The purpose of this book is to present a research summary on solid mechanics at large strain, including the treatment of bifurcation and instability phenomena. The framework is crucial to the understanding of failure mechanisms in ductile materials, as connected to material instabilities, such as, shear banding.

I have employed Chapters 2 through 5 as a textbook for a graduate course on nonlinear elasticity that I have offered at the University of Trento since 1999, whereas Chapters 8, 10, 11 and 13 have been the basis for a course held at CISM (no. 414, ‘Material Instabilities in Elastic and Inelastic Solids’, H. Petryk, ed.). Chapters 6, 7, 9, 12, 14 and 15 have been added to present the elasticity and the yield critera in detail, including a treatment on elastic bifurcation and instability, wave propagation and multiple shear banding. This material has been taught during seminars for graduate students at various universities. Chapter 16 is devoted to the perturbative approach to material instability, developed by me in a series of articles in cooperation with D. Capuani, M. Brun, F. Dal Corso, M. Gei, A. Piccolroaz and J. R. Willis. Finally, I have to admit that the Introduction of the book is overlong; in fact, I have used it for a 20-hour graduate course on stability and bifurcation. The hope is to attract attention to the main topics presented in the book.

Local and global uniqueness and stability criteria were introduced in Chapters 10 and 11, with reference to non-associative elastoplasticity (Chapter 8). The incremental non-linearity of the rate-constitutive equations of plasticity and the lack of symmetry connected to the flow-rule non-associativity strongly complicate the bifurcation and instability analyses with respect to the case of incremental elasticity. Therefore, despite interest in the applications to bifurcation problems for geological and quasibrittle materials, there have been only a few attempts to apply the comparison solids methodology to bifurcation problems (Bruhns and Raniecki, 1982; Kleiber, 1984; 1986; Tomita et al., 1988; Bigoni, 2000), so our interest in this chapter is to provide examples of the methodologies explained in Chapters 10 and 11. We will use the simplest constitutive setting, which is that of small strain Drucker-Prager elastoplasticity with deviatoric associativity, a context in which we will limit examples to local stability criteria, whereas the use of Raniecki comparison solids will be presented for a simple elastoplastic non-associative model at large strain.

Broadly speaking, constitutive laws set a bridge between strain and stress, so they have to keep into account the behaviour of the specific material under consideration. This behaviour can be fully reversible and described by elasticity (e.g., in the case of rubber) or can be partially irreversible and described by elastoplasticity (e.g., in the case of mild steel), or it may be time-dependent and described by viscous laws (e.g., in the case of a Newtonian fluid). We exclude for the moment the elastoplastic behaviour, and we present constitutive laws in this chapter for anisotropic elastic materials and for incrementally deformed solids (elastoplasticity is deferred to Chapter 8).

The introduction of constitutive laws for materials subject to large strains requires preliminary statements of general principles, such as the so-called material frame indifference and indifference with respect to rigid-body rotations of the reference configuration. When these concepts are specified, the development of constitutive equations to model different materials is greatly simplified. After introduction of these general principles, we conclude the present chapter with the formulation of elastic constitutive laws, generalising concepts proposed in Chapter 4, now to includemicro-structural anisotropy.

The description of the motion, deformation and stress of a solid body subject to external actions is the focus of solid mechanics, a science that was initiated more than four centuries ago by G. Galilei (1564–1642). Solid mechanics is articulated into five main parts: (1) kinematics and the concept of deformation, (2) mass conservation, (3) forces and stress, (4) the constitutive equations and (5) the setting of the boundary value problem. We will be concerned in this chapter with the preceding points (1) through (3), whereas constitutive equations and the setting of the boundary value problem will be deferred to chapters 4 and 6 through 9. As a complement to the material that will be presented in this chapter, we suggest the exhaustive treatments by Truesdell and Noll (1965), Truesdell (1966), Chadwick (1976), Gurtin (1981), Ogden (1984), and Podio Guidugli (2000).

