1 Abstract. 2 Introduction

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1 Fractal interfaces in masonry structures. Methods of calculation O.K. Panagouli*, E.S. Mistakidis*, P.D. Panagiotopoulos*, A. Liolios^ "Institute of Steel Structures, Aristotle University, GR Thessaloniki, Greece ^Institute of Structural Mechanics, Democritus University of Thrace, Xanthi, Greece 1 Abstract Masonry structures involving interfaces with fractal geometry are analysed here as a sequence of classical interface subproblems. These classical subproblems result from the consideration of the fractal interface as the unique "fixed point" of a given Iterative Function System (I.F.S.). On the interface nonmonotone contact and friction conditions are assumed to hold. The methods developed consist an extension of the classical FEM to the case of fractal interfaces. The case of nonconvex energy problems is investigated, and some results from static analysis of masonry walls with prescribed fractal cracks are included in order to illustrate the theory. 2 Introduction Fractal geometry provides both a general description and a mathematical model for many complex forms found in nature. This fact makes fractal geometry central to the various fields of science such as chemistry, physics, biology, geology and materials science. The property of fractality appears very often in structural analysis and in applied mechanics. For instance the cracks and the interfaces in rocks, bones, in composite materials, in concrete and in masonry structures are of fractal type [1], [2]. Following here the definition of Mandelbrot [1] as it is completed by Wallin [3], we consider that a set F C R is a fractal set if F has fractional "HausdorfP dimension, or if the dimension of F is an integer strictly larger than its topological dimension. Many geometrical concepts that transcend the power of description of the traditional geometry have been developed by mathematicians but few of them [4], [5] are suitable for the engineering sciences. The approach we follow here is based on Barnsley's approach

2 292 Dynamics, Repairs & Restoration [5] which is characterized by a single deterministic framework and it uses classical geometrical entities (for example affine transformations, scaling and rotations) to express relations between parts of generalized geometrical objects, namely "fractal subsets" of the Euclidean plane. Using only these relations, the Iterated Function System (I.F.S.) theory is very suitable for the engineering sciences since it permits the satisfactory study of the whole problem with reliable numerical calculations. As it is well-known from the relevant literature [6], [7] the crack interfaces in masonry structures may either have a zig-zag shape following the joints of the stones (or the blocks) or pass through the blocks. The first kind of cracks appears in the case of low-strength mortars and lightly compressed walls. On the contrary the second kind of cracks appears in the case of masonry walls constructed with mortars of better quality and especially in highly compressed regions of the structure. New theories concerning the geometry of the cracks in masonry and brickwork structures [2] have been developed lately. According to these theories, a propagating crack may either depart from its original straight trajectory to curve or split into two or more branches or, under a high state of stress, may be divided into a river delta crack pattern. This fragmentation may often be a succession of multiple branching of what was initially a single crack [8]. In the present paper our attention is focused on the fractal geometry of the interfaces where their behaviour is modelled by means of a nonmonotone contact and friction mechanism. In the method proposed here the fractal interfaces are approximated by classical curves and each one of the resulting classical problems is solved by using tools from nonsmooth mechanics [9], [10] for the analysis of the whole structure. In the sequel the influence of the interface fractality on the displacement and stress fields is investigated. At the limit the solution of the corresponding problem is obtained. 3 Formulation of mathematical models for the approximation of fractal interfaces Fractal phenomena in nature are often rather complicated to describe, therefore various assumptions and approximations are required in order to analyze a mathematical model. Throughout this section the development of I.F.S. as a practical tool for the production of fractal sets is concerned. As we have mentioned above the Hausdorff dimension of a set F, denoted here by dimf, characterizes the fractality of this set. This dimension can be defined for any shape and is a natural generalization of both the empirical dimension and the similarity dimension. For the definition of the Hausdorff dimension we refer to Barnsley [5] and Falconer [4] but the reader could understand it as a notion analogous to the classical dimension. Let {X, d} be a complete metric space with the metric d. We denote

