G. Frantziskonis, F.F. Tang and C.S. Desai, "Borehole Scale Effects and Related Instabilities," Engr. Fracture
Mechs., 39, 377-389, 1991
A new mechanics based approach is proposed for scale effects and instabilities on borehole problems. In borehole type of structural systems, two types of instabilities can take place. The first is due to surface degradation growth and results into spalling of layers at the hole wall. The second is due to damage progression, and results into globally unstable response of the structure. The hole size has been found experimentally to be an important parameter in breakout instability initiation. Laboratory size holes may overestimate instability initiation properties by a large factor. At the same time, material properties such as peak stress depend largely on the size and shape of a specimen subjected to uniaxial or triaxial compression. This work attempts to incorporate size and scale effects into the instability initiation conditions. The important task of transferring information from laboratory experiments to actual large scale engineering problems is analyzed and discussed. The potential of the theory is demonstrated. The need for further experimental and theoretical work is identified. 
G. Frantziskonis and I. Vardoulakis, "On the Micro-Structure of Surface Effects and Related Instabilities,"
Europ. J. Mechs. A/Solids, 11, 21-34, 1992
Based on Mindlin's theory for material micro-structure interesting surface effects under conditions of equilibrium are studied in this paper. The governing field equations for uniaxial plane deformations are established; surface instability analysis shows non uniform deformations for a layer of specified distance from the surface. Experimental as well as fracture mechanics based considerations show that this surface-layer is extremely thin for metals while for brittle materials its magnitude is of the order of 1 cm. Material micro-structure introduces a singular perturbation to the original Hill and Hutchinson problem; here we introduce a single perturbation parameter and we obtain a "dispersion" law for the surface buckling load. It is found that surface degradation and skin effects can be attributed to localized surface buckling instabilities. Experimental information on skin effects can provide an estimation of the internal material length. 
I. Vardoulakis and G. Frantziskonis, "Micro-Structure in Kinematic-Ha-rdening Plasticity," Europ. J. Mechs.
A/Solids, 11, 467-486, 1992
A gradient regularization of the classical kinematic-hardening plasticity is presented. The underlying continuum model is formally related to Mindlin's elasticity theory with micro-structure. The evolution law for the back stress is identical to Mindlin's higher order equilibrium equation. For consistency reasons the flow rule of classical plasticity is modified by incorporating the Laplacian of the plastic multiplier. The variational formulation of the problem with appropriate boundary conditions is given and an expression for the dissipated energy is established. Shear-band analysis shows that the theory provides the band thickness, and regularizes the governing equations. Micro-structure introduces a singular perturbation to the classical surface instability analysis, and the internal length l is the perturbation parameter. In addition, micro-structure effects tend to reduce the wavelength at onset of surface instability. 
D. Daniewicz and G. Frantziskonis, "On Edge Delamination in Laminated Compos-ites," Composite Struct., 21,
141-153, 1992
A damage evolution theory for the effects of the rather complicated edge delamination phenomena on composite laminate response is developed. The theory is a mechanics-based formulation which quantifies the damage development of a laminate under general loading, and incorporates it directly into the laminate constitutive equations. The theoretical development attempts to describe the basic features of the edge delamination phenomena by introducing a single laminate-specific constant ö. The methodology of ö selection is presented for a [45\0\90\] s T300 graphite\epoxy laminate. Edge effects are demonstrated by comparing the state of the above laminate with and without edge delamination damage. The capability of the theory in predicting the volume scaling effect (the phenomenon whereby the onset of edge delamination is observed to be a function of ply thickness, when the stacking sequence is invariant) is shown through consideration of the failure strengths and strains of the [0\90 n] s laminate series. 
G. Frantziskonis, C.S. Desai, F.F. Tang, and D. Daniewicz, "Degradation Mechanisms in Brittle Materials
Investigated by Ultrasonic Scanning," Engr. Fracture Mechs., 42, 347-369, 1992
The paper reports on an experimental study of degradation mechanisms and patterning in brittle materials. Specimens of a cast-in brittle material were subjected to external mechanical load. At various pre-peak load levels, through transmission ultrasonic measurements were taken at several locations of the sample. The damage evolution and its patterning is studied through analysis of the attenuation of the transmitted wave. For monotonically increasing load significant attenuation was observed at low load levels, while for unloading the attenuation changes were relatively small. The test results indicate random and non-symmetric distribution of attenuation within the sample. However, in general, higher attenuation was observed close to the load-free surfaces than in the interior of the sample. This observation signifies the so-called skin effect. Occasionally, at a few locations of various samples the energy of the received wave increased with increasing external load. This may indicate unloading of partially formed cracks. Measured surface strains are compared to the overall ones and the degradation patterning within the samples. The ultrasonic measurements are examined with respect to estimating the internal material length. The relation between ultrasonic wave attenuation and mechanical dissipated energy is examined. 
