## Abstract

A weak line inclusion model in a nonlinear elastic solid is proposed to analytically quantify and investigate, for the first time, the stress state and growth conditions of a finite-length shear band in a ductile prestressed metallic material. The deformation is shown to become highly focused and aligned coaxial to the shear band—a finding that provides justification for the experimentally observed strong tendency towards rectilinear propagation—and the energy release rate to blow up to infinity, for incremental loading occurring when the prestress approaches the elliptic boundary. It is concluded that the propagation becomes ‘unrestrainable’, a result substantiating the experimental observation that shear bands are the preferential near-failure deformation modes.

## 1. Introduction

Localized deformations in the form of shear bands emerging from a slowly varying deformation field are known to be the preferential near-failure deformation modes of ductile materials (e.g. Fenistein & van Hecke 2003; Bei *et al*. 2006; Lewandowski & Greer 2006; Rittel *et al*. 2006). Therefore, shear band formation is the key concept to explain failure in many materials and, according to its theoretical and ‘practical’ importance, it has been the focus of an enormous research effort in the last 30 years. From the theoretical point of view, this effort has been mainly directed in two ways,1 namely the dissection of the specific constitutive features responsible for strain localization in different materials2 and the struggle for the overcoming of difficulties connected with numerical approaches.3 Although these problems still seem far from being definitely solved, the most important questions in this research area have only marginally been approached and are therefore still awaiting explanation. They are as follows.

The highly inhomogeneous stress/deformation state developing near a shear band tip is unknown from an analytical point of view (and numerical techniques can hardly have the appropriate resolution to detail this).

It is not known if a shear band tip involves a strong stress concentration.

The fact that shear bands grow quasi-statically and

*rectilinearly*for remarkably long distances under mode II loading conditions, while the same feature is not observed in the akin problem of crack growth, remains unexplained.Finally, and most importantly, the reason why shear bands are preferential failure modes for quasi-statically deformed ductile materials has no justification.

Surprisingly, analytical investigation of the above problems and even of the stress field generated by a finite-length shear band, possibly including near-tip singularities, has never been attempted. Moreover, shear band growth has been considered only in a context pertaining to slope-stability problems in soil mechanics (Palmer & Rice 1973; Rice 1973), an approach recently developed by Puzrin & Germanovich (2005).

A full-field solution is given here for a finite-length shear band4 in an anisotropic, prestressed, nonlinear elastic material, incrementally loaded under mode II and revealing: stress singularity; high inhomogeneity of the deformation and its focusing parallel and coaxially aligned to the shear band. Moreover, the incremental energy release rate is shown to blow up when the stress state approaches the condition for strain localization (i.e. the elliptic boundary). These general findings are applied to the so-called ‘*J*_{2}-deformation theory material’, the most important constitutive model for the plastic response of ductile metals,5 and provide justification to the above-mentioned aspects of shear banding in ductile materials.

## 2. The shear band model

A shear band of finite length, formed inside a material at a certain stage of continued deformation, is a very thin layer of material across which the normal component of incremental displacement and of nominal traction remain continuous, but the incremental nominal tangential traction vanishes, while the corresponding displacement becomes unprescribed (figure 1). Therefore, it results spontaneous to model such a shear band as a weak surface along which neighbouring materials can freely slide, but are constrained to remain in contact. Note that this slip surface is different from a crack since it can carry normal tractions, so that only under special symmetry conditions on the prestress state it may behave as a crack when subjected to shear parallel to it (the so-called ‘mode II’ loading in fracture mechanics).

Models of slip bands similar to the weak line model have been proposed in metal plasticity (e.g. Cottrell 1953, §10) and geomechanics (Palmer & Rice 1973; Rice 1973; Puzrin & Germanovich 2005), although the neighbouring materials assumed in these models are free of prestress and linear elastic so that there is no correlation between the shear band and the surrounding stress that has generated it.

