A percolation-type fracture criterion for composites with randomly oriented fibres

A fracture model is built up for a solid composed of brittle fibres randomly oriented in the matrix volume. The fracture process includes a stable growth of microcracks caused by fibre breaking under the load and formation of an infinite cluster of the microcracks. Both upper and lower bounds for ultimate stress in a fibre system are found as functions of the fibre volume fraction. The calculation of the ultimate stresses are performed by using the percolation theory and the theory of branching processes. At the present stage of the theory under consideration, only two types of the microcracks are appraised, namely that of a delamination type which corresponds to a weak fibre/matrix interface, and that of a penny shape which corresponds to a strong fibre/matrix interface. A particular solid contains only one type of the microcracks. In both cases, non-linear dependencies of the ultimate composite strength on fibre volume fraction are obtained.     

Consistently linearized constitutive equations of micromechanical models for fibre composites with evolving damage

The numerical analysis of engineering structures is usually based upon the assumptions of a homogeneous as well as a continuous medium. These simplifications are maintained also for structures made of fibre reinforced composite materials which possess by definition a heterogeneous finescale architecture. Furthermore in the course of the loading of such structures void nucleations might arise out of the debonding of the embedded fibres or the growth of microcracks inside the matrix phase. Hence, the assumption of a continuous and homogeneous medium is not valid from a microscopical point of view. Nevertheless, it is numerically advantageous to keep up these simplifying assumptions on the macrolevel. Therefore, the knowledge of the so called macroscopic or effective material behaviour is needed. The overall properties can be described in terms of volume averaged quantities that smear the heterogeneities of the microscopic structure and the influence of its defects. Since the evolution of damage within composite materials means a rather complex process, a purely phenomenological approach is hardly feasible. Hence, the average properties are to be obtained from a micromechanical analysis of the discontinuous and damaged finescale structure. The efficiently reformulated version of the micromechanically based Generalized Method of Cells (GMC) provides the macroscopic tangential constitutive tensor in closed-form. The numerical efficiency of the approach allows for the use of the GMC as the constitutive model for nonlinear finite element analyses. Two-scale simulations of macroscale composite structures considering process depending damage evolution on the microscale of heterogeneous media becomes feasible.     

Heuristic model of pseudo-macrocracking in composites with brittle matrix

Considered in this work is a heuristic model of pseudo-macrocracking of a fiber reinforced composite with a brittle matrix. By preserving the qualitative feature of the fracture process, crack extension corresponds to the formation of microcracks ahead of the main macrocrack. Critical stress intensity factor can be obtained from a simple energy balance. Pseudo-macrocracking can also be observed in a non-homogeneous solid which does not contain macrocracks. The results can be applied for evaluating the load carrying capacity of fiber-reinforced composites.     

Crack paths formed by multiple debonds in LFRP composites

Onset and growth of debonds at fibre-matrix interfaces in a bundle of fibres subjected to transverse loads are studied numerically. In particular, the crack path formed by debonded neighbour fibres is analysed. The Linear Elastic–Brittle Interface Model (LEBIM) is used to model the fibre-matrix interface behaviour. This simple model of a Long Fibre Reinfoced Polymer (LFRP) composite includes ten parallel fibres embedded in a matrix cell whose external dimensions are much larger than the fibre radius. The advantage of the present LEBIM formulation of the so-called matrix cracking lies in its ability to make quantitative predictions about the concurrent fibre-matrix debond onset and mixed-mode interface crack growth in a fibre bundle. The numerical analysis predicts failure loads producing the first and subsequent debond onsets, leading to a crack path. A discussion on the position where debond occurs is also included. Finally, the effect of the load biaxiality on the crack path is studied in detail.     

Toughness in obliquely-stressed fibrous composites

An earlier treatment of toughness for continuous, uniform, fibre reinforced materials given by M.R. Piggott (1970) is extended to the case where the stress is not parallel to the fibres. Experiments on pairs of fibres crossing cracks obliquely are used to reveal the effect on fibre strength of fibre flexure at the crack. The theory indicates that, so long as splitting parallel to the fibres does not occur, the fracture surface energy γ_φ for a material stressed at an angle φ to the fibres is given with sufficient accuracy for brittle fibres by the approximate formula γ_φ = γ_o(1?2.4A tan φ), where γ_o is the surface energy for fracture normal to the fibre direction, and A is a non-dimensional parameter depending on the force exerted by the matrix on the fibres, and involving, in particular, the ratio of matrix flow stress to the fibre ultimate tensile strength. For ductile fibres, the work of fracture increases with the angle φ at a rate depending on fibre breaking stress. The form of fracture surface and the onset of splitting are also discussed.     

