Publications

Yao Qiao, Marco Salviato


This work proposes an investigation of the fracturing behavior of polymer nanocomposites. Towards this end, the study leverages the analysis of a large bulk of fracture tests from the literature with the goal of critically investigating the efects of the nonlinear Fracture Process Zone (FPZ). It is shown that for most of the fracture tests the effects of the nonlinear FPZ are not negligible, leading to signifcant deviations from Linear Elastic Fracture Mechanics (LEFM) sometimes exceeding 150% depending on the specimen size and nanofiller content. To get a deeper understanding of the characteristics of the FPZ, fracture tests on geometrically-scaled Single Edge Notch Bending (SENB) specimens are analyzed leveraging a cohesive zone model. It is found that the FPZ cannot be neglected and a bi-linear cohesive crack law generally provides the best match of experimental data.


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Marco Salviato, Michele Zappalorto, Lucio Maragoni


In this work, the exact solution for the stress fields ahead of cracks initiaed at sharp notch tips under antiplane shear and torsion loadings is derived in close form, leveraging conformal mapping and the complex potential method for antiplane elasticity. Based on the stress field distributions, relevant expressions for the mode III crack stress intensity factors are derived and their accuracy is discussed in detail taking advantage of a bulk of results from FE analyses.


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Seunghyun Ko, Kenrick Chan, Reed Hawkins, Rohith Jayaram, Christopher Lynch, Reda El Mamoune, Minh Nguyen Nicolay Pekhotin, Natania Stokes, Daniel N. Wu, Mark E. Tuttle, Jinkyu Yang, Marco Salviato


In this paper, we investigate the intra-laminar size effect of discontinuous fiber composites (DFCs) with three different unidirectional prepreg platelet sizes (75×12, 50×8, and 25×4 mm). Experimentally, we test five different sizes of single edge notched specimens, geometrically scaled (1:2/3:1/3:1/6:1/20), with the constant thickness. We observe notch insensitivity meaning that the crack initiate away from the notch, when the structure sizes are small (from the ratio 1/20 to 1/6). However, the crack always initiate for the ratio of 2/3 and 1. Bazants size effect law is used to analyze such unconventional fracturing behaviors. The experimental results are fitted using the linear regression analysis follow by the size effect law. The transition behavior of the DFCs from the strength based criteria to the energy based criteria is clearly observed. Also, as the platelet size increases, the fracture behaviors shift away from the energy based criteria, which implies a decrease in brittleness. To obtain the intra-laminar fracture energy, Gf , we have developed a finite element model based on the stochastic laminate analogy. The platelet size of 75×12 mm shows 96.8% increase in the fracture energy compared to the platelet size of 25×4 mm while behaves less brittle way. In conclusion, this study examines the effect of the platelet sizes of the DFCs in the presence of the notch. In this process, capturing the quasi-brittleness of the material using the nonlinear fracture mechanics is essential and we accomplish this using the simple size effect law. This work expands on an earlier SAMPE conference proceeding [1], and thus, there is a significant overlap in texts and figures between this and the SAMPE conference proceedings.



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Seunghyun Ko, Kenrick Chan, Reed Hawkins, Rohith Jayaram, Christopher Lynch, Reda El Mamoune, Minh Nguyen Nicolay Pekhotin, Natania Stokes, Daniel N. Wu, Mark E. Tuttle, Jinkyu Yang, Marco Salviato


We investigate experimentally and numerically the size effect of discontinuous fiber composites (DFC) for two different unidirectional prepreg chip sizes (50×8 and 75×12 mm). We tested geometrically similar five different sizes of single-edge notched specimens with a constant thickness. We observed that fracture may happen away from the notch depending on the specimen sizes and chip sizes. To analyze such transitional notch sensitivity in the DFC, we use Bazant’s size effect law. Experimental results are well fitted with the law, clearly showing the transition behavior of the DFC from being notch insensitive to sensitive. The transition of fracturing behavior implies that the design criteria of the DFC is shifting from strength-based to energy-based approaches. To obtain the fracture energy, Gf, we developed a finite element model based on the stochastic laminate analogy. The resulting Gf for the DFC with the 75×12 mm chips is 22.2% higher than 50×8 mm chips. In conclusion, we show both the chip sizes and the structure size play an important role in fracturing behavior of the DFC.



