A DISCRETE NONLOCAL FORMULATION USING LOCAL CONSTITUTIVE LAWS

Transcription

1 International Journal of Fracture 130: L175 L182, Kluwer Academic Publishers. Printed in the Netherlands. A DISCRETE NONLOCAL FORMULATION USING LOCAL CONSTITUTIVE LAWS Elena Ferretti Alma Mater Studiorum University of Bologna, Faculty of Engineering, DISTART Scienza delle Costruzioni, Viale Risorgimento 2, Bologna, Italy elena.ferretti@mail.ing.unibo.it Abstract. Nonlocality is discussed in differential and discrete formulations. When modeling heterogeneous materials, a length scale must be introduced into the material description of the differential formulation. This happens since metrics is lost in performing the limit process. Avoiding the limit process, that is, using a discrete formulation, the length scale is intrinsically taken into account. Moreover, nonlocality seems to characterize global variables rather than material. This made it possible to move the length scale from constitutive to governing equations. Keywords: nonlocality, differential formulation, discrete formulation, Cell Method, constitutive relationships, size-effect, softening. 1. Introduction. One of the main earch fields in last years concerns the modeling of heterogeneous materials. For these materials, the classical local continuum concept, with the sts at a given point uniquely depending on the current values, and possibly also the previous history, of deformation and temperature at that point only, does not seem to be adequate. Actually, modeling the size effect is impossible in the context of the classical plasticity, both in problems with strain-softening and in those with no strain-softening at all. The reason for this was found to be the local nature of the constitutive relationships between sts and strain tensors, which is not adequate for describing the mechanical behavior of solids in the classical differential formulation, since no material is an ideal continuum, decomposable into a set of infinitesimal material volumes, each of which can be described independently. All materials, natural and man-made, are characterized by microstructural details whose size ranges over many order of magnitude (Bažant and Jirásek, 2002). Beginning with Krumhansl (1965), Rogula (1965), Eringen (1966), Kunin (1966), and Kröner (1968), the idea was promulgated that heterogeneous materials should properly be modeled by some type of nonlocal continuum, in which the sts at a certain point is a function of the strain distribution over a certain repentative volume of the material centered at that point (Bažant and Chang, 1984). This idea led to nonlocal models, in which the classical continuum description is improved with an internal length parameter, introduced in the constitutive laws. 2. Locality and nonlocality in differential formulations. According to the mathematical definition of nonlocality given by Rogula (1982), the operator A in

2 L176 the abstract form of the fundamental equations of any physical theory, Au = f, is called local when, if u( x) = v( x) for all x in a neighborhood of point x 0, then Au ( x) Av( x) =. As pointed out by Bažant and Jirásek (2002), differential operators satisfy this condition, because the derivatives of any arbitrary order do not change if the differentiated function changes only outside a small neighborhood of the point at which the derivatives are taken. On the basis of this statement, it may be asserted that any formulation using differential operators is intrinsically local. That is, the differential formulation is not adequate for describing nonlocal effects. The reason for this lies in the basics itself of the differential formulation, performing the limit process. The differential formulation requi field functions, which have to depend on point position, x, y, z, and instants, t. If the field functions are not directly described in terms of x, y, z, and t, they are obtained from global variables, by performing densities and rates. Global variables are domain variables, depending on x, y, z, and t, but also on line extensions, L, areas, S, volumes, V, and time intervals, Δt. The density finding process is carried out with the intention of formulating the field laws in an exact form. However, with the reduction of global variables to point and instant variables we loose metrics and, thus, we loose the possibility of describing more than 0-dimensional (nonlocal) effects. Metrics must be reintroduced a-posteriori, if we want to model the nonlocal effects. One may ask, now, where metrics must be reintroduced. In nonlocal approaches, a length scale is incorporated in constitutive laws, but there is not evidence for this choice to be the only, or the physically most appealing one. 3. The length scale. In order to answer the question on where to reintroduce the length scale, let us examine the physical variables and their classification. Among all the possible classifications, the one we will adopt in the following distinguishes between configuration and source variables (Hallen, 1962; Penfield and Haus, 1967; Tonti, 1972). Configuration variables describe the field configuration (displacements), while source variables describe the field sources (forces). Those equations relating configuration variables by each other and source variables by each other are structure equations, while those relating configuration with source variables are constitutive equations. Each physical phenomenon occurs in space, and space has a multidimensional geometrical structure. If the physical variables are able to describe phenomena in space, such as they actually are, then the physical variables themselves have a multi-dimensional geometrical content. As far as Solid Mechanics is concerned (Fig. 1), volume forces, which are source variables, are associated with a length scale in dimension 3, since their geometrical referent is a volume. Analogously, surface forces, which are source variables, have a two-

