Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.     

A modified effective crack-length formulation in elastic-plastic fracture mechanics

In examining the performance of standard effective crack-length formulations, the authors noted quantitative accuracy up to “high” fractions of limit load under loading conditions for which the elastic T-stress was non-negative, while a pronounced deviation from the corresponding continuum elastic-plastic plane-strain finite-element solutions was seen in shallow-cracked geometries having negative T-stress. This trend can be rationalized by noting that, under modified boundary layer (K_I and T) loading, the maximum plastic zone radius strongly increases as the T-stress decreases from zero (J.R. Rice (1974), J. Mech. Phys. Solids 22, 17–26; S.G. Larsson and A.J. Carlsson (1973), J. Mech. Phys. Solids 21, 263–277; N.P. O'Dowd and C.F. Shih (1991), J. Mech. Phys. Solids 39(8), 989–1015.) Accordingly, we formulate a modified effective crack length to account for the effects of the elastic T-stress.

The new formulation consistently extends the load range for which accurate predictions of compliance, J-integral, and crack-tip constraint are obtained in several plane strain specimen geometries. The magnitude of influence of the T-stress varies with specimen type and relative crack depth. The greatest “improvement” to standard effective crack length approximations occurs in specimens of “moderately” negative T-stress.     

The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers

In this paper semi-analytical expressions are derived for the energy release rate and the stress intensity factors of edge-cracked homogeneous and orthotropic layers subject to arbitrary generalized end forces. The expressions are accurate for long and short cracks. Following the work of Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] for isotropic bi-material layers, the derivation extends the method proposed by Suo [Suo ZG. Delamination specimens for orthotropic materials. J Appl Mech 1990;57(3):627-34] for axial forces and bending moments in order to include the contribution of the shear forces. The shear contribution to the fracture parameters depends on the shear deformations along the layer and the elastic near tip deformation of the material. Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] derived semi-analytical expressions for the fracture parameters that depend on the crack tip stress resultants, the elastic constants and five numerically-determined constants globally describing the effect of shear. In this paper analogous constants are derived for orthotropic layers and defined by semi-analytical expressions that highlight their physical significance and allow separation of the different contributions. The derivation is based on the assumption that the near tip deformation can be described by means of relative rotations between the cross sections of the different sub-layers at the crack tip (root rotations). The root rotations depend linearly on the crack tip stress resultants through compliance coefficients that are derived numerically in the paper for a wide range of orthotropic materials. Applications to different mixed mode delamination and peeling problems, for which accurate two-dimensional finite element solutions can be found in the literature, highlight the accuracy of the proposed expressions.     

On the effective electroelastic properties of microcracked generally anisotropic solids

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration.     

Composite materials whose phases have partially equal elastic moduli

Hill [J. Mech. Phys. Solids 11 (1963) 357, 12 (1964) 199] discovered that, regardless of its microstructure, a linearly elastic composite of two isotropic phases with identical shear moduli is isotropic and has the effective shear modulus equal to the phase ones. The present work generalizes this result to anisotropic phase composites by showing and exploiting the fact that uniform strain and stress fields exist in every composite whose phases have certain common elastic moduli. Precisely, a coordinate-free condition is given to characterize this specific class of elastic composites; an efficient algebraic method is elaborated to find the uniform strain and stress fields of such a composite and to obtain the structure of the effective elastic moduli in terms of the phase ones; sufficient microstructure-independent conditions are deduced for the orthogonal group symmetry of the effective elastic moduli. These results are applied to elastic composites consisting of isotropic, transversely isotropic and orthotropic phases.     

Elastic property prediction by finite element analysis with random distribution of materials for tungsten/silver composite

In the present numerical study, we introduce a finite element analysis for heterogeneous materials via a random distribution of materials to predict effective elastic properties. With this random distributing strategy, a large scale parametric analysis via finite element becomes feasible for the multi-phase heterogeneous solids. Taking a well-documented tungsten–silver bi-continuous material as an example, the numerical prediction provided here for the effective properties is checked by experimental testing data available in open publication. Discussions on the present finite element prediction and other approaches are also made by comparing with Hashin and Shtrikman (J Mech Phys Solids 11:127–140, 1963) bounds in the composite mechanics.     

Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method

Making use of the Displacement Discontinuity Boundary Integral Equation Method (DDBIEM), the dimension of the plastic zone at the tip of a penny-shaped crack in a three-dimensional elastic medium is determined by the application of the Dugdale model; Furthermore, the solutions for a penny-shaped crack in three-dimensional piezoelectric media are obtained by the use of the Dugdale-like model proposed by Gao et al.[Gao H, Zhang T, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997;45:491–510], in which the electrical polarization is assumed to reach a saturation limit in a thin annular region in front of a crack while the mechanical stresses have the ordinary singularity.     

Effects of grain sizes, shapes, and distribution on minimum sizes of representative volume elements of cubic polycrystals

In the literature the concept of representative volume element (RVE) was introduced to correlate the effective or macroscopic properties of materials with the properties of the microscopic constituents and microscopic structures of the materials. However, to date little quantitative knowledge is available about minimum RVE sizes of various engineering materials. In our recent paper [J. Mech. Phys. Solids 50 (2002) 881], a new definition of minimum RVE size was introduced based on the concept of nominal modulus. Numerical experiments using the finite element method (FEM) were then carried out for determining the minimum RVE sizes of more than 500 cubic polycrystals in the plane stress problem, under the assumption that all grains in a polycrystal have the same square shape––called the simple polycrystal model. The major finding is that the minimum RVE sizes for effective elastic moduli have a roughly linear dependence on crystal anisotropy degrees. The present paper takes into account the effect of grain sizes, shapes, and distribution on the minimum RVE sizes for real cubic polycrystals that are formed by crystallization processes. Similar roughly linear dependence is found again, with the slope about 19% lower than that in the simple polycrystal model. This finding is interesting and useful because numerical experiments on minimum RVE sizes for a large number of crystals are quite time-consuming and the simple polycrystal model reduces significantly the FEM pre- and post-processing works. This should be particularly true in numerically testing minimum RVE sizes for three-dimensional polycrystals and for nonelastic properties in future works. With a maximum relative error 5%, all the polycrystals tested have a minimum RVE size of 16 or less times the grain size.     

Imperfection sensitive variation of critical loads at hilltop bifurcation point

Variation of critical loads due to initial imperfections at the hilltop bifurcation point is described by elastic stability theory. We derive a system of bifurcation equations for a potential system expressing local behavior at this bifurcation point, which is a double critical point occurring as a coincidence of a simple pitchfork bifurcation point and a limit point. The piecewise linear law of imperfection sensitivity of critical loads in Thompson and Schorrock [J. Mech. Phys. Solids 23 (1975) 21] is revised by extending initial imperfections to be considered in the bifurcation equations. Based on this sensitivity law, a procedure to determine the most influential (worst or optimum) initial imperfection is formulated. As the most essential development of this paper, under the assumption that initial imperfections are subject to a multi-variate normal distribution, we derive the probability density function of critical loads that follows a Weibull-like distribution. The validity of theoretical developments is assessed through its application to elastic truss structures.     

A 3D fourth order fabric tensor approach of anisotropy in granular media

We propose a micromechanical approach for granular media, with a particular account of the texture-induced anisotropy and of the strain localization rule. The approach is mainly based on the consideration of a fourth order fabric tensor able to capture general anisotropy which can be induced by complex distribution of contacts. Incorporation of this fourth order fabric tensor in a suitable homogenization scheme allows to determine the corresponding macroscopic elastic properties of the granular material. For this purpose, in addition to the classical Voigt upper bound, a new kinematics-based localization rule is proposed. It generalizes the one formulated by Cambou et al. [B. Cambou, Ph. Dubujet, F. Emeriault, F. Sidoroff, Eur. J. Mech. A/Solids 14 (1995) 225–276] in the case of an isotropic contact distribution. The results of the complete model compare well to numerical simulations results when available [C.S. Chang, C.L. Liao, Appl. Mech. Rev. 47 (1 Part 2) (1994) 197–207] (case of isotropic distribution of contacts). Finally, the interest of the fourth order fabric tensor based approach combined with the proposed localization rule is shown for different distributions of contacts by comparing its predictions to those given by a second order fabric tensor approach.     