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Antonio Alessandro Deleo, Marco Salviato


An extensive parametric study sweeping both material and geometrical parameters is performed using size effect law (SEL) and Cohesive Zone Modeling (CZM). These simulations and recent experimental results suggest that the cohesive law of composites and nanocomposites is bi-linear and that the Fracture Process Zone (FPZ) may be almost fully developed already for laboratory-sized specimens. As a consequence, the scaling of the fracturing behavior may be influenced significantly by not only the total fracture energy but also the parameters describing the shape of the bi-linear cohesive law. The goal of this work is to formulate a master size effect curve that can be used for quick, yet accurate, estimation of energy and cohesive law parameters that can be used for the design of composite aircraft structures.


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Sean E. Phenisee, Sung Lin Tien, Marco Salviato


The use of unidirectional (UD) composites for primary and secondary structures is becoming broader and broader with applications including aeronautics and astronautics, automotive and bioengineering. However, the efficient use of these materials requires the development of proper computational tools for design. Leveraging the spectral stiffness decompostition of the stiffness tensor and a microplane formulation, the present contribution proposes a computational model to capture the main damage and fracturing mechanisms in UD composites. Preliminary simulations of uniaxial tests are compared to experimental data taken from the “World Wide Failure Exercise” to investigate the predictive capability of the model.


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Kevin Guo, Yao Qiao, Marco Salviato


This work proposes an investigation on the scaling of fatigue crack growth in pristine epoxy. Towards this end, fatigue fracture tests on geometrically scaled Single Edge Notch Bending (SENB) specimens were conducted. It is shown that Paris-Erdogan law exhibits strong size effect on both slope and threshold for pure epoxy. This indicates that the fatigue Fracture Process Zone (FPZ) size is significantly larger compared to the quasi-static one. In fact, fracture tests conducted by Mefford et al. [1] and a comprehensive literature study [2] showed that the FPZ has a negligible effect on the scaling of the fracturing behavior of pure epoxy under quasi-static loading condition. By introducing a fatigue size effect model based on the energetic-equivalence framework, the thresholds in the Paris-Erdogan curves representing geomaticallyscaled specimens can be successfully adjusted whereas the slopes still exhibit size effect. This latter aspect needs to be studied further to enable the application of the Paris-Erdogan law to quasi-brittle structures of different sizes and geometries.


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Yao Qiao, Marco Salviato


This work proposes an investigation on the fracturing behavior of polymer nanocomposites. Towards this end, the study leverages the analysis of a large bulk of fracture tests from the literature with the goal of critically investigating the effects of the nonlinear Fracture Process Zone (FPZ). It is shown that for most of the fracture tests, the effects of the nonlinear FPZ are not negligible, leading to significant deviations from Linear Elastic Fracture Mechanics (LEFM). By means of Size Effect Law (SEL) on the assumption of a linear cohesive crack law, the fracture tests in the literature were re-analyzed. As the data indicate, this aspect needs to be taken into serious consideration since the use of LEFM to estimate mode I fracture energy, which is common practice in the literature, can lead to an error as high as 157% depending on the specimen size and nanofiller content. This was further confirmed by matching the data by means of a cohesive zone modeling featuring a linear cohesive law. Taking advantage of size effect tests on thermoset-based graphene nanocomposites, it was also found that these materials are better described by a bilinear cohesive law. It was shown that, while the use of a linear cohesive law provides a good approximation, a bilinear cohesive law provides a superior description of the fracturing behavior for different sizes.