3 L177 dimensional geometrical referent (the surface), strains, which are configuration variables, have a one-dimensional geometrical referent (the line), and displacements, which are configuration variables, have a zero-dimensional geometrical referent (the point). Figure 1. Association between global variables and cell complexes in Solid Mechanics (Tonti, 2001). Thus, it seems that dimensional scales and nonlocal effects are associated with the variables directly, and not with the equations relating the variables by each other. That is, nonlocality seems to be a property of the global variables, and not of the constitutive laws. Consequently, reintroducing or perving nonlocality in governing equations is physically more correct than reintroducing nonlocality in constitutive equations. When speaking of reintroduction, we deal with a differential formulation, while, when speaking of pervation, we deal with a discrete formulation. The difference is not negligible, since, in order to reintroduce a length scale, it is necessary to develop an adequate approach, while, in order to perve the length scale, it is sufficient to avoid the limit process, using discrete approaches, and a nonlocal formulation is automatically obtained. In conclusion, obtaining a nonlocal formulation by using local constitutive laws and discrete operators seems to be possible, besides than physically appealing. If this were the case, the new nonlocal formulation would be advantageous from the numerical point of view, since the numerical solution is achieved faster by using discrete operators rather than differential operators. 4. The Cell Method. In a discrete nonlocal formulation, all operators must be discrete and the limit process must be avoided at each level of the formulation itself. The direct or physical approach initially used in the Finite Element Method (Huebner, 1975; Livesley, 1983; Fenner, 1996) is not suitable to this aim, since it starts from point-wise conservation equations and the discrete formulation is induced by the differential formulation. Even the Finite Volume Method (FVM) and the Finite Differences Method (FDM) are based on a differential formulation. The Cell Method (CM) is a new method (Tonti, 2001) that could seem very similar to the physical approach and the vertex-based scheme of the FVM. Nevertheless, the similarity is apparent, since the CM is not based on a differential formulation and can be used for building a discrete nonlocal formulation. The CM uses cell complexes, which does not simply are the ult of a domain discretization, needed by the numerical analysis. They substitute the coordinate

4 L178 systems when we need to describe not only points, but also lines, surfaces, and volumes. 0- and more than 0-dimensional quantities are described directly, avoiding the limit process and the subsequent discretization, by associating them with nodes, edges, surfaces, and volumes of the cell complexes. Once chosen a set of points, the primal nodes P (black points in Fig. 2), the lines connecting these nodes define a spatial mesh, said the primal cell complex. Edges, areas, and volumes of the primal cell complex are, pectively, primal sides L, surfaces S, and volumes V. Now, let us consider the surfaces, locus of the points which are equidistant from each pair of primal nodes (gray surfaces in Fig. 2). Let us use these surfaces for building a second spatial mesh, the dual cell complex. Points, edges, areas, and volumes of the dual cell complex are dual nodes P, sides L, surfaces S, and volumes V. Figure 2. Primal and dual cell complexes (Tonti, 2001). As discussed in Tonti (2001), the objects of the primal cell complex are natural geometrical referents of configuration variables, while the objects of the dual cell complex are natural geometrical referents of source variables (Fig. 1). 5. Constitutive assumptions. The last point to consider in building a discrete nonlocal formulation using local constitutive laws is how to formulate the constitutive law in order it is actually local, with the nonlocality of governing equations not automatically extending to constitutive relationships by scale change. In order to answer this question, let us examine the identification process of the material properties for concrete in uniaxial compsion. Usually, concrete specimens in uniaxial compsion are considered to fail with propagation of subvertical macro-s. Nevertheless, the actual failure mechanism of concrete specimens develops internally, since internal macro-s propagate through the specimens from the very beginning of the compsion test. In cylindrical specimens, the internal s isolate a istant inner core of bi-conic shape (Fig. 4), while the outer part is expulsed along the radial direction and splits into several portions (scheme in Fig. 3). Thus, the sub-vertical s on the external surface are a secondary effect of the actual failure mechanism, which gradually modifies the istant structure of the specimen. The modification of the istant structure involves a modification of the istant area, A. Due to the gradual modification of the istant structure, also A modifies gradually from the very beginning of the test. Reductions of A must be considered when the constitutive law is derived from the experimental load-displacement curves, N-u. This means that we can, and we actually must separate the material from the structure scale. That is, the constitutive behavior is

