Interactions between multiple rigid lamellae in a ductile metal matrix: shear band magnification and attenuation in localization patterns

A ductile matrix material containing an arbitrary distribution of parallel and stiff lamellar (‘rigid-line’) inclusions is considered, subject to a prestress state provided by a simple shear aligned parallel to the inclusion lines. Because the lamellae have negligible thickness, the simple shear prestress state remains uniform and its amount can be high enough to drive the matrix material on the verge of ellipticity loss. Close to this critical stage, a uniform remote Mode I perturbation realizes shear band formation, growth, interaction, thickening or thinning. This two-dimensional problem is solved through the derivation of specific boundary integral equations, holding for a nonlinear elastic matrix material uniformly prestressed; the related numerical treatment is specifically tailored to capture the stress singularity present at the inclusion tips. Results show how complex localized deformation patterns form, so explaining features related to the failure mechanisms of ductile materials reinforced with stiff and thin inclusions. In particular, the influence of the inclusion distribution on the shear bands pattern is disclosed. Conditions for the magnification (the attenuation) of the localized deformations are revealed, fostering the progress (the setback) of the failure process.


Introduction
A distribution of hard inclusions enhances stiffness of a soft and ductile matrix, thus originating a material able to combine rigidity with toughness. This combination would not only realize the dream of medieval sword makers, but is highly requested in a number of advanced technical applications. For this reason, a great research effort has been directed to the development of metal matrix composites (MMC), where the matrix is a compliant metal, for instance, aluminum, magnesium, or titanium, while the reinforcement, in the form of particles, platelets, short or continuous fibres, may be another metal or a different, stiffer and stronger material, such as a ceramic (in which case the composite is called 'cermet' [46]). In this field, stiff lamellar inclusions are common and can be present as parallel distributions, as for instance in materials mimicking nacre [17]. Due to their high-contrast stiffness with the matrix and their high slenderness, the lamellae are also called 'rigid-line inclusions'.
Although beneficial for stiffness (and other properties such as hardness and resistance to decomposition by heat or chemical attack), the introduction of stiff lamellar inclusions plays a complex role on the effective strength. It has been proven both theoretically [3,6,11,13,21,30,32,39] and experimentally [27,28,35,37] that stiff lamellar inclusions create a strong stress concentration in the matrix material, which may lead to premature failure by shear band or crack nucleation and growth [47]. Therefore, the study of shear banding represents a key for the design of materials with superior mechanical properties.
The framework introduced by Dal Corso and Bigoni [12] is extended in the present article to the analysis of interactions between multiple rigid-line inclusions, all aligned parallel to a simple shear deformation of arbitrary amount, applied to an infinite matrix material. Under the assumed distribution of inclusions, the prestress state introduced through shearing remains uniform and can be increased up to the verge of ellipticity loss. A new boundary integral equation is obtained, governing the incremental response of a uniformly prestressed elastic material containing the rigid-line lamellae and wherê∞ 1,1 and̂∞ 2,2 define the two normal incremental strains at infinity, which are independent variables when the material is compressible, but become constrained bŷ∞ 2,2 = −̂∞ 1,1 in the incompressible case, eqn (7). The presence of the rigidline inclusions introduce further constraints on the perturbation of the mechanical fields. In particular, the incremental displacement field along each line inclusion is constrained to suffer an incremental rigid-body movement 1 (̂1,̂2) = ( ) 1 , , where ( ) 1 and ( ) 2 denote the incremental translation of the inclusion center in thê1-̂2 coordinate system, while ( ) is the incremental rigid-body rotation of the inclusion.
Under quasi-static conditions, the incremental stress field̂obeys the incremental equilibrium equationŝ and generates tractions on each inclusion providing null values for the incremental resultant forces and moment, respectively, where the brackets [[⋅]] denote the jump of the relevant quantity across the upper and lower face of the rigid-line inclusion, Finally, it is instrumental to recall that the incremental displacement , the nominal stress incremenṫ, and the constitutive tensor (expressed in the principal reference system 1 − 2 ) are related to the respective quantities (̂,̂,̂) in the reference system̂1 −̂2 through the following rotation rules, where are the components of the rotation tensor Before moving to the boundary integral equations (presented in the next section), it is worth noting that a square root singularity may arise in the nominal stress field at the two tips of each rigid inclusion. Following the stress intensity factor normalization based on strain measures as in [11,12,13,26,27,28,37], the incremental Mode I stress intensity factors (SIFs)̂( ) , at the left and̂( ) , at the right tip of the -th rigid inclusion are given bŷ where is the incremental shear modulus for a shear parallel to the 1 -axis, and the plus/minus sign is related to right/left tip. Finally, it is recalled that in the case of an isolated rigid-line inclusion ( =1, (1) = ) embedded in an unbounded matrix, uniformly prestressed and subject to a uniform remote incremental loading, the two tips display the same incremental stress intensification,̂( 1) , =̂( 1) , =̂( =1) , corresponding tô

