A unified fractal approach for the interpretation of the anomalous scaling laws in fatigue and comparison with existing models

In this paper, a critical reexamination of the fractal models for the analysis of crack-size effects in fatigue is proposed. The enhanced ability to detect and measure very short cracks has in fact pointed out the so-called anomalous behavior of short cracks with respect to their longer counterparts. The crack-size dependencies of both the fatigue threshold and the Paris’ constant C are only two notable examples of these anomalous scaling laws. In this context, a unified theoretical model seems to be missing and the behavior of short cracks can still be considered as an open problem. A new generalized theory based on fractal geometry is herein proposed, which permits to consistently interpret the short crack-related anomalous scaling laws within a unified theoretical framework. The proposed model is used to interpret relevant experimental data related to the crack-size dependence of the fatigue threshold in metals. As a main result, the model gives an explanation to the experimentally observed variability in the slope of the asymptote of the scaling law for the fatigue threshold in the short crack regime.     

Cracks of fractal geometry with unilateral contact and friction interface conditions

Structures involving cracks of fractal geometry are studied here, on the assumption that at the interface unilateral contact and friction boundary conditions hold. Approximating the fractal by a sequence of classical surfaces or curves and combining this procedure with a two-level contact-friction algorithm based on the optimization of the potential and of the complementary energy, we get the solution of the problem after some appropriate transformations relying on the S.V.D.-decomposition of the equilibrium matrix. Numerical examples, using singular elements for the consideration of the crack singularity, illustrate the theory.     

Finite element analysis of magnetic disk with a horizontal subsurface crack

Sliding contact of a rigid rough surface with a semi‐infinite medium including a horizontal subsurface crack was investigated by using linear elastic fracture mechanics and finite element method (FEM). The fractal geometry was used to characterize the rigid rough surface. The propagation of crack was studied with the shear and tensile stress intensity factors. The effect of surface roughness, crack length to depth ratio and friction at the contact and crack interfaces was investigated by using the FEM. It was shown that increasing friction coefficient at the contact interface increases both K_II and K_I.     

Scaling phenomena due to fractal contact in concrete and rock fractures

Concrete-to-concrete friction contributes in many cases to the stability of a structure. At different scales, the slope stability of rock joints is deeply influenced by the surface morphology and shows a marked size-dependence. In this paper, the closure and sliding-dilatant behaviour of cracks in concrete and rocks is investigated by means of a coupled numerical/experimental approach. These natural interfaces show self-affine properties in the relevant scale range. Attention has been focused on the stress transfer mechanism across the interfaces, showing that the sets of contact points possess the self-similar character of lacunar fractal sets. Scaling laws come into play and the size-effects on the shear strength of rough interfaces, and on their closure deformability, can be explained. This revised version was published online in July 2006 with corrections to the Cover Date.     

An extended (fractal) Overlapping Crack Model to describe crushing size-scale effects in compression

The inherent microstructural disorder strongly influences the mechanical behaviour of heterogeneous materials such as concrete and rocks. Tensile and compression tests, in fact, evidenced a localization of strain and dissipated energy in the post-peak softening branch, with a consequent scale dependence of the stress–strain response. For this reason, the well-known Cohesive Crack Model and the recently proposed Overlapping Crack Model are useful tools for describing the size effects in tension and compression, respectively. In general, strain localization, damage and fracture, which are phenomena affecting the failure of concrete, are not rigorously interpretable in the framework of continuum mechanics. On the other hand, since the flaw and the aggregate distributions in quasibrittle materials are often self-similar (i.e. they look the same at different magnification levels), the microstructure may be correctly modelled by fractal sets. In this paper, the approach based on fractal geometry, that has profitably been applied for the tensile behaviour, is applied to obtain a fractal overlapping law from uniaxial compression tests. According to this approach, it is assumed that energy dissipation, stress and strain are not defined with respect to the canonical physical dimensions, though on fractal sets presenting noninteger physical dimensions. As a consequence, these classical parameters should be substituted by fractal quantities, which become the true material properties.     