Effective properties of power-law solids containing elliptical inhomogeneities. Part I: Rigid inclusions

The plane strain deformation of a power-law material containing a dispersion of rigid elliptical inclusions is investigated. Accurate constitutive relations are established for dilute concentrations of inclusions over a wide range of martrix hardening exponents and inclusion aspect ratios. The essential step in the analysis is the solution of the kernel problem for an isolated inclusion, and this is obtained using a Ritz procedure in which trial displacements are derived from a displacement potential in elliptic-cylindrical coordinates. Approximate constitutive relations for nondilute concentrations of inclusions are then established using a differential self-consistent scheme. The study is primarily concerned with randomly oriented inclusions, but limited results are also presented for aligned inclusions. As part of the investigation, the procedure introduced by Ponte Castaneda (J. Mech. Phys. Solids (1991)) is evaluated. This is a variational procedure which exploits information about a linear solid to obtain bounds or estimates for the overall response of nonlinear solids with the same microgeometry of second phase. Finally, connection is made with recent results for spheroidal inclusions given by Lee and Mear (J. Mech. Phys. Solids (1991); Int. J. Solids Struct. (1991)), and an ad hoc estimate for the overall response of solids containing randomly oriented rigid, spheroidal inclusions is proposed.     

Crack tip fields in a viscoplastic solid: monotonic and cyclic loading

The asymptotic stress and strain field near the tip of a plane strain Mode I stationary crack in a viscoplastic material are investigated in this work, using a unified viscoplastic model based on Chaboche (Int J Plast 5(3):247–302, 1989). Asymptotic analysis shows that the near tip stress field is governed by the Hutchinson–Rice–Rosengren (HRR) field (Hutchinson in J Mech Phys Solids 16(1):13–31, 1968; Rice and Rosengren in J Mech Phys Solids 16(1):1–12, 1968) with a time dependent amplitude that depends on the loading history. Finite element analysis is carried out for a single edge crack specimen subjected to a constant applied load and a simple class of cyclic loading history. The focus is on small scale creep where the region of inelasticity is small in comparison with typical specimen dimensions. For the case of constant load, the amplitude of the HRR field is found to vanish at long times and the elastic K field dominates. For the case of cyclic loading, we study the effect of stress ratio on inelastic strain and find that the strain accumulated per cycle decreases with stress ratio.     

Interaction of dynamic mode-I cracks with inclined interfaces

In this paper, we report on an experimental study of the deflection/penetration behavior of dynamic mode-I cracks propagating at two different crack velocities (slower and faster) toward inclined weak interfaces of three dissimilar angles (α): 30°, 45° and 60°. A simple wedge-loading specimen configuration as proposed by Xu et al. [Xu LR, Huang YY, Rosakis AJ. Dynamic crack deflection and penetration at interface in homogenous materials: experimental studies and model predictions. J Mech Phys Solids 2003;51:461-86], made of brittle Homalite-100, is used. A modified Hopkinson bar setup is used to achieve well-controlled impact loading conditions. Dynamic photoelasticity in conjunction with high-speed photography is used to capture real-time isochromatics associated with deflected/penetrated cracks.     

Analysis of in-plane transonically propagating interface crack with a finite contact zone

The report of Lambros and Rosakis [(1995) J Mech Phys Solids 43(2): 169–188] has focused attention on steady-state transonic interfacial crack growth accounting for the phenomenon of crack face contact in elastic/rigid bimaterial but could not handle issues relating to energy transmission across the interface. The present paper attempts to provide a complete explicit expression of the asymptotic fields induced by transonically propagating interfacial crack in elastic/elastic bimaterial for in-plane case. The energy distribution on the contact area, crack tip and two singular characteristic lines is analysed thoroughly and compared with the dynamic separated J-integrals. The length of the contact zone is also discussed briefly by establishing energy fracture criterion that satisfies contact condition. The two-dimensional in-plane asymptotic deformation field surrounding the contact area of a crack propagating transonically along an elastic/elastic bimaterial interface is observed and discussed thoroughly.     

The manufacture and characterisation of a novel,low modulus,negative Poisson’s ratio composite

Relatively few negative Poisson’s ratio (auxetic) composites have been manufactured and characterised and none with inherently auxetic phases [Milton G. J. Mech. Phys. Solids 1992;40:1105–37]. This paper presents the use of a novel double-helix yarn that is shown to be auxetic, and an auxetic composite made from this yarn in a woven textile structure. This is the first reported composite to exhibit auxetic behaviour using inherently auxetic yarns. Importantly, both the yarn and the composite are produced using standard manufacturing techniques and are therefore potentially useful in a wide range of engineering applications.     