Asymptotic study of geometrically non-linear elastic strip with regular system of fibres

Two-dimensional composite material consisting of linear elastic fibres embedded in a non-linear elastic strip has been studied. That inhomogeneous medium models a mechanical behaviour of a rubber-like textile reinforced with threads distributed in parallel. The analytical solution relating the displacement field to fibre width, fibre distance, fibre length and fibre and matrix elastic moduli has been derived. To this end we used a special perturbation and homogenization technique. Anlustrative example is also provided.     

On the permeability of fibre-reinforced porous materials

Biological tissues can be considered as composite materials comprised of a porous matrix filled with interstitial fluid and reinforced by impermeable collagen fibres. Motivated by studies on fluid flow in articular cartilage, we would like to quantify the undeformed configuration permeability of fibre-reinforced composite materials. If there is a sufficient scale separation between the internal structure of the porous matrix and the arrangement of the fibres, the matrix can be taken as a porous continuum at the fibre scale. In this case, the fibres can be treated as inclusions in a porous continuum, and the overall permeability of the composite can be evaluated using homogenisation procedures. For an isotropic homogeneous matrix, the symmetry of the system is governed by the orientation of the fibres. Here, we propose to retrieve the overall permeability through geometrical considerations and directional averaging methods. The special case of transverse isotropy is discussed in detail, with particular attention to the sub-cases of aligned fibres and fibres lying on a plane.     

Viscoelastic Deformation of a Reinforced Plate with a Crack

The deformation of a viscoelastic reinforced composite is studied. The composite has an axis of elastic symmetry and consists of transversally isotropic fibers and a viscoelastic matrix, which differ by the volume concentration and mechanical characteristics. The material is modeled by a transversely isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. A plate fabricated from the composite in question is weakened by a through mode I crack and is subjected to constant tensile forces. The viscoelastic properties of the matrix material are described by a convolution operator. The Volterra principle is used to derive expressions for the viscoelastic characteristics and crack opening. The irrational function of the integral operator that describes the crack opening is expanded into an operator continued fraction and is represented as the sum of base operators     

Necessary and sufficient fracture criteria for a composite with a brittle matrix. Part 1. Weak reinforcement

Fracture of a composite medium with a brittle matrix is studied. The brittle or plastic material of the reinforcing elements is highly deformable. For normalrupture macrocracks, necessary criteria of brittle strength and sufficient criteria of quasibrittle strength are proposed. Simple analytical dependences of the macrocrack length on the loading parameter, structural, rigidity, and strength parameters of the medium, and damage parameters of the material of the components are obtained. The critical loads for these criteria may differ substantially even if the reinforcement coefficient is small and the material of reinforcing elements is highly deformable. If the necessary criterion is satisfied, crack extension occurs and microcracks are formed in the bonds of the structure located ahead of the macrocrack tip. The number of damaged bonds depends on the macrocrack opening and characteristics of postcritical deformation of the damaged bonds.     

Stress analysis of notched anisotropic finite plates under mechanical and hygrothermal loads

Summary Notch-induced stress concentrations in anisotropic composite materials depend on their directional material properties, especially for uniaxially reinforced composites with high-modulus fibres. The design of notched high-performance composites requires therefore a special proof of their notched strength, which includes the structural parameters of the fibre/matrix combination, fibre orientation and layer arrangement. The assessment of the effects of the finite outer boundary is of practical importance when dimensioning critical notched regions. An anisotropic plate with finite dimensions and a hole in its center will be used here to model stress concentrations. The calculation is based on conformal mappings combined with complex-valued stress functions. The outer boundary is described using point-matching and the least-squares method. The solutions are conducive to the assessment of the essential influencing factors of material properties, geometry and loads. Notched finite plates made of fibre/matrix composites, mainly carbon-fibre reinforced polymers, will be presented as illustrations. Received 29 June 1998; accepted for publication 22 October 1998     