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Cory Hage Mefforda, Yao Qiaoa, Marco Salviato


This work proposes an investigation on the scaling of the structural strength of polymer/graphene nanocomposites. To this end, fracture tests on geometrically scaled Single Edge Notch Bending (SENB) specimens with varying contents of graphene were conducted to study the effects of nanomodification on the scaling. It is shown that, while the strength of the pristine polymer scales according to Linear Elastic Fracture Mechanics (LEFM), this is not the case for nanocomposites, even for very low graphene contents. In fact, small specimens exhibited a more pronounced ductility with limited scaling and a significant deviation from LEFM whereas larger specimens behaved in a more brittle way, with scaling of nominal strength closer to the one predicted by LEFM. This behavior, due to the significant size of the Fracture Process Zone (FPZ) compared to the specimen size, needs to be taken into serious consideration. In facts, it is shown that, for the specimen sizes investigated in this work, neglecting the non-linear effects of the FPZ can lead to an underestimation of the fracture energy as high as 113%, this error decreasing for increasing specimen sizes.


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Chiara Ceccatoa, Marco Salviato, Carlo Pellegrinoa, Gianluca Cusatis


Fiber Reinforced Polymers (FRP) have been widely used in different civil engineering applications to enhance the performance of concrete structures through flexural, shear or compression strengthening. One of the most common and successful use of FRP sheets can be found in the confinement of existing concrete vertical elements which need rehabilitation or increased capacity in terms of strength and ductility. However, efficient design of FRP retrofitting urges the development of computational models capable of accurately capturing (a) the interaction between the axial strains and lateral expansion of concrete with the corresponding stress increase in the external jacket; and (b) the fracturing behavior of the FRP jacket. In this study, experimental data gathered from the literature and relevant to FRP-confined columns are simulated by adopting the Lattice Discrete Particle Model (LDPM) and the Spectral Microplane Model (SMPM), recently developed to simulate concrete failure and fracture of anisotropic materials, respectively. LDPM models the meso-scale interaction of coarse aggregate particles and it has been extensively calibrated and validated with comparison to a large variety to experimental data under both quasi-static and dynamic loading conditions but it has not been fully validated with reference to low confinement compressive stress states, relevant to the targeted application. This task, along with the calibration of SMPM for FRP, is pursued in the present research. The results show that, with the improvement of the existing LDPM constitutive equations to account for low confinement effects, LDPM and SMPM are able to predict the concrete material response governed by the nonlinear interaction of confined vertical members strengthened by means of externally bonded FRP composites.


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Congrui Jin, Marco Salviato, Weixin Li, Gianluca Cusatis


This contribution investigates the extension of the microplane formulation to the description of transversely isotropic materials such as shale rock, foams, unidirectional composites, and ceramics. Two possible approaches are considered: (1) the spectral decomposition of the stiffness tensor to define the microplane constitutive laws in terms of energetically orthogonal eigenstrains and eigenstresses and (2) the definition of orientation-dependent microplane elastic moduli. The first approach, as demonstrated previously, provides a rigorous way to tackle anisotropy within the microplane framework, which is reviewed and presented herein in a clearer manner; whereas the second approach represents an approximation which, however, makes the formulation of nonlinear constitutive equations much simpler. The efficacy of the second approach in modeling the macroscopic elastic behavior is compared to the thermodynamic restrictions of the anisotropic parameters showing that a significant range of elastic properties can be modeled with excellent accuracy. Further, it is shown that it provides a very good approximation of the microplane stresses provided by the first approach, with the advantage of a simpler formulation. It is concluded that the spectral stiffness decomposition represents the best approach in such cases as for modeling composites, in which accurately capturing the elastic behavior is important. The introduction of orientation-dependent microplane elastic moduli provides a simpler framework for the modeling of transversely isotropic materials with remarked inelastic behavior, as in the case, for example, of shale rock.


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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.


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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.


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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 composite tubes.
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.


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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.


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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.


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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.


