Marco Salviato & Michele Zappalorto
In this work, a unified solution approach is proposed for the analytical evaluation of the stress fields close
to notches under antiplane shear and torsion loadings, which allows a large variety of notch problems to
be tackled. The method is based on the complex potential approach for antiplane elasticity combined
with the use of proper conformal mappings. In particular, it is shown that a well defined analytical link
does exist between the complex potential to be used to determine stresses and the first derivative of the
conformal mapping used to mathematically describe the notch profile. This makes some methodologies
such as Schwarz-Christoffel transformation, which allows describing any polygonal domain automatically,
very attractive for the direct solution of notch problems. A bulk of solutions are provided to support this
finding, from cracks and pointed notches, to radiused notches. In addition, the accuracy of each proposed
solution is discussed in detail taking advantage of a bulk of results from FE analyses.
Marco Salviato, Viet T. Chau, Weixin Li, Zdeněk P. Bažant & Gianluca Cusatis
Static and dynamic analysis of the fracture tests of fiber composites in hydraulically servo-controlled testing machines currently in use shows that their grips are much too soft and light for observing the postpeak softening. Based on static analysis based on the second law of thermodynamics, confirmed by dynamic analysis of the test setup as an open system, far stiffer and heavier grips are proposed. Tests of compact-tension fracture specimens of woven carbon-epoxy laminates prove this theoretical conclusion. Sufficiently, stiff grips allow observation of a stable postpeak softening, even under load-point displacement control. Dynamic analysis of the test setup as a closed system with proportional-integrative-differential (PID)-controlled input further indicates that the controllability of postpeak softening under crack-mouth opening displacement (CMOD) control is improved not only by increasing the grip stiffness but also by increasing the grip mass. The fracture energy deduced from the area under the measured complete load-deflection curve with stable postpeak is shown to agree with the fracture energy deduced from the size effect tests of the same composite, but the size effect tests also provide the material characteristic length of quasibrittle (or cohesive) fracture mechanics. Previous suspicions of dynamic snapback in the testing of stiff specimens of composites are dispelled. Finally, the results show the stress- or strain-based failure criteria for fiber composites to be incorrect, and fracture mechanics, of the quasibrittle type, to be perfectly applicable.
Marco Salviato, Kedar Kirane, Shiva Esna Ashari, Zdenek P. Bazant, Gianluca Cusatis
Design of large composite structures requires understanding the scaling of their mechanical properties,
an aspect often overlooked in the literature on composites.
This contribution analyzes, experimentally and numerically, the intra-laminar size effect of textile
composite structures. Test results of geometrically similar Single Edge Notched specimens made of [0+]8
epoxy/carbon twill 2 2 laminates are reported. Results show that the nominal strength decreases with
increasing specimen size and that the experimental data can be fitted well by Bazant's size effect law,
allowing an accurate identification of the intra-laminar fracture energy of the material, Gf.
The importance of an accurate estimation of Gf in situations where intra-laminar fracturing is the main
energy dissipation mechanism is clarified by studying numerically its effect on crashworthiness of
Simulations demonstrate that, for the analyzed geometry, a decrease of the fracture energy to 50% of
the measured value corresponds to an almost 42% decrease in plateau crushing load. Further, assuming a
vertical stress drop after the peak, a typical assumption of strength-based constitutive laws implemented
in most commercial Finite Element codes, results in an strength underestimation of the order of 70%.
The main conclusion of this study is that measuring accurately fracture energy and modeling correctly
the fracturing behavior of textile composites, including their quasi-brittleness, is key. This can be
accomplished neither by strength- or strain-based approaches, which neglect size effect, nor by LEFM
which does not account for the finiteness of the Fracture Process Zone.
Marco Salviato, Shiva Esna Ashari, Gianluca Cusatis
This contribution proposes a general constitutive model to simulate the orthotropic stiffness, pre-peak
nonlinearity, failure envelopes, and the post-peak softening and fracture of textile composites.
Following the microplane model framework, the constitutive laws are formulated in terms of stress and
strain vectors acting on planes of several orientations within the material meso-structure. The model
exploits the spectral decomposition of the orthotropic stiffness tensor to define orthogonal strain modes
at the microplane level. These are associated to the various constituents at the mesoscale and to the
material response to different types of deformation. Strain-dependent constitutive equations are used
to relate the microplane eigenstresses and eigenstrains while a variational principle is applied to relate
the microplane stresses at the mesoscale to the continuum tensor at the macroscale.
The application of the model to a twill 2 2 shows that it can realistically predict its uniaxial as well as
multi-axial behavior. Furthermore, the model shows excellent agreement with experiments on the axial
crushing of composite tubes, this capability making it a valuable design tool for crashworthiness applications.
The formulation is computationally efficient, easy to calibrate and adaptable to other kinds of composite
architectures such as 2D and 3D braids or 3D woven textiles.
Kedar Kirane, Marco Salviato, Zdenek P. Bazant
A multiscale model based on the framework of microplane theory is developed to predict the elastic and fracturing
behavior of woven composites from the mesoscale properties of the constituents and the weave architecture. The
effective yarn properties are obtained by means of a simplified mesomechanical model of the yarn, based on a mixed
series and parallel coupling of the fibers and of the polymer within the yarns. As a novel concept, each of the several
inclined or aligned segments of an undulating fill and warp yarn is represented by a triad of orthogonal microplanes,
one of which is normal to the yarn segment while another is normal to the plane of the laminate. The constitutive law
is defined in terms of the microplane stress and strain vectors. The elastic and inelastic constitutive behavior is
defined using the microplane strain vectors which are the projections of the continuum strain tensor. Analogous to the
principle of virtual work used in previous microplane models, a strain energy density equivalence principle is employed
here to obtain the continuum level elastic and inelastic stiffness tensors, which in turn yield the continuum level stress
tensor. The use of strain vectors rather than tensors makes the modeling conceptually clearer as it allows capturing the
orientation of fiber failures, yarn cracking, matrix microcracking, and interface slip. Application of the new
microplane-triad model for a twill woven composite shows that it can realistically predict all the orthotropic elastic
constants and the strength limits for various layups. In contrast with the previous (nonmicroplane) models, the formulation
can capture the size effect of quasi-brittle fracture with a finite fracture process zone (FPZ). Explicit finite-element
analysis gives a realistic picture of progressive axial crushing of a composite tubular crush can initiated by a divergent
plug. The formulation is applicable to widely different weaves, including plain, twill, and satin weaves, and is easily
extensible to more complex architectures such as hybrid weaves as well as two-and three-dimensional braids.
Kedar Kirane, Marco Salviato, Zdenek P. Bazant
An accurate prediction of the orthotropic elastic constants of woven composites from the
constituent properties can be achieved if the representative unit cell is subdivided into
a large number of finite elements. But this would be prohibitive for microplane analysis of
structures consisting of many representative unit cells when material damage alters the elastic
constants in each time step in every element. This study shows that predictions almost as accurate
and sufficient for practical purposes can be achieved in a much simpler and more efficient manner
by adapting to woven composites the well-established microplane model, in a partly similar way as
recently shown for braided composites. The undulating fill and warp yarns are subdivided into
segments of different inclinations and, in the center of each segment, one microplane is placed
normal to the yarn. As a new idea, a microplane triad is formed by adding two orthogonal microplanes
parallel to the yarn, one of which is normal to the plane of the laminate. The benefit of the
microplane approach is that it is easily extendable to damage and fracture. The model is shown to
give realistic predictions of the full range of the orthotropic elastic constants for plain, twill,
and satin weaves and is extendable to hybrid weaves and braids.