5 L179 not the mirror image of a structural problem at a lower scale. In Ferretti (2004), the effective sts, σ eff, and the effective strain, ε eff, have been introduced as constitutive parameters describing the material behavior when the reduction of A is considered, with σ eff = N A. Consequently, the identification procedure proposed by Ferretti (2004) does not consist of a mere change of scale, and N-u and σ ε curves are not identical in shape. eff - eff Figure 3. Concrete specimen at the end of the test, and scheme of slitting on the middle cross-section. Figure 4. Concrete specimen at the end of the test, after removal of the outer part. Effettive sts σeff [MPa] Figure Effettive strain ε eff [με] σ eff average curve - ε eff dispersion range for variable slenderness and average curve. In particular, a softening N-u diagram does not necessarily involve a softening σ eff -ε eff diagram. A monotone strictly nondecreasing material law (Fig. 5) has been identified by Ferretti (2004) for concrete specimens in uniaxial compsion, whose N-u diagrams are softening, by means of experimental evaluation of A during loading. This material law is also size-effect insensitive (Fig. 5). Only in the assumption of scales separation it is possible to associate the nonlocality with the governing equations only, without automatic extension to the constitutive laws. Thus, the material law we used in the discrete nonlocal D 2R L=(1.5 4)D H=(3 8)R

6 L180 numerical analysis is the one following from the identification procedure proposed in Ferretti (2004). 6. The Cell Method code. In the hypothesis of scales separation, the numerical analysis must be able to reproduce the failure path shown in Fig. 4. This means that the code must be able to update the domain when the propagates. To this aim, a CM code operating by means of a nodal relaxation technique with intra-element propagation (Fig. 6) has been developed (Ferretti, 2003). The CM association between source variables and dual cells ( 4) allows evaluation of the sts field for the finite neighborhood of the tip, on the basis of the sts acting on the dual sides around the tip. A special hexagonal element has been studied (Fig. 7), for regularizing the shape of the dual cell on the tip and, thus, compute the sts field at a prefixed distance from the tip. Inter element propagation Propagation direction Computed direction Intra element propagation Propagation direction Computed direction Figure 6. Inter and intra element propagation for the nodal relaxation technique. hexagonal element Delaunay Voronoi Figure 7. Special element for sts analysis around the tip. 7. Numerical ults. The CM code developed by Ferretti (2003) has been used in conjunction with the local material law proposed by Ferretti (2004), in order to model the N-u behavior of concrete compsed cylinders with several L/D ratios. Due to the cylindrical geometry, the numerical analysis has been performed on one quarter of the longitudinal section only (Fig. 8c). The numerical initiation point and propagation direction (about 70 with pect to the horizontal axis) are in good agreement with the experimental findings (Figs. 4, 8b,c). The longitudinal sts field for a short propagation is shown in Fig. 8a. The large perturbation of the sts field in Fig. 8a validates the main assumption on the basis of the identification procedure proposed by Ferretti (2004), according to which even for short s the difference between nominal and istant area is significant. The numerical N-u curves are plotted in Fig. 9 for several L/D ratios. These curves are softening, even if the material law we used is monotone. This happens since, due to its ability to update the domain and evaluate the sts redistribution after propagation, the code is able to estimate the decrease of structural stiffness following from ing. In other words, Fig. 8a gives an explanation of

