
E
Rn+"',
(30)
is defined by
f(%+i) =
(31)
Figure 3. Nonconvex friction law which admits the convex decomposition of (33).
Details of numerical algorithms based on this general approach can be borrowed from parallel recent developments in convex elastoplasticity (see, among others [21-251). Let us extend the previously outlined model to include nonconvex, quasidifferentiable yield sets described by R = {S E R” ] 4(S) < 0). (32) To give a concrete example, we assume a model case where Cl is the union of a finite number of convex subsets Ri, i E Z = (1, . . . , m}, i.e., (see, e.g., Figure 3) R=
u i-ii, %={S E Rm 1&(S)
5 0))
(33)
iEI where CJ$are classical convex and differentiable yield functions (cf. (23)). The cone of feasible directions (Bouligand cone) to a set defined by (32) is characterized means of the directional derivatives of $(.) as
rl(S) = {g E R”’ I ti’(s,g) 5 0). By means of the definition of a quasidifferentiable rl(s)
by
(34)
function rr is equivalently expressed by
= {g E Rm I (wl,g) + (w2,g) 5 0, VW E 3$(s),
forSOme w
E
&J(S)} v
(35)
Quasidifferential
or
h(s) =
u
n
w,E&(S)
Modelling
463
u
T~,+~,(s)=
WlE&(S)
TS+(v)+w2(S),
(36)
w2EWS)
where the following shortcuts have been used (see also Figure 2): Tw+wa(S) Z39(s)+w2
($1
=
(9
E
R”’
=
{g
E
R”
I h,d
I
+
hr
d
+
b-ad
(~2,
IO),
d
5 0,
vwl
E
&#G)} .
(37)
One should mention here that (36) represents a systematic convex decomposition of the tangential cone to R at the point S, due to the fact that the sets defined in (36),(37) are convex. Thus, a representation of the normal cone to R at S by using convex constituents is possible. It has the form J%(S) = n u T:1+W2(S) = n $+Cs)+W2 (S) w2al5(S)
=
we!6(s)
f-)
w2aws)
{ -T~w)+w,
(S))
*
wzfk(S,
Here, the polar cone to T, denoted by To, and the conjugate cone of T, denoted by T+, are defined by To = -T+ = {g E R” 1 (g,s) 5 1, Vx E T} . (39) In the R”+m space, we have, respectively
1’ p(s) =[&)1’ r(s) =
[
Rn ri(S)
(40)
The definitions of the previous section and the optimality conditions for a quasidifferential optimization problem give rise to the following variational expression in compact form: find s~+~ such that %,+J(s,+i)
# 0,
n T+(s,+l)
(41)
Qw2 E &J(S,+,).
Relation (41) is written equivalently as %+,
f(Sn+l
1 fl wh+-n+l))#
%,+J(sn+d
n { -T’(sn+d}
#
0,
VW2
E W&I+1),
‘it
VW2
E &m+d,
or
(42) (43)
where f(s,+i) has been defined in (31). In terms of plastic multipliers, we first write the flow rule (21) as [I$”
E Ah(S)
=
US&@(S)~j(Sj
+ W2)l
Xj 2
0,
VW2
E
&(S).
(44)
Note here that, in contrast to what happens in the convex case, the plastic multipliers Xj depend on the choice of w2 E ad(S). By using the time discretization scheme introduced previously, we get the incremental relation (cf. (26))
4nSn+l
-
bln+l
+
[4F+1
+
u ~j~~~(~~+l)
FOG+1-
‘%,I
=
0,
Xj(Sj
+ 202)
=
0,
VW2 E &(S,+r).
(45)
464
G. E. STAVROULAKIS
The above expressions can further be simplified if the sets & and 84 are polyhedral. Then, only the vertices Sj E &(&+I) an d wz E &(S,+l) need to be considered in (44),(45) and, u is replaced by C, as it will be shown in the concrete example in the sequel. For the yield surfaces (33), the above general scheme results in the following formulae: function 4(S) of (32) can be written as a min-type function by using the yield loci of subsets Ri in (33), i.e., (46) For the min-type function of (46), which is quasidifferentiable, ential and quasidifferential are given by the relations
we may assume that its subdiffer-
(47) where the following active index set To(S) = {i E 1, s.t., d(S) = q&(S)} is used. A more general approach to nonconvex elastoplasticity based on star-shaped yield sets and the theory of [ZS] has also been proposed [16,17]. Instead of following this line of generalization, we present in the next section a more restricted model which is appropriate for certain adhesive unilateral contact problems.
5. MODIFIED L.C.P. APPROACH TO THE ADHESIVE UNILATERAL CONTACT PROBLEM In thii section, a model for adhesive unilateral contact problems which has been introduced in [27] is reviewed. It depends on an additive decomposition of unilateral contact effects and adhesive effects, where the unilateral contact problem for linearly elastic structures is formulated ss a linear complementarity problem and the additional adhesive effects are supposed to be produced by nonconvex yield sets in a way similar to the one followed in the plasticity model of the previous section. Let us consider a discretized unilateral contact problem. Let discrete contact forces at a unilateral boundary or interface be denoted by ri, i = 1, . . . , n and be assembled in vector r. External forces are denoted by vector p. Superscript T denotes ss usual the transpose of a matrix or vector. Let moreover at the unilateral boundary or interface discrete gaps be denoted by vector y . Let us assume that contact forces and gaps are decomposed in a unilateral contact part and an adhesive part (denoted by subscripts u and a, respectively) as follows r=rU+ra,
Y =yu+ya.