The key to the analysis of the stress/deformation fields near a shear band and its advance under load increments is a perturbative approach similar to that proposed by Bigoni & Capuani (2002, 2005) and Piccolroaz *et al*. (2006). In particular, an infinite, incompressible, nonlinear elastic material is considered, homogeneously deformed under the plane strain condition. According to the Biot (1965) theory, the response to an incremental loading is expressed in terms of the nominal (unsymmetrical) stress increment , related to the gradient of incremental displacement ∇** v** (satisfying the incompressibility constraint tr ∇

**=0) through the linear relation(2.1)where T denotes the transpose; is the incremental in-plane mean stress; and the fourth-order tensor is a function of the current state of stress (expressed through the principal components of Cauchy stress,**

*v**σ*

_{1}and

*σ*

_{2}) and material response to shear (

*μ*for shear parallel and

*μ*

_{*}for shear inclined at

*π*/4 with respect to

*σ*

_{1}) describing orthotropy (aligned parallel to the current principal stress directions; see appendix A for details). All parameters defining and representing the current state of the material can be condensed into the following dimensionless quantities:(2.2)where

*μ*and

*μ*

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

The differential equations governing incremental equilibrium can be classified according to the values assumed by parameters *ξ* and *k*, to distinguish between elliptic imaginary (EI), elliptic complex (EC), parabolic (P) and hyperbolic (H) regimes (figure 2*a*).

According to the conventional approach (Biot 1965; Rudnicki & Rice 1975), shear bands are understood as planes across which incremental velocity gradient becomes discontinuous and may emerge only in a continuous deformation path as soon as either the EC/H or EI/P boundary is ‘touched’. Two equally inclined (with respect to the principal stress directions) bands are generated in the former case (figure 2*c*), while only one band forms aligned with the principal maximum tensile stress, say *σ*_{1}, in the latter case (figure 2*b*). Following the alternative approach (Bigoni & Capuani 2002, 2005; Piccolroaz *et al*. 2006), shear bands spontaneously emerge as the response to a perturbation applied inside the elliptic regime, but in the vicinity of either the EI/P or EC/H boundary. Since experiments suggest that shear banding is strongly influenced by the presence of randomly distributed defects (Xue & Gray 2006), it is assumed that during homogeneous deformation of an infinite medium subjected to remote stress with *k*>0, a defect is present in the form of a thin zone of material that has touched the EI/P or EC/H boundary and has been transformed into a shear band of length 2*l* (in other words, the weak line inclusion in the proposed modelling), leaving the surrounding material uniformly deformed/stressed and still in the elliptic regime, although near the elliptic boundary. (This uniform state of stress has to satisfy the Hill exclusion condition, to avoid ‘spurious’ interfacial instabilities; see appendix A.) The shear band of length 2*l* is inclined with respect to the *x*_{1}-axis at an angle *ϑ*_{0} that can be determined from a known formula (Hutchinson & Tvergaard 1981), thus providing the inclination of the weak line inclusion with respect to the material orthotropy *x*_{1}-axis (figure 3). Taking this configuration as the initial state, the response of the shear band to an incremental perturbation is analysed.

According to the weak line model, under an incremental mode I perturbation the shear band does not alter the incremental response of the surrounding material (so that it is ‘neutral’), but under a mode II perturbation the shear band behaves as a slip surface of length 2*l* (prestressed both longitudinally and transversely), and strongly non-uniform and singular fields are generated.

## 3. Analytical solution for a shear band of finite length loaded incrementally

The analytical solution for a finite-length crack incrementally loaded by a uniform mode II far field in a prestressed material similar to that described by equation (2.1) was available only when the crack is aligned parallel or orthogonal to the orthotropy axes, a situation corresponding to a shear band forming at the EI/P boundary, where symmetry implies that a crack behaves as a slip surface, so that the crack and the weak line become equivalent models. The solution for an inclined crack in a prestressed material (obtained in appendix B) is interesting in itself (since it shows features of interactions between shear bands and crack tip fields) and of fundamental importance for the understanding of the shear band problem addressed here.

For shear bands occurring at the EC/H boundary, the solution for a weak line inclusion inclined with respect to the orthotropy axes, not previously available for the material under consideration, is given here. Developing this solution for constitutive equations (2.1) and employing it to analyse a shear band, a singularity is found. Moreover, *a full-field representation is obtained for the incremental stress/strain field near a shear band of finite length*.

The solution for an inclined shear band in an infinite medium can be expressed in a – reference system located at the shear band centre, with the -axis aligned parallel to the shear band, and rotated at an angle *ϑ*_{0} with respect to the reference system in which constitutive equations (2.1) are expressed (figure 3).

The stress components in the – reference system can be obtained through a rotation of the components in the prestress principal reference system *x*_{1}–*x*_{2}, so that, since the two systems are rotated at an angle *ϑ*_{0} (taken positive when anticlockwise), we have(3.1)so that the nominal stress increment, incremental displacement and its gradient can be expressed in the – reference system as(3.2)while the constitutive equations (2.1) transform to(3.3)where the transformed fourth-order tensor is given by(3.4)where the indices range between 1 and 2.