Introduction of damage evolution in a scale transition approach for highly-filled particulate composites

The present paper deals with a non-conventional scale transition for modelling the behaviour of highly-filled particulate composites, starting from a methodology initially proposed by Christoffersen [Christoffersen J. Bonded granulates. J Mech Phys Solids 1983;31:55–83] and recently extended by Nadot et al. [Nadot C, Dragon A, Trumel H, Fanget A. Damage modelling framework for viscoelastic particulate composites via a scale transition approach. J Theor Appl Mech 2006;44(3):553–83] in presence of damage. The model thus obtained is here completed with several ingredients allowing to describe damage evolution and in particular a defect nucleation criterion as well as a closure criterion. These criteria are formulated in terms of displacement, and so as to ensure continuity in terms of macroscopic stress. They are finally introduced in an iterative numerical solving procedure which allows to follow damage evolution as a discrete sequence of interfacial debonding including also eventual closure of defects.     

Dynamic crushing of honeycombs and features of shock fronts

The in-plane dynamic crushing of 2D hexagonal-cell honeycombs has been simulated using finite elements to explore the dynamic response of cellular materials and to investigate the features of the crushing front and to examine the assumptions employed in a one-dimensional shock theory [Reid SR, Peng C. Dynamic uniaxial crushing of wood. Int J Impact Eng 1997;19:531–70; Tan PJ, Reid SR, Harrigan JJ, Zou Z, Li S. Dynamic compressive strength properties of aluminium foams. Part II – shock theory and comparison with experimental data and numerical models. J Mech Phys Solids 2005;53:2206–30]. It has been demonstrated that progressive cell crushing is observed to propagate through the material in a ‘shock’ like manner when the crushing velocity exceeds a critical value. The simulations show that there exists a zone at the shock front across which there are essentially discontinuities in the material ‘particle velocity’, ‘stress’ and ‘strain’ as defined herein. At supercritical crushing velocities the thickness of this zone remains about one cell size, which varies little with the crushing velocity and the relative density. Densification strain increases as crushing velocity increases and asymptotes to a limit once a shock front forms. It has also been shown that the one-dimensional shock theory [Reid SR, Peng C. Dynamic uniaxial crushing of wood. Int J Impact Eng 1997;19:531–70; Tan PJ, Reid SR, Harrigan JJ, Zou Z, Li S. Dynamic compressive strength properties of aluminium foams. Part II – shock theory and comparison with experimental data and numerical models. J Mech Phys Solids 2005;53:2206–30], which was based on an equivalent rigid-perfectly plastic-locking stress–strain curve, tends to overestimate slightly the crushing stress and energy absorbed.     

A failure criterion for brittle elastic materials under mixed-mode loading

The failure criterion of Leguillon at reentrant corners in brittle elastic materials (Leguillon 2002, Eur J Mech A/Solids 21: 61–72; Leguillon et al. (2003), Eur J Mech A—Solids 22(4): 509–524) validated in (Yosibash et al. 2004, Int J Fract 125(3–4): 307–333) for mode I loading is being extended to mixed mode loading and is being validated by experimental observations. We present an explicit derivation of all quantities involved in the computation of the failure criterion. The failure criterion is validated by predicting the critical load and crack initiation angle of specimens under mixed mode loading and comparison to experimental observations on PMMA (polymer) and Macor (ceramic) V-notched specimens.     

Size effect in micro-scale cantilever beam bending

When the thickness of metallic cantilever beams reduces to the order of micron, a strong size effect of mechanical behavior has been found. In order to explain the size effect in a micro-cantilever beam, the couple-stress theory (Fleck and Hutchinson, J Mech Phys Solids 41:1825?C1857, 1993) and the C-W strain gradient theory (Chen and Wang, Acta Mater 48:3997?C4005, 2000) are used with the help of the Bernoulli?CEuler beam model. The cantilever beam is considered as the linear elastic and rigid-plastic one, respectively. Analytical results of the cantilever beam deflection under strain gradient effects by applying these two kinds of theories are obtained, from which we find an explicit relationship between the intrinsic lengths introduced in the two kinds of theories. The theoretical results are further used to analyze the experimental observations, and predictions by both theories are further compared. The results in the present paper should be useful for the design of micro-cantilever beams in MEMS and NEMS.