G.
Frantziskonis, F.F. Tang and C.S. Desai, "Borehole Scale Effects and Related
Instabilities," *Engr. Fracture*
*Mechs*., **39**, 377-389, 1991
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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
__Summary__
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
__Abstract__
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
__Abstract__
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
__Abstract.__
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
__Abstract__
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
__Abstract__
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
__Abstract__
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.