(48)
We recall that a frictionless unilateral contact problem with no adhesion effects (i.e., r,, = 0 and y,, = 0) can be formulated as a Linear Complementarity Problem LCP of the form [2,28,29] w-Fz=d,
w I 0,
z Z 0,
wTz=o,
(49)
where w=wu=yu,
z = z, = ru.
(50)
Here F is the normal compliance matrix and d is the initial openings matrix (see, e.g., [28,29]). LCP (49) is by definition equivalent to the following variational inequalities: find z, 2 0 such that (Fz, + d)T(v - zU) > 0,
vvzo,
(51)
w,=Fz,+d.
(52)
and find w,, 2 0 such that zz(w:
- w,) 2 0,
Vw;>O,
Qua&differential
Modelling
465
Adhesive effects with nonconvex yielding characteristics can be included in LCP (49) as follows: let y = yU, r = r. + rU. Let adhesive tractions ra be derived by means of an appropriate nonempty closed and possibly nonconvex subset C of R”
-ra E %(Y),
(53)
or equivalently I& (y, y’ - y) 1 -r;T (y’ Adding relation (54) and variational hemivariational inequality find w E R”+ such that,
inequality
1; (w,w*-w)+zT
y) ,
t/y* E R”.
(54)
(52) (recall that w = w, here), we get the
(w*-w)
> 0,
Fz+d=w,
VW* E R”+. (55)
Analogously, locking effects with a general yield function that can be described by a possibly nonconvex, nonempty closed subset Ci of Rn can be tackled. In this case, we have y = y,, + y,,, r = r,,. A nonconvex locking criterion is considered such that it can be described by
or I& (r, r* - r) > -y,T (r* - r) ,
Vr’ E Rn+m.
From (57) and the variational inequality (51)) we get the hemivariational find z E Rn such that,
(57) inequality
(Fz + d)T (z* - z) + 1;. (z, z* - 1;) = WI (z* - z)
+I&, (z, z* - z) L 0,
(58)
t/z*-zeRn.
In an analogous way, frictional contact problems can be formulated as nonsymmetric LCP’s [29]. Adhesive and locking effects can also be introduced in this more general case by following the same procedure [27]. This model has initially been proposed in [27] for the study of frictionless and frictional adhesive gripper problems in robotics. In the same reference, existence results are given based on [30].
6. NONSMOOTH
STRUCTURAL
ANALYSIS
ALGORITHMS
A number of structural analysis algorithms appropriate for the solution of the adhesive contact problems formulated in this paper will be mentioned here. Due to lack of space and due to the review nature of this work, concrete numerical applications will not be presented here. They can be found in the references. Convex analysis applications in structural analysis will not be reviewed here. This is the case of unilateral contact or convex, associated plasticity problems which lead to the solution of variational inequalities or convex minimization problems. The reader is referred to [2,14,15] among others. Our interest will be focused on the treatment of nonconvexity which is the primary problem introduced by adhesive effects. Nonsmooth numerical optimization techniques have been used in a restricted extend for the solution of the nonsmooth potential energy optimization problem which describes the structural analysis problem. A recent review can be found in 1311. The need for controlling complicated numerical accuracies and the inefficiency of general purpose algorithms to cope with large scale problems have restricted the development in this direction. Adhesive joints and nonmonotone skin effects have been formulated ss a hemivariational inequality problem and have been solved by an iterative method in [32]. A hemivariational inequality describes both local minima and certain critical points of the potential energy function [2,4].
466
G. E. STAVROULAKIS
Substationarity optimization approaches for the solution of adhesive and delamination effects are proposed in [33,34]. A common feature of these approaches is the formulation and the solution of appropriate convex optimization subproblems, which in an iterative scheme lead to the solution of the initial problem. The quasidifferential optimization approach gives rise to a systematic convex decomposition of the nonconvex optimization problem into nested convex subproblems. Since each convex subproblem leads to the solution of an equivalent convex variational inequality, we speak about systems of variational inequalities. A simpler introduction to this method is achieved for potential energy functions which are differences of convex functions [5,18]. The extension to the general Q.D. case is discussed in [12]. Nonmonotone adhesive friction effects coupled with unilateral contact effects are solved in [18]. Adhesive skin effects are presented as an example in [12]. Comparisons of the Q.D. approach with the parallel developments in the field, which have been briefly mentioned previously, are given in [4,35,36]. Extension for the solution of the nonconvex elastoplasticity model and the LCP model with adhesion is currently under development. Hints are given in the relevant publications [16,17,27].
REFERENCES A hemivariational inequality approach to the analysis of 1. C.C. Baniotopoulos and P.D. Panagiotopoulos, composite material structures, In Engineering Applications of New Composites, (Edited by S.A. Paipetis and G.C. Papanicolaou), Omega Pub]., London, (1987). Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy 2. P.D. Panagiotopoulos, finctions, Birkhiiuser Verlag, Basel, (1985). 3. Z. Naniewicz, On some nonconvex variational problems related to hemivariational inequalities, Nonlinear Analysis, Theory, Methods and Applications 13, 87-100 (1989). Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer4. P.D. Panagiotopoulos, Verlag, Berlin, (1993). 5. G.E. Stavroulakis, Convex decomposition for nonconvex energy problems in elsstostatics and applications, European J. of Mechanics A/Solids 12, l-20 (1993). V.F. Demyanov and L.N. Vasiliev, Nondifferentiable Optimization, Optimization Software, New York, (1985). Calculus, Optimization Software, New York, (1985). 7. V.F. Demyanov and A.M. Rubinov, Quasidiflerentiable Calculus. Mathematical Programming Study, a. V.F. Demyanov and L.C.W. Dixon, Editors, Quasidifferential Vol. 29, North-Holland, Amsterdam, (1986). 9. V.F. Demyanov and A.M. Rubinov, Introduction to Constructive Nonsmooth Analysis, Peter Lang Verlag, Frankfurt a.M., pp. 414, (1995). Nonconvex superpotentials and hemivariational inequalities. Quasidifferentiability in 10. P.D. Panagiotopoulos, mechanics, In Nonsmooth Mechanics and Applications CZSM Lect. Notes Nr. 309, (Edited by J.J. Moreau and P.D. Panagiotopoulos), Springer-Verlag, Wien, (1988). 6.
