## Abstract

A ductile metal matrix (modelled as a nonlinear elastic material) containing a dilute suspension of an iso-oriented lamellar stiff phase (modelled as stiffeners, i.e. zero thickness, rigid inclusions) is subject to a simple shear of finite amount, parallel to the inclusion orientation, and subsequently perturbed through an incremental Mode I loading, uniform at infinity. Solution to this problem permits analytical investigations of the emergence of shear bands and their interaction with a rigid inclusion (involving a stress square-root singularity at its tip) and discloses the mechanisms of ductile failure in reinforced materials (explaining for instance the experimental evidence that shear bands tend to nucleate and grow parallel to thin hard inclusions). Finally, investigated beyond the elliptic range, the obtained solution becomes non-unique and reveals non-decay and singularity of the fields, facts that provide analytical justification for the difficulties associated with numerical treatment of shear bands.

## 1. Introduction

Experimental evidence in ductile matrix materials containing a second stiff phase shows that localized deformations in the form of shear bands nucleate at inclusion boundaries, where they grow and subsequently develop complex interactions between themselves and the second phase. Therefore, the presence of defects is crucially important in the understanding of failure in ductile matrix composites, particularly when defects involve stress concentrations, as is the case of the lamellar suspensions experimentally investigated in metals (Öztürk *et al*. 1991), plastics (Bigoni *et al*. 2008) and rocks (Misra & Mandal 2007). In these composites, only an analytical solution can detail the structure of singularities related to the sharpness of both lamellae and shear bands, while numerical approaches can hardly have the necessary resolution. The objective of the present article is to derive one such solution, through a generalization of results obtained by Dal Corso *et al*. (2008), and to employ it to investigate the complex interaction between singularities and shear bands at the inclusion tip. To this purpose, a rigid line inclusion, a so-called ‘stiffener’, is analysed embedded in an elastic incompressible infinite material (in the examples we will refer to a *J*_{2}-deformation theory material) deformed under plane strain. The medium is assumed to be subject to a simple shear deformation of finite amount parallel to the inclusion line, which generates a uniform state of strain and stress, both having principal directions *inclined* with respect to the stiffener line. Taking this configuration as reference in a relative Lagrangian description, the response is analysed to an incremental uniform Mode I loading at infinity,1 so that the solution can be explored near the border of ellipticity loss, where the incremental fields are shown to self-organize along shear bands.

The obtained Mode I loading analytical solution shows the development of two shear bands, with, for low hardening, the one closest in alignment with the stiffener being the more pronounced.2 Therefore, a stiffener embedded in a low hardening matrix tends to focus the deformation parallel to its line, a finding substantiated by the experimental results of Misra & Mandal (2007), referring to geological formations and to models prepared with polymethylmethacrylate containing metal inclusions.3

Finally, accepting the fact that decay of the incremental fields does not occur and incremental strain and stress becomes infinite along certain shear band directions, the obtained solution is extended and investigated outside ellipticity, namely, in the parabolic and hyperbolic regimes. Here, it is shown that the solution is not unique, so that one, two, three or four (one or two) differently inclined shear bands are predicted to become possible in the hyperbolic (parabolic) range. This result substantiates with an analytical solution the well-known difficulties connected to the numerical treatment of ill-posed (i.e. non-elliptic) boundary-value problems.

## 2. Stiffener neutrality during simple shear of an elastic material

With reference to figure 1, a block of a material is subject to a simple shear deformation when two displacement components are null (along axes and ) and the other component, , depends linearly only on , namely(2.1)so that, if a point at horizontally displaces of *s*, we can determine the dimensionless parameter *γ*=*s*/*h*, controlling the amplitude of shear deformation.

The left () and right () Cauchy–Green deformation tensors can be calculated from the displacement field (2.1) and result in(2.2)where and are the two unit vectors parallel to the and axes, respectively.

The eigenvectors of and define the principal Eulerian and Lagrangian axes, so that in the plane these are(2.3)(where *j*=1, 2) respectively, while the principal stretches result is given by(2.4)where *j*=1, 2. Note that(2.5)so that if *γ*>0 (<0) then *λ*_{1}>*λ*_{2} (*λ*_{1}<*λ*_{2}).