Growth of a penny-shaped crack with a nonsmall fracture process zone in a composite

The paper studies the stress rupture behavior of a reinforced viscoelastic composite through which a penny-shaped mode I crack propagates under a constant load. The composite has hexagonal symmetry and consists of elastic isotropic fibers and viscoelastic isotropic matrix. The material is modeled as a transversely isotropic homogeneous viscoelastic medium with effective characteristics. The crack is in the isotropy plane. The ring-shaped fracture process zone at the crack front is modeled by a modified Dugdale zone with time-dependent stresses. The viscoelastic properties of the matrix are characterized using a resolvent integral operator. Use is made of Volterra's principle, the method of operator continued fractions, and the theory of precritical crack growth in viscoelastic bodies. The problem is reduced to nonlinear integral equations. Numerical results are obtained for certain components of the composite, constant volume fractions, and different fracture strengths Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 45–51, August 2008.     

Brush-like modes of fatigue fracture in unidirectional fibre composites

Fatigue fracture of unidirectional fibre composites under tension along the fibres is discussed with account of the interaction between various mechanisms of damage such as single and multiple fibre ruptures, matrix cracking, and matrix-fibre debonding. The case of brittle fibres and a comparatively weak and ductile matrix is considered that exposes non-conventional modes of fracture, named “brush-like” cracks. Growth of such cracks under cyclic quasistatic loading is studied, and the effect of various factors on the crack growth rate is investigated.     

Wave propagation in layer with two preferred directions

This article is concerned with overall or macroscopic properties of a composite material with no distinction made between the fibres and the matrix which they are embedded in. All the properties with dimensions larger than the fibre diameter and spacing are regarded as averaged over a volume of material. The systems of particular interest here are in the fibre reinforced composites with the fibres being very much stiffer and stronger than the matrix.Laminated plates of fibre-reinforced material are often fabricated from prepreg tapes, laid up according to some specific arrangement of fibre orientation and then bonded together. An angle-ply laminate is formed by alternating plies so that the families in adjacent laminas are inclined by angle ϕ and −ϕ to given direction alternately. The process of fabricating a multilayered plate of this material gives rise to a laminate in which the plies are separated by resin rich layer, and when this layer is thin enough that its thickness is negligible it may be regarded as plate reinforced by two families of fibres. Problems shall be considered in three dimensions, but attention shall be restricted to linear elasticity theory. The plate under consideration is reinforced by two mechanically equivalent families of fibres, but with no other preferred directions, so that it is locally orthotropic with respect to the plane of the fibres and to the two planes that orthogonally bisect the fibres.In this article linear elastic stress–strain relation is employed to derive dispersion curves for plane harmonic waves propagating in a plate of finite thickness but of infinite lateral extent. Attention is restricted to waves propagating in the plane parallel to stress free plate faces where waves travelling at any angle to one of the families of very strong fibres are examined. The dispersion equations, relating the phase velocity to the wavelength, are obtained. The fundamental modes are examined for symmetric as well as for anti-symmetric deformations. This leads to full understanding of displacement field as well as stress field.     

Mechanical response of neo-Hookean fiber reinforced incompressible nonlinearly elastic solids

The mechanical behavior of an incompressible neo-Hookean material, directionally reinforced by neo-Hookean fibers, is examined under homogeneous deformations. A composite model for this transversely isotropic material is developed based on a multiplicative decomposition of the deformation gradient which considers interaction between the fiber and the matrix. The so-called standard reinforcing model exhibits non-monotonic behavior in compression. The present composites-based approach leads to a modification of the standard reinforcing model in which monotonic behavior in compression is observed. This stems from the micromechanical basis of the model in which the fiber is treated as a neo-Hookean material. The conditions for loss of monotonicity and positivity in the stress-shear behavior in off-axis simple 2D shear are also obtained.     