3 Dynamics, Repairs & Restoration 293 by H(X] the space of the compact subsets of X. If d(a, B) is the distance between the sets A C X and B C X denned by the formula (cf. Barnsley [5]) d(a, B] = maxmind(z,2/), (1) x A y B then the space H(X) endowed with the metric /t(a, B) = max{(f(a, B), cf(b, A)} VA, B G ^(%) (2) is a complete metric space and is called space of deterministic fractals. An iterated function system (I.F.S.) on X consists of n contractive mappings Wi : X» X with contractivity factors 0 < Si < 1, i = 1, 2,..., n, i.e. d(^(a:),^(2/)) < 5^(2,%) V%,2/ E X, 0 < 5^ < 1. (3) It can be easily shown that if the new set-valued function Wi : H(X) > H(X) is defined by setting Wi(B) = {wi(x);x B} VBeH(X) (4) and W(B) = Wi(B)VW2(B)V...\JWn(B) VB G H(X\ (5) then this set-valued function is a contraction mapping on H(X) with contractivity factor s max{5i, 52,..., } The unique fixed "point" of W is the set A C H(X) such that A = ^(A) = Q I^(A) (6) 1=1 and is given by the relation A = m lim >oo W^\B) V5 G H(X\ (7) where ^( )(%) = X, %f( )(X) = ^(^"-^(%)), m = 1, 2,... (8) are the forward iterates of W. The set A is called the "attractor" of the I.F.S. {X]Wi, i = 1,2,...,n} and is the deterministic fractal of the considered I.F.S. For simplicity we restrict attention to "hyperbolic" I.F.S. of the form {IR/*; Wi : i = 1, 2,..., n} where each mapping is an afnne transformation of the special structure. As an example, Table 1 provides the I.F.S. code for the fractal tree of Fig. 1. It is important to notice here that for experimental purposes the boxcounting dimension is used. Fundamental to the definition of box-counting dimension, as to the most definitions of dimension, is the idea of "measurement at scale #" and its behaviour as 8 > 0. The dimension of a fractal set Fj denoted here by D, is determined in this case by the power law #(&) - constant «T^ (10) where N(6] denotes the number of boxes of side length S which intersect

4 294 Dynamics, Repairs & Restoration the set F = Ur=i Wi(F). CLi 0.10 bi - Ci - di Ci fi Table 1. Parameters characterizing the example of Figure 1. Figure 1. Sixth approximation of the fractal set resulting from the I.F.S. code of Table 1. 4 Variational formulation of the problem and methods of solution Let us consider a linear elastic structure occupying a subset 0 of R^ in its undeformed state having a boundary F which is decomposed into two mutually disjoint parts IV and IV- It is assumed that on IV (resp. IV) the displacements (resp. the tractions) are given. In the structure 0 some cracks with interfaces $ of fractal type are formed. As it has been mentioned above, these cracks in brittle materials frequently propagate along one or more irregular ways (e.g. in a zig-zag where a large zig-zag is composed of small zig-zags). In this case the fracture system may be considered to be a cluster of branches propagating in such a way that new branches in the i-fl-th step are successively created from a former branch at the i-th step (see e.g. Fig. 1). In other words the fracture system can be modelled by an I.F.S. procedure. Regarding now the behaviour of the interfaces <&, we assume that nonmonotone, possibly multivalued laws describe the behaviour of each interface in the normal and tangential directions. More specifically, for the normal (resp. the tangential) to the interface direction we assume that a nonmonotone, possibly multivalued law g (resp. h) holds between the normal traction SN and the normal relative displacements [UN] (resp. between the tangential traction ST and the relative tangential displacements [UT])> Both functions #, h result from functions which are locally essential bounded (i.e. from L^.(R)) by filling in the jumps. Moreover we will have that there_ exist locally Lipschitz functions j'#, JT for which we consider the djn and djt-> where 8 denotes Clarke's generalized gradient [11]. Then the following boundary conditions hold: - SN E BJN(UN, %), (11) -STE9jT(wT,z). (12) Then according to Panagiotopoulos [10], an equilibrium position of 0 is characterized by the following hemivariational inequality problem: Find

5 Dynamics, Repairs & Restoration 295 u VQ such as to satisfy the inequality a(u,v-u)+ I jtf(un,vtr-un)d$+ I ^(UT^VT-UT^ > (l,v-u) \/v G VQ J$ J< (13) where VQ is the kinematically admissible set, i.e. VQ = {u Ui = 0 on IV}, a(.,. ) the linear elastic strain energy, (/, u) the work of the external forces, JN(>), JT(-) are the nonconvex superpotentials of the nonmonotone laws g and h respectively, and j^, j%, are the directional derivatives of Clarke. We have to notice here that although the formulation of the problem as a hemivariational inequality has a lot of the advantages regarding the mathematical study, it leads to a lot of difficulties concerning the numerical treatment. For this reason, the method developed in [10] for the numerical treatment of hemivariational inequalities is used. According to this method (see also [10], [12]) the nonconvex superpotentials JN and j? are replaced by the appropriately chosen convex superpotentials j'^ and j? respectively and the hemivariational inequality (13) is written in the form: Find u G VQ such that where a(u, v-u)+ I JN(VN) - j^(u^d^ -h / JT(VT) ~ JT(^T)^ J$ J $ >(/,%-%) + #(?,,%) VvE% (14) - I J^(UT,VT - u^d (15) v/$ Then the following iterative scheme is proposed for the solution of the problem: Find T>) G VQ such that >(/,v-^)) + 7Z(i;,^-")) VrGV[) (16) Note that relation (16) is a variational inequality (V.I.), since R is a constant calculated using the results of the previous step. The solution of this V.I. can be obtained using a quadratic programming (Q.P.) algorithm. The advantages of this method, that reduces the initial hemivariational inequality problem to a sequence of V.I. problems, are obvious because nowadays there exist a lot of fast and robust algorithms for the solution of the resulting Q.P. problems. In our case, where a fractured body fi with fractal interfaces $ is studied, the arising nonmonotone problem (13) holds in the sense of a limit process. As it has been mentioned above, $ is the fixed point of a given transformation denoted by W ', i.e. $ = W3> and $<'+*) = W^\ $^ -> $, as j -> oo. (17) Thus, for each approximation $^0 of the fractal interface $ we have to solve a structure l('\ Since $k') is an interface set with classical geometry the solutions T/J) and a^ (where u^ and a^ are the corresponding displace-