11. P.D. Panagiotopoulos quasidifferentiability, 12.
G.E. Stavroulakis, mechanics, Journal
and G.E. Stavroulakis, New types of variational Acta Mechanicu 94, 171-194 (1992). V.F. Demyanov and L.N. Polyakova, of Global Optimization 6 (4), 327-354
principles
Quasidifferentiability (1995).
based on the notion of
in nonsmooth,
nonconvex
Quasidiflerentiability and 13. V.F. Demyanov, G.E. Stavroulakis, L.N. Polyakova and P.D. Panagiotopoulos. and Economics, Kluwer Academic Press, Dordrecht, Nonsmooth Modelling in Mechanics, Engineering (1996). 14. G. Duvaut and J.L. Lions, Les Znequations en Mechanique et en Physique, Dunod, Paris, (1972). 15. N. Kikuchi and J.T. Oden, Contact Problems in Elasticity. A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia, PA, (1988). G.E. Stavroulakis, Variational problems for nonconvex elastoplssticity bssed on the quasidifferentiability concept, In Proc. dth Creek National Congress on Mechanics, Vol. 1, Mechanics of Solids, (Edited by P.S. Theocaris and E.E. Gdoutos), pp. 527-534, Democritus University of Thrace and Hellenic Society of Theoretical and Applied Mechanics, (1995). 17. G.E. Stavroulakis, Quasidifferentiable and star-shaped sets. Application in nonconvex, finite dimensional elsstoplssticity, Communications on Applied Nonlinear Analysis 2 (3), 23-46 (1995). 18. G.E. Stavroulakis and P.D. Panagiotopoulos, Convex multilevel decomposition algorithms for nonmonotone problems, Znt. J. Num. Math. Engng. 36, 1945-1966 (1993).
16.
19. J. Salencon, Calcul ci la Rupture Paris, (1983).
et Analyse
Limite, Presses de l’ecole nationale des ponts et chaus&es,
Quasidifferential 20.
21. 22. 23. 24. 25.
26. 27. 28. 29. 30. 31. 32.
33.
34.
35. 36.
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J.J. Telega, Topics on unilateral contact problems in elasticity and inelasticity, In Nonsmooth Mechanics and Applications, pp. 341-462, (Edited by J. J. Moreau and P.D. Panagiotopoulos), CISM Lect. Notes No. 302, Springer-Verlag, Wien, (1988). J.C. Simo, J.G. Kennedy and S. Govindjee. Nonsmooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical analysis, Int. J. Num. Meth. Engng. 26, 2161-2185 (1988). R. Glowinski and P. Le Tallec, Augmented Lagrangian and Opemtor-Splitting Methods for Nonlinear Mechanics, SIAM, Philadelphia, PA, (1989). P. LeTallec, Numerical Analysis of Viscoelastic Problems, MsssonfSpringer-Verlag, Paris/Berlin, (1990). B.D. Reddy, Algorithms for the solution of internal variable problems in plasticity, Computer Methods in Applied Mechanics and Engineering 93, 253-273 (1991). J.C. Simo, Recent developments in the numerical analysis of plasticity, In Progress in Computational Analysis of Inelastic Structures. 115-174, (Edited by E. Stein), CISM Courses and Lectures No. 321, Springer-Verlag. Wien, (1993). A.M. Rubinov and A.A. Yagubov, The space of star-shaped sets and its applications in nonsmooth optimization, Mathematical Programming Study 29, 176-202 (1986). G.E. Stavroulakis, D. Goeleven and P.D. Panagiotopoulos, New models for a class of adhesive grippers: The hemivariational inequality approach, Archives of Applied Mechanics 67 (l/2), 50-61 (1996). A. Klarbring and G. BjGrkman, A mathematical programming approach to contact problem with friction and varying contact surface, Computers and Structures 30, 1185-1198 (1988). B.M. Kwak and S.S. Lee, A complementarity problem formulation for two-dimensional frictional contact problems, Computers and Structures 28, 469-480 (1988). S. Adly, D. Goeleven and M. ThBra. Recession mappings and noncoercive variational inequalities, Nonlinear Analysis Theory Methods and Applicationsn (1995) (to appear). M. Miettinen, M. Mkel and J. Haslinger, On numerical solution of hemivariational inequalities by nonsmooth optimization methods, Journal of Global Optimiration 6 (4). 401-425 (1995). E. Koltsakis. Theoretical and numerical study of structures with nonmonotone boundary conditions. Application to adhesion joints, Doct. Dissertation, Dept. of Civil Engineering, Aristotle University, Thessaloniki, Greece, (1991). M.Ap. Tzaferopoulos and P.D. Panagiotopoulos, Delamination of composites as a substationarity problem: Numerical approximation and algorithms. Computer Methods in Applied Mechanics and Engineering 110 (l/2), 63-86 (1993). E.S. Mistakidis and P.D. Panagiotopoulos, Numerical treatment of nonmonotone (zig-zag) friction and adhesive contsct problems with debonding. Approximation by monotone subproblems, Computers and Structures 47, 33-46 (1993). G.E. Stavroulakis and ES. Mistakidis. Numerical treatment of hemivariational inequalities in mechanics: Two methods based on the solution of convex subproblems, Computational Mechanics 10, 406-416 (1995). E.K. Koltsakis, E.S. Mistakidis and M.Ap. Tzaferopoulos, On the numerical treatment of nonconvex energy problems of mechanics, Journal of Global Optimization 6 (4), 427-448 (1995).
