Multiscale Crack Research Articles

A model for predicting the multiscale crack growth due to an impact in heterogeneous viscoelastic solids

A two-scale model for predicting the multiple crack growth in viscoelastic solids due to an impact is presented. The cracks are considered only at the local scale through the use of a micromechanical viscoelastic cohesive zone model. The multiscale model has been implemented in a finite-element code. In order to minimize the computation time, the local finite-element meshes are solved in parallel by multiple processors. An example problem is given in order to demonstrate the capabilities of the model.

Phenomenology of seismic macrofracture formation

A phenomenological scheme is proposed for the formation of a seismic macrofracture in the crust through multiscale avalanche-like crack coalescence of the inverse cascade type. Relations connecting fractal dimensions of crack sets and the concentration criterion of coalescence of cracks that are required for the onset of multiscale fracture are obtained. It is shown that the enlargement of cracks from micro-to macroscales includes a relatively small number of hierarchical coalescence stages.

Simultaneous occurrence of double micro/macro stress singularities for multiscale crack model

It is rare to find two distinct stress singularities with different order at the same crack tip while the local displacements remained regular. Such a discovery was made when developing a dual scale micro/macro crack model and validated from a closed form asymptotic analytical solution for a micro/macro crack subjected to mixed boundary conditions. The two different orders of the stress singularities are 0.75 and 0.25 for an incompressible material. The existence of multi-singularities with different orders at the crack point is an unexplored area of research. For a sharp notch wedge, multi-stress singularities at the wedge point can prevail and they depend on the notch angle and Poisson’s ratio. Physical implications of such solutions when viewed at the microscopic scale are particularly significant as they tend to suggest that the multiple combinations of geometries and materials can yield so many different combinations of distortional and dilatational effects. No wonder microscopic irregularities seem to be the rule. Discontinuous numerically solutions, however, may or may not correspond to singularities. In view of the recent research emphases on multiscaling, the multiple singularity solution at a point seems to signify the inherent characteristics of micro-defects. They are irregular and defy self-similarity. The double singularity alone implies that a microcrack can extend in two different directions from the same location, known as branching in the very rapid propagation of a macrocrack. At the microscopic scale, double singularity implies the possibility that branching can occur under macroscopic static load. Such a behavior of static branching of microcracks associated with double singularity is being established. A fundamental aspect of this model is the inherent existence of restraining stress from the material that is essential for the microcrack and only sometimes for the macrocrack. As a rule, the microcrack being smaller would react to the material restraint more readily. Such a feature is necessary in the model in addition to the effects of micro/macro material interaction and the macrocrack characteristic length with reference to the macrocrack length. In normalized form, the minimum of three parameters, say μ ∗, σ ∗ and d ∗, would be necessary for the formulation of the dual scale micro/macro crack model. The combined effects of μ ∗ (micro/macro material), σ ∗ (load and restraint) and d ∗ (characteristic length) cannot be over emphasized in multiscale modeling. In the spirit of classical fracture mechanics, a micro/macro crack intensification factor K micro macro is defined and obtained in closed form. It contains three basic parameters μ ∗, d ∗ and σ ∗ being unique to all microcracks while two parameters ϕ 1 and ϕ 2 can be used to describe additional features of the microstructure. More precisely, the specific material microstructure can be modeled by two or more parameters associated with the eigenvalues selected rather than using the constitutive relations in the traditional manner. The asymptotic form of K micro macro is exact such that the case specific nature of microcracking can be distinguished by two constants. The contribution of this work is to establish the potential use of the dual singularity solution. The basic concept of the method applies to other multiscale problems in mechanics and physics.

Simultaneity of multiscaling for macro–meso–micro damage model represented by strong singularities