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Yewang Su, Zdenˇek P. Bažant, Youxuan Zhao, Marco Salviato, Kedar Kirane


While dynamic comminution is of interest to many processes and situations, this work is focused on the projectile impact onto concrete walls, in which predictions have been hampered by the problem of the so-called ‘dynamic overstress’. Recently, in analogy with turbulence, Bažant and Caner modeled the overstress as an additional viscous stress generated by apparent viscosity that accounts for the energy dissipation due to kinetic comminution of concrete into small particles at very high shear strain rate. Their viscosity estimation, however, was approximate since it did not satisfy the energy balance exactly. Here their model is extended and refined by ensuring that the drop of local kinetic energy of high shear strain rate of forming particles must be exactly equal to the energy dissipated by interface fracture of these particles. The basic hypothesis is that the interface fracture occurs instantly, as soon as the energy balance is satisfied. Like in the preceding work, this additional apparent viscosity is a power function of the rate of the deviatoric strain invariant. But here the power exponent is different, equal to −7/3, and the apparent viscosity is found to be proportional also to the time derivative of the rate of that invariant, i.e., to the second derivative of the shear strain. It is assumed that the interface fracture that comminutes the material into small particles occurs instantly, as soon as the local kinetic energy of shear strain rate in the forming particles becomes equal to the energy required to form interface fractures. The post-comminution behavior, including subsequent further comminution and clustering into bigger particle groups to release the kinetic energy that is being dissipated by inter-group friction, is discussed and modeled. The present formulation makes it possible to eliminate the artificial damping of all types, which is normally embedded in commercial finite element codes but is not predictive since it is not justified physically. With the aforementioned improvements, and after implementation into the new microplane model M7 for fracturing damage in concrete (which includes the quasi-static rate effects), the finite element predictions give superior agreement with the measured exit velocities of steel projectiles penetrating concrete walls of different thicknesses and with the measured depths of penetration into concrete blocks by projectiles of different velocities. Finally it is pointed out that the theory presented may also be used to predict proximate fragmentation and permeability enhancement of gas shale by powerful electric pulsed-arc explosions in the borehole


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Zdenek P. Bazant, Marco Salviato, Viet T. Chau, Hari Visnawathan, Aleksander Zubelewicz


Although spectacular advances in hydraulic fracturing, also known as fracking, have taken place and many aspects are well understood by now, the topology, geometry, and evolution of the crack system remain an enigma and mechanicians wonder: Why fracking works? Fracture mechanics of individual fluid-pressurized cracks has been clarified but the vital problem of stability of interacting hydraulic cracks escaped attention. First, based on the known shale permeability, on the known percentage of gas extraction from shale stratum, and on two key features of the measured gas outflow which are (1) the time to peak flux and (2) the halftime of flux decay, it is shown that the crack spacing must be only about 0.1 m. Attainment of such a small crack spacing requires preventing localization in parallel crack systems. Therefore, attention is subsequently focused on the classical solutions of the critical states of localization instability in a system of cooling or shrinkage cracks. Formulated is a hydrothermal analogy which makes it possible to transfer these solutions to a system of hydraulic cracks. It is concluded that if the hydraulic pressure profile along the cracks can be made almost uniform, with a steep enough pressure drop at the front, the localization instability can be avoided. To achieve this kind of profile, which is essential for obtaining crack systems dense enough to allow gas escape from a significant portion of kerogen-filled nanopores, the pumping rate (corrected for the leak rate) must not be too high and must not be increased too fast. Furthermore, numerical solutions are presented to show that an idealized system of circular equidistant vertical cracks propagating from a horizontal borehole behaves similarly. It is pointed out that one useful role of the proppants, as well as the acids that promote creation of debris in the new cracks, is to partially help to limit crack closings and thus localization. To attain the crack spacing of only 0.1 m, one must imagine formation of hierarchical progressively refined crack systems. Compared to new cracks, the system of pre-existing uncemented natural cracks or joints is shown to be slightly more prone to localization and thus of little help in producing the fine crack spacing required. So, from fracture mechanics viewpoint, what makes fracking work?–the mitigation of fracture localization instabilities. This can also improve efficiency by fracturing more shale. Besides, it is environmentally beneficial, by reducing flowback per m3 of gas. So is the reduction of seismicity caused by dynamic fracture instabilities (which are more severe in underground CO2 sequestration).