Modelling Vol. 28. No. 4-8, pp. 455-467, 1998 @ 1998 Elsevier Science Ltd. All rights reserved
Printed in Great Britain 0895-7177/98 $19.00 + 0.00
PII: SO8957177(98)00135-6
Quasidifferential Modelling of Adhesive Contact Institute
Abstract-A
G. E. STAVROULAKIS of Applied Mechanics, Department of Civil Engineering Technical University of Braunschweig, D-38106 Braunschweig, Germany [email protected]
number of phenomenological models for the adhesive contact problem are presented
in this paper. The nonmonotone nature of the adhesive contact laws and the inequalities that are introduced by unilateral contact effects lead to nonsmooth and nonconvex potential energy optimization problems. For discretised problems the potential energy function is in general assumed to be qua&differentiable and/or the sets that describe the inequality subsidiary conditions are also assumed to be described by quasidifferentiable functions. The structural analysis problem is described in a systematic way by the optimality conditions of the quasidifferentiable potential energy. The arising variational problems generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a computationally efficient explication of hemivariational inequalities of nonconvex, non smooth mechanics. Nonlinear structural analysis methods are proposed for the solution of the problems by using elements of quasidifferential optimization. @ 1998 Elsevier Science Ltd. All rights reserved. Keywords-Quasidifferentiability, lit&, Hemivsziational inequalities.
Nonconvex energy, Nonsmooth mechanics, Variational inequr+
1. INTRODUCTION Adhesive contact effects and adhesive frictional effects which are possibly coupled by unilateral contact effects arise mainly in structures which include interfaces (e.g., in layered composite plates or in repaired cracked structures). A phenomenological quasistatic modelling of the mechanical behaviour of these effects leads to the adoption of a nonmonotone and possibly multivalued traction-displacement law for the adhesive interface. Nonmonotonicity copes with strength-degradation effects and multivaluedness of the law, e.g., the appearance of complete vertical branches in the graph of the law copes with locking and instantaneous cracking effects. Laws in the normal to the interface direction, the tangential direction and coupled laws can be used [1,2]. The structural analysis problem for structures which involve nonmonotone, possibly multivalued laws, can be described in certain cases by means of a nonsmooth and nonconvex superpotential energy function. Critical points, substationary points and of course local minima of the potential describe all stable and unstable equilibrium points (i.e., the solutions) of the examined structural system. They are in general solutions of hemivariational inequality problems [2-41. For the systematic study of these problems and for the construction of algorithms for their efficient solution, separate treatment of convexity and concavity information in connection with Part, of this
work has been completed in the RWTH Aachen, Germany, with the financial support of the Alexander von Humboldt Foundation, Bad Godeeberg, Bonn.
455
456
G. E. STAVROULAKIS
techniques of nonsmooth analysis has been found to be fruitful [5]. Fortunately, this information is known at the elementary level of the constitutive laws or the boundary conditions which are adopted for the mechanical modelling of the structural elements and can be traced up to the level of the structure by means of appropriate calculus rules. These steps can be performed by using the notion of the quasidifferentiability and the corresponding quasidifferential calculus [6-91. Thus, quasidifferential modelling and optimization techniques are introduced in structural analysis [lo-13). We recall here that for convex, nondifferentiable problems the adoption of the convex subdifferential give rise to the formulation and study of variational inequality problems in mechanics [2,14]. Unilateral contact and Coulomb friction effects have been tackled in a systematic way by means of convex optimization techniques and the theory of variational inequalities (see, e.g., [2,14,15] and the references therein). The convex analysis approach to contact problems will not be reviewed in details in this paper. Instead it will be used as a subproblem for the treatment of the nonconvex case. For nonconvex, nondifferentiable problems the explication of the arising optimality conditions gives rise to systems of variational inequalities [5,11] which can be solved either by quasidifferential optimization algorithms, or in some specific cases by appropriate multilevel decomposition techniques. Elements of quasidifferential modelling in mechanics of adhesive contact effects are given in this paper. The purpose of this work is to present various models which have been developed for analogous applications elsewhere, to discuss their appropriateness for the modelling of adhesive effects in a phenomenological elastostatic way and to address the reader’s attention to appropriate algorithms for their numerical solution. Details of the various presented models, the algorithms and numerical examples have not been included, since they are available in previous publications. The layout of the paper is as follows: quasidifferential energy optimization problems, both constrained and unconstrained ones and the corresponding optimality conditions are formulated in the next section. The adhesive contact problem is formulated in Section 3. In Section 4, a nonconvex qussidifferential plasticity theory is outlined which can be used for the modelling of certain adhesive and nonconvex locking effects. In Section 5, a simplified linear complementarity problem formulation with adhesive unilateral contact effects is described. It can be considered as a special case of the nonconvex plasticity approach with the advantage that it leads to more simple problems. In the last section, appropriate algorithms for the solution of the arising problems are listed and references to companion publications are given where the details of the algorithmic implementation and some representative numerical examples can be found.