H. Dai and G. Frantziskonis, "Heterogeneity, Spatial Correlations, Size Effects and Dissipated Energy in Brittle
Materials," Mechanics of Materials, 18, 103-118, 1994
The present study was inspired from recent experiments performed by the second author and co-workers where heterogeneity in a brittle material was investigated by ultrasonic scanning. The experiments showed an irregular pattern of material heterogeneity even before any external load was applied on the specimens. Under increasing uniaxial compressive load, the initial heterogeneity pattern evolved, and finally a macro crack network formed. In a previous study the finite element method was used in conjunction with a theory for distributed damage to study the effects of material heterogeneity numerically. Both experiments and finite element analysis showed that initially "strong" regions dissipated energy at a much higher rate than "weak" ones. However, the FEM is more suitable and efficient when material response can be homogenized and deformation gradients are not prominent. If this is not the case, or if one is interested in understanding and/or modeling the effects of heterogeneity and crack network formation, a lattice approach may be more suitable. In this study, the predictions of the lattice based numerical approach are compared to experimental data on crack formation. Recently, a branch of statistical physics has focused on statistical modeling of materials. Here, it is attempted to "connect" this approach to continuum solid mechanics theories. Important connections are believed to be spatial correlations and localization phenomena in materials, as discussed herein. Using random fields to represent initial heterogeneity, the effects of spatial correlations on size effects, dissipated energy and on crack formation are studied. Results from two random field generation algorithms are reported, and, surprisingly, dissipated energy and crack network characteristics were dependent on the algorithm. Both provide a good description of the well-known size effect observed in brittle materials. However, notable differences between the two algorithms with respect to localization of deformation were identified. 
G. Frantziskonis, "On Scaling Phenomena in Fracture of Heterogeneous Solids," Europ. J. Mechs. A/Solids, 13, 73-92, 1994
Several studies address the scaling geometrical properties of fracture surfaces/networks in solid materials. This paper examines the implications of the underlying heterogeneous fields before fracture on the properties of the fracture network. Three loading conditions are examined analytically as two-dimensional problems - uniaxial load, pure shear, and dilatation. The relevant kinematic quantities before (macro) fracture are described statistically as random fields or as scaling fields. It is shown that fracture networks with scaling properties can develop. Notably, some similarity is identified between the case of fast crack propagation using scaling fields, and the case of "slow" crack propagation. Under low rate force or displacement controlled external load application, the slow crack propagation case is the only one consistent with relevant physical reasoning. The present analytical solutions can be used for identification of material properties through post-mortem information, i.e. by studying the geometrical properties of fracture networks/surfaces. They also provide a better understanding of the mechanisms contributing to fracture, and show the important contribution of material micro-structure on fracture toughness. 
G. Frantziskonis, "On the Possibly Multifractal Properties of Dissipated Energy in Brittle Materials," Appl. Mech. Rev., 47, 1(p2), S132-S140, 1994
The paper reports an analytical study on the properties of fracture networks in brittle materials. Micro-deformation gradients are considered random fields and/or scaling fields. Under dynamic crack propagation conditions the possibly fractal properties of the (macro) crack pattern are governed by the interplay of fluctuations and spatial correlations. For "slow" crack propagation they are governed by the kinematic fields in the vicinity of crack or notch tips. The spatial distribution of dissipated energy, due to fracture, is evaluated. It is shown that there is a strong possibility that the dissipated energy is multifractal. Here, its properties are characterized in a fashion similar to the so-called p-model where p herein denotes normalized dissipated energy. For the three cases analyzed - uniaxial tension, pure shear, and dilatation - the dissipated energy under pure shear shows the strongest disorder, the one under dilatation the weakest, and the tension case is always between these two. 
