## Abstract

The non-classical problem of fracture mechanics of composites compressed along the layers with interfacial cracks is analysed. The statement of the problem is based on the model of piecewise homogeneous medium, the most accurate within the framework of the mechanics of deformable bodies as applied to composites. The condition of plane strain state is examined. The layers are modelled by a transversally isotropic material (a matrix reinforced by continuous parallel fibres). The frictionless Hertzian contact of the crack faces is considered. The complex fracture mechanics problem is solved using the finite-element analysis. The shear mode of stability loss is studied. The results are obtained for the typical dispositions of cracks. It was found that the interacting crack faces, the crack length and the mutual position of cracks influence the critical strain in the composite.

## 1. Introduction

An experiment, like every other event which takes place, is a natural phenomenon; but in a scientific experiment the circumstances are so arranged that the relation between a particular set of phenomena may be studied to the best advantage(Maxwell 1876).

The classical work of Griffith (1921) defined fracture mechanics as a new scientific area in natural sciences. In the latter half of the twentieth century, fracture mechanics was one of the most actively developed fundamental and applied areas in mechanics. At present, mechanics and fracture mechanics of micro- and nanocomposites (Guz & Guz 2006; Guz *et al.* 2007) and multiscale mechanics (Wing *et al.* 2006) are the topical and the most actively developed areas in engineering and physical sciences.

The work of Dow & Gruntfest (1960) was the first to describe the microbuckling phenomenon of fibres as a form of fracture of a unidirectional composite under compression. Since then, the beginning of the fracture process under compression is usually associated with the buckling of the microstructure of the material when the critical load is determined by parameters characterizing the microstructure of the composite rather than by the dimensions and shape of the specimen or structural member, i.e. with the internal or the surface instability phenomena according to Biot (1965). Later, other authors (Biot 1965; Rosen 1965; Guynn *et al.* 1992; Budiansky & Fleck 1994) described quantitatively this phenomenon using various approximate models. A detailed review of different approaches was given by Guz (1992). It was concluded by Guynn *et al.* (1992), Guz (1992, 1999), Soutis & Turkmen (1995) and Guz & Guz (2006) that approximate models are not very accurate when compared with experimental measurements and observations. The most accurate (exact) approach to study fracture due to the internal instability is based on the model of a piecewise homogeneous medium, when the behaviour of each component of the material is described by the three-dimensional equations of solid mechanics, provided certain boundary conditions are satisfied at the interfaces.

All works mentioned above considered perfectly bonded layers only. However, in practical cases, the assumption of perfect bonding between the adjacent layers in composite does not correspond to reality. Different cases of reduced adhesion in composites may occur during the fabrication process or in service. The classical Griffith–Irvin criterion of fracture or its generalizations are inapplicable for laminated composites compressed along layers and, therefore, along interlaminar defects, since in such a case all stress intensity factors and crack opening displacements are equal to zero.

Compressive behaviour of materials with interfacial defects were analysed within the scope of several approaches. The problems of the interaction of two parallel internal cracks and a periodic array of internal parallel cracks were studied within the framework of beams and shells theory. This approach was called the beam approach and was first used by Obreimoff (1930).

In the literature, one can find only the first step on the way to the exact solution of the problem of stability in the compression of composites with interlaminar defects. The exact solution to the problem of two half-planes compressed along interfacial cracks was given for the wide range of material models by Guz (1999) and Guz & Guz (2000). Guz (1997) introduced classification of cracks, and Guz (1998) and Guz & Herrmann (2003) suggested and substantiated the bounds for critical loads. The recent studies addressed the problem of the interaction of two parallel cracks (Winiarski & Guz 2006, 2007*a*,*b*).

Here, we investigate the effect of the interaction of cracks in a periodic array of interfacial cracks using the approach developed by Guz (1997). The following concept of fracture is adopted: the onset of fracture coincides with local loss of stability of the equilibrium state of the material that surrounds the crack. The shear mode of stability loss is considered. The models of ‘classical crack’ and a crack with Hertzian contact interaction of crack faces are compared.

## 2. Mathematical formulation

The composite under consideration consists of alternating layers. The periodic array of cracks is located on the different coplanar interfaces (figure 1). The layers are simulated by a compressible elastic transversally isotropic solid representing a matrix reinforced by continuous parallel fibres in continuum approximation. The elastically equivalent directions are perpendicular and parallel to the interfaces. Henceforth, all values referred to these layers will be labelled by indices *r* (0°-plies) and *m* (90°-plies).

The composite is in the condition of plane strain state under uniaxial compression along layers by ‘dead’ loads applied at infinity in such a manner that equal deformations along all layers are provided in the direction of loading (‘0’ means a pre-critical state)(2.1)

The static method of investigation of static problems of the three-dimensional theory of stability of deformable bodies can be used, since the conditions sufficient for the applicability of this method were proved to be satisfied in the case of compression by dead loads (Guz 1999).

The following eigenvalue problem must be solved. The stability equations for each of the layers are (Guz 1999)(2.2)where *t*_{ij} are components of a non-symmetric Kirchhoff stress tensor.