2. QUASIDIFFERENTIABLE POTENTIAL ENERGY OPTIMIZATION AND OPTIMALITY CONDITIONS A function f defined on an open set X E Rn is called quasidifferentiable at z E X if it is directionally differentiable [6,9] at z and its directional derivative f’(z,g) along the direction g E Rn can be written as
Here U and V are convex compact sets in Rn. The ordered pair of(z) = [V, V] is called the quasidifferential of f at z, U is called the subdifferential of f at t and is denoted by &f(z) and V is the superdifferential of f at z, denoted by at(z). Practically all convex and nonconvex, possibly nondifferentiable, finite-dimensional potential energy functions which arise in structural analysis are quasidifferentiable [12,13]. For the calculation of the quasidifferential of given structured functions, (i.e., functions which are produced from classical differentiable components and the rules of addition, subtraction, multiplication, division, finite combinations of maximum and minimum operators and of composite functions) calculus rules have been developed [7-9,131.
Quaeidifferential Modelling
457
Accordingly, a set A c Rn which is defined by a finite number of equalities and inequalities involving quasidifferential functions is called quasidifferential. For these kind of generally nonconvex sets with nonsmooth boundaries, normal and tangential vectors are exactly defined by the quasidiierentiability concept. Thus, mechanical theories which involve normals and tangentials to these sets can be constructed without the classical restriction of convexity (see, e.g., the model proposed in [16,17] for nonconvex plasticity, which will be used in the sequel for the treatment of adhesive contact problems). For quasidifferentiable optimization problems necessary and in some cases sufficient optimality conditions can be written and will be used for the description of appropriate structural analysis problems. First-order necessary (respectively, sufficient) conditions for a quasidifferentiable function f to admit a local minimum (respectively, a strict local minimum) at point t’ E Rn read (respectively, int 2 (z*)) . -df (Z*) c 2L (z*) (2) For constrained optimization, we have analogous inclusions (see [6,9,13] and the subsequent sections). For comparison with classical linearization techniques, we recall here that the notion of the quasidifferential gives rise to the following local approximation of f(z) around point x* (qussilinearization) :
with (oZ(crA)/a) t 0, as cr 1 0, VA E Rn. For an extension of the outlined theory to cover Hausdorff continuous operators (which are more tractable from the numerical point of view), the notion of the codifferentiability has been proposed (see, e.g., [9,13]). Since for the theoretical presentation of the models, no additional information is introduced by this notion, this extension will not be reviewed here.
3. STRUCTURAL ANALYSIS SUPERPOTENTIAL
PROBLEM FORM
IN Q.D.
Let us assume a structure with adhesive contact interfaces and let us assume that the phenomenological elastostatic analysis problem of this structure admits a potential characterization, i.e., the governing relations of the problem can be derived by appropriate differentiation of a potential energy function. For the modelling of interface adhesive effects, we adopt nonmonotone, possibly multivalued (i.e., with complete vertical branches) interface traction-relative displacements laws (see, e.g., [1,4], F’g 1 ure 1). A finite element discretization of the direct stiffness method with nodal displacements as the primary variables of the problem is assumed here. Let u be the n-dimensional vector of displacement degrees of freedom and e is the m-vector of element deformations. A fairly general discrete potential energy optimization problem in elastostatics reads (4) where II(e) is the elastic energy stored in the system due to deformation, O(u) is the potential that counts for various boundary, interface or skin effects and p(u) is the potential that generates the external loading vector. In general, relative interface displacements are involved in the definition of @(u) and, respectively, interface tractions appear as their work dual variables in the virtual work expression which follow. The more economical notation is preferred here. For technical details see [1,5,13,18]. The geometric compatibility transformation is written in the form of a generally nonlinear but differentiable operator d(u) : Rn + R”‘, e = d(u).
(5)
458
G. E. STAVROULAKIS
Multibranch elasticity or holonomic plasticity models with ascending and descending complete vertical branches that count for crashing, cracking, and locking effects in a phenomenological way are covered by this model, by using nonsmooth and possibly nonconvex superpotential energy functions II(e) in (4). Analogously, nonmonotone relations, like stick-slip boundary or interface laws, frictional or softening frictional laws, etc., introduce nonsmooth and nonconvex potential functions Q(u) in (4) (see, [2,4,13] among others). The set of kinematically admissible displacements is in general a quasidifferentiable set, defined by Uad = {U E R” :'H(U) 5 0, G(U)=
o},
(6)
where O(u) : Rn + Rnl, ‘H(u) : R” --+ Rnz, and nr + 712 < n are general nonlinear and possibly nondifferentiable (but quasidifferentiable) equality and inequality constraints. Bilateral and unilateral contact effects and locking behaviour in a large displacement setting leads to relations of the (6) type. For simplicity only, unconstrained structural analysis problems, i.e., u ad = Rn are considered in this section.
S
de
[ul
(b)
(4
(c) Figure 1. Monotone tentials.
and nonmonotone
interface laws with corresponding
superpo-
Variational formulations for the elastostatic analysis problem described by (4) will be produced in the sequel by writing down the optimality conditions for this qussidifferentiable minimization problem and using the quasidifferential calculus for the derivation of the quasidifferential of the complete function a(u). Variational equalities for classical smooth problems, variational inequalities for nonsmooth, subdifferentiable problems (cf. [2,4,14]), and systems of variational inequalities for general quasidifferentiable problems (cf. [4,5,11,13]) are thus derived in a systematic way.
Quasidifferential
Modelling
459
We recall that for classical nonlinear elastostatic analysis problems with Ud = Rn, the structural analysis problem is written in the strong form: find u E Rn such that
Vrqu) = 0,
(7)
or in the weak (variational equality) form: find u E Rn such that l?(u,Au)
= Vl=I(~)~vu
=
1
F
T %A,
[ + [FITAu+
[F]TAu=O,
VAUER”.