On the numerical treatment of the delamination problem in laminated composites under cleavage loading

A method for the effective numerical treatment of the delamination problem in laminated composites under cleavage loading is herein proposed. The interlaminar interface mechanical behaviour is described by means of the so-called complete laws which are non-monotone and possibly multivalued force/ displacement laws including jumps (or in general, decreasing branches) corresponding to the discontinuous strength reduction. These complete laws that take into account the development of delamination phenomena in a quasistatic way are derived by non-convex energy functions, called delamination superpotentials which in turn, lead to the formulation of the principle of virtual work for the laminated composite in a hemivariational inequality form and to the generalisation of the principle of minimum potential energy as a substationarity principle. Applying an appropriate finite element discretisation scheme to the laminated composite, the respective discrete problem is formulated which describes the response of the structure taking into account the development of the delamination phenomenon. The numerical treatment of the latter problem is successfully performed by applying a new algorithm that approximates the nonmonotone law by a sequence of monotone ones. The performed numerical applications presented in the last part of the paper and several analogous numerical experiments exhibit very good convergence properties.     

The ‘fractal assumption’ in cracks: Bilateral calculation methods

The paper considers the fractal nature of cracks by assuming the fractal as the fixed point of a given transformation. This concept enables the study of a sequence of classical-geometry crack problems whose limit gives the solution of the fractal-geometry crack problem. The method is illustrated by a numerical example. It was shown that the fractal nature of the cracks influences the value of the stress intensity factors.     

Effect of interfacial friction during forging of solid powder discs of large slenderness ratio

The paper reports an investigation into the effect of interfacial friction law during the forging of a powder circular disc with large slenderness ratio(L/D), between two flat dies. The deformation pattern during the operation is influenced by many factors, which interact with each other in a complex manner. The relative velocity between the work piece material and the die surface, together with high interfacial pressure and/or deformation modes, creates the conditions essential for adhesion in addition to sliding. The decisive factors are the interfacial conditions, initial relative density of the preform and geometry of the preform. An attempt has been made to determine the most realistic interfacial friction law and die pressures developed during such forging using an upper bound approach. The results so obtained are presented graphically and discussed critically to illustrate the interaction of various interfacial friction laws involved.     

Diffraction from fractal grating Cantor sets

In this paper, we have generalized the F^α-calculus by suggesting Fourier and Laplace transformations of the function with support of the fractals set which are the subset of the real line. Using this generalization, we have found the diffraction fringes from the fractal grating Cantor sets.     

Micropolar continuum mechanics of fractal media

This paper builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a technique based on a dimensional regularization, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established.     

On the existence of macro- and nanostructures with phonon spectra of low fractal dimensions

It is shown that structures possessing predominantly low-frequency vibrational spectra and phonon spectra of low fractal dimensions d _f < 1, which have been predicted previously by the fractal theory of heat capacity, can exist among natural objects representing the cluster state of matter. This conclusion is based on an analysis of the frequency spectra of molecular vibrations in water clusters containing 10–22 molecules, which have been obtained using ab initio numerically simulations. It has been found that fractal dimensions of the corresponding phonon spectra fall within d _f = 0.3−0.5. The temperature dependence of the heat capacity C(T) of these clusters differs from that of bulk samples of the same chemical composition (e.g., macroscopic samples of ice with d _f ≈ 1.4). These differences are even more pronounced in comparison to substances that obey the traditional Debye law (d _f = 3).     

Temperature dependence of fractal dimension of grain boundary region in SnO2 based ceramics

Fractal dimensions of grain boundary region in doped SnO₂ ceramics were determined based on previously derived fractal model. This model considers fractal dimension as a measure of homogeneity of distribution of charge carriers. Application of the derived fractal model enables calculation of fractal dimension using results of impedance spectroscopy. The model was verified by experimentally determined temperature dependence of the fractal dimension of SnO₂ ceramics. Obtained results confirm that the non-Debye response of the grain boundary region is connected with distribution of defects and consequently with a homogeneity of a distribution of the charge carriers. Also, it was found that C−T ⁻¹ function has maximum at temperature at which the change in dominant type of defects takes place. This effect could be considered as a third-order transition.     

Statistics of contacts and the dependence of their total length on the normal force for fractal surfaces with different Hirsch indices

In many applied problems, the main role is played by the total length L of real contacts between two rough surfaces rather than by the contact area. The dependence of the total contact length on the pressing force F for arbitrary fractal surfaces has been studied for the first time and it is established that this dependence is described by scaling relation L ∝ F ^3/4.     

