## Abstract

Several methods for the analysis of composite materials with periodic microstructure in which localized effects (such as concentrated loads, cracks and stationary/progressive damage) occur are resented. Owing to the loss of periodicity caused by these localized effects, it is no longer possible to identify and analyse a repeating unit cell that characterizes the periodic composite. For elastostatic problems, these methods are based on the combination of the representative cell method (RCM), the higher-order theory for functionally graded materials and often the high-fidelity generalized method of cells (HFGMC) micromechanical model. For elastodynamic problems, the combination of the dynamic RCM with a theory for wave propagation in heterogeneous media is used for the prediction of the time-dependent response of the periodic composite with localized effects. In the framework of the RCM, the problem for a periodic composite that is discretized into numerous identical cells is reduced to a problem of a single cell in the discrete Fourier transform domain. In the framework of the higher-order theory and the theory of wave propagation in composites, the resulting governing equations and interfacial conditions in the transform domain are solved by dividing the single cell into subcells and imposing the latter in an average (integral) sense. The HFGMC is often used for the prediction of the proper far-field boundary conditions based on the response of the unperturbed composite. The inverse of the Fourier transform provides the real elastic field at any point of a composite with localized effects. This research summarizes a series of investigations for the prediction of the behaviour of periodic composites with localized loading, fibre loss, damage and cracks subjected to static and dynamic loadings under isothermal and full thermomechanical coupling conditions.

## 1. Introduction

The behaviour of composite materials with periodic microstructure can be micromechanically predicted by identifying a repeating unit cell that can be analysed based upon the constitutive response of the phases, their volume fraction and by considering their detailed interactions. However, in the presence of local effects, such as distributed loads over a limited region, some fibre loss, cracks under static or dynamic loading and damage, the periodic character of the composite is lost and micromechanical analyses of periodic composites are no longer applicable.

The present paper summarizes a series of investigations in which the behaviour of periodic composites with several types of localized effects have been analysed and predicted. In general, these analyses are based on the combination of three approaches. In the first one, the representative cell method (RCM) [1,2] is used. Here, the composite is discretized into numerous identical cells, and the discrete Fourier transform is applied on the governing equations, interfacial and localization conditions. This reduces the original problem to a single-cell problem in the transform domain, where specific interfacial and boundary conditions are imposed. The second approach forms the solution of this cell problem in the transform domain, which is performed using the higher-order theory for functionally graded materials (HOTFGM) [3] in the static case, and wave propagation in composites theory [4,5] in the dynamic case. It is often necessary also to use the high-fidelity generalized method of cells (HFGMC) micromechanical model [3,6], which is based on the homogenization technique for the prediction of the far-field unperturbed periodic composite. The inverse of the Fourier transform provides the elastic field at any point of the composite with localized effects.

It should be emphasized that the direct use of a numerical method (such as the finite element procedure or the HOTFGM) to analyse a composite with localized effects is very difficult and cumbersome. This is due to the fact that, in the absence of the existence of a repeating unit cell, only the far-field boundary conditions are known. Hence, one needs to simulate a perturbed composite region occupying a large number of repeating unit cells, anticipating that the effect of the imposed inaccurate boundary conditions will diminish in the vicinity of the perturbations, which should be located quite far away from the external boundaries. The present approach, on the other hand, allows one to model an infinite region but requires multiple analyses of the single representative cell only, thus significantly reducing the computational cost of the modelling.

This paper is organized as follows. Section 2 presents the theory in its most general form with a localized thermomechanical loading that forms a generalization of Ryvkin & Aboudi [7]. This is follows by §3 that summarizes the analysis of a composite with fibre loss, the details of which have been presented by Ryvkin & Aboudi [8]. Section 4 presents the analyses for periodically layered materials with a transverse crack [9], fibre-reinforced composites with a penny-shaped crack [10] and a crack in porous materials [11]. Section 5 considers the localization problems of cracks in periodically layered composites subjected to dynamic loadings, under isothermal [5] and full thermomechanically coupled conditions [12,13]. Finally, §6 presents the effect of localized stationary and progressive damage [14], including cracks in composites [15].

## 2. Localization due to thermomechanical loadings in composites

Consider a composite with a periodic microstructure that is subjected to a localized thermomechanical loading. As has been discussed above, the periodicity of the problem is lost and the composite behaviour can no longer be determined from a micromechanical theory of composites with periodic stress field where a repeating unit cell can be identified and analysed.