Let II(e) of (4) be quasidifferentiable and let DII(e) = [aII(e),aII(e)] E Rm x R”‘. Dfi(e) E R”’ x R” can be constructed by using the rules of the quasidifferential calculus (see, [9, p. 127; 121) for the composite function II(e(u)). If moreover ‘P(u) and p(u) are differentiable, we get
Di=I(u) = [&=I(u),RI(U)] = [g-I(u) + W(u)
+ VP(U),
The optimality conditions for the quasidifferentiable unconstrained find u E Rn such that -@I(u) c El(u).
all(u)] .
optimization
(9) problem read: (IO)
They lead to the system of variational inequalities: find u E Rn such that VW E --Xl(u).
w c &I(u),
(II)
An interesting case arises if the potential energy is a difference convex (d.c.) function. For instance, let a small displacement problem be considered, i.e., (5) replaced by d(u) = GTu, with GT an (m x n) matrix and accordingly p(u) = pTu, with p the n-dimensional loading vector. Let moreover II(e) in (4) be convex and differentiable, e.g., consider for instance the linear elasticity problem with II(e) = 1/2eTKce, where Kc is the (m x m) natural stiffness matrix of the structure. Let the only cause of nonconvexity and nondifferentiability in the problem be introduced by a difference convex boundary potential energy function, i.e., a(u) is written as a difference of convex, possibly nondifferentiable terms [5,18] G(u)
In this case, by using the quasidifferential
=
@‘1(u)
-
@2(u).
(12)
calculus, we get
@I(u) = Ku + p + a@,(u),
Bl=I(u) = -z&(u).
(13)
Optimality conditions for the d.c. potentially lead to the following system of variational inequalities that describe the nonsmooth and nonconvex structural analysis problem: find u E Rn such that O~Ku+p+d@i(u)-w, (14) for each w E Rn with w E a@,(U).
(15)
Problem (14),(15) can be solved iteratively, where in each iteration cycle two convex subproblems are solved (see, [5,18]). The above decomposition is explicitly defined due to the global d.c. rep resentation of the potential energy. In the general qussidifferential case, the system of variational inequalities (11) is implicitly defined.
460
G. E. STAVROULAKIS
4. Q.D. PLASTICITY-BASED MODELLING ADHESIVE EFFECTS In this section,
we outline
a model for the adhesive
for the treatment
of nonconvex
in [11,13,16,17].
In general,
convex analysis
approach
elastoplasticity framework
to standard
yield sets are introduced.
In turn,
the rules of the quasidifferential of plasticity. by appropriately
By means
normal calculus
of this approach,
formulated
systems
(see, [16,17] for more details). This theory will be specialized
effects which is based on a general
problems
of elastoplasticity generalized
OF
with quasidifferentiable and in particular
elastoplasticity,
and tangential
nonconvex
elastoplasticity
of convex variational
here for the treatment
sets introduced by extending
inequalities of adhesive
the convex
problem
the
quasidifferentiable
cones to the yield sets are produced
and they are used to generalize the general
approach
by
flow rules
can be described
as it will be outlined
here
effects, where the adhesive
interface (or boundary) is modelled by means of a thin layer. At this layer, relative displacement and traction vectors are denoted by [u] and S, respectively. By following the general framework of elastoplasticity, a yield locus C in the S space can be postulated and in turn adhesive effects can be described like the flow rules of plasticity. Convex models for interfaces, which include among others unilateral contact effects and Coulomb static friction effects with given normal traction, can be modelled in this way (see [19, pp. 100-103; 201). Adhesive effects require the consideration of nonconvex yield surfaces (see Figure 2). Quasidifferential modelling provides us with a tool for the treatment of this case in finite-dimensional (discretized) problems, as it will be shown here. Only associated plasticity based models will be considered since they give rise to energy optimization problems which permit a direct application of the quasidifferential optimization results of the previous section to the structural analysis problem. Nonassociated models can be constructed by following the lines of [20].
Figure 2. Nonconvex yield surface and associated
flow rule
Let for a discretized problem the only internal variables, in the standard elastoplasticity terminology, be the inelastic relative displacements [u]‘“. Let moreover the elastic region in the is used: interface traction space be a closed convex subset C in Rm. The following notation elastic strains ee, inelastic strains ein, enlarged stress vector s, and natural stiffness matrix K. The above quantities
are defined
by
Quasidifferential
Modelling
461
Here, as usual e, cr, Ko refer to the elastic substructure, i.e., all the finite elements of the structure except of the inelastic quantities of the adhesive interface. Interface quantities are denoted by [u], [u]~“,S, Kin, where Kin is the elastic stiffness matrix of the interface and [,lin counts for inelastic effects. For notational simplicity, vectors and matrices are not written boldface in this section. Moreover, as in standard elastoplasticity, we assume the existence of a convex free energy potential W (e - e’“) = f (e - ein)T KO (e - e’“) ,
(17)
and a convex, nonnegative, sublinear, and positively homogeneous of order one internal dissipation potential
v = I; ([a]‘“)f
mg
{ST[tip} .
(18)
Here I6 is the support function of the convex set C (see, e.g., [2, p. 551). The governing relations for the structural analysis problem are derived by differentiating (respectively, subdierentiating) (17) (respectively, (18)) with respect to e (respectively, [ti]‘“):
[;I=[7 Iin] [[u]‘Iu]~~]~ s = aI&
(19) (20)
([ti]‘“) .
The latter relation can be inverted to give the flow rule, in local variables
[ii]‘” E
aIc(s)
= NC(S),
(21)
= J+(s),
(22)
or in global variables P E &(a).
with 6 = R” x C. Here Ic denotes the indicator function of C and NC is the normal cone to the set C. If C is described by the convex yield locus F(S) as C = {S E R”’ 1F(S) 5 0))
(23)
relation (21) is equivalent to the convex variational inequality
[ti]‘”T (T
- S) 5 0,
VT E C, S E C, or VT s.t. F(T) 5 0.