7 L181 why structural and material scales must be separated, since the sts field is not homogeneous, and comparison between Figs. 5 and 9 shows that it is realistic to consider that softening is not a material but a structural property. It is worth noting that, not only the softening, but also the size-effect is captured. Actually, the numerical decreasing of stiffness and strength, with increasing L/D ratio, is in good agreement with the experimental evidence (Fig. 9), even if a size-effect insensitive law has been used in modeling. The nonlocal nature of the CM code with local material law is proved just by its ability of modeling the size-effect, which is impossible for differential local approaches. Delaunay Voronoi a) tip b) N y c) tip N Initiation point Figure 8. a) Sts field after short ing; b) Crack-path; c) Loading scheme and modeled domain. 600 x Load [kn] D L=(1.5 4)D Numerical curves Experimental curves 0 L/D Displacement [mm] Figure 9. Comparison between numerical and experimental ults for compsed specimens. 8. Conclusions. Nonlocal modeling by means of local constitutive laws and discrete operators is possible and provides ults in good agreement with the experimental findings. It is thus possible to move the length scale from the constitutive laws to the governing equations. Two arguments are on the basis of the new nonlocal formulation: it is physically more appealing having a length

8 L182 scale in the governing equations rather than in the constitutive laws, since nonlocality attains to global variables and not to constitutive laws, and the material behavior can be separated from the structural behavior, since the sts field is not homogeneous and modifies at each stage of the load carrying process, letting impossible to establish a scale factor between the two behaviors. If compared with the differential nonlocal formulation, the discrete nonlocal formulation is advantageous in two senses: we do not need to discretize the field equations, since they are obtained in discrete form directly, and we do not need to calibrate any material parameter. As far as this last point is concerned, the absence of material parameters to calibrate on the specific test follows from the local nature of the material law we have used. All the structural effect are automatically computed by the code, on the basis of the sts redistribution following from propagation. It is the propagation which depends on the specific test, since the limit condition is reached for different loads if the specimen geometry is changed, and the CM code is able to model this geometrical dependence due to the intrinsic nonlocality of the CM. The most important ult achieved by means of this code concerns the ability of modeling softening and size-effect as structural behaviors, starting from a monotone and size-effect insensitive material law. References Bažant, Z. P., and Chang, T. P. (1984). Is Strain-Softening Mathematically Admissible? Proc., 5 th Engineering Mechanics Division 2, Bažant, Z. P., and Jirásek, M. (2002). Nonlocal Integral Formulations of Plasticity and Damage: Survey of Progs. J. Eng. Mech. 128(11), Eringen, A. C. (1966). A Unified Theory of Thermomechanical Materials. Int. J Eng. Sci. 4, Fenner, R. T. (1996). Finite Element Methods for Engineers. Imperial College Ps, London. Ferretti, E. (2003). Crack Propagation Modeling by Remeshing using the Cell Method (CM). CMES, 4(1), Ferretti, E. (2004). Experimental Procedure for Verifying Strain-Softening in Concrete. International Journal of Fracture (Letters section) 126(2), L27-L34. Hallen, E. (1962). Electromagnetic Theory, Chapman & Hall. Huebner, K. H. (1975). The Finite Element Method for Engineers. Wiley. Kröner, E. (1968). Elasticity Theory of Materials with Long-Range Cohesive Forces. Int. J. Solids Struct. 3, Krumhansl, J. A. (1965). Generalized Continuum Field Repentation for Lattice Vibrations. In Lattice Dynamics, R. F. Wallis ed., Pergamon, London, Kunin, I. A. (1966). Theory of Elasticity with Spatial Dispersion. Prikl. Mat. Mekh. (in Russian) 30, 866. Livesley, R. K. (1983). Finite Elements, an Introduction for Engineers. Cambridge University Ps. Penfield, P., and Haus, H. (1967). Electrodynamics of Moving Media, M.I.T. Ps. Rogula, D. (1965). Influence of Spatial Acoustic Dispersion on Dynamical Properties of Dislocations. I. Bulletin de l Académie Polonaise des Sciences, Séries des Sciences Techniques 13, Tonti, E. (1972). On the Mathematical Structure of a Large Class of Physical Theories. Rend. Acc. Lincei 52, Tonti, E. (2001). A Direct Discrete Formulation of Field Laws: the Cell Method. CMES, 2(2),