For simplicity, figure 1 illustrates the case of a doubly periodic fibre-reinforced material in which the circular elastic fibres are oriented in the 1-direction, surrounded by the elastic matrix material. In accordance with the RCM [1], the periodic composite shown in figure 1 is viewed as an assemblage of bonded identical cells labelled by the two indices (*K*_{2},*K*_{3}), where *K*_{2},*K*_{3}=0,∓1,∓2,…. In the most general case of a composite with triply periodic microstructure, its region is divided into identical cells extending in three directions, labelled by (*K*_{1},*K*_{2},*K*_{3}), where *K*_{1},*K*_{2},*K*_{3}=0,∓1,∓2,…. The composite is described with respect to global coordinates (*x*_{1},*x*_{2},*x*_{3}). In addition, in each cell, local coordinates are introduced whose origin is located at its centre. It is assumed that, on the boundaries of the cell (0,0,0), a traction is applied. Figure 1, for example, exhibits again the doubly periodic case where the applied traction is directed in the normal 2-direction. It is, however, possible to apply other types of loading such as a biaxial, transverse shear, axial shear or a temperature deviation *θ* or any thermomechanical combinations. These applied loadings give rise to periodic composites with localized loading effects. In all these cases, owing to the application of the localized thermomechanical loading, the periodicity of the elastic field is lost.

In the following, the governing equations of the considered problem are presented. In the absence of body forces, the equilibrium equations in any cell (*K*_{1},*K*_{2},*K*_{3}), whose dimensions are *D*, *H* and *L* in the 1-, 2- and 3-directions, respectively, are given by
2.1
where ** σ** is the stress tensor. The constitutive equations for a thermoelastic material are
2.2
where

**and**

*C***are the stiffness and thermal stress tensors of the materials occupying any cell (**

*Γ**K*

_{1},

*K*

_{2},

*K*

_{3}),

**is the strain tensor and**

*ϵ**θ*is the temperature deviation (from a reference temperature) that is applied in cell (0,0,0) only, reflecting a localized thermal loading due to the existence of the Kronecker delta

*δ*

_{l,m}. The generalization to a temperature localization over more cells is straightforward. The strains are related to the displacement gradients in the standard form: 2.3

Let the vectors be defined by
2.4
These vectors assemble the three displacement components and three traction components , *j*=1,2,3, on a plane perpendicular to the *x*_{p}-axis at the cell (*K*_{1},*K*_{2},*K*_{3}) boundaries.

In addition to the requirements that the tractions and displacements should be continuous at the interfaces separating the fibre and matrix regions, continuity of displacements and tractions (*j*,*p*=1,2,3) between adjacent cells should be imposed. Thus,
2.5
where (*p*=1,2,3) define any possible discontinuity in the displacements and tractions on the cell's boundaries, in the 1-, 2- and 3-directions, respectively.

In the special doubly periodic case shown in figure 1, where the self-equilibrated normal tractions are acting in the 2-direction and applied on the boundaries between cells (0,0), (−1,0) and (1,0), the corresponding discontinuity vector is given by
2.6
where *σ*_{0} is an amplitude factor. Similar expressions can be established for other types of combined mechanical loadings, where *θ*=0.

For a thermal loading, on all boundaries are equal to zero, and *θ*≠0. Obviously, various combined thermomechanical loadings can be applied. It should be noted that, although the present problem formulation describes tractions that are applied on the boundaries of cell (0,0,0), it is also possible to apply such tractions at the boundaries and interior of several cells.

Let us apply on the formulated problem (2.1)–(2.5) the triple discrete Fourier transform. This provides the transformed component of the displacement, for example, in the form 2.7 As a result, we obtain in the transform domain a representative cell problem in the region , and . The governing equations for this cell, which correspond to equations (2.1)–(2.5), are 2.8 2.9 2.10 and 2.11 In the aforementioned two-dimensional case, equation (2.6) takes the form 2.12 Equations (2.11) are referred to as the Born–von Karman type of boundary conditions.

It is readily seen that one needs to solve equations (2.8)–(2.11) in the transform domain, where the identity of the cell (*K*_{1},*K*_{2},*K*_{3}) has disappeared. This solution can be obtained using the higher-order theory [3]. In the framework of this theory, the cell in the transform domain is divided into several subcells (figure 1*b,c*), in every one of which the displacement vector is expanded into a second-order polynomial. The unknown coefficients of this expansion (microvariables) are determined by imposing the equilibrium equations (2.8), the constitutive equations (2.9), the continuity of displacements and tractions at the interfaces between the various constituents as well as the Born–von Karman conditions (2.11). All these equations and conditions are implemented in the framework of this theory in the average (integral) sense.