(24
The rate elastoplastic analysis problem is discretized by appropriate schemes and by the assump tion that within the time step At = t,,+l - t, the behaviour is holonomic. Thus, a stepwise holonomic problem arises. For instance, by an implicit Euler scheme (21) yields in
%+1
-
At
ek E
dIc(sn+l)
* eF+,;, - ek E X&s,+l).
(25)
Relations (25) and (19) written for time step (n + 1) (with F = K-l) relation 0 E Fs,,+l - e,+l + ek + a+ (sn+l) , or equivalently to the minimization min &t+rEC
Jn+i(sn+l)
lead to the incremental (26)
(stepwise holonomic) problem = ;s;+iFs,+i
-
e,T+lsn+l
in
+ e,
T
(27)
G. E. STAVROULAKIS
462
Solutions of problem (27) are also solutions of the differential inclusion (multivalued equation)
or %+J(%+1
) fl {-q+?I+1)}
#
0.
(29)
An equivalent convex variational inequality problem reads: find s,+r such that Vf(Sn+l)(T
where f(s,+r)
-
Sn+l)
+ I&)
-
A&+1)
2 a
t/r
E
Rn+"',
(30)
is defined by
f(%+i) =
(31)
Figure 3. Nonconvex friction law which admits the convex decomposition of (33).
Details of numerical algorithms based on this general approach can be borrowed from parallel recent developments in convex elastoplasticity (see, among others [21-251). Let us extend the previously outlined model to include nonconvex, quasidifferentiable yield sets described by R = {S E R” ] 4(S) < 0). (32) To give a concrete example, we assume a model case where Cl is the union of a finite number of convex subsets Ri, i E Z = (1, . . . , m}, i.e., (see, e.g., Figure 3) R=
u i-ii, %={S E Rm 1&(S)
5 0))
(33)
iEI where CJ$are classical convex and differentiable yield functions (cf. (23)). The cone of feasible directions (Bouligand cone) to a set defined by (32) is characterized means of the directional derivatives of $(.) as
rl(S) = {g E R”’ I ti’(s,g) 5 0). By means of the definition of a quasidifferentiable rl(s)
by
(34)
function rr is equivalently expressed by
= {g E Rm I (wl,g) + (w2,g) 5 0, VW E 3$(s),
forSOme w
E
&J(S)} v
(35)
Quasidifferential
or
h(s) =
u
n
w,E&(S)
Modelling
463
u
T~,+~,(s)=
WlE&(S)
TS+(v)+w2(S),
(36)
w2EWS)
where the following shortcuts have been used (see also Figure 2): Tw+wa(S) Z39(s)+w2
($1
=
(9
E
R”’
=
{g
E
R”
I h,d
I
+
hr
d
+
b-ad
(~2,
IO),
d
5 0,
vwl
E
&#G)} .
(37)
One should mention here that (36) represents a systematic convex decomposition of the tangential cone to R at the point S, due to the fact that the sets defined in (36),(37) are convex. Thus, a representation of the normal cone to R at S by using convex constituents is possible. It has the form J%(S) = n u T:1+W2(S) = n $+Cs)+W2 (S) w2al5(S)
=
we!6(s)
f-)
w2aws)
{ -T~w)+w,
(S))
*
wzfk(S,
Here, the polar cone to T, denoted by To, and the conjugate cone of T, denoted by T+, are defined by To = -T+ = {g E R” 1 (g,s) 5 1, Vx E T} . (39) In the R”+m space, we have, respectively
1’ p(s) =[&)1’ r(s) =
[
Rn ri(S)
(40)
The definitions of the previous section and the optimality conditions for a quasidifferential optimization problem give rise to the following variational expression in compact form: find s~+~ such that %,+J(s,+i)
# 0,
n T+(s,+l)
(41)
Qw2 E &J(S,+,).
Relation (41) is written equivalently as %+,
f(Sn+l
1 fl wh+-n+l))#
%,+J(sn+d
n { -T’(sn+d}
#
0,
VW2
E W&I+1),
‘it
VW2
E &m+d,
or
(42) (43)
where f(s,+i) has been defined in (31). In terms of plastic multipliers, we first write the flow rule (21) as [I$”
E Ah(S)
=
US&@(S)~j(Sj
+ W2)l
Xj 2
0,
VW2
E
&(S).
(44)
Note here that, in contrast to what happens in the convex case, the plastic multipliers Xj depend on the choice of w2 E ad(S). By using the time discretization scheme introduced previously, we get the incremental relation (cf. (26))
4nSn+l
-
bln+l
+
[4F+1
+
u ~j~~~(~~+l)
FOG+1-
‘%,I
=
0,
Xj(Sj
+ 202)
=
0,
VW2 E &(S,+r).
(45)
464
G. E. STAVROULAKIS
The above expressions can further be simplified if the sets & and 84 are polyhedral. Then, only the vertices Sj E &(&+I) an d wz E &(S,+l) need to be considered in (44),(45) and, u is replaced by C, as it will be shown in the concrete example in the sequel. For the yield surfaces (33), the above general scheme results in the following formulae: function 4(S) of (32) can be written as a min-type function by using the yield loci of subsets Ri in (33), i.e., (46) For the min-type function of (46), which is quasidifferentiable, ential and quasidifferential are given by the relations
we may assume that its subdiffer-
(47) where the following active index set To(S) = {i E 1, s.t., d(S) = q&(S)} is used. A more general approach to nonconvex elastoplasticity based on star-shaped yield sets and the theory of [ZS] has also been proposed [16,17]. Instead of following this line of generalization, we present in the next section a more restricted model which is appropriate for certain adhesive unilateral contact problems.