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Marco Salviato, Kedar Kirane, Zdeněk P. Bažant


In preceding studies, the type of cumulative probability distribution functions (cdf) of strength and of static lifetime of quasibrittle structures, including their tails, was mathematically derived from atomistic scale arguments based on nano-scale cracks propagating by many small, activation energy-controlled, random breaks of atomic bonds in the nanostructure. It was shown that a quasibrittle structure (of positive geometry) must be modeled by a finite (rather than infinite) weakest-link model, and that the cdf of structural strength as well as lifetime varies from nearly Gaussian to Weibullian as a function of structure size and shape. Excellent agreement with the observed distributions of structural strength and static lifetime was demonstrated. Based on the same theoretical framework, the present paper formulates the statistics of the residual structural strength, which is the strength after the structure has been subjected to sustained loading. A strength degradation equation is derived based on Evans' law for static crack growth during sustained loading. It is shown that the rate of strength degradation is not constant but continuously increasing. The cdf of residual strength of one RVE is shown to be closely approximated by a graft of Weibull and Gaussian (normal) distributions. In the left tail, the cdf is a three-parameter Weibull distribution consisting of the (nþ1)th power of the residual strength, where n is the exponent of the Evans law and the threshold is a function of the applied load and load duration. The finiteness of the threshold, which is typically very small, is a new feature of quasibrittle residual strength statistics, contrasting with the previously established absence of a threshold for strength and lifetime. Its cause is that there is a non-zero probability that some specimens fail during the static preloading, and thus are excluded from the statistics of the overload. The predictions of the theory are validated by available test data on glass–epoxy composites and on borosilicate and sodalime silicate glasses. The size effect on the cdf of residual strength is also determined. The size effect on the mean residual strength is found to be as strong as the size effect on the mean initial strength.


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Marco Salviato, Zdeněk P. Bažant


The fiber bundle model is widely used in probabilistic modeling of various phenomena across different engineering fields, from network analysis to earthquake statistics. In structural strength analysis, this model is an essential part of extreme value statistics that governs the left tail of the cumulative probability density function of strength. Based on previous nano-mechanical arguments, the cumulative probability distribution function of strength of each fiber constituting the bundle is assumed to exhibit a power-law left tail. Each fiber (or element) of the bundle is supposed to be subjected to the same relative displacement (parallel coupling). The constitutive equations describing various fibers are assumed to be related by a radial affinity while no restrictions are placed on their particular form. It is demonstrated that, even under these most general assumptions, the power-law left tail is preserved in the bundle and the tail exponent of the bundle is the sum of the exponents of the power-law tails of all the fibers. The results have significant implications for the statistical modeling of strength of quasibrittle structures.


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M. Quaresimin, M. Salviato, M. Zappalorto


In the present work, a multi-scale modelling strategy to assess the fracture toughness of nanoparticle filled thermosetting polymers is presented. The model accounts for the main damaging mechanisms arising in this kind of materials, i.e. nanoparticle debonding, plastic yielding of nanovoids and plastic shear banding of the polymer. Further, the proposed analytical framework considers the influence of an interphase around nanoparticles, a particular feature of nanocomposites.
Comparison of the theory to a bulk of experimental data from the literature shows a very good agreement.


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Michele Zappalorto, Marco Salviato, Marino Quaresimin


In this work the mixed mode fracture behaviour (I + II) of an epoxy/nanoclay nanocomposite system is analysed, discussing the results from Single Edge Notch Bending tests. It is found that nanomodification significantly enhances the fracture toughness of the epoxy resin on the entire range of mixed mode loadings (from pure mode I to pure mode II), improvements depend however on the mode mixity.
Experimental results are compared to theoretical predictions based on different criteria for mixed mode fracture in brittle homogeneous materials. As expected, it is found that, while the data from pure epoxy are satisfactorily predicted almost independently of the adopted approach, the agreement is much worse in the case of nanomodified materials. Explanations of this behaviour can be found in the emergence of additional microscale and nanoscale toughening mechanisms due to nanomodification not properly described by conventional models.