G. Frantziskonis, P. Karpur, T. Matikas, S. Krishamurthy & P. Jero, "Fiber Matrix Interface - Information from Experiments via Simulation," Composite Struct. 29, 231-247, 1994
This study explores a novel procedure for obtaining quantitative information on the mechanical properties of the fiber-matrix interface in composite materials. The method, based on lattice discretization of a medium, simulates actual experiments in detail, including fiber breakage, matrix yield and/or cracking, and interface failure. The paper concentrates on two experiments performed commonly, the so-called fragmentation test for metal matrix, and the pushout/pullout test for metal as well as ceramic matrix composites. Based on the documented capability of the method to simulate actual experimental data, reliable values of (homogenized) interface properties can be obtained. In addition, the simulations provide further understanding of the mechanisms involved during the relevant testing. Although this study presents results from basic problems, the method is general enough to include effects of residual stress, of high temperature environment, of dynamic crack propagation, as well as three-dimensional details of the interface failure process. The potential exists for simulating nondestructive wave based techniques aimed at evaluating interface properties. 
G. Frantziskonis, "Heterogeneity and Implicated Surface Effects - Statistical, Fractal Formulation and Relevant
Analytical Solution," Acta Mechanica, 108, 157-178, 1995
The aim of this paper is to examine the implications of material heterogeneity on brittle material response and on relevant surface effects. Statistical and fractal concepts are used for this purpose. In the statistical formulation the displacement gradients of the micro-medium are considered to be random fields characterized by stationary exponential or Gaussian auto-covariance and by the relevant correlation length or scale of fluctuation. Through Taylor series expansion around the mean of the random field, an important analogy is found between the statistical formulation and the micro-structural theory, originally introduced by Mindlin, where higher order gradients of deformation appear in the constitutive equations. The analogy is valid only when fluctuations are small, so that some simplifications are allowed. It is found that the so-called internal length appearing in the micro-structural formulation is analogous to the correlation length in the statistical one. In the statistical approach there are no extra boundary conditions in the formulation, as is the case when higher order gradients are introduced. However, what is known as "conditioning" of the random field at the boundaries effects its behavior near/on them. The statistical approach can provide further information in the form of higher order moments not captured by the gradient theory. Material/structure response is strongly dependent on the aforementioned scale. Its effect is most pronounced near the boundaries of a structure where its role on surface related phenomena is paramount. In order to study heterogeneity at a hierarchy of scales, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant power decay laws are considered. The formulation introduces the fractal dimension of the heterogeneous displacement gradient of the micro-medium, a length describing the overall size of the structure, and the lower cutoff of the scaling law. The physical interpretation of the lower cutoff is the lower limit of applicability of the power scaling law. Mathematically it is important since in this case the fractal can be "followed" in the spatial domain. Similarly to the statistical case, an analogy between the fractal formulation and gradient theories is identified. No extra boundary conditions appear in the fractal formulation. However, there are still open questions with respect to the behavior of a fractal after conditioning, as is the case on boundaries. The analytical solution of a relevant surface instability problem for the gradient, statistical, and fractal formulation is presented. The solution was obtained through symbolic computations by computer because the analytical work is tedious and error prone. The analytical solution provides significant insight into the problem of heterogeneity and skin effects in brittle materials, internal length estimation, and the role of fractal scaling properties. Finally, the concepts introduced herein are discussed with respect to experimental information and numerical implementation.

G. Hong, A. Yalizis & G. Frantziskonis, "Hygrothermal Degradation in Glass-Epoxy - Evaluation through Stress Wave Factors," Composite Struct., 30, 407-418, 1995
The paper documents an experimental study on nondestructive evaluation of hygrothermally degraded glass fiber - epoxy matrix composite laminates. Specimens were subjected to accelerated Life Tests, i.e. high temperature and high pressure steam. The efficiency of the stress (ultrasonic) wave propagation through the specimens is evaluated quantitatively, at various stages of degradation, via the so-called Stress Wave Factors (SWFs). Scanning electron micrographs were taken at some stages of degradation. In addition, the specimens were tested in bending mode. All measurement are correlated with the SWFs. 
G. Frantziskonis & B. Loret, "Scale Dependent Constitutive Relations - Information from Wavelet Analysis and Application to Localization Problems," Eur. J. Mechs. A/Solids, 14, 873-892, 1995
This study explores wavelet analysis as a tool to include the influence of scale on material behavior. The wavelet representation of the failure state of a one-dimensional problem provides the relevant scale as a very important parameter in the material law. The conse-quences are paramount since the local scale involved, and its influence on material response, is decided from the global deformation field, rather than being defined as a fixed quantity independent of deformation. The scale dependent constitutive relations are examined in detail with respect to localization problems in the context of viscoplasticity. 