The eigenvectors (2.3) yield the inclination of the Eulerian, *ϑ*_{E}, and Lagrangian, *ϑ*_{L}, axes with respect to the -axis(2.6)

With reference to the Cauchy–Green deformation tensors (2.2), we note that a fibre parallel to the -axis, singled out by the unit vector remains unstretched, since(2.7)so that the fibres parallel to the -axis in figure 1 are so-called ‘zero elongation lines’ (Weissenberg 1948). As a conclusion, a stiffener parallel to the shear deformation leaves the material unperturbed.

For a material isotropic in the reference configuration, the Cauchy stress is coaxial to the left Cauchy–Green deformation tensor , so that in the Eulerian principal reference system we have the spectral representation(2.8)where, in the reference system, the unit vectors *v*_{1} and *v*_{2}, equations (2.3)_{1}, are now defined by the components(2.9)Therefore, in the reference system, we have(2.10)from which the well-known ‘universal relation’(2.11)is obtained.

Although all the subsequent developments are fully general, we will refer in the examples to a *J*_{2}-deformation theory material defined by the power-law strain energy density(2.12)where *K* is a positive stiffness parameter; *N*∈]0,1] is the strain-hardening exponent; and *ϵ*_{e} is the effective strain that, for the plane strain case, reduces to(2.13)in which *ϵ*=*ϵ*_{1}=log *λ*_{1} is the logarithmic stretch, a function of *γ* through equation (2.4). For a *J*_{2}-deformation theory material, we obtain(2.14)where *p* is the mean stress and *μ* and *μ*_{*} are incremental shear moduli that depend on the current stretch in the following way:4(2.15)where(2.16)

Considering small shear amplitude *γ*, the constitutive relations (2.14) become(2.17)

The deviatoric stress and the shear stress, made dimensionless through division by *K*, are reported in figure 2 for a *J*_{2}-deformation theory material, at (*a*) low (*N*=0.1) and (*b*) high (*N*=0.8) strain hardening. Note that it is evident from equation (2.17), that for *γ*=0, the curve representing the deviatoric stress has a horizontal tangent and the curve representing the shear stress has a vertical tangent. Although the vertical tangent is not visible in figure 2*b*, it is easy to check its existence; in fact, the incremental equations corresponding to equation (2.17) become(2.18)where represents the shear (mean stress) increment and *μ*_{*} tends to infinity when *γ* tends to zero. Moreover, the curve relative to *N*=0.1 coincides with that plotted by Harren *et al*. (1989, fig. 2).

## 3. Mode I perturbation of an elastic infinite medium prestressed through a finite shear parallel to an embedded stiffener

A rigid-line inclusion is analysed embedded in an elastic material, homogeneously prestressed within the elliptic regime by a simple shear and subject to an incremental Mode I deformation uniform at infinity (for uniform Mode II loading, the stiffener leaves the medium unperturbed). We start by briefly introducing the incremental constitutive equations.

### (a) Incremental constitutive equations

The incremental response of an incompressible, nonlinear elastic material homogeneously deformed under a plane strain condition, can be described through the linear relation (Bigoni & Dal Corso 2008)(3.1)where (.)^{T} denotes the transpose; is the increment of the nominal (unsymmetrical) stress tensor ; ∇** v** is the gradient of incremental displacement (satisfying the incompressibility constraint tr ∇

**=0); and is the incremental in-plane mean stress. The fourth-order constitutive tensor , describing orthotropy (aligned parallel to the current principal stress directions), is a function of the current state of stress, and its components (for instance given by Bigoni & Dal Corso 2008) are functions of the following dimensionless quantities:(3.2)where**

*v**T*

_{1}and

*T*

_{2}are the principal components of Cauchy stress and

*μ*and

*μ*

_{*}describe the material response to shear (

*μ*for shear parallel and

*μ*

_{*}for shear inclined at

*π*/4 with respect to

*T*

_{1}), and may be arbitrary functions of the current stress and/or strain.

For a *J*_{2}-deformation theory material, the dimensionless constants (3.2)_{1,3} are related to the principal stretch in the *x*_{1}-direction, namely *λ*_{1}, through the relations(3.3)while the in-plane stress parameter *η* remains unprescribed.