In the – reference system, the so-called ‘perturbed problem’ is solved in which the traction at infinity is applied with reversed sign along the shear band surfaces. In terms of perturbed stream function , defined to provide the incremental displacements as(3.5)the full-field solution for a shear band of length 2*l* can be written as(3.6)where(3.7)and (*Ω*_{j} is purely imaginary in EI and complex in EC and Re denotes the real part of its argument)(3.8)The unknown complex constants (*j*=1, 2) in equation (3.6) can be determined by imposing boundary conditions at the shear band surfaces, namely

null incremental nominal shearing tractions(3.9)

continuity of the incremental nominal normal traction(3.10)

continuity of normal incremental displacement(3.11)

where the brackets denote the jump of the relevant argument across the shear band.

Employing equation (3.6) and imposing the boundary conditions (3.9)–(3.11) at the sliding surface yields the following algebraic system for the unknown constants :(3.12)where coefficients *c*_{ij} are defined by equations (B 6). The determinant of the coefficient matrix in equation (3.12) vanishes both when the surface bifurcation condition, equation (A 18), is met and at the EC/H boundary.

Similar to the crack solution, the asymptotic fields near the shear band tip result to be given in polar coordinates (centred at the shear band tip ) by(3.13)for the incremental nominal stress ahead of the tip, where(3.14)while for the incremental displacements we have (where constants have been neglected)(3.15)holding at the shear band surfaces, for ‘small’ Δ*l*.6

In the particular case of a shear band aligned parallel to the prestress principal direction *σ*_{1} (i.e. *ϑ*_{0}=0), solution (3.6) simplifies to(3.16)

## 4. Rectilinear shear band growth is a preferred failure mode

Solution (3.6) is employed to obtain results shown in figure 4, where level sets of incremental deviatoric strain are reported for a shear band (inclined at 29.3°) in a ductile low-hardening metal, modelled through the *J*_{2}-deformation theory with *N*=0.3, at null prestrain *ϵ*=0 (figure 4*a*) and at a prestrain *ϵ*=0.548 (figure 4*b*), taken close to the EC/H boundary. It can be noted from the figure (additional results are reported in appendix C) that, while at null prestrain (far from the elliptic boundary in figure 4*a*) the incremental strain field is not particularly developed and does not evidence focusing, near the elliptic boundary (figure 4*b*) *the incremental strain field is localized and elongated, and evidences a strong focusing in the direction aligned parallel to the shear band*. This finding suggests that, *while mode II rectilinear crack propagation in a homogeneous material does not usually occur* (*since in first approximation cracks deviate from rectilinearity following the maximum near-tip hoop stress inclination*), *shear band growth is very likely to occur aligned with the shear band itself*. This observation explains the strong tendency that shear bands evidence towards the rectilinear propagation for long (compared with their thickness) distances (e.g. Korbel & Bochniak 2004; Bei *et al*. 2006). Moreover, the focusing of incremental deformation and the stress singularity strongly promotes shear band growth.

To further analyse shear band growth, *the incremental energy release rate for an infinitesimal shear band advance* (see appendix B*c*) can be calculated for an orthotropic prestressed material, equation (2.1), by employing the asymptotic near-tip representations (3.13) and (3.15) in equation (B 19)7(4.1)where the complex constants are the solutions of equation (3.12). Equation (4.1) becomes, for a shear band aligned parallel to the principal direction of prestress *σ*_{1} (*ϑ*_{0}=0),(4.2)In equations (4.1) and (4.2) constant is the incremental mode II stress intensity factor, defining the intensity of the singularity in terms of applied incremental loading and equal to(4.3)for a rectilinear shear band of length 2*l* in an infinite material (inclined with respect to the orthotropy and prestress axes).

The release rate (4.1) represents the energy released for an infinitesimal advance of the shear band and has the typical behaviour shown in figure 5, referred to the same material considered in figure 4 (there are no qualitative changes when other values of the hardening exponent *N* are considered, see appendix C).

It is assumed in fracture mechanics that a crack advances under small-scale yielding when the energy release rate exceeds a critical threshold, believed to be a characteristic of the material. Whether this criterion can be generalized to the present context or not can still be a matter of discussion, but the important point is that *the incremental energy release rate blows up to infinity when the elliptic boundary is approached*. In these conditions, a shear band can drive itself on and overcome possible barriers; in other words, it can grow ‘unrestrainable’, a finding which, together with the previous results on near-tip stress/deformation states, legitimizes for the first time the common experimental observation that shear bands are the preferred near-failure deformation modes.