5. MODIFIED L.C.P. APPROACH TO THE ADHESIVE UNILATERAL CONTACT PROBLEM In thii section, a model for adhesive unilateral contact problems which has been introduced in [27] is reviewed. It depends on an additive decomposition of unilateral contact effects and adhesive effects, where the unilateral contact problem for linearly elastic structures is formulated ss a linear complementarity problem and the additional adhesive effects are supposed to be produced by nonconvex yield sets in a way similar to the one followed in the plasticity model of the previous section. Let us consider a discretized unilateral contact problem. Let discrete contact forces at a unilateral boundary or interface be denoted by ri, i = 1, . . . , n and be assembled in vector r. External forces are denoted by vector p. Superscript T denotes ss usual the transpose of a matrix or vector. Let moreover at the unilateral boundary or interface discrete gaps be denoted by vector y . Let us assume that contact forces and gaps are decomposed in a unilateral contact part and an adhesive part (denoted by subscripts u and a, respectively) as follows r=rU+ra,
Y =yu+ya.
(48)
We recall that a frictionless unilateral contact problem with no adhesion effects (i.e., r,, = 0 and y,, = 0) can be formulated as a Linear Complementarity Problem LCP of the form [2,28,29] w-Fz=d,
w I 0,
z Z 0,
wTz=o,
(49)
where w=wu=yu,
z = z, = ru.
(50)
Here F is the normal compliance matrix and d is the initial openings matrix (see, e.g., [28,29]). LCP (49) is by definition equivalent to the following variational inequalities: find z, 2 0 such that (Fz, + d)T(v - zU) > 0,
vvzo,
(51)
w,=Fz,+d.
(52)
and find w,, 2 0 such that zz(w:
- w,) 2 0,
Vw;>O,
Qua&differential
Modelling
465
Adhesive effects with nonconvex yielding characteristics can be included in LCP (49) as follows: let y = yU, r = r. + rU. Let adhesive tractions ra be derived by means of an appropriate nonempty closed and possibly nonconvex subset C of R”
-ra E %(Y),
(53)
or equivalently I& (y, y’ - y) 1 -r;T (y’ Adding relation (54) and variational hemivariational inequality find w E R”+ such that,
inequality
1; (w,w*-w)+zT
y) ,
t/y* E R”.
(54)
(52) (recall that w = w, here), we get the
(w*-w)
> 0,
Fz+d=w,
VW* E R”+. (55)
Analogously, locking effects with a general yield function that can be described by a possibly nonconvex, nonempty closed subset Ci of Rn can be tackled. In this case, we have y = y,, + y,,, r = r,,. A nonconvex locking criterion is considered such that it can be described by
or I& (r, r* - r) > -y,T (r* - r) ,
Vr’ E Rn+m.
From (57) and the variational inequality (51)) we get the hemivariational find z E Rn such that,
(57) inequality
(Fz + d)T (z* - z) + 1;. (z, z* - 1;) = WI (z* - z)
+I&, (z, z* - z) L 0,
(58)
t/z*-zeRn.
In an analogous way, frictional contact problems can be formulated as nonsymmetric LCP’s [29]. Adhesive and locking effects can also be introduced in this more general case by following the same procedure [27]. This model has initially been proposed in [27] for the study of frictionless and frictional adhesive gripper problems in robotics. In the same reference, existence results are given based on [30].
6. NONSMOOTH
STRUCTURAL
ANALYSIS
ALGORITHMS
A number of structural analysis algorithms appropriate for the solution of the adhesive contact problems formulated in this paper will be mentioned here. Due to lack of space and due to the review nature of this work, concrete numerical applications will not be presented here. They can be found in the references. Convex analysis applications in structural analysis will not be reviewed here. This is the case of unilateral contact or convex, associated plasticity problems which lead to the solution of variational inequalities or convex minimization problems. The reader is referred to [2,14,15] among others. Our interest will be focused on the treatment of nonconvexity which is the primary problem introduced by adhesive effects. Nonsmooth numerical optimization techniques have been used in a restricted extend for the solution of the nonsmooth potential energy optimization problem which describes the structural analysis problem. A recent review can be found in 1311. The need for controlling complicated numerical accuracies and the inefficiency of general purpose algorithms to cope with large scale problems have restricted the development in this direction. Adhesive joints and nonmonotone skin effects have been formulated ss a hemivariational inequality problem and have been solved by an iterative method in [32]. A hemivariational inequality describes both local minima and certain critical points of the potential energy function [2,4].
466
G. E. STAVROULAKIS
Substationarity optimization approaches for the solution of adhesive and delamination effects are proposed in [33,34]. A common feature of these approaches is the formulation and the solution of appropriate convex optimization subproblems, which in an iterative scheme lead to the solution of the initial problem. The quasidifferential optimization approach gives rise to a systematic convex decomposition of the nonconvex optimization problem into nested convex subproblems. Since each convex subproblem leads to the solution of an equivalent convex variational inequality, we speak about systems of variational inequalities. A simpler introduction to this method is achieved for potential energy functions which are differences of convex functions [5,18]. The extension to the general Q.D. case is discussed in [12]. Nonmonotone adhesive friction effects coupled with unilateral contact effects are solved in [18]. Adhesive skin effects are presented as an example in [12]. Comparisons of the Q.D. approach with the parallel developments in the field, which have been briefly mentioned previously, are given in [4,35,36]. Extension for the solution of the nonconvex elastoplasticity model and the LCP model with adhesion is currently under development. Hints are given in the relevant publications [16,17,27].
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