Introducing a stream function with the property(3.4)(so that the incompressibility condition is automatically satisfied), the differential equations governing incremental equilibrium yield the values of the roots *Ω*_{j} satisfying(3.5)According to the nature assumed by roots *Ω*_{j} as a function of parameters *ξ* and *k*, the incremental problem can be classified as elliptic (E), parabolic (P) and hyperbolic (H) (see for instance Bigoni & Dal Corso 2008, fig. 2). In particular, defining(3.6)where is the imaginary unit and and (Re[.] and Im[.] denote the real and imaginary part of the relevant argument), the elliptic regime corresponds to(3.7)We define the elliptic imaginary (EI) regime as(3.8)where there are four imaginary conjugate roots *Ω*_{j}, so that(3.9)and the elliptic complex (EC) regime as(3.10)where there are four complex conjugate roots *Ω*_{j}, so that(3.11)

Considering now a reference system rotated at an angle *ϑ*_{0} with respect to the axes defining the prestress directions and the orthotropy axes (figure 3), the linear constitutive relation (3.1) becomes(3.12)where the nominal stress increment, incremental displacement and its gradient, and the constitutive tensor can be expressed in the reference system as(3.13)through the rotation ** Q** defined as(3.14)In the reference system, the stream function employed in equation (3.4) becomes , where(3.15)and the property (3.4) now becomes(3.16)from which the incremental strain and stress fields can be obtained directly in the reference system.

### (b) Incremental perturbed solution

A rigid line inclusion of length 2*l* is considered embedded in an elastic material, which is subject to a uniform state of prestress produced by a simple shear parallel to the inclusion and defined by the shear amount *γ*. The homogeneous state of prestress has the principal stress axes inclined by *ϑ*_{E}, equation (2.6), with respect to the inclusion line and is taken as the reference state5 on which a perturbation corresponding to remote uniform Mode I incremental deformation is superimposed.

A reference system located at the stiffener centre, taken with the inclusion line parallel to the -axis, is inclined at an angle *ϑ*_{0} (taken equal to *ϑ*_{E}) with respect to the *x*_{1}–*x*_{2} system, defining the principal stress directions, as shown in figure 3.

The kinematical boundary conditions for a stiffener express the fact that this can only suffer an incremental rigid-body motion(3.17)so that , and *ω*_{S} represent unknown quantities to be determined as a part of the solution, by imposing boundary conditions to ensure equilibrium of the stiffener in terms of incremental ‘global’ axial and shear forces, and incremental moment, respectively,(3.18)where the brackets 〚.〛 denote the jump in the relevant argument, taken across the stiffener.

Owing to central symmetry considerations (with respect to the stiffener centre) involved in the far-field loading problem under analysis and the specific form of solution sought in the following, the boundary conditions (3.17) and (3.18) can be reduced to the following homogeneous incremental displacement gradient conditions:(3.19)plus the requirement that the normal stress increment be continuous across the stiffener(3.20)a condition allowing determination of *ω*_{S}.

Prescribing an incremental deformation at infinity and proceeding in analogy to the crack and shear band problems analysed by Bigoni & Dal Corso (2008), the stream function of the perturbed problem can be sought in the form6(3.21)satisfying automatically the decaying condition on the velocity, incremental strain and stress at infinity (in the elliptic regime) and providing a stress square-root singularity at the stiffener tips.

Imposing the stream function (3.21) to satisfy the boundary conditions along the stiffener line (3.19) and (3.20) yields the following linear problem for the complex constants *D*_{1} and *D*_{2}:(3.22)where the real constants *c*_{1j} and *c*_{2j} (*j*=1, …, 4) are defined as(3.23)and depend on the stiffener inclination *ϑ*_{0} and on the prestress and orthotropy parameters *ξ*, *k* and *η*.

We introduce the normalized stiffener rotation *Γ* as(3.24)and since analytical proof looks awkward, we have numerically checked that (i) the solution of the present problem is independent of the in-plane stress parameter *η*, (ii) the coefficients *D*_{j} (*j*=1, 2) solving system (3.22) satisfy the following two equations:(3.25)and (iii) the normalized stiffener rotation *Γ* (3.24) satisfies the conditions(3.26)

Defined in terms of incremental velocity gradient as in Dal Corso *et al*. (2008), the incremental stress intensity factor for Mode I loading is(3.27)resulting independent of the prestress parameters *ξ*, *k* and *η* (but *μ* may depend on the full set of current state variables) and of the angle *ϑ*_{0} between the stiffener and the directions of principal stress *T*_{1}.