Once this solution has been achieved, the actual thermoelastic field can be readily determined by applying the inverse transform, which is given by
2.13
at any location in cell (*K*_{1},*K*_{2},*K*_{3}). In practice, the solution of equations (2.8)–(2.11) is determined for a spectrum of −*π*≤*ϕ*_{1}≤*π*, −*π*≤*ϕ*_{2}≤*π* and −*π*≤*ϕ*_{3}≤*π*, and the triple integrals in (2.13) are approximated by Gauss numerical integration (say), yielding
2.14
where , and are the Gauss roots, and *w*_{m1,m2,m3} are the corresponding weighting factors. Verifications of the method of analysis in the two-dimensional case of a composite with localized applied loadings together with some results are given by Ryvkin & Aboudi [7].

As has been mentioned above, the triple integration in formula (2.13) can be carried out using, for example, Gauss numerical integration (2.14). This requires the evaluation of the integrand at a sufficient number of points within the integration domain −*π*≤*ϕ*_{p}≤*π*, *p*=1,2,3. Furthermore, in the vicinity of the point (*ϕ*_{1},*ϕ*_{2},*ϕ*_{3})=(0,0,0), the integrand possesses high gradients, thus requiring dense integration points. An alternative efficient procedure is based on a shifted finite discrete Fourier transform. To this end, the infinite composite domain is replaced by a sufficiently large finite domain on the boundaries of which the elastic field perturbations caused by the localized effects are negligibly small. The finite domain consists this time of (2*M*_{1}+1)(2*M*_{2}+1)(2*M*_{3}+1) cells such that *K*_{p}=0,±1,±2,…,±*M*_{p} with *p*=1,2,3. Accordingly, equation (2.7) is replaced by
2.15
where
2.16

The inverse transform that corresponds to equation (2.15) is given by the triple sum: 2.17 Both the shifted transform (2.15) as well as its non-shifted version in which 2.18 can be used.

The computational efficiency of the present approach for the localization analysis can be illustrated by considering a doubly periodic composite with localized effects. Suppose that the cell is divided in the framework of HOTFGM into 40×40 subcells (which has been found to provide accurate results in most cases). The HOTFGM analysis requires the solution of 15 unknowns in each subcell (i.e. 30 unknowns in the transform domain). This discretization requires for each combination of , and the solution of a sparse system of 48 000 algebraic equations. Let us assume that the region is divided into 21×21 cells (i.e. *M*_{2}=*M*_{3}=10). A direct numerical solution with the same number of degrees of freedom would require solving a system of 40×40×30×21×21≈21×10^{6} equations, which is very large.

## 3. Localization due to fibre loss in composites

Consider a periodic fibre-reinforced composite that is subjected to a far-field loading. It is assumed that, owing to damage or a production defect, several fibres are missing. Consequently, the periodicity of the composite has been lost. This problem has been analysed by combining three approaches. In the first one, the non-periodic elastic field is generated by a summation of a series of auxiliary problems (Green's functions), every one of which is a solution of a problem of a periodic composite in which all fibres exist, the far-field loading is absent and the stress field is generated by jumps in the tractions at certain locations of the fibre–matrix interface. For a fibre–matrix interface within cell (0,0), for example, of a doubly periodic fibre-reinforced composite, these jumps can be expressed by 3.1

where are the components of the traction vector, , **n** is the unit normal vector at a point located on the fibre–matrix interface of cell (0,0), and the vector represents the applied traction that accounts for the jump of the traction at this point. The method of solution of this problem is based on the implementation of the RCM that has been fully described in §2, in conjunction with the HOTFGM, which forms the second approach. In the third one, the periodic elastic field in a periodic composite in which all fibres exist and are perfectly bonded to the matrix needs to be established. This field can be determined from the HFGMC micromechanical analysis. The superposition of the last two solutions generates the requested thermoelastic field of the composite with a missing fibre, the boundaries of which are traction-free. This localization problem can be generalized for the analysis of composites with several missing fibres. Verifications of this approach and results for a single fibre loss have been given by Ryvkin & Aboudi [8].

## 4. Localization due to cracks in composites

In the present section, the analysis of periodic composites in which cracks are embedded is presented. As crack existence destroys the periodicity of the material, it is shown how the RCM approach in conjunction with the HOTFGM allows one to analyse the behaviour of such composites.