The state of affairs near a crack tip is very different when viewed at the different scales, say from the macroscopic to the atomic. These different appearances have intrigued solid state physicists, mechanists and material scientists for decades. Although they are physically connected, it is not obvious what roles would their mathematical relationships play? Investigations associated with scaling in size and time, evolution of material damage, multiscaling of macro-, meso- and micro-mechanics theories are presumably topics that could provide insights to the behavior of the internal structures of matters and their interactions at the different scales. When seeking for analytical models that can address multiscaling, continuum mechanics seems to give way to the particle approach when the scale reduces to that for the atoms. It is also not clear whether the particle and continuum view points should be regarded as separate disciplines for their differences have generated much discussion with little to gain. There is the general feeling that a re-examination of what has been taken for granted may be in order. Based on the comments made earlier, a multiscale crack damage model is developed by duplicating the physics much like the capability of an optical or electronic microscope. The model contains the macroscopic, mesoscopic and microscopic damage represented by singularities. The results at the different scale levels are connected via stress and displacement compatibility conditions in continuum mechanics and they can switch from one scale to another in a discrete manner where equilibrium is assumed within the range of an arbitrarily defined scale. The sizes involved at the different scales depend on the material, load and geometry and they are determined by solving two highly non-linear equations. Simplicity is gained by using singularities to represent the geometrical discontinuities such as intergranular or transgranular failure. This discussion is limited to a particular group of material damage that can be modelled by the so-called strong singularities as compared to the r −0.5 singularity at the continuum level. The strong singularities would correspond to 0.25 < λ* < 0.5 where λ* = 0.5 corresponds to the singular stress behavior for a line crack in an elastic medium. The lowest real part of the eigenvalue λ is denoted by λ*. In passing, it should be pointed out that the 1/ r singularity (i.e. λ* = 0) corresponds to an edge dislocation which will not be included in this discussion. In order not to cluster the presentation with unnecessarily complex notations of distinguishing the difference between the macroscopic and microscopic constants, the same set of constants will be assumed. Besides, there are fundamental issues with reference to the definition of microscopic material properties whether they should be obtained from testing micro-size specimens or considered as local material properties as part of the bulk. These two contrasting views are not the same. The choice to use one or the other remains debatable. The emphasis here is placed on relating strong singularity group to the geometry of a micro-notch where one notch edge is free and the other is fixed. It is connected to a transition or mesoscopic zone. The notch angle can be varied to model different degrees of intergranular opening. The microscopic non-homogeneous features are reflected by the rotations of the lines with different notch angles in a log–log plot of the strain energy density function with distance. Keep in mind that when the same near crack tip region is viewed at the macroscopic scale, the state of affairs are homogeneous. This in essence is what a microscope does. The objective of this work is to demonstrate that such a capability can be built into an analytical model with all of the complexities of the microstructure, the description of which are made possible by the singularity representation approach.

Mesofracture mechanics: a necessary link

Classical fracture mechanics is based on the premise that small scale features could be averaged to give a larger scale property such that the assumption of material homogeneity would hold. Involvement of the material microstructure, however, necessitates different characteristic lengths for describing different geometric features. Macroscopic parameters could not be freely exchanged with those at the microscopic scale level. Such a practice could cause misinterpretation of test data. Ambiguities arising from the lack of a more precise range of limitations for the definitions of physical parameters are discussed in connection with material length scales. Physical events overlooked between the macroscopic and microscopic scale could be the link that is needed to bridge the gap. The classical models for the creation of free surface for a liquid and solid are oversimplified. They consider only the translational motion of individual atoms. Movements of groups or clusters of molecules deserve attention. Multiscale cracking behavior also requires the distinction of material damage involving at least two different scales in a single simulation. In this connection, special attention should be given to the use of asymptotic solution in contrast to the full field solution when applying fracture criteria. The former may leave out detail features that would have otherwise been included by the latter. Illustrations are provided for predicting the crack initiation sites of piezoceramics. No definite conclusions can be drawn from the atomistic simulation models such as those used in molecular dynamics until the non-equilibrium boundary conditions can be better understood. The specification of strain rates and temperatures should be synchronized as the specimen size is reduced to microns. Many of the results obtained at the atomic scale should be first identified with those at the mesoscale before they are assumed to be connected with macroscopic observations. Hopefully, “mesofracture mechanics” could serve as the link to bring macrofracture mechanics closer to microfracture mechanics.

Experimental Study of Self-Similar Crack Distributions

This paper presents laboratory experiments on polyester resin samples subjected to special regimes of curing under which multiple shrinkage cracks with wide distributions of sizes were produced. Loading these samples produced stress-strain curves that followed a power law within certain stress ranges. The results of the tests were analysed using the mechanics of materials with self-similar crack distributions. The analysis showed that the observed power law in the stress-strain relationship indicated that the crack distribution was approximately self-similar. Thus, the described experiments serve as a verification procedure for the proposed formation mechanism of multiscale self-similar crack distribution in brittle materials. This mechanism is based on: (1) the action of self-equilibrating stress fluctuations as initiators and propagators of cracks and (2) the size discriminating action of crack interaction. The concentration factors for the crack distributions have been retrieved from the experimental data.