6 296 Dynamics, Repairs & Restoration ment and stressfields)are obtained using the iterative scheme of (16). We repeat this procedure several times by increasing j\ at the limit j > oo, u^ and en-?) give the solution of the fractal interface problem with nonmonotone contact and friction interface conditions. 5 Numerical Applications As an application we consider here the structure of Fig. 2 submitted to loading in its plane. Linear elasticity and geometrical linearity are assumed. The modulus of Elasticity is E = 2.1xlQ*kN/rn? and the Poisson's ratio i/=0.33. The thickness is taken as equal to 0.10m. Two interfaces 4>i and $2 of fractal geometry are formed in the structure as it is shown in Fig. 2a. $1 and $2 are defined to be the attractors of the I.F.S.s {R^jiyi,^} and {R^; 1^3,1^4} respectively, with the parameters given in Table 2. Wi Wi W2 W3 W^ CLi , Ci di Ct fi Table 2. Parameters characterizing the example of Figure 2a. F F F F F F F F F F SK, (kn), N F F F F F F 5.00m F F F F 4 (a) (c) Figure 2. (a) A multifractured masonry structure. (b),(c) The interface laws

7 Dynamics, Repairs & Restoration 297 2nd approximation 3rd approximation ^i- 4th approximation Major stress: fti E+02 [" ""] E+02 ~c~ E+02 QTJ E+02 m-1.ooooe+02 [~F~ E+01 [~G~1 O.OOOOE+00 [~j 1.OOOOE+02 ~j"" E+02 ~K~l E+02 [T~ E+02 [~M~ E+02 [~N~ E+02 [~Q~ E+02 [~p~ E+02 [~Q~ E+02 ~R~ E+02 ~s~ E+02 J^\ Figure 3. Major stress for the second to fifth approximations of the fractal interfaces. For the normal to the interface direction, the nonmonotone law of Fig. 2b is assumed to hold, whereas for the tangential to the interface direction, the normalized friction law of Fig. 2c holds. In order to obtain the frictional force ST we have to multiply the ordinate of the diagram by fi * j)#, where /z is the friction coefficient and in our case // = 0.3. Based on the above I.F.S.s, the different structures corresponding to the consecutive approximations of the fractal interfaces are calculated for the same kinematic conditions, the same loading and the interface laws of Fig.2b,c.

8 298 Dynamics, Repairs & Restoration The algorithm presented in the previous section was applied to analyse each arising problem on a HP755 workstation. Triangular constant stress elements have been used for the discretization. In Fig. 3 the major stresses for the second to fifth approximations of the fractal interfaces $1 and $2 are given. It is observed that near to the interface 4>i the stress field becomes stable from the third approximation. On the contrary, it is obvious that near to the interface $2 the same stress field becomes stable after the fourth approximation. This is due to the fact that this interface has many singularities which make necessary the use of higher order approximations in order to have convergence to the final solution of the problem. Note that the irregularities of a fractal boundary increase when its fractal dimension is not near to its topological dimension. In the example we study here, the fractal dimension of the interface $1 is D = 1.10, whereas for the interface $2 we have D = References 1. Mandelbrot, B., The Fractal Geometry of Nature, W. H. Freeman and Co., N. York, Takayasu, H., Fractals in Physical Sciences, Manchester Univ. Press, Manchester, Wallin, H., Interpolating and orthogonal polynomials on fractals, Consir. Approx., 1989, 5, Falconer, K. J., The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge, second edition, Barnsley, M., Fractals Everywhere, Academic Press, Boston, N. York, Stavroulakis, G.E., On the static behaviour of cracked masonry walls, Eur. J. Mech., A/ Solids, 1990, 9(4), MacLeod, J.A. and Abu-el-Magd, S.A., The behaviour of brick walls under conditions of settlement, The Structural Engineer, 1980, 58A(9), Xie, H., The fractal effect of irregularity of crack branching on the fracture toughness of brittle materials, Int. Journal of Fracture, 1989, Panagiotopoulos, P. D., Inequality Problems in Mechanics. Convex and Nonconvex Energy Functions, Birkhauser Verlag, Boston Basel, Panagiotopoulos, P. D., Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag, Berlin Heidelberg, Clarke, F.H., Optimization and Nonsmooth Analysis, J.Wiley, New York, Mistakidis, E.S and Panagiotopoulos, P.D., On the Approximation of Nonmonotone Multivalued Problems by Monotone Subproblems Computer Methods in Applied Mechanics and Engineering, 1994, 114,