### (a) A transverse crack in periodically layered composites

Consider a periodically layered composite subjected to a far-field tensile normal loading , with an embedded transverse crack, oriented in the normal direction to the layering (figure 2). The composite consists of alternating stiff and soft layers. The behaviour of the cracked composite can be analysed and established by using the RCM that has been discussed in §2, specified for the doubly periodic composite case, together with the HOTFGM. The RCM in conjunction with HOTFGM is used for the construction of a series of Green's functions for the displacement jumps in *N* segments along the crack's line (*i*,*j*=1,…,*N*). The applied unit displacement jumps in the -direction on a segment −*Δ*/2≤*x*_{3}≤*Δ*/2 at *x*_{2}=0 can be represented by
4.1

A traction-free crack is achieved by superimposing the applied far field as
4.2
with *j*=1,…,*N*, and being the far-field normal stress in the stiff and soft layers, respectively. Verifications of this analysis, together with various types of results, can be found in Ryvkin & Aboudi [9].

### (b) A penny-shaped crack in periodically fibre-reinforced composites

The previous analysis of a transverse crack in periodically layered composites can be extended to investigate the behaviour of periodically fibre-reinforced composites, in which circular fibres are oriented in the 1-direction, with an embedded penny-shaped crack at *x*_{1}=0 in the *x*_{2}–*x*_{3} plane, thus resulting in a broken fibre. Here, the full RCM three-dimensional analysis of §2 is used to construct Green's functions in *N* rectangular domains at *x*_{1}=0 (*i*,*j*=1,…,*N*) in the *x*_{2}–*x*_{3} plane. These Green's functions are generated at , by applying a unit displacement jump on rectangle *j*. This applied unit displacement jump in the *x*_{1}-direction in the rectangle *R*_{i}, located at the plane *x*_{1}=0 at cell *K*_{1}=*K*_{2}=*K*_{3}=0, can be represented by the relations
4.3
The traction-free crack surface is obtained by the following superposition of Green's functions:
4.4
where is the remote normal stress in the fibre. Various verifications of this analysis and results are given by Ryvkin & Aboudi [10]. A similar approach has been presented by Ryvkin & Aboudi [11] for the analysis of an infinite elastic plane weakened by a doubly periodic system of square voids.

## 5. Localization due to cracks in composites subjected to dynamic loadings

### (a) The isothermal case

Just like the previously discussed cases, a crack in a periodic fibre-reinforced composite that is subjected to dynamic loadings destroys the periodic character of the composite. In this time-dependent problem, however, the previous analysis needs to be generalized first by extending the RCM method to the dynamic case by the construction of time-dependent Green's functions in the transform domain. In addition, the resulting dynamic equations need to be solved by establishing an appropriate theory for elastic wave propagation in composite materials.

In the present dynamic case, equation (2.1) takes the form
5.1
where *ρ* is the mass density of the material and a dot denotes differentiation with respect to time *t*. Under the present dynamic circumstances, the other equations in §2 have the same form, except that all variables are time-dependent.

In Aboudi & Ryvkin [5], the two-dimensional case of the sudden appearance of a transverse crack in a periodically layered composite has been considered. There, the shifted finite Fourier transform (2.15) has been used. The dynamic counterpart to the applied jump conditions of equation (4.1) in the *x*_{2}-direction at the segment −*Δ*/2≤*x*_{3}≤*Δ*/2 at *x*_{2}=0 that are presently required to generate the dynamic Green's functions are given by
5.2
where Hv(*t*) is the Heaviside step function and *j*=1, *j*=2 and *j*=3 correspond to loadings in mode III, I and II, respectively.

The theory for transient wave propagation in composites that has been presented in Aboudi & Ryvkin [5] is a generalization of the approach that was originally formulated by Aboudi [4,16]. The original theory was limited to deal with specific types of impactive loadings that preserve certain symmetries of the wave motion. The extended version that has been presented by Aboudi & Ryvkin [5] is very general, as it is able to accommodate any type of applied boundary conditions irrespective of whether a symmetry exists or not. As can be expected, the application of this theory requires an incremental procedure in time.

### (b) The full thermomechanically coupled case

Consider a periodic composite subjected to dynamic loadings with a transverse crack. When the full thermomechanically coupled equations are used, it is possible to predict, in particular, the induced temperature in the vicinity of the crack's tip and its surrounding. In this non-isothermal case, in addition to the elastodynamic equations, the coupled energy (heat) equation and the continuity of temperature and heat flux at the boundaries of the cell must be incorporated. In conjunction with the RCM method, the energy equation is given by
5.3
where *c*_{v} and ** q** are the specific heat and heat flux vector, respectively, and

*T*

_{0}is a reference temperature. The heat flux is given according to Fourier's law by 5.4 with

*κ*being the thermal conductivity of the material. The continuity of the temperature and heat flux between adjacent RCM cells are given by equation (2.5) in which the temperature and heat flux have to be included in the elements of vectors , i.e. 5.5 where and are the value of the temperature and thermal flux on a plane perpendicular to the

*x*

_{p}-axis at the cell (

*K*

_{1},

*K*

_{2},

*K*

_{3}) boundaries.