Kinematics

Bodies occupy configurations, which are regions of the three-dimensional Euclidean point space. Obviously, a body should not be confused with its configuration, for the same reason that the center-line of a cantilever beam should not be confused with the points occupied by the elastica.

Elastic behaviour is characterised by the immediate reversibility of the deformation on release of the stress. Though common at small strain, this behaviour becomes ‘rare’ for materials subjected to large strain. In fact, elastic strain is limited to 1% for crystalline materials and amorphous materials in their rigid state and decreases to 0.1% and less for steel and to 0.001% for granular materials. In practice, the only materials behaving elastically at large strain are rubber, where extensibility can reach 500% to 1000%, and biological soft tissues. However, our interest in elastic modelling is not only limited to materials really behaving elastically; rather, it is also important to describe the loading branch of the constitutive behaviour of elastoplastic materials (roughly speaking, the behaviour exhibited when unloading is never involved). We will see, in fact, that bifurcation and instability analyses for elastoplastic materials are usually reduced to the analysis of so-called elastic comparison solids (Chapter 10).

The objective of this chapter is to introduce the constitutive equations for elastic solids isotropic in their unloaded configuration in the simplest way, deferring the detailed treatment of elastic anisotropy and of a general constitutive framework to Chapter 6.

All analyses considered up to this point refer to situations of perfect systems loaded in perfect conditions. We are now in a position to judge the merits and the limits of the previous approach and to set up a new methodology that may capture aspects remaining undetected within the previously given framework. These aspects can be illustrated with reference to two examples: the local instability criteria of ellipticity loss and flutter instability. In particular, we know that both refer to homogeneously deformed infinite bodies and that the former is linked to the appearance of shear bands (or strain localisation), whereas the latter is linked to the generation of ‘blowing-up incremental solutions’.

Wave propagation in solids is a topic strictly connected with stability and bifurcation. It will be shown in this chapter that the condition for incremental bifurcation analysed in chapter 12 for elastic solids is equivalent to the condition of vanishing propagation speed for an incremental wave mode, whereas instability corresponds to a blow up of the wave mode amplitude during propagation.

The simple example of small-amplitude vibrations of a beam superimposed on a given axial stress (‘pre-stress’) is sufficient to clarify the above-mentioned issues. To this purpose, we reconsider the beam illustrated in Section 10.2.3, subjected to axial load F corresponding to a longitudinal Cauchy stress - σ parallel to the beam axis x1.

When a ductile material such as, for instance, mild steel (Fig. 8.1) is deformed in a sufficiently severe way, irreversible or, in other words, ‘plastic’ strain occurs.

In the case of steel, the irreversible deformation is the ‘global effect’ of dislocation activity which initiates at a certain threshold stress. More in general, plastic flow is always related to the activation of some irreversible micro-mechanism, such as micro-cracking in rock and concrete and sliding between grains in granular matter. From the point of view of constitutive modelling, the ‘activation stress’ is decided on the basis of a suitable yield function of the type (7.1), which is the first ‘building block’ of a plasticity or damage theory.

Analysis of kinematics and balance laws was given in Chapter 3, whereas constitutive equations were detailed in Chapters 4, 6 and 8, with a digression on yield functions presented in Chapter 7. We are now in a position to ‘collect the equations’ and set boundary value problems in finite and rate forms for solids loaded on the boundary. However, we have until now assumed certain hypotheses of regularity that we want to relax, so before setting boundary value problems, a digression on moving singularities in solids becomes instrumental. This also will be useful in the development of acceleration waves and shear band analysis.

Moving discontinuities in solids

Until now, we have more or less tacitly assumed that all the fields are ‘sufficiently’ regular, which is to say smooth. However, there are many situations in which smoothness or even continuity is lost. For instance, displacement, deformations or stresses can suffer jumps across a fracture or a so-called imperfect interface (such as those considered by Bigoni et al., 1997, 1998), or across a rigid thin inclusion (such as that considered by Dal Corso et al., 2008, and Dal Corso and Bigoni, 2009), or simply at an interface separating two different solids, for instance, in a multilaminated material.