Integral equation formulation
The incremental displacement̂at the point̂of a prestressed hyperelastic solid can be evaluated through the following integral equation [5,34,38] valid for a simply connected body ℬ, subject to mixed conditions on its boundary ℬ of unit outward normal̂. In equation (17), the tensor̂is the incremental nominal stress produced by the incremental displacement field̂whileî s the incremental nominal stress related to the infinite body displacement Green function̂ [5] (̂) = − 1 in whicĥ( ) is the acoustic tensor pertaining to the elastic body under consideration and is a unit vector, When an infinite body is subject to a remote linearly-varying displacement̂∞ and contains inclusions of domain ( ) , the incremental displacement̂at any point̂inside the medium, can be obtained from equation (17), where the boundary of the integral is reduced to the inclusion interfaces ( ) ( = 1, ..., ) wherêis the unit normal at the matrix-inclusion interface, outward to the matrix and therefore inward to the inclusion. Equation (20) is valid for any (compressible or incompressible) elastic material, uniformly prestressed [7, ?].
Considering that each inclusion has the shape of a zero-thickness line located as described in the previous section, the inclusion interface ( ) reduces to where ( )+ and ( )− are the two major sides of the -th inclusion of length ( ) and assumed of vanishing thickness 2 with corresponding unit normal̂defined aŝ In the limit of vanishing inclusion thickness, → 0, and considering the definition of the line domain ( ) , equation (3), of the unit normal̂, equation (23), and of the jump operator, equation (12), the integral equation (20) simplifies tô Published Recalling that across a rigid-line inclusion the displacement field is continuous, equation (24) finally reduces tô(̂) representing the boundary integral equation valid for an infinite medium, uniformly prestressed, containing rigid-line inclusions, incrementally loaded through remote incremental linear displacement̂∞. According to the rotation rule for second-order tensors, equation (13) 2 , the transformation between the displacement Green functions,̂in thê1 −̂2 reference system and in the 1 − 2 reference system, is given bŷ so that the boundary integral equation (26) can be rewritten aŝ where the superscript denotes the transpose operator. The jump in the nominal stress across the rigid-line inclusions and the 3 rigid-body displacements ( ( ) 1 , ( ) 2 , and ( ) ) are unknown at this stage. These quantities can be obtained by substituting the integral equation (28) into the 3 equations of incremental equilibrium of each rigid-line inclusion, eqns (11), and into the incremental rigid-body movement constraint (9), which can be rewritten aŝ for̂∈ ( ) and̂∈ ( ) with { , } = 1, ..., . Finally, towards the evaluation of the incremental Mode I stress intensity factor, equation (15), a differentiation of equation (26) in thê1 −̂2 reference system provides the gradient of incremental displacement According to the rotation rules, equation (13), the transformation involving the Green functions for gradient of the displacement iŝ, where the subscript after a comma applied to the Green function denotes differentiation with respect to the -axis, so that the boundary integral equation (30) can be rewritten aŝ Therefore, the incremental SIFŝunder Mode I, equation (15), are given bŷ Published