The calculation of the incremental energy release rate and the incremental axial force in the stiffener (deferred to appendix A) generalizes results for a stiffener aligned parallel to the principal stress directions obtained by Bigoni *et al*. (2008) and Dal Corso *et al*. (2008). Those results can be now recovered by setting *ϑ*_{0}=0 and thus obtaining(3.28)showing that in the case of a stiffener aligned parallel to prestress principal axes, there is no rigid rotation of the line due to the symmetry of the problem. The rigid rotation is null also in another case corresponding to *k*=0 (and {*ϑ*_{0}, *η*}≠0)(3.29)

The normalized stiffener rotation *Γ* (3.24) is reported in figure 4, for a *J*_{2}-deformation theory material, showing an antisymmetric behaviour with respect to the shear parameter *γ*. Note that results reported in figure 4 are independent of the hardening parameter *N*, except for the fact that the curve terminates at failure of ellipticity *γ*^{E}=*γ*^{E}(*N*).

## 4. Incremental strain field near an inclined stiffener

The previously obtained solution can now be employed to analyse the incremental strain field near the stiffener. In particular, level sets of the modulus of perturbed incremental deviatoric strain for *J*_{2}-deformation theory of plasticity are reported in figure 5 for (*a*,*b*) low *N*=0.1 and (*c*,*d*) high *N*=0.8 strain hardening.

For a *J*_{2}-deformation theory material, the loss of ellipticity occurs at *γ*^{E}=0.654 for *N*=0.1 and at *γ*^{E}=2.452 for *N*=0.8. Two values of amount of shear *γ* have been considered in figure 5, namely, *γ*=0 corresponding to a material with null prestress7 and *γ*≃0.95*γ*^{E}, namely, close to the boundary of ellipticity loss, before the Mode I perturbation is applied.

When the perturbation is applied at high prestrain, the incremental deformation fields appear strongly focused along the near-tip directions of the shear bands formally possible at ellipticity loss, thus confirming the results of Bigoni *et al*. (2008). Moreover, the results pertaining to low strain hardening (*N*=0.1, figure 5*b*) show that the bands closest to the stiffener line are privileged, so that something resembling a ‘thick’ shear band parallel to the stiffener appears, a finding in qualitative agreement with the experimental results of Misra & Mandal (2007).

## 5. Incremental solution in the parabolic and hyperbolic regimes

Accepting non-decaying of the solution and an infinite strain (and stress) increment along certain shear band lines, the previously obtained solution can be extended to the parabolic and hyperbolic regimes. Since beyond the elliptic range a problem is known to be ill-posed, the analysis of the solution within the parabolic and hyperbolic regimes is instructive to reveal features related to ill-posedness.

To obtain a solution (which need not be unique) beyond the elliptic range, we have to go back to the representation of the stream function (3.21), where the indices now have to range from 1 to *n*, so that(5.1)where *n* indicates the number of non-conjugate roots (i.e. *n*=4 (*n*=3) in H (P)).

Outside the elliptic regime, the characteristic lines defined as(5.2)become real and correspond to four (two) different families in the hyperbolic (parabolic) regime and, in particular, their inclinations correspond to the following shear band inclinations:(5.3)where *j*=1, …, 4 in H (*j*=1, 2 in P).

The decaying of solution (5.1) is lost along the characteristic lines emanating from the stiffener tips, where, additionally, the increment of strain, and consequently stress, becomes infinite (while for the other characteristic lines cutting the stiffener the solution remains always bounded). By contrast, incremental displacements remain continuous and finite everywhere, even along characteristics.

The system of linear equations(5.4)replaces system (3.22) to determine the *n* complex constants *D*_{j}, providing the solution. Note that the determination of these 2*n* real constants depend on four equations, so that ∞^{2n−4} solutions are possible.

Focusing attention to the hyperbolic regime, where *n*=4 and the roots *W*_{j} are real, the system (5.4) simplifies to(5.5)so that the general solution of linear system (5.5) can be written as(5.6)where *ζ*_{1}, *ζ*_{2}, *ζ*_{3} and *Χ* are arbitrary real constants.