The stress and displacement perturbations are related as (Guz 1999)(2.3)

The components of *ω* tensor depend on the material properties and the load.

The layer interfaces consist of zones of perfectly bonded layers, *S*_{pb}, and cracks, *S*_{cr}(2.4)

(2.5)

The boundary conditions should be written for each interface as follows. On the interfaces with perfect bonds, the conditions of stress and displacement continuity are(2.6)Formulating the boundary condition on the crack surfaces, the following two possible cases are taken into consideration. In the first case, the modes of stability loss with an ‘open’ shape for the cracks are analysed(2.7)In the second case, the shear-type modes of stability loss are studied, where the possible closing of crack surfaces and their interaction is investigated. It should be noted that the sizes of the contact zones are not known in advance; they are part of the solution. Therefore, the crack surface, consists of zones of ‘open cracks’, and contact zones, (2.8)Here, the frictionless Hertzian contact of the crack faces is assumed for the contact zones(2.9)In the case of classical cracks, where interpenetration of crack faces is allowed, the conditions (2.7) are applied to both considered cases of stability loss.

Besides that, in all cases, the condition of the attenuation of perturbations when moving away from the cracks must be satisfied(2.10)

## 3. Results and discussion

In order to solve the above-mentioned problem, the commercially available finite-element code Abaqus was used to conduct the numerical analysis. The finite-element model assembly of the composite structure, asymmetry planes (the shear mode) and symmetry planes (the extensional mode) are shown in figure 1. The crack length *a* is related to lamina thickness by parameter *a*/*h* (‘the crack size’). The mutual location of the cracks is characterized by parameter *b*/*a* (‘the crack distance’).

The convergence of the finite-element solution to the exact solution was examined by means of *p*-convergence and *h*-convergence procedures following Fung & Tong (2001).

The results of the eigenvalue analysis for the array of parallel cracks are shown in figure 2*a*,*b*. The experimental mechanical properties for each layer are shown in table 1.

Figure 2*a* shows the critical strain versus the crack size, *a*/*h*, when *b*/*a*=0 and the layer volume fraction *V*_{L}=0.2. The highest critical strain (2.78%) coincides with the critical strain value for a microcrack calculated analytically by Guz & Guz (2000). Here, the surface instability phenomenon is the governing mechanism of the onset of fracture. The influence of frictionless Hertzian contact of the crack faces on the critical compressive strain is clearly visible. The strain level is higher than for the case of classical cracks, where interpenetration of crack faces is permitted. The results for classical cracks and cracks with interacting crack faces coincide for crack sizes *a*/*h*<0.1, where the surface instability phenomenon is the governing mechanism of the onset of fracture and the microcrack-like buckling behaviour of cracks occurs (Winiarski & Guz 2006). Therefore, for microcrack-like buckling behaviour of cracks, the frictionless Hertzian contact of the crack faces does not alter the critical compressive strain level. The effect of contact of the crack faces on the critical strain for the volume fraction of 0°-plies *V*_{L}=0.2 is the most significant for crack sizes 0.1>*a*/*h*>15. For the crack sizes *a*/*h*>10, the influence of interacting crack faces on the strain is decreasing, and for crack sizes *a*/*h*>50 the results coincide.

Figure 2*b* shows the critical strain versus the layer volume fraction, *V*_{L}, when *b*/*a*=0 and *a*/*h*=20. The lowest critical strain level corresponds to the layer volume fraction *V*_{L}=0.5. It is worth noting that the results for both approaches to crack modelling coincide for the layer volume fraction within the range 0.3<*V*_{L}<0.78. Here, the internal instability phenomenon is the governing mechanism of the onset of fracture. Whereas for very small and large volume fractions of 0°-plies, the influence of frictionless Hertzian contact of the crack faces on the critical strains is significant, the strain level is over 100% higher than for the case of classical crack. The surface instability phenomenon is the governing mechanism of the onset of fracture for 0.01≥*V*_{L}≥0.99. The results for the crack distance *b*/*a*>0 reveal that critical compressive strains are increasing, except for the microcrack-like buckling regions (Winiarski & Guz 2006, 2007*a*,*b*).

## 4. Conclusions

The effect of the interaction of cracks in a periodic array of interfacial cracks in a layered material under compressive loading was investigated for the shear mode of stability loss using the models of classical crack and a crack with Hertzian contact interaction of the crack faces.

The effect of the crack size and the crack distance was determined. The critical strain decreases with the increasing crack size. The highest critical strain for very small crack sizes coincides with the critical strain value for a microcrack. The influence of frictionless Hertzian contact of the crack faces was clarified. For the model of interacting crack faces, the critical strains are equal to or greater than the model of classical cracks. In the case of surface instability, the interacting crack faces do not alter the critical strain.

The particular governing mechanisms for the onset of fracture (internal or surface instability) were established for different volume fractions of the layers.

## Footnotes

One contribution of 20 to a Theme Issue ‘James Clerk Maxwell 150 years on’.

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