Numerical treatment of the boundary integral equation
The collocation method is exploited to numerically solve the integral equations for the incremental equilibrium (11) and rigid-body displacement (29) of the inclusion. A special technique has to be applied to treat the stress singularity at the inclusion tips [19,20]. In particular, the mixed boundary element method [10,19,31] is implemented here with the use of discontinuous elements [9,42], necessary to overcome the difficulty connected with the singularity of the traction occurring at corner points, which correspond to the lamellae tips in our case [15,29]. Before considering a specific collocation node positioning, the discretization for the lamellae and the related stress jumps is introduced in general terms (Fig. 2). The -th lamella is subdivided into ( ) elements, each of length Δ ( ) ( = 1, ..., ( ) ), therefore where the -th element of -th line inclusion is defined by the following coordinate subset (in the deformed state) wherê( Each element of the -th inclusion is characterised by three nodes, with the -th one characterized by the coordinate Figure 2: Discretization of the -th lamella in ( ) elements (upper part) with the specification of collocation points and shape functions (lower part). Elements are continuous quadratic except at the two tips, where semi-discontinuous elements are adopted to capture the square-root singularity of the incremental stress.
where ( )( ) ∈ [0, 1] is the dimensionless coordinate of the -th node. By introducing the nodal value of stress jump Published By means of representation (38), the incremental equations of rigid-body motion (29) are discretized at the -th node of the -th element along the -th inclusion ( = 1, 2, 3; = 1, ..., ( ) ; = 1, ..., ) aŝ and the equations of incremental equilibrium (11) for the -th inclusion ( = 1, ..., ) as Equations (39) and (40) From the practical point of view, the collocation nodes are considered to be located in such a way that two adjacent elements share their terminal node, so that the last node of every element coincides with the first node of the subsequent element,̂( and the discretization is symmetric along each line inclusion More specifically, the considered discretization involves continuous quadratic elements along the whole line inclusion except at the two tip elements ( = 1 and = ( ) ), where semi-discontinuous elements have been used. Considering that the quadratic shape functions ( ) for a master element are where ∈ [−1, 1] is the master element coordinate and ( = 1, 2, 3) are the collocation points, by assuming the collocation points located at 1 = −1, 2 = 0, 3 = 1 and through the change of variable = 2 − 1 ∈ [0, 1], the collocation points and the shape functions for the continuous quadratic elements in the inner part of the lamella follow as Differently, in order to capture the singularity at the inclusion tips, specific quarter point semi-discontinuous elements are considered for the two tip elements ( = 1 and = ( ) ) of each inclusion in order to properly display the square-root stress at the tips [9,18,45]. For the first element ( = 1), by assuming the collocation points located at 1 = −3∕4, 2 = 0, 3 = 1 and through the (nonlinear) change of variable = 2 √ − 1, the collocation points and the shape functions are and, because of symmetry, for the last element ( = ( ) ) are In all of the numerical evaluations presented in the next section, each rigid-line inclusion is discretized adopting a minimum number of elements ( ) = 45, through the following symmetric scheme. Two identical refined uniform meshes, with a minimum number of 20 elements each, are adopted near each tip for a length ( ) ∕5. A coarse uniform mesh, with a minimum number of 5 elements, is used in the central part of the inclusion (of remaining length 8 ( ) ∕5).
For the numerical computation of the incremental SIFs, a distance = 10 −3 ( ) from the inclusion tips, found to be sufficient for accurate estimation, has been considered. 5 Application to ductile metals: the J 2 -deformation theory of plasticity Incompressibility. Adopting incompressibility, the fourth-order elasticity tensor (which satisfies the major symmetry = ) assumes under broad hypotheses the form [4] where and * are incremental shear moduli depending on the state of prestress through the in-plane deviatoric stress , and the mean stress Π defined as being 1 and 2 the two principal values of the Cauchy prestress. In the present case, the latter is generated by a simple shear, aligned parallel to the 1 − 2 reference system, which in turn is inclined at the angle with respect to thê1 −̂2 reference system (wherê1 is parallel to the inclusion line). Due to incompressibility, the uniform Mode I loading applied at infinity, eqn (8), is subject to the constraint The incremental Green's function has been obtained by Bigoni and Capuani [5] for incompressible materials under plane strain conditions and is not repeated here for conciseness.
It is also interesting to note that the mean stress Π remains unprescribed in the following because, from the equation (48), it affects only the relation between incremental stress and incremental displacement gradient. Indeed, the unknowns of the boundary integral formulation are the incremental stress jumps from which the incremental displacement componentŝ1 and̂2 can be evaluated through equation (26). J 2 -deformation theory of plasticity. Henceforth, the interactions between shear bands and rigid-line inclusions is analyzed by restricting the attention to a matrix material obeying the J 2 -deformation theory of plasticity [25], whose nonlinear constitutive equations, when plane strain prevails, reduce to where is a stiffness parameter, ∈ (0, 1] is the hardening exponent, while 1 is the principal stretch along the 1 -axis, related to the shear prestrain through equation (4). Under the assumptions of plane strain and incompressibility, the incremental moduli and * for the J 2 -deformation theory material can be obtained as [4] = 2 For the non-linear elastic material (51), failure of ellipticity and shear band formation is determined as the solution of the nonlinear equation = ln 1 tanh ( for the critical stretch 1 at the ellipticity loss, corresponding to the critical level of shear prestrain . When the elliptic boundary is approached, two shear bands are predicted to emerge for a J 2 -deformation theory material, inclined at the angles ± 0 with respect to the principal axis 1 Finally, it is worth to note that therefore, one shear band direction is perfectly aligned with the inclusion axis, while the other is inclined twice the (anti-clockwise) angle with respect to inclusion line.