## 5. Conclusions

The modelling of a finite-length shear band in an infinite prestressed material presented in this article keeps into account stress-induced and inherent anisotropy, and large strain effects. Quasi-statically loaded ductile materials have been addressed exhibiting incompressible flow, typically metals, and the modelling permits the first explicit closed-form evaluation of all mechanical fields near a shear band of finite length and provides justification to the fact that shear bands are preferred modes growing rectilinearly for long distances, as experimentally found by Korbel & Bochniak (2004) among others. A number of features in the modelling of the material (the possibility of elastic unloading outside the shear band, dynamic loading and thermal effects and, for granular materials and soils, pressure sensitivity of yielding and plastic flow dilatancy) and of the shear band (the possibility of introducing cohesive tangential forces between the weak line surfaces) have been sacrificed for mathematical tractability, although their incorporation can certainly be pursued. In particular, elastic unloading near the shear band and thermal effects have been found to be important (the latter when dynamic loading is involved, Guduru *et al*. (2001), while the former even for quasi-static loading, Gajo *et al*. (2004)) and the development of weak cohesive forces at the shear band surfaces might prelude the extreme loss of (incremental) stiffness assumed in our model. However, considering our previous treatment of various perturbations in materials prestressed near the boundary of ellipticity loss (Bigoni & Capuani 2002, 2005; Piccolroaz *et al*. 2006), we believe that the results presented in this article have general validity and can be extended to include much more complicated effects.

## Acknowledgments

Financial support of Trento University is gratefully acknowledged.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2008.0029 or via http://journals.royalsociety.org.

↵Features of strain localization occurring

*after*its onset have scarcely been theoretically explored. For instance, there is almost nothing about post-localization behaviour. Research devoted to this topic has been developed by Hutchinson & Tvergaard (1981), Tvergaard (1982), Petryk & Thermann (2002) and Gajo*et al*. (2004).↵This line of research has been initiated by Rudnicki & Rice (1975) and developed in a number of directions (including gradient effects (Aifantis 1987; Aifantis & Willis 2005), temperature effects (Gioia & Ortiz 1996; Benallal & Bigoni 2004), anisotropy effects (Bigoni & Loret 1999) and yield-vertex effects (Petryk & Thermann 2002)).

↵Reviews on the numerical work developed in these years have been given by Needleman & Tvergaard (1983) and Petryk (1997).

↵In addition to the shear band solution, we provide in appendix B the full-field solution for a finite-length crack in a prestressed material loaded incrementally under modes I and II. This solution is new in the case when the crack is inclined with respect to the material's orthotropic axes and is fundamental to the understanding of the shear band problem. Although based on the assumption that dead loading tractions are present inside the crack to equilibrate the assumed prestress state, this solution is interesting in itself, when used near the boundary of ellipticity loss, since it reveals features related to the interaction between shear bands and crack tip fields, so that it may explain experimental observations relative to crack growth in ductile materials (McClintock 1971; Hallbäck & Nilsson 1994).

↵The finite-strain

*J*_{2}-deformation theory of plasticity has been proposed by Hutchinson & Neale (1979). This theory accounts for the most important features of plastic flow in metals (except for the possibility of elastic unloading, which is*a priori*ruled out) and correctly predicts the onset of shear banding (see Hutchinson & Tvergaard 1981).↵The following properties of function

*ϒ*↵have been proven, while the properties

↵have been numerically found to hold, from which the identities

↵follow with the help of a symbolic manipulator.

↵Note that the perturbed solution for the shear band model can be alternatively obtained providing a mixed-mode loading to an inclined crack (see appendix B

*a*). The mode I loading component is ‘calibrated’ with respect to the mode II component in such a way as to eliminate the jump in normal incremental displacement along the crack faces generated by a pure mode II loading, in other words, to satisfy condition (3.11). All these procedures bear on the special feature found in the solution of the crack problem that a mode I uniform loading along the crack faces is sufficient to eliminate a mode II transversal mismatch in incremental displacements. In particular equation (4.1) can be obtained from equation (B 23), considering a mixed mode defined by , so that the condition of continuity of transversal incremental displacement yields↵and the constants defining the crack and shear band solutions are related through

↵Therefore, the difference between the crack and shear band problems lies in a uniform nominal normal stress increment applied at the crack surfaces.

- Received January 24, 2008.
- Accepted March 12, 2008.

- © 2008 The Royal Society