Analysis of the behaviour of a material element after shear banding has occurred is crucial for an understanding of induced softening and size-effect phenomena, which may be relevant for different purposes, for instance, predicting possible catastrophic failure of structural elements (see the Introduction). Moreover, post-shear banding may involve multiple shear band formation, a phenomenon observed in different materials: foams (Moore et al., 2006), ductile metals (Hall, 1970), honeycombs (Papka and Kyriakides, 1999), sand (Finno et al., 1997), shape memory alloys (Shaw and Kyriakides, 1997) and the stacks of drinking straws shown in the Introduction (Section 1.6).

The mechanical modelling of the behaviour of materials subject to large strain is a concern in a number of engineering applications. During deformation, the material may remain in the elastic range, as, for instance, when a rubber band is stretched, but usually inelasticity is involved, as, for instance, when a metal staple is bent. The achievement of severe deformations involves the possibility of the nucleation and development of non-trivial deformation modes—including localized deformations, shear bands and fractures—, emerging from nearly uniform fields. The description of the conditions in which these modes may appear, which can be analysed through bifurcation and stability theory, represents the key for the understanding of failure of materials and for the design of structural elements working under extreme conditions. Bifurcation and instability modes occur in a variety of geometrical forms (as can be shown with the example of a cylinder subject to axial compression) and may explain the so-called ‘size effect’, ‘softening’ and ‘snap-back’ even when fracture, damage and inelasticity are excluded. Shear banding can occur as an isolated event, leading to global failure, or as a repetitive mechanism of strain ‘accumulation’ (as can be shown through the examples of chains with softening elements). Features determining bifurcation loadings and modes strongly depend on the constitutive features of the materials involved (as can be shown with the example of the Shanley model for inelastic column buckling).

From the uniqueness and stability criteria considered in the preceding chapter, local conditions may be derived, which are treated herein. The importance of local conditions lies in the connection to material instabilities, namely, to instabilities which can develop from a point in a continuum and therefore result independent of the boundary conditions. For instance, we will see that loss of ellipticity corresponds to shear band formation. The following five local criteria will be analysed in this chapter:

• Positive definiteness of the constitutive operator (PD)

• Non-singularity of the constitutive operator (NS)

• Strong ellipticity (SE)

• Ellipticity (E)

• Flutter (F)

To begin providing an example of the preceding criteria in the simple case of the infinitesimal theory of isotropic elasticity, we recall the condition of positive definiteness (PD) [Eqs. (2.192)], which is the positiveness of the eigenvalues 2μ and λ + 2μ/3 of the elastic fourth-order tensor. Non-singularity (NS) corresponds to the condition of non-vanishing of these eigenvalues, and in a similar vein, strong ellipticity (SE) corresponds to the positiveness of the eigenvalues μ and λ + 2μ of the acoustic tensor (2.176), whereas ellipticity (E) corresponds to the non-vanishing of the same eigenvalues.

In Chapters 10 and 11we introduced sufficient conditions for uniqueness and stability of elastic and elastoplastic solids. In engineering applications the usual problem is to find bifurcation loads and modes during continued deformation of a solid body subjected to a prescribed loading program.

The purpose of this Chapter is to formulate and solve several bifurcation problems for incompressible materials deformed incrementally in plane strain or axisymmetrically. In particular, the following five bifurcation problems are addressed: (1) a homogeneously stressed half space, loaded parallel to the free surface, (2) two homogeneously stressed elastic half spaces, loaded parallel to the surface separating them, (3) a homogeneously stressed block loaded parallel to two edges (a problem that will be generalized to include bifurcations of a layer on an elastic foundation, a layer on an elastic half space and a generic stack of layers), (4) a cylinder loaded under uniaxial compression parallel to its axis, and (5) an elastic layer subject to finite bending.