M.J. Meisner and G. Frantziskonis, Multifractal Fracture-toughness Properties of Brittle Materials, J. Physics A, 29, 2657-2670, 1996
The paper documents a study of novel fracture-toughness properties of brittle heterogeneous materials. Through simulations of the rupture process based on a lattice discretization of the material, the spatial variation of dissipated energy due to fracture is evaluated. Under certain conditions, its distribution is characterized by a multifractal spectrum f("). Importantly, f(") depends not only on the initial heterogeneity present in the material but also on the nature of the externally applied load. This provides a renewed load-path dependent definition of fracture toughness material properties. It avoids the difficulties associated with Atraditional@ continuum/fracture mechanics definitions where the macroscopic fracture mode must be known a priori. 
M. J. Meisner and G. Frantziskonis, Dissipated Energy as a Function of Material Microstructure: Towards Optimum Fracture-toughness, J. Mech. Beh. of Matls., 6, 285-300, 1996
The paper addresses material properties related to fracture-toughness, the focus being on brittle disordered materials. Through numerical lattice discretization based simulations of the rupture process the spatial variation of dissipated energy due to fracture is evaluated. Under certain conditions, its distribution is characterized by a multifractal spectrum f("). As it happens, the multifractal spectrum, f("), depends not only on the initial disorder present in the material, but also on the nature of the externally applied load. It is explicitly demonstrated that there is a direct connection between the multifractal spectrum of the dissipated energy and the total dissipated energy, due to micro-crack formation during the rupture process. In other words, when the disorder of the material increases, a relevant parameter decreases, and the relative total dissipated energy decreases. The converse situation is also true. Finally, it is stressed that this work is amenable to experimental and further numerical verifications -- some paths toward this goal are discussed. 
G. Frantziskonis, "Heterogeneity and Its Implications - Micromechanical, Statistical, Fractal Approach and their
Similarity," In Damage in Composite Materials, G. Voyiadgis Editor, Elsevier, New York, 1993
The implications of material heterogeneity on composite material response is examined. Micro-structure considerations yield a general theory where the so-called internal material length and higher order deformation gradients enter the formulation. The statistical approach to heterogeneity displays the importance of the internal material length, which is identified as the fluctuation scale of the relevant random field. Through series expansion it is shown that introducing higher order gradients of deformation in the constitutive equations is analogous to introducing the statistical properties of heterogeneity. Common between the two approaches is the internal length or scale of fluctuation. It is also shown that the statistical approach can provide further information, in the form of higher order statistical moments not captured by the gradient theory. The fluctuation scale has a strong influence on the damage evolution and failure mode of the structure (specimen) and rigorous methods for its estimation are suggested. In order to study the hierarchy of scales of heterogeneity, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant scaling laws are employed. This introduces the fractal dimension of the heterogeneous micro-deformation gradient, and the lower and upper cutoffs of the fractal scaling. An analogy between the fractal formulation and gradient theories is identified. The analytical solution of a relevant surface instability problem for the micro-mechanical, gradient, statistical, and fractal formulation is achieved. Further a phenomenological approach to the problem of edge delamination in laminated composites is presented. 
G. Frantziskonis, "Crack Pattern Related Universal Constants," In Probabilities and Materials, NATO-ASI
Series, D. Breysse Editor, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994
The implications of material heterogeneity on the crack pattern formed during progressive failure of materials are examined. The displacement gradients of the micro-medium are considered to be random stationary fields or fractal fields. Both formulations yield interesting and, surprisingly, analogous results. Two variables are found to effect the crack pattern that will form under loading. The first one is the fluctuations, i.e. coefficient of variation of the displacement gradient fields. The second one is correlations, i.e. correlation length of the random fields as compared to the size of the structure/specimen, or lower, upper cutoffs and fractal dimension of the fractal fields. The (macro) crack pattern that will develop is governed by the interplay of fluctuations and correlations, independent of the elastic constants (for deterministic Poisson ratio) and fairly independent of the crack formation criterion. Universal constants are found for the coefficients of variation. For values above the universal ones the material will develop a highly irregular crack pattern with multiple crack intersections and/or self-intersecting cracks. This is typically observed in testing of high strength concretes, high strength rocks, metals of ultra high hardness and certain ceramics, composites. Under dynamic load and/or under load control the highly irregular crack pattern is accompanied by an "explosive" burst of chunks of material and high energy release. For values below the universal ones, a single crack (may be accompanied by branches) will develop. This is typical of low strength concretes, soft rocks, low hardness metals, certain ceramics, composites. Energy release is relatively low - characteristic of low strength materials.