Constitutive behavior of a microcracking brittle solid in cyclic compression

A constitutive formulation is presented to determine the “driving force” for Mode I fatigue crack growth in notched plates of brittle solids stressed in uniaxial cyclic compression. For the particular case of a microcracking medium, it is demonstrated that residual tensile stresses are induced ahead of the notch during unloading from the maximum far-field compressive stress. We propose that it is this region of residual tensile stresses at the notch-tip which promotes fatigue crack growth in ceramics along the notch plane in a direction normal to the compression axis. The predictions of the analysis are compared with new experimental results on compression fatigue in brittle solids. Specifically, it is shown that the numerical estimates of the near-tip tensile zone size for microcracking ceramics compare favorably with the experimentally measured distance of stable Mode I fatigue crack growth after the first compression cycle. Experimental information on the threshold stress for microcracking, transition stress for the inducement of residual tension during unloading, and the effect of mean stress on fracture, as well as direct observations of microcracks and crack growth in compression fatigue, corroborate the assumptions and implications of the analysis.     

The bimodal theory of plasticity: A form-invariant generalisation

The bimodal plasticity model of fibre-reinforced materials is currently available and applicable only in association with thin-walled fibrous composites containing a family of straight fibres which are conveniently assumed parallel with the x₁-axis of an appropriately chosen Cartesian co-ordinate system. Based on reliable experimental evidence, the model suggests that plastic slip in the composite operates in two distinct modes; the so-called matrix dominated mode (MDM) which depends on a matrix yield stress, and the fibre dominated mode (FDM) which depends also on the fibre yield stress. Each mode is activated by different states of applied stress, has its own yield surface (or surfaces) in the stress space and has its own segment on the overall yield surface of the composite. This paper employs theory of tensor representations and produces a form-invariant generalisation of both modes of the model. This generalisation furnishes the model with direct applicability to relevant plasticity problems, regardless of the shape of the fibres or the orientation of the co-ordinate system. It thus provides a proper mathematical foundation that underpins important physical concepts associated with the model while it also elucidates several technical relevant issues. A most interesting of those issues is the revelation that activation of the MDM plastic regime is possible only if the applied stress state allows the fibres to act like they are practically inextensible. Moreover, activation of the more dominant, between the two MDM plastic slip branches is possible only if conditions of material incompressibility hold, in addition to the implied condition of fibre inextensibility. A direct mathematical connection is thus achieved between basic, experimentally verified concepts of the bimodal plasticity model and a relevant mathematical model originated earlier from the theory of ideal fibre-reinforced materials. An additional issue of discussion involves the number of independent yield stress parameters that the bimodal theory needs to take into consideration. Moreover, an analytical expression is provided of a relatively simple mathematical surface that possesses all known features of the FDM yield surface; currently captured with the aid of both experimental and computational means. The present study is guided by the existing relevant experimental evidence which, however, is principally associated with the plastic behaviour of solids reinforced by strong fibres. Nevertheless, several of the outlined developments are expected to be applicable to composite materials containing a single family of more compliant or even weak fibres.     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.     

Analysis for enhancement of composite resistance to matrix cracking and interfacial debonding with rubber-coated reinforcement

An analytical model is presented for a unidirectional composite with a matrix crack straddling across rubber-coated fiber reinforcements. An expression is derived for the energy released in matrix cracking. A penny-shaped matrix crack configuration is chosen as an example. With the aid of Hankel's transform, a linear integral equation is derived and solved numerically for the reinforcement stress and energy release in terms of a parameter λ that depends on the composite material and crack geometry. The maximum stress intensity factor for a matrix crack in the unidirectional composite increases monotonically with λ, attaining the largest value for a crack in a homogeneous matrix material.     

Theory of “hot” cracks

During the solidification of a welding seam, there sometimes develop in it so-called “hot” cracks, leading to a defect in the article. An analogous phenomenon of the formation of “pits” and voids in ingots is observed during the metallurgical process. There is considered below a theoretical model, within whose framework the problem of the formation and development of a hot crack can be solved. The solution of this problem permits comparing different thermal conditions and selecting the most favorable. A statement of the problem is given and the fundamental assumptions are formulated. A study is made of the kinetics of the growth of a hot crack. The question of the asymptotic dimension of hot cracks with t →∞ is discussed, and simple sufficient conditions are given, with whose satisfaction a hot crack is not formed. A study is made of the development of a crack in the mathematically similar problem of brittle failure from local heating.