Validation of the boundary integral formulation
The developed boundary integral method is validated here through a comparison with available analytical solutions referring to two simple cases in which the matrix material is prestressed, but there is only one inclusion, or there are two inclusions, but the matrix material is in an unloaded state.

An isolated rigid-line inclusion in a prestressed material
The numerical methodology developed from the presented formulation is compared with the analytical solution obtained in [12] for an isolated rigid-line inclusion ( =1), embedded in an infinite incompressible material, prestressed through a simple shear, and subject to a uniform Mode I perturbation.
The mechanical state near the lamella is characterized in terms of its rigid-body rotation Γ (1) = (1) ∕̂∞ 1,1 and of the incremental nominal shear stress jump ]] ∕(̂∞ 1,1 ) measured across it, as the result of the applied perturbation. These two quantities are reported in Figure 3 for a J 2 -deformation theory material with hardening exponent = 0.4, for which ellipticity is lost at an amount of shear ≈ 1.462, and the shear band inclination is 0 ≈ 0.150 . In the figure, the stress jump is reported as a function of the coordinatê1∕ , for a prestrain = 0.992 ≈ 1.45. The rigid-body rotation Γ (1) is reported as a function of the prestrain ∕ ∈ [−1, 1]. The numerical solution (dashed line) practically coincides with the analytical one (continuous line), except at the node shared by the two external elements, characterized by different shape functions and representing the transition from the semi-discontinuous to the continuous quadratic element. This reveals that the latter struggles to fit in with the sharp turn due to the square-root singularity at the tip (see the detail at the centre of Figure 3). Although this visible gap delays the solution convergence only at that node, it can be reduced by further decreasing the mesh size there.

Two collinear rigid-line inclusions in a linear isotropic incompressible material (at null prestress)
A second validation of the proposed technique is performed through a comparison referring to the case of a linear elastic isotropic incompressible material at null prestress, which can be recovered from the J 2 -deformation theory by assuming an hardening exponent equal to the unity, = 1, and a null prestrain, = 0. The presence of two, = 2, collinear rigid-line inclusions of equal length is considered following the scheme presented in Fig. 4, now to be considered only in the case of the undeformed configuration (on the left in the figure). The geometry is specified by where > 2 defines the distance between the inclusion centroids (1) and (2) , the angle Φ is the inclination of the line joining the inclusion centers with respect to thê1-axis, and is the semi-length, equal for both inclusions. material subject to a simple shear of amount parallel to the horizontal axis. The relative position of two lamellae is described by the distance between their centroids, points (1) and (2) , and the angle Φ in the undeformed configuration, transformed respectively into , (1) , (2) , and after deformation.
Due to the symmetry of both geometry and loading, the incremental Mode I SIFs, eqns (15), referred to the four tips, satisfy the following identitieŝ (   2) , =̂ ( 1) , , are the incremental SIF at the inner and outer tips respectively, obtained analytically as [41] ( =2) , The SIFŝ( =2) , and̂( =2) , are plotted in Figure 5, normalized through division by the corresponding quantitŷ( =1) holding for the single inclusion, eqn (16). An excellent agreement is shown between the analytical values (continuous line) from eqn (58) and the numerical evaluation (spots) from the present boundary integral technique.

Interactions between shear bands and rigid lamellae
The developed formulation is exploited to analyze the incremental mechanical response to a remote Mode I perturbation, uniform at a large distance from the lamellae, when the matrix material (characterized by hardening exponent = 0.4) is prestrained up to a level close to ellipticity loss, = 0.992 . Consideration of this prestress state allows for the analysis of shear band formation and interaction. It is recalled that, in the case of an isolated rigid-line inclusion, the shear band direction aligned with the lamella axis is dominant, while the other, developed in the conjugate direction, remains 'weaker' [12,13]. This pattern is considered in the following as a reference state to disclose the influence on shear bands of various distributions of lamellae. To this purpose, recalling that the incremental strain̂is the symmetric part of the incremental displacement gradient,̂= (̂, +̂, )∕2, and it is deviatoric because of incompressibility (dev̂=̂), the results are reported in terms of the modulus of the perturbed incremental strain,