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Marco Salviato, Michele Zappalorto, Marino Quaresimin


The most appealing feature of nanofilled polymers is the perspective of obtaining surprisingly high mechanical properties at low nanofiller volume fractions. The knowledge of nanostructure–property relationships is however essential for the design of these materials.
In the present work, a model for the critical hydrostatic tension related to nanoparticle debonding is presented. The model accounts for some important issues inherently related to the nanoscale with particular reference to surface elastic stresses on the nanoparticle periphery and the emergence of a zone of altered chemistry surrounding the nanoparticle. The analytical solution suggests that the range of nanoparticle radii where interfacial effects do affect the solution is limited to the nanometer scale. In more details, considering the interphase and surface elastic properties used in the analysis, it has been found that for stiff particles with radius between 10 nm and 100 nm (silica, alumina and other metal oxide nanoparticles) the prominent role is played by the interphase elastic properties. Surface elastic constants were found to have, instead, only a negligible effect.


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Zappalorto M., Salviato M., Pontefisso A. Quaresimin M


In this work, an experimental investigation of the notch effect on clay-modified epoxy resins is carried out, iscussing the results from Single Edge Notch Bending tests and Double Edge Notch Tension tests on notched components. It is found that when the notch root radius is greater than a limit value, which depends on the clay content, the brittle failure of notched nanomodified specimens is controlled by the material strength. Under this circumstance nanomodification, while enhancing the polymer fracture toughness, might have a detrimental effect on the strength of notched components. This study brings to light a new feature of nanomodification according to which particular care should be used when using nanomodified resins for structural applications in the presence of notches or holes.


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Marco Salviato, Michele Zappalorto, Marino Quaresimin


In this paper a multiscale model is provided to assess the toughening improvements in nanoparticle filled polymers caused by the formation of localised plastic shear bands, initiated by the stress concentrations around nanoparticles. The model quantifies the energy absorbed at the nanoscale and accounts for the emergence of an interphase zone around the nanoparticles. It is proved that the elastic properties of the interphase, which are different from those of the matrix, due to chemical interactions, highly affect the stress field rising around particles and the energy dissipation at the nanoscale.


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Marco Salviato, Kedar Kirane, Zdeněk P. Bažant


In preceding studies, the type of cumulative probability distribution functions (cdf) of strength and of static lifetime of quasibrittle structures, including their tails, was mathematically derived from atomistic scale arguments based on nano-scale cracks propagating by many small, activation energy-controlled, random breaks of atomic bonds in the nanostructure. It was shown that a quasibrittle structure (of positive geometry) must be modeled by a finite (rather than infinite) weakest-link model, and that the cdf of structural strength as well as lifetime varies from nearly Gaussian to Weibullian as a function of structure size and shape. Excellent agreement with the observed distributions of structural strength and static lifetime was demonstrated. Based on the same theoretical framework, the present paper formulates the statistics of the residual structural strength, which is the strength after the structure has been subjected to sustained loading. A strength degradation equation is derived based on Evans law for static crack growth during sustained loading. It is shown that the rate of strength degradation is not constant but continuously increasing. The cdf of residual strength of one RVE is shown to be closely approximated by a graft of Weibull and Gaussian (normal) distributions. In the left tail, the cdf is a a three-parameter Weibull distribution consisting of the (n + 1)th power of the residual strength, where n is the exponent of the Evans law and the threshold is a function of the applied load and load duration. The finiteness of the threshold, which is typically very small, is a new feature of quasibrittle residual strength statistics, contrasting with the previously established absence of a threshold for strength and lifetime. Its cause is that there is a non-zero probability that some specimens fail during the static preloading, and thus are excluded from the statistics of the overload. The predictions of the theory are validated by available test data on glass-epoxy composites and on borosilicate and soda- lime silicate glasses. The size effect on the cdf of residual strength is also determined. The size effect on the mean residual strength is found to be as strong as the size effect on the mean initial strength.


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Michele Zappalorto, Marco Salviato, Marino Quaresimin


The high fracture toughness improvements exhibited by nanofilled polymers is commonly thought of as due to the large amount of energy dissipated at the nanoscale.
In the present work, a multiscale modelling strategy to assess the nanocomposite toughening due to plastic yielding of nanovoids is presented. The model accounts for the emergence of an interphase with mechanical properties different from those of the matrix


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Michele Zappalorto, Marco Salviato, Marino Quaresimin


A closed form solution for the stress fields around a rigid nanoparticle under uniaxial tensile load is provided. The work explicitly accounts for the presence, around the nanoparticle, of an interphase of thickness comparable to the particle size and different elastic properties from those of the matrix. The solution allows one to determine, in closed form, the stress concentration around nanoparticles relevant for fracture and strength assessments of polymer nanocomposites.


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Marino Quaresimin, Marco Salviato, Michele Zappalorto


The assessment of nanocomposite mechanical properties is a challenging task. Due to their hierarchical structure, which spans from nano to macro length-scales, a different way of thinking from traditional approaches is needed to account for the characteristic phenomena of each length-scale and bridge their effects from the smaller scale to the macroscale.
In the present work, some important issues of nanocomposite modelling are discussed. Then, a classification of the available modelling strategies is proposed, according to the scale from which the problem is addressed. This comprehensive analysis is thought as a necessary tool for the development of new effective approaches.


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Marino Quaresimin, Marco Salviato, Michele Zappalorto


The present work illustrates the experimental results of a project aiming to assess the benefits deriving from the matrix nanomodification of composite laminates made by vacuum infusion of woven glass fabrics. The following properties have been investigated: mode I fracture toughness and crack propagation resistance for neat and clay-modified epoxy, interlaminar shear strength, mode I delamination resistance for base and clay-modified epoxy laminates.
Available results indicate a significant improvement in the fracture toughness and crack propagation threshold of clay-modified epoxy. However, due to the nanofiller morphology, the behaviour of clay-modified laminates is still almost comparable to that of the base laminates.


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Marco Salviato, Michele Zappalorto, Marino Quaresimin


With the aid of an energy analysis and the surface elasticity theory, this work provides a closed form solution for the critical debonding stress of a rigid nanoparticle embedded in an elastic matrix subjected to a remote hydrostatic stress. It is proved that the debonding stress depends on the particle radius, the matrix elastic properties and the fracture energy per unit surface. The solution allows quantifying the effects of surface elastic constants, also showing that the smaller the particle size the more significant those effects are.



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Michele Zappalorto, Marco Salviato, Marino Quaresimin


One of the most appealing features concerned with nanomodification of polymeric resins for structural applications is the perspective of obtaining high toughness even at low nanofiller volume fractions. Such performances are related to the energy dissipated through the damage mechanisms taking place at the nanoscale. Among these, nanoparticle debonding could take an important role either as a mechanism itself or as a trigger for phenomena like plastic void growth or matrix shear yielding. In the present work, a model for the hydrostatic tension related to debonding is presented. The model accounts for some important issues inherently related to the nanoscale with particular reference to the emergence of an interphase surrounding the nanoparticle. Results can be useful in view of a multi-scale modelling of the problem.



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Marco Salviato, Michele Zappalorto, Marino Quaresimin


Nanocomposites hierarchical structure, ranging from nano to macro length-scales, urges to account for the characteristic phenomena of the different involved length-scales and to bridge their effects up to the macroscale. This makes the assessment of nanocomposite mechanical properties a challenging task.
In the present work, a model to assess the nanocomposite toughening due to plastic yielding of nanovoids is presented. The model accounts for the emergence of an interphase, created by the inter- and supra-molecular interactions arising at the nanoscale, with mechanical properties different from those of the matrix.



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Michele Zappalorto, Marco Salviato, Marino Quaresimin


The high mechanical performances achievable by nanomodification of polymeric resins at low nanofiller volume fractions are related to the energy dissipated through the damage mechanisms taking place at the nanoscale. Among the others, nanoparticle debonding could take an important role either as a mechanism itself or as a trigger for mechanisms like plastic void growth or matrix shear yielding. In the present work, a model for the assessment of the extension and the shape of the debonding region (DBR) and the toughness improvement due to nanoparticle debonding is presented. The model takes into account some important issues inherently related to the nanoscale with particular reference to the emergence of an interphase surrounding the nanoparticle. Results can be useful in the scope of a multi-scale modelling strategy to the problem.



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