Since system (5.5) admits ∞^{4} solutions, we have chosen to represent in figure 6 the deformed shape of an area near the stiffener, for the four solutions corresponding (from left to right and from the upper to the lower part) to index *h* ranging between 1 and 4 so that8(5.7)

We can note from figure 6 that for given uniform Mode I remote loading of a stiffener embedded in a medium uniformly prestrained beyond the elliptic range:

an infinite number of solutions is possible,

these solutions do not decay at infinity,

they correspond to infinite incremental strain and stress along shear bands, and

these shear bands emanate from the tips of the stiffener.

The above conclusion, based on an analytical solution, explains the difficulties typically encountered in the numerical analyses of ill-posed boundary-value problems.

## 6. Conclusions

An incremental solution has been given for uniform Mode I perturbation of a rigid, thin inclusion in an infinite (incompressible) elastic material, subject to uniform prestress with principal axes inclined with respect to the inclusion line. The obtained results conclude an investigation initially limited to coaxiality between the prestress principal axes and the inclusion line (Bigoni *et al*. 2008; Dal Corso *et al*. 2008). It is shown, in particular, that the initial state of uniform prestress can be obtained through a finite simple shear parallel to the inclusion line and that the amount of shear may be sufficient to bring the state of stress as close as possible the elliptic boundary and even beyond it, namely, within the hyperbolic or parabolic regimes. The obtained results therefore allow analytical investigations of the interactions between a rigid inclusion and shear bands and reveal a tendency towards a focusing of the shear bands parallely to the stiffener line. Finally, extending the solution beyond the elliptic regime, it has been shown that the fields do not decay, carry an infinite strain rate within the shear bands and that there exist ∞^{4} (∞^{2}) solutions in the hyperbolic (parabolic) regime. These results analytically highlight the well-known numerical difficulties encountered in the analysis of boundary-value problems beyond the elliptic range.

## Acknowledgments

Financial support of MIUR-Prin 2007 (prot. 2007YZ3B24_002) is gratefully acknowledged.

## Footnotes

↵Since the state of prestress is arbitrarily inclined with respect to the inclusion line, the solution obtained in the present article generalizes that previously given by Dal Corso

*et al*. (2008), where a stiffener aligned parallel to the principal axes of prestress has been considered. Note that Mode I loading is defined with respect to the axes parallel and orthogonal to the stiffener.↵Differently from results presented by Bigoni

*et al*. (2008), now the two axial symmetries are lost, so that the shear bands are not symmetrical with respect to the stiffener line and one of the two becomes a preferential failure mode.↵We include in appendix B of this article, the solution where a stiffener is forced to incrementally rotate (through application of an external incremental bending moment) by a given amount, a problem modelling the mechanical fields generated through the so-called ‘vane test’ in soil mechanics. Differently from the case when the inclusion is embedded in a matrix and loaded at infinity, in this situation, we show that the two shear bands more inclined with respect to the stiffener become the preferred failure modes.

↵Note that the

*J*_{2}-deformation theory material is a nonlinear elastic material isotropic in the unloaded state. Owing to the fact that the incremental shear moduli*μ*and*μ*_{*}tend to infinity when*ϵ*tends to zero (and*N*<1), the ratio*μ*_{*}/*μ*tends to*N*(and therefore is different from 1).↵The analysis will be carried out with respect to a generic, uniform state of prestress (with principal values inclined at

*ϑ*_{0}different from*ϑ*_{E}), not necessarily generated through a simple shear deformation.↵The stream function (3.21) enforces continuity in displacement (or in tractions for fracture mechanics problems) along the branch cut (, ), when appropriate values of the complex constants

*D*_{j}are chosen. This property remains valid independently of the regime considered.↵At null prestress, the

*J*_{2}-deformation theory of plasticity becomes incrementally rigid (since both*μ*and*μ*_{*}tend to infinity, but their ratio*ξ*tends to*N*). It is expedient, therefore, to plot results normalized through division by*μ*, so that they tend to results pertaining to an incompressible orthotropic material deformed in small strain. The axes of orthotropy are therefore inclined at 45° with respect to the stiffener line in figure 5.- Received June 9, 2008.
- Accepted August 19, 2008.

- © 2008 The Royal Society