In addition, the theory of isothermal wave propagation in heterogeneous media that has been presented by Aboudi & Ryvkin [5] has to be generalized to include thermomechanical coupling effects. A full thermomechanically coupled analysis has been presented by Aboudi [12,13] for a spatially two-dimensional case of a transverse crack in periodically layered composites subjected to dynamic loadings. The resulting analysis provides the temperature variations along the crack's line and its distribution around the crack, which are shown in various circumstances.

## 6. Localization due to damage in composites

Consider a periodically fibre-reinforced composite in which stationary or evolving damage takes place in a confined region (figure 3*a*). Owing to this localized phenomenon, the periodic character of the composite has obviously been lost. Here, too, the RCM approach in conjunction with the HOTFGM can be used to determine the behaviour of the damaged composite.

The constitutive equation of the thermoelastic phase is given by
6.1
where *D* is the current value of the isotropic damage variable. At any instant, the expression for the damage is given by [17]
6.2
where 〈*x*〉=*x*, if *x*>0 and 〈*x*〉=0 for *x*<0. In this equation, at any instant of loading, *Y* _{max} denotes the resulting maximum value of the damage energy release rate *Y* , observed during the incremental procedure. The latter is given by
6.3
where *E* and *ν* are the Young's modulus and Poisson's ratio of the material, and
6.4
being the equivalent and hydrostatic stresses, respectively. In equation (6.2), *Y* _{D} is the value of *Y* at the initiation of damage, and *S* and *s* are material parameters.

In the framework of RCM, the composite region is divided into cells denoted by (*K*_{1},*K*_{2},*K*_{3}), as shown in figure 3*b* in the two-dimensional case. The constitutive equation (6.1) can be written in the form
6.5
where *σ*^{e(K1,K2,K3)} can be referred to as the eigenstress that is given by
6.6
In this relation, undamaged regions are associated with *D*=0.

In the present analysis, the RCM is used without the need to generate a series of Green's functions, in conjunction with the superposition of the far field. For a unidirectional composite that is analysed in the region −*H*≤*x*_{2}≤*H* and −*L*≤*x*_{3}≤*L* (figure 3*b*), the boundaries of this region must be far away from the localized damage. It has been shown by Aboudi & Ryvkin [14] that the corresponding boundary conditions can be established in the finite discrete Fourier transform domain. In the more general three-dimensional case, these boundary conditions, applied over the representative cell region , and , can be expressed as
6.7
6.8
and
6.9
Here *m*_{p}=−*M*_{p},…,*M*_{p}, *p*=1,2,3, and *δ*_{p} denotes the vector of the displacement differences at the faces of the *p*-boundaries of the domain, which can be established from
6.10
and , and are the average strains of the unperturbed periodic composite, which have to be determined from its effective compliance ** S*** and thermal expansion coefficient

*** tensors, in conjunction with the appropriate imposed traction boundary conditions and temperature deviation**

*α**θ*: 6.11 The effective compliance and thermal expansion coefficient tensors of the (unperturbed) composite can be determined by the HFGMC micromechanical model [6]. More details, verifications, discussions and results for stationary damage, evolving damage and cracks in periodically layered composites are given by Aboudi & Ryvkin [14]. This approach has been recently extended by Ryvkin & Aboudi [15] for the analysis of periodically layered thermoelastic composites in which systems of transverse and interfacial cracks have been developed in its brittle constituents. In particular, the shielding effect of interface debonding for cracking in the soft layers in the composite is quantified, and the specific case of an incomplete layer is addressed. The same approach has also been followed for the prediction of the electromagnetothermoelastic field distributions of smart materials with localized cracks, inclusions and cavities [18].

## 7. Conclusions

It has been shown that the behaviour of periodic composites with localized effects can be analysed and predicted. The composite consists of thermoelastic phases and is subjected to either static or dynamic loading. The localized effects appear in the form of concentrated loadings, loss of several fibres, transverse and various systems of cracks, and stationary and evolving damage. The analysis is based on the combination of two and more often three distinct theories, namely, the RCM, HOTFGM and HFGMC. Future investigations should address the incorporation of inelastic effects.

## Footnotes

One contribution of 17 to a Theme Issue ‘A celebration of mechanics: from nano to macro’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.