Prestressed matrix with two rigid-line inclusions
The interaction between two rigid lamellae ( = 2), characterized by different length and embedded in a ductile metal matrix is considered, when a Mode I incremental perturbation is applied, superimposed upon a certain prestress induced by a simple shear loading of finite amount. The position of the two rigid inclusions is defined by introducing the following relations between the coordinates of the inclusion centroids in the undeformed configuration (Fig. 4, left) (2) where is the distance between the two centers of the rigid inclusions and Φ is the angle measuring the inclination of the line joining the two centers with respect to thê1-axis. Subject to a simple shear deformation, the configuration of the rigid inclusions changes, so that the centroid coordinates 1 , 2 , singling out the deformed configuration, become ( Figure  4, right) where and are respectively the distance and the inclination, related to the initial values and Φ through the amount of shear as .
Because of the polar symmetry, the fields calculated for the initial angle Φ − are identical to those calculated for the angle Φ and therefore the investigation is restricted to Φ ∈ [0, ). Furthermore, elementary geometrical considerations lead to the observation that when > (1) + (2) (when < (1) + (2) ) the mechanical response is continuous (is discontinuous) at Φ = 0 and Φ = . The above described layout is analyzed in the following for specific geometries, so to dissect the respective influences of the various parameters. Two rigid-line inclusions with centers aligned on the same vertical axis in the undeformed state (Φ = ∕2). The modulus of the perturbed incremental strain | |̂−̂∞ | | is reported in Fig. 6 in the surroundings of two rigid-line inclusions. The latter have the inclusion centers aligned vertically in the undeformed state, so that the configuration is described by equation (60) with Φ = ∕2. The prestrain level is ≈ 0.992 and the inclusions are analyzed at varying distance = {0.04, 0.2, 0.5} (decreasing from the upper to the lower part of the figure). The cases of inclusions with same length ( (1) = (2) = ) and with different length ( (1) = 2 (2) = ) are reported on the left and on the right, respectively. Both cases show that, for a prestrain state close to the loss of ellipticity, the deviatoric strain is localized into two principal shear bands parallel to the inclusion lines. These two shear bands, when the two inclusions are sufficiently close, 'merge' in one thick shear band, completely enclosing both inclusions. It is interesting to note that the values of inclination Φ and distance also define the position of one inclusion in relation to the two shear bands inclined at 2 0 with respect tô1 and originating at the tips of the other inclusion in the simple shear deformation state. In particular, three special inclinations,Φ, Φ (−) , and Φ (+) , can be defined as those doi: https://doi.org/10.1016/j.jmps.2022.104925 and discriminating the following two interesting cases.
• When Φ =Φ, both tips of each inclusion touch the inclined shear bands originated from the tips of the other inclusion. Therefore, each inclusion is enclosed inside the domain defined by the inclined shear bands originated from the tips of the other inclusion; corresponding to prestressed configurations (at the ellipticity loss) where each inclusion lies partially inside ( (+) and  (−) ) or completely outside ( (+) and  (−) ) the strip region defined by the inclined shear bands originated from the tips of the other inclusion. Due to polar symmetry, the incremental rigid-body rotations of the two inclusions coincide where Γ ( =2) is the dimensionless rotation, and two of the four incremental SIFs at the lamellae tips, eqn (15), are coincident̂(

2)
, The influence of the inclusions' position with respect to the inclined shear bands is highlighted through the map of the modulus of the perturbed incremental strain | |̂−̂∞ | | in Fig. 9 and through the sketch of the incrementally deformed configurations in Fig. 10. In particular, the latter figure reports the incremental deformed configurations due to the Mode I perturbation for a mesh of identical squares in the undeformed state. In both figures, the centroids distance is constant, = 2 , while three values for the angle Φ are considered: Φ (+) = 0.399 (left), Φ = 0.550 ∈ (Φ (+) ,Φ) (center), and Φ = 0.700 (right).
For Φ = 0.550 (Figs. 9 and 10, central parts) an inclined shear band, emanating from right tip of the lower rigidline, intersects the central part of the upper lamella. The intersecting shear band results to be thin, so that a shear band intersecting a rigid lamella suffers a reduction in 'intensity'.
An opposite situation is visible for Φ (+) andΦ (Figs. 9 and 10, on the left and right). Here, the tips of the upper and lower lamellae are respectively aligned along one or two inclined shear bands, resulting in a strong amplification. More specifically, the case of the alignment of one tip pair (Φ (+) ) is substantially different from that of two tip pairs (Φ). Indeed, the aligned tips have opposite incremental displacement in the former case, becoming concordant in the latter. This implies that the incremental rigid-body rotation Γ ( =2) for the geometryΦ is much larger than that pertaining to the case characterized by Φ (+) . doi: https://doi.org/10.1016/j.jmps.2022.104925  To further confirm this behaviour, the incremental rotation Γ ( =2) (normalized through division by the corresponding rotation calculated for an isolated lamella, Γ ( =1) = ( =1) ∕̂∞ 1,1 ) is reported in Fig. 11, as a function of the distance (at fixed angle Φ, left) and of the angle Φ (at fixed distance , right). The special distributions corresponding to the alignment of one tip pair (Φ (+) and Φ (−) ), or two tip pairs (Φ), are also reported as dashed curves. In the former case, the rigid-body rotation attains its maximum value, while in the latter the rotation reaches a value close to a relative minimum. Interestingly, when the two lamellae are collinear (Φ = 0), the incremental rotation results unaffected by the distance > 2 and assumes the same value corresponding to that characteristic of the isolated inclusion, Γ ( =2) (Φ = 0) = Γ ( =1) . The above discussion is completed with the analysis of the two incremental SIFŝ( 1) , and̂( 1) , displayed in Fig. 12 (respectively on the upper and lower part), as functions of the distance (at a specific angle Φ, left) and of the angle Φ (at a specific distance , right). These curves are complemented by those reported as dashed curves and corresponding to the special geometries involving the alignment of one tip pair (Φ (+) and Φ (−) ) and of two tip pairs (Φ). From these curves, it can be observed that̂ ( 1) , (̂( 1) , ) is almost constant and close to the valuê( =1) pertaining to the isolated inclusion, when Φ ∈ (Φ, ) (Φ ∈ (0,Φ)). This behaviour is consequent to the fact that the singular field at the relevant tip is unaffected by those belonging to the other lamella, which is shielded by the inclined shear band.
Finally, the incremental SIFs display a strong variation near Φ =Φ, because the corresponding geometry is close to the alignment of two tip pairs.

Amplification, reduction, and shielding of shear bands pattern for multiple inclusions
Interactions between lamellae are now investigated for four different polar symmetric distributions of = 4 and = 5 inclusions, as reported in Fig. 13 Because of polar symmetry, the following identities hold for the incremental rotations and stress intensity factors Maps of the modulus of the perturbed incremental strain | |̂−̂∞ | | are reported in Fig. 14 for the four distributions of lamellae shown in Fig. 13, while the corresponding values of incremental rotations Γ ( ) and incremental stress intensity factorŝ( ) , are listed in Table 1, normalized through division by the corresponding quantity calculated for one lamella of length 2 , Γ ( =1) and̂( =1) . From the figure and the table it can be concluded that the mechanical response is strongly affected by the inclusion distribution, and in particular:   Fig. 13, normalized through division by the corresponding value (Γ ( =1) and ( =1) ) for the isolated lamella of length 2 . Moreover, the modulus of the perturbed incremental strain | |̂−̂∞ | | is almost constant in the region enclosed by the two outer inclined shear bands. It follows that, roughly speaking, the portion of solid enclosed between the two outer inclined shear bands, and encompassing all the five lamellae, approximately displays an incremental rigid-body rotation.

Conclusions
A specifically-derived boundary integral equation has been shown to provide a highly detailed description of the incremental strain fields near complex distributions of rigid-line inclusions, embedded in a matrix material. The latter is modelled according to the J 2 -deformation theory of plasticity and is subject to a simple shear of finite amount, before the application of a Mode I perturbation. The amount of shear can be high enough to bring the matrix material near failure of ellipticity, where the nucleation and growth of shear bands can be analyzed. This analysis reveals a series of new features related to the failure of a ductile material through shear bands, in particular, mechanisms of weakening or strengthening of shear bands caused by their interactions with the lamellae.
Although developed under quasi-static conditions, the present approach is more general than it might appear. Indeed, it can be extended to analyse the dynamic interaction between multiple lamellae and also to three-dimensional distributions [1,43,44], exploiting the Green's function provided by Argani et al. [2].   can be specified only once the collocation node positions are defined, with the source node located along the -th inclusion and the field node along the -th inclusion.