Experimental investigation of the contact mechanics of rough fractal surfaces

The nonstationary character of roughness is a widely recognized property of surface morphology and suggests modeling several solid surfaces by fractal geometry. In the field of contact mechanics, this demands novel investigations attempting to clarify the role of multiscale roughness during physical contact. Here we review the results we recently obtained in the characterization of the contact mechanics of fractal surfaces by depth-sensing indentation. One class of experiments was conducted on organic thin films, load-displacement curves being acquired by atomic force microscopy using custom-designed tips. Another class of experiments focused on well-defined crystalline and mechanically polished ceramic substrates probed by a traditional nanoindenter. We observed the first-loading cycle to be considerably affected by surface roughness. Plastic failure was found to dominate incipient contact while contact stiffness increased on decreasing fractal dimension and roughness. Our findings suggest fractal parameters to drive contact mechanics whenever the penetration depth is kept below the interface width.     

Discrete fractal fracture mechanics

A modification of the classical theory of brittle fracture of solids is offered by relating discrete nature of crack propagation to the fractal geometry of the crack. The new model incorporates all previously considered theories of fracture processes, in particular the Griffith [Griffith AA. The phenomenon of rupture and flow in solids. Philos Trans Roy Soc Lond 1921;A221:163-398] theory, its contemporary extension known as LEFM and the most recently developed Quantized Fracture Mechanics (QFM) by Pugno and Ruoff [Pugno N, Ruoff RS. Quantized fracture mechanics. Philos Mag 2004;84(27):2829-45]. Using an equivalent smooth blunt crack for a given fractal crack, we find that assuming that radius of curvature of the blunt crack is a material property, the crack roughens while propagating. In other words, fractal dimension at the crack tip is a monotonically increasing function of the nominal crack length, i.e., the presence of the Mirror-Mist-Hackle phenomenon is analytically demonstrated.     

The mechanics of self-similar and self-affine fractal cracks

In this paper we study the mechanical attributes of the fractal nature of fracture surfaces. The structure of stress and strain singularity at the tip of a fractal crack, which can be self-similar or self-affine, is studied. The three classical modes of fracture and the fourth mode of fracture are discussed for fractal cracks in two-dimensional and three- dimensional solid bodies. It is discovered that there are six modes of fracture in fractal fracture mechanics. The J-integral is shown to be path-dependent. It is explained that the proposed modified J-integrals in the literature that are argued to be path-independent are only locally path-independent and have no physical meaning. It is conjectured that a fractal J-integral should be the rate of potential energy release per unit of a fractal measure of crack growth. The powers of stress and strain singularities at the tip of a fractal crack in a strain-hardening material are calculated. It is shown that stresses and strains have weaker singularities at the tip of a fractal crack than they do at the tip of a smooth crack.     

The fourth mode of fracture in fractal fracture mechanics

This paper offers a systematic approach for obtaining the order of stress singularity for different self-similar and self-affine fractal cracks. Mode II and Mode III fractal cracks are studied and are shown to introduce the same order of stress singularity as Mode I fractal cracks do. In addition to these three classical modes, a Mode IV is discovered, which is a consequence of the fractal fracture. It is shown that, for this mode, stress has a weaker singularity than it does in the classical modes of fracture when self-affine fractal cracks are considered, and stress has the same order of singularity when self-similar cracks are considered. Considering this new mode of fracture, some single-mode problems of classical fracture mechanics could be mixed-mode problems in fractal fracture mechanics. By imposing a continuous transition from fractal to classical stress and displacement fields, the complete forms of the stress and displacement fields around the tip of a fractal crack are found. Then a universal relationship between fractal and classical stress intensity factors is derived. It is demonstrated that for a Mode IV fractal crack, only one of the stress components is singular; the other stress components are identically zero. Finally, stress singularity for three-dimensional bodies with self-affine fractal cracks is studied. As in the two-dimensional case, the fourth mode of fracture introduces a weaker stress singularity for self-affine fractal cracks than classical modes of fracture do.     

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.     

Advances in computational contact mechanics

Great strides have recently been made in the application of computational mechanics to the design of highly complex engineering systems. It has now become abundantly clear that advanced modelling techniques are central to the competitiveness of the industrialised nations. Excellent examples of this assertion are the computer-integrated design of the recent Boeing 777 aircraft, the collapsible foam-filled structures for the car of the next century and new prosthetic implants for Rheumatoid Arthritis. It is with this in mind that the author focuses his attention to a class of problems where contact mechanics plays a major role in dictating the mechanical integrity of the component/system. Three aspects of the current study are accordingly examined. The first is concerned with the development of the appropriate dynamic variational inequalities expressions, which are capable of the accurate and consistent representation of contact problems. The second is concerned with the development of robust solution algorithms that guarantee the accurate imposition of the kinematic contact constraint and avoid interpenetration. The third is concerned with the application of the developed algorithms to realistic design problems involving intricate mechanical and biomechanical systems.     

From Griffith's theory to the fractal fracture mechanics

Conclusion The above analysis shows the necessity for further development of fractal fracture mechanics at micro-, meso-, and macrolevels, using fractal theory and the general principle of synergetics.A. A. Baikov Institute of Metallurgy, Russian Academy of Sciences, Moscow. Published in Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 29, No. 3, pp. 101–106, May–June, 1993.     

The fractal geometry of Si3N4 wear and fracture surfaces

Good mechanical properties and chemical stability at high temperatures make silicon nitride a good candidate as an advanced engine material. Much research has been done to characterize the mechanical strength and resistance of crack propagation in this material. In this paper, we use fractal analysis to study the geometry of Si3N4 fracture and wear surfaces. We found that the geometries of the failure surfaces as characterized by the fractal dimensional increment, D*, under different failure stress states are similar for the same brittle material, but different for different brittle materials. The similar D* in an identical brittle material implies that the failure process in the material is the same regardless of loading mode, i.e., mode I or mixed-mode stress. The fractal technique is shown to be useful for correlating the fractal dimension to the material properties and fracture-surface topography. This revised version was published online in November 2006 with corrections to the Cover Date.     

Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation

The present paper deals with two interrelated subjects: the fractal geometry and the fractal behaviour in unilateral contact problems. More specifically, throughout this paper both the interfaces and the friction laws holding on these interfaces are modelled by means of the fractal geometry. It is important to notice here that the fractality of the induced friction laws takes into account the randomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, both the fractal law and the fractal interface are considered to be graphs of two different fractal interpolation functions which are the “fixed points” of two contractive operators. Using this method, the fractal friction law is approximated by a sequence of nonmonotone possibly multivalued classical C ⁰-curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone problem by a sequence of monotone subproblems. Numerical applications from the static analysis of cracked structures with a prescribed fractal geometry and fractal interface laws are included in order to illustrate the theory.     

Application of fractal fracture mechanics to polymers and polymeric composites

It is shown that the fundamental concepts of fractal fracture mechanics can be applied both to polymers and polymeric composites and to metals and ceramics. The critical crack opening displacement can be chosen as a scale of fracture of polymeric materials. The results obtained for polymers and polymeric composites are described by the same sigmoidal dependence, which means that the regularities of fracture processes in these materials are common.Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 4, pp. 53–57, July–August, 2004.     

The use of contact reflectometry in the investigation of dental surfaces

Many studies have demonstrated not only the value of including an abrasive in the formulation of toothpaste, but also the damage caused by some abrasives to dental surfaces. The techniques which could be used to evaluate the abrasive power and polishing capacity of a toothpaste or dentifrice are generally difficult to apply and no progress has been made in $\text{in vitro}$ determinations. A novel method, based on contact reflectometry is described. The reproducibility of the method is demonstrated, as are the good correlation obtained with established methods and the ease with which it can be employed.     

On the fractal dimension of fracture surfaces of concrete elements

The problem of the relation between the fractal dimension of a fractured surface and the fracture toughness expressed by the stress intensity factor is investigated. The theoretical conditions for such assumptions are discussed. Collected experimental results and new tests performed onconcrete specimens subjected to Mode II fracture seem to confirm that relation within the scope of materials tested and with certain necessary restrictions.     

Elastic-plastic adhesive contact of non-Gaussian rough surfaces

The paper describes an analysis of adhesion at the contact between non-Gaussian rough surfaces using the Weibull distribution with skewness as the key parameter to characterize asymmetry. The analysis uses an improved elastic-plastic model of contact deformation that is based on accurate Finite Element Analysis (FEA) of an elastic-plastic single asperity contact. Large range of interference values is considered starting from fully elastic through elastic-plastic to fully plastic regime of contacting asperities. The well-established elastic and plastic adhesion indices are used to consider the different conditions that arise as a result of varying load and material parameters. The loading and unloading behaviour for different combinations of the adhesion indices and skewness values are obtained as functions of mean separation between the surfaces. Transitional values of adhesion indices and skewness at which the influence of surface forces becomes insignificant are found to depend on material and surface parameters. Comparison with studies using previous elastic-plastic model that was based on some arbitrary assumptions shows significant differences in loading behaviour.     

Micro and macro contact mechanics for interacting asperities

Contact of rough surfaces at micro and macro scales is studied in this paper. The asperities at micro scale are characterised by small radius of curvature whereas the waviness is characterised by large radius of curvature. When two rough surfaces come in contact, on the micro scale, of asperities contacts in a very small area leave large gaps between the surfaces; whereas on the macro scale the surfaces conform to each other under the application of load without gaps. Contact at micro scale is modelled by superposition of Hertzian stress fields of individual asperity contacts and the waviness at macro scale is modelled as a mixed boundary problem of rough punch indentation where displacements of uneven profile are prescribed along the region of contact. In both the cases for simplification the roughness is assigned to one surface making the other surface perfectly flat an assumption often made in contact mechanics of rough bodies. The motivation for modelling the asperities at micro scales comes from the preliminary results obtained from photoelastic experiments. Numerical results are presented based on the analytical results available for Hertzian contacts. The motivation for modelling the asperities at macro scales comes from the results available in literature for flat contacts from solving mixed boundary elasticity problems. A condition of full stick is assumed along the contact which is a common assumption made for rough contacts. The numerical results are presented for both the cases of rough contact at micro and macro scales.     

The effect of the compression of wavy surfaces on contact thermal resistance

In this article we obtain expressions which relate the contact thermal resistance between surfaces with spherical and cylindrical undulations to the external load imposed on them. It is assumed that the amplitudes of the waves obey a Gaussian distribution and that the microscopic asperities have constant radius.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.19, No. 2, pp. 327–331, August, 1970.     

The effect of crack propagation mechanism on the fractal dimension of fracture surfaces in steels

The paper reports on the results of a fractal analysis of fracture surfaces of Ni–Cr steel in two different states of heat treatment simulating embrittlement. The change in the fractal dimension of the fracture surface demonstrates a wavy character and dispersion depending on the microstructural state of the tested steel. The results of the fractal analysis in the crack growth direction and across the entire crack front were used as the basis for a reconstruction of the geometry of the fracture surface, providing a new geometric tool for fractographic analysis. The competing effects of transgranular and intergranular brittle fracture may lead to increased roughness of the fracture surface and its fractal dimension. The threshold value of the fractal dimension of the sections perpendicular to the fracture surface, indicating the transition from transgranular to intergranular fracture, is 1.12.