Scaling and fractality in subcritical fatigue crack growth: Crack‐size effects on Paris' law and fatigue threshold

The present contribution investigates the crack‐size effects on Paris' law in accordance with dimensional analysis and intermediate asymptotics theory, which makes it possible to obtain a generalised equation able to provide an interpretation to the various empirical power‐laws available in the Literature. Subsequently, within the framework of fractal geometry, scaling laws are determined for the coordinates of the limit‐points of Paris' curve so that a theoretical explanation is provided to the so‐called short cracks problem. Eventually, the proposed models are compared with experimental data available in the literature which seem to confirm the advantage of applying a fractal model to the fatigue problem.     

Contact problems with nonmonotone friction: discretization and numerical realization

The paper deals with the formulation, approximation and numerical realization of a constrained hemivariational inequality describing the behavior of two elastic bodies in mutual contact, taking into account a nonmonotone friction law on a contact surface. The original hemivariational inequality is transformed into a problem of finding substationary points of a nonconvex, locally Lipschitz continuous function representing the discrete total potential energy functional. The resulting discrete problem is solved by using a nonsmooth variant of the Newton method. Numerical results of a model example are shown.     

Numerical treatment of hemivariational inequalities in mechanics: two methods based on the solution of convex subproblems

Hemivariational inequality problems describe equilibrium points (solutions) for structural systems in mechanics where nonmonotone, possibly multivalued laws or boundary conditions are involved. In the case of problems which admit a potential function this is a nonconvex, nondifferentiable one. In order to avoid the difficulties that arise during the calculation of equilibria for such mechanical systems, methods based on sequential convex approximations have recently been proposed and tested by the authors. The first method is based on ideas developed in the fields of quasidifferential and difference convex (d.c.) optimization and transforms the hemivariational inequality problem into a system of convex variational inequalities, which in turn leads to a multilevel (two-field) approximation technique for the numerical solution. The second method transforms the problem into a sequence of variational inequalities which approximates the nonmonotone problem by an iteratively defined sequence of monotone ones. Both methods lead to convex analysis subproblems and allow for treatment of large-scale nonconvex structural analysis applications.The two methods are compared in this paper with respect to both their theoretical assumptions and implications and their numerical implementation. The comparison is extended to a number of numerical examples which have been solved by both methods.     

Pressure-restoration curve for a fractal cracked medium

The problem on flow of a single-phase, weakly compressible fluid in a cracked medium with the fractal geometry of cracks has been considered. The problem on the pressure-restoration curve has been solved in the approximation of an axisymmetric flow near the well. The corrections to the formula obtained in the classical theory of filtration for the pressure-restoration curve have been computed for the case where the dimension of a system of cracks is close to the dimension of a confining volume. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 79, No. 2, pp. 76–80, March–April, 2006.     

Coarsening of δ-Ni<Subscript>2</Subscript>Si precipitates in a Cu–Ni–Si alloy

The coarsening behavior of rod-shaped and spherical δ-Ni₂Si precipitates in a Cu–1.86 wt% Ni–0.45 wt% Si alloy during aging at 823–948 K has been investigated by measuring both precipitate size by transmission electron microscopy (TEM) and solute concentration in the Cu matrix by electrical resistivity. The rod-shaped δ precipitates have an elongated shape along 〈[`5] 5 8 \overline{5} 5 8 〉_m and a {110}_m habit-plane facet. The coarsening theory of a spherical precipitate in a ternary alloy developed by Kuehmann and Voorhees (KV) has been modified to a case of rod-shaped precipitates. The coarsening kinetics of average size of the rod-shaped and spherical δ precipitates with aging time t obey the t ^1/3 time law, as predicted by the modified KV theory. The kinetics of depletion of the supersaturation with t are coincident with the predicted t ^−1/3 time law. Application of the modified KV theory has enabled calculation of the energies of sphere, {110}_m and rod-end interfaces from the data on coarsening alone. The energy of the {110}_m interface having a high degree of coherency to the Cu matrix is estimated to be 0.4 J m⁻², the incoherent sphere-interface energy 0.6 J m⁻², and the rod-end interface energy 5.2 J m⁻².     

Two-dimensional analysis of anisotropic crack problems using coupled meshless and fractal finite element method

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with fractal the finite element method (FFEM) for analyzing homogeneous, anisotropic, and two dimensional linear-elastic cracked structures subjected to mixed-mode (modes I and II) loading conditions. FFEM is adopted for discretization of domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprises both the element-free Galerkin and the finite element shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM-FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary and no post-processing (employing any path independent integrals) is needed to determine fracture parameters such as stress-intensity factors (SIFs) and T − stress. The numerical results based on all four orthotropic cases show that SIFs and T − stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions.