It is well known that the biological composites have superior mechanical properties due to their exquisite multilevel structural hierarchy. However, the underlying mechanisms of the roles of this hierarchical design in the toughness of the biocomposites remain elusive. In this paper, the deformation and fracture mechanism of multilevel hierarchical structures are explored by molecular dynamics simulations. The effects of the multilevel design on fracture toughness, nonlinear deformation of soft matrix, and the crack path pattern were quantitatively analyzed. We showed that the toughness of composites is closely associated with the pattern of the crack path and the nonlinear deformation of the matrix. Additionally, the structure with a higher level of hierarchy exhibit higher toughness, which is less sensitive to the geometrical change of inclusions, such as the aspect ratio and the staggered ratio. This work provides more theoretical evidence of the toughening mechanism of the multilevel hierarchy in fracture toughness of biological materials via new methods of analyzing fracture of multilevel structures and provides guidelines for the design of high-performance engineering materials.

The problem of toughening heterogeneous materials with a stiff matrix and compliant inclusions is investigated through numerical simulations and experiments. Specifically, the problem of optimizing a combination of effective toughness and effective elastic modulus in the context of a square array of compliant inclusions in a stiff matrix is explored. Crack propagation in the heterogeneous material is simulated using a variational phase-field approach. It is found that the crack can meander between or get attracted to and trapped in the inclusions. Composite specimens with a stiff matrix and compliant circular inclusions were 3D printed, and their fracture toughness was measured using a specially designed loading fixture. The experimental results show agreement with the numerical predictions by demonstrating the attraction and trapping of cracks in the inclusions. This study demonstrates the potential for significant enhancement of toughness through elastic compliance contrast between the matrix and the inclusion without notably compromising the effective elastic modulus of the composite material.

The improvement of fracture strength by insertion of thin, soft interlayers is a strategy observed in biological materials such as deep-see sponges. The basic mechanism is a reduction of the crack driving force due to the spatial variation of yield strength and/or Young's modulus. The application of this “material inhomogeneity effect” is demonstrated in this paper. The effectiveness of various interlayer configurations is investigated by numerical simulations under application of the configurational force concept. Laminated composites, made of high-strength tool steels as matrix materials and low-strength deep-drawing steel as interlayer material, were manufactured by hot press bonding. The number of interlayers and the interlayer thickness were varied. Fracture mechanics experiments show crack arrest in the first interlayer and significant improvements in fracture toughness, even without the occurrence of other toughening mechanisms, such as interface delamination. The application of the material inhomogeneity effect for different types of matrix materials is discussed.

Polydimethylsiloxane (PDMS) has a good elasticity but with a pretty low fracture toughness, which limits its use in practical applications. This paper presents a simple and low-cost approach to manufacture a PDMS/fabric composite through incorporating the commercially available stretchy plain weft-knitted nylon fabric into the PDMS matrix. The fracture toughness of the composite is much higher than that of pure PDMS with an increase up to 700%. The toughening mechanism, which can be attributed to the deformation localization induced fiber stretch and damage propagation in the PDMS matrix, is fully investigated. During cyclic loadings, the composite may exhibit a linear elastic response or a significant hysteresis depending on the stretch level. These results provide physical insights into the deformation mechanism of a soft fabric-reinforced composite and may offer practical routes to realize robust crack-insensitive PDMS.

One observation from interface defeat experiments with thick ceramic targets is that confinement and prestress becomes less important if the test scale is reduced. A small unconfined target can show similar transition velocity as a large and heavily confined target. A possible explanation for this behavior is that the transition velocity depends on the formation and growth of macro cracks. Since the crack resistance increases with decreasing length scale, the extension of a crack in a small-scale target will need a stronger stress field, viz., a higher impact velocity, in order to propagate. An analytical model for the relation between projectile load, corresponding stress field, and the propagation of a cone-shaped crack under a state of interface defeat has been formulated. It is based on the assumption that the transition from interface defeat to penetration is controlled by the growth of the cone crack to a critical length. The model is compared to experimentally determined transition velocities for ceramic targets in different sizes, representing a linear scale factor of ten. The model shows that the projectile pressure at transition is proportional to one over the square root of the length scale. The experiments with small targets follow this relation as long as the projectile pressure at transition exceeds the bound of tensile failure of the ceramic. For larger targets, the transition will become independent of length scale and only depend on the tensile strength of the ceramic material. Both the experiments and the model indicate that scaling of interface defeat needs to be done with caution and that experimental data from one length scale needs to be examined carefully before extrapolating to another.

This article presents a computational constitutive model for glass subjected to large strains, high strain rates and high pressures. The model has similarities to a previously developed model for brittle materials by Johnson, Holmquist and Beissel (JHB model), but there are significant differences. This new glass model provides a material strength that is dependent on the location and/or condition of the material. Provisions are made for the strength to be dependent on whether it is in the interior, on the surface (different surface finishes can be accommodated), adjacent to failed material, or if it is failed. The intact and failed strengths are also dependent on the pressure and the strain rate. Thermal softening, damage softening, time-dependent softening, and the effect of the third invariant are also included. The shear modulus can be constant or variable. The pressure-volume relationship includes permanent densification and bulking. Damage is accumulated based on plastic strain, pressure and strain rate. Simple (single-element) examples are presented to illustrate the capabilities of the model. Computed results for more complex ballistic impact configurations are also presented and compared to experimental data.

Bone is similar to fiber-reinforced composite materials made up of distinct phases such as osteons (fiber), interstitial bone (matrix), and cement lines (matrix-fiber interface). Microstructural features including osteons and cement lines are considered to play an important role in determining the crack growth behavior in cortical bone. The aim of this study is to elucidate possible mechanisms that affect crack penetration into osteons or deflection into cement lines using fracture mechanics-based finite element modeling. Cohesive finite element simulations were performed on two-dimensional models of a single osteon surrounded by a cement line interface and interstitial bone to determine whether the crack propagated into osteons or deflected into cement lines. The simulations investigated the effect of (i) crack orientation with respect to the loading, (ii) fracture toughness and strength of the cement line, (iii) crack length, and (iv) elastic modulus and fracture properties of the osteon with respect to the interstitial bone. The results of the finite element simulations showed that low cement line strength facilitated crack deflection irrespective of the fracture toughness of the cement line. However, low cement line fracture toughness did not guarantee crack deflection if the cement line had high strength. Long cracks required lower cement line strength and fracture toughness to be deflected into cement lines compared with short cracks. The orientation of the crack affected the crack growth trajectory. Changing the fracture properties of the osteon influenced the crack propagation path whereas varying the elastic modulus of the osteon had almost no effect on crack trajectory. The findings of this study present a computational mechanics approach for evaluating microscale fracture mechanisms in bone and provide additional insight into the role of bone microstructure in controlling the microcrack growth trajectory.

Bimaterial systems in which two dissimilar materials are adhesively joined by a thin adhesive interlayer have been widely used in a variety of modern industries and engineering structures. It is well known that interfacial fracture is the most common failure mode for these bimaterial systems. Particularly, the interface fracture is a mixed mode in nature mode-I (pure peel) and mode-II (pure shear) due to the disrupted symmetry by the bimaterial configuration. Obviously, characterizing individual fracture modes, especially mode-I fracture, is essential in understanding and modeling the complex mixed mode fracture problems. Meanwhile, the J -integral is a highly preferred means to characterize the interfacial fracture behaviors of a bimaterial system because it cannot only capture more accurate toughness value, but also facilitate an experimental characterization of interfacial traction-separation laws (cohesive laws). Motivated by these important issues, a novel idea is proposed in the present work to realize and characterize the pure mode-I nonlinear interface fracture between bonded dissimilar materials. First, a nearly pure mode-I fracture test can be simply realized for a wide range of bimaterial systems by almost eliminating the mode-II component based on a special and simple configuration obtained in this work. Then, the concise forms of the J -integral are derived and used to characterize the interfacial fracture behaviors associated with classical and shear deformation beam theories. The proposed approach may be considered as a promising candidate for the future standard mode-I test method of bimaterial systems due to its obvious accuracy, simplicity, and applicability, as demonstrated by the numerical and experimental results.

The tensile fracture strength of a sandwich panel, with a center-cracked core made from an elastic-brittle diamond-celled honeycomb, is explored by analytical models and finite element simulations. The crack is on the midplane of the core and loading is normal to the faces of the sandwich panel. Both the analytical models and finite element simulations indicate that linear elastic fracture mechanics applies when a K -field exists on a scale larger than the cell size. However, there is a regime of geometries for which no K -field exists; in this regime, the stress concentration at the crack tip is negligible and the net strength of the cracked specimen is comparable to the unnotched strength. A fracture map is developed for the sandwich panel with axes given by the sandwich geometry. The effect of a statistical variation in the cell-wall strength is explored using Weibull theory, and the consequences of a stochastic strength upon the fracture map are outlined.

In a recent investigation of microstructural effects in finite periodic multilayers, we have shown that under Mode I loading, the crack-opening displacement approaches that of the same crack in an equivalent homogenized material as the microstructure comprised of alternating stiff and soft layers becomes increasingly finer. In contrast, Mode I stress intensity factor asymptotically converges to values that depend on the stiffness of the cracked layer. Preliminary calculation of Mode I strain energy release rate as a function of the microstructural refinement suggested that this may be a better fracture mechanics parameter for assessing fracture toughness of periodic layered media. Herein, we extend the above investigation by considering both Mode I and II loading to study the effect of layer modulus ratio on fracture mechanics parameters as a function of microstructural refinement. The previously introduced concept of partial homogenization of the microstructure sufficiently far from the crack is also pursued in order to gauge its efficiency in correctly capturing fracture mechanics parameters with a minimum of computational effort. The fracture mechanics parameters are shown to be influenced by the local microstructure to an extent that depends on the layer modulus mismatch. An accurate calculation of these parameters requires the retention of several layers adjacent to the affected cracked layer whose number depends on the modulus mismatch and loading mode.

Four-point-bending V-notched specimens of lead zirconate titanate (PZT) poled parallel to the long axis are fractured under conditions of controlled crack growth in a custom-made device. In addition to the mechanical loading electric fields, up to 500 V ∕ mm are applied parallel and anti-parallel to the poling direction, i.e., perpendicular to the crack surface. To determine the different contributions to the total energy release rate, the mechanical and the piezoelectric compliance, as well as the electrical capacitance of the sample, are recorded continuously using small signal modulation/demodulation techniques. This allows for the calculation of the mechanical, the piezoelectric, and the electrical part of the total energy release rate due to linear processes. The sum of these linear contributions during controlled crack growth is attributed to the intrinsic toughness of the material. The nonlinear part of the total energy release rate is mostly associated to domain switching leading to a switching zone around the crack tip. The measured force-displacement curve, together with the modulation technique, enables us to determine this mechanical nonlinear contribution to the overall toughness of PZT. The intrinsic material toughness is only slightly dependent on the applied electric field (10% effect), which can be explained by screening charges or electrical breakdown in the crack interior. The part of the toughness due to inelastic processes increases from negative to positive electric fields by up to 100%. For the corresponding nonlinear electric energy change during crack growth, only a rough estimate is performed.

The fracture behavior of polycrystalline silicon in the presence of atomically sharp cracks is important in the determination of the mechanical reliability of microelectromechanical system (MEMS) components. The mode-I critical stress intensity factor and crack tip displacements in the vicinity of atomically sharp edge cracks in polycrystalline silicon MEMS scale specimens were measured via an in situ atomic force microscopy/digital image correlation method. The effective (macroscopic) mode-I critical stress intensity factor for specimens from different fabrication runs was 1.00 ± 0.1 MPa √ m , where 0.1 MPa √ m is the standard deviation that was attributed to local cleavage anisotropy and grain boundary effects. The experimental near crack tip displacements were in good agreement with the linearly elastic fracture mechanics solution, which supports K dominance in polysilicon at the scale of a few microns. The mechanical characterization method implemented in this work allowed for direct experimental evidence of incremental (subcritical) crack growth in polycrystalline silicon that occurred with crack increments of 1 - 2 μ m . The variation in experimental effective critical stress intensity factors and the incremental crack growth in brittle polysilicon were attributed to local cleavage anisotropy in individual silicon grains where the crack tip resided and whose fracture characteristics controlled the overall fracture process resulting in different local and macroscopic stress intensity factors.

The paper considers the problem of a plane-strain fluid-driven fracture propagating in an impermeable elastic solid, under condition of small (relative) solid toughness or high (relative) fracturing fluid viscosity. This condition typically applies in hydraulic fracturing treatments used to stimulate hydrocarbons-bearing rock layers, and in the transport of magma in the lithosphere. We show that for small values of a dimensionless toughness K , the solution outside of the immediate vicinity of the fracture tips is given to O ( 1 ) by the zero-toughness solution, which, if extended to the tips, is characterized by an opening varying as the ( 2 ∕ 3 ) power of the distance from the tip. This near tip behavior of the zero-toughness solution is incompatible with the Linear Elastic Fracture Mechanics (LEFM) tip asymptote characterized by an opening varying as the ( 1 ∕ 2 ) power of the distance from the tip, for any nonzero toughness. This gives rise to a LEFM boundary layer at the fracture tips where the influence of material toughness is localized. We establish the boundary layer solution and the condition of matching of the latter with the outer zero-toughness solution over a lengthscale intermediate to the boundary layer thickness and the fracture length. This matching condition, expressed as a smallness condition on K , and the corresponding structure of the overall solution ensures that the fracture propagates in the viscosity-dominated regime, i.e., that the solution away from the tip is approximately independent of toughness. The solution involving the next order correction in K to the outer zero-toughness solution yields the range of problem parameters corresponding to the viscosity-dominated regime.

Based on a dynamic fracture mechanics analysis, the stress field in a continuous film (called a web) during slitting (or cutting) is investigated. For a homogeneous, isotropic and linearly elastic web, the steady-state dynamic stress field surrounding the slitter blade can be related to the interacting traction between the moving web and the blade, and to the far-field tension that is parallel to the slitting direction. The interaction between the moving web and the blade also includes friction that is considered to be a Coulomb type. By solving an integral equation, the normal traction between the web and the blade can be expressed as a function of the blade profile and the web speed. Numerical calculations are performed for an ideal razor blade with the wedge shape. The analysis presented in this study indicates that the contact between the moving web and the blade does not start at the tip of the blade but rather starts at some distance behind the blade tip. Moreover, it is found that the distance from the point where the web begins to separate to the point where the blade and the web start to have contact, is controlled by the toughness of the web material and also by the web speed. Some characteristic nature of the dynamic stress field surrounding the slitter blade is investigated based on the dynamic fracture mechanics analysis results.

Mathematical models, for the stress analyses of unidirectional end notch flexure and end notch cantilever specimens using classical beam theory, first, second, and third-order shear deformation beam theories, have been developed to determine the interlaminar fracture toughness of unidirectional composites in mode II. In the present study, appropriate matching conditions, in terms of generalized displacements and stress resultants, have been derived and applied at the crack tip by enforcing the displacement continuity at the crack tip in conjunction with the variational equation. Strain energy release rate has been calculated using compliance approach. The compliance and strain energy release rate obtained from present formulations have been compared with the existing experimental, analytical, and finite element results and found that results from third-order shear deformation beam theory are in close agreement with the existing experimental and finite element results.

A permeable crack model is proposed to analyze crack growth in a piezoelectric ceramic. In this model, a permeable crack is modeled as a vanishing thin, finite dimension, rectangular slit with dielectric medium inside. A first-order approximation solution is derived in terms of the slit height, h 0 . The main contribution of this paper is that the newly proposed permeable crack model reveals that there exists a realistic leaky mode for electrical field, which allows applied electric field passing through the dielectric medium inside a crack. By taking into account the leaky mode effect, a correct estimation of electrical and mechanical fields in front of a crack tip in a piezoelectric ceramic is obtained. To demonstrate this new finding, a closed-form solution is obtained for a mode III permeable crack under both mechanical as well electrical loads. Both local and global energy release rates are calculated based on the permeable crack solution obtained. It is found that the global energy release rate derived for a permeable crack is in a broad agreement with some known experimental observations. It may be served as a fracture criterion for piezoelectric materials. This contribution reconciles the outstanding discrepancy between experimental observation and theoretical analysis on crack growth problem in piezoelectric materials.

A simple relationship is obtained between the external force F and the fracture toughness R for thin sheets in steady state elastoplastic combined tearing and peeling along self-similar paths. The relationship depends only on the material properties ( E , σ y , and α for an elastoplastic material with linear hardening) and strip cross section (B and H ). An earlier analysis (which incorporates transient tearing and peeling) requires lengthy computations over the whole length of the strip. The present analysis avoids that complication. Experiments in steady-state agree with the theory.

Ceramic coatings deposited on metal substrates generally develop significant compressive stresses when cooled from the temperature at which they are processed as a result of thermal expansion mismatch. One of the main failure modes for these coatings is edge delamination. For an ideally brittle interface, the edge delamination of a compressed thin film involves mode II interface cracking. The crack faces are in contact with normal stress acting across the faces behind the advancing tip. Frictional shielding of the crack tip has been shown to increase the apparent fracture toughness. Roughness effects associated with the separating faces can also contribute to the apparent toughness. A model of mode II steady-state edge delamination that incorporates combined friction and roughness effects between the delaminated film and substrate is proposed and analyzed. This model is used to assess whether frictional shielding and surface roughness effects are sufficient to explain the large apparent mode II fracture toughness values observed in experiments.

Axisymmetric buckling, initial post-buckling and growth of a circular delamination bridged by nonlinear fibers in three-dimensional composites are studied by a perturbation method. The through-thickness fibers are assumed to provide nonlinear restoring traction resisting the deflection of the delaminated layer. A closed-form solution for the central deflection of the delamination due to on applied compressive stress during initial post-buckling is obtained. In addition, some simple formulas for calculating the strain energy release rate and the mixed mode stress intensity ratio (i.e., Mode II versus Mode I) at the delamination crack tip are also established. Some interesting conclusions arising directly from the perturbation solutions are drawn. These include: (1) initial post-buckling behavior of a circular delamination is unstable for a softening bridging model; this may result in initial delamination growth for some materials with lower fracture toughness when the delamination buckles rather than post-buckles. However, stable growth is obtained for a hardening bridging model; (2) with an increase of the nonlinear fiber bridging parameter β ¯ , the residual stiffness of a three-dimensional composite structure with a circular delamination increases gradually; (3) bridging force changes the catastrophic nature of the delamination growth and increases the stability of the delamination. The range and the dynamic effect of the unstable delamination growth diminish or disappear as the bridging parameters increase; (4) for the bridged delamination, the higher the material fracture toughness, the higher the stability of the delamination growth, and the smaller the range and dynamic effect of its unstable growth. [S0021-8936(00)03203-7]