Two-phase piecewise homogeneous plane deformations of a fibre-reinforced neo-Hookean material with application to fibre kinking and splitting

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Abstract

Two-phase piecewise homogeneous plane deformations are examined in respect of a neo-Hookean matrix material reinforced with embedded aligned fibres characterized by a single stiffness parameter. The deformations are interpreted in terms of fibre kinking and fibre splitting. Previous work has shown that such a transversely isotropic material can lose ellipticity if the reinforcing stiffness is sufficiently large and the fibre direction is sufficiently compressed. In particular, it was shown that the associated failure modes are characterised by the emergence of weak surfaces of discontinuity that are normal to the fibre direction (the onset of fibre kinking) or parallel to the fibre direction (the onset of fibre splitting). Here, the analysis of strong surfaces of discontinuity, developing from weak ones, is studied. The considered model can give rise to piecewise smooth plane deformations separated by a plane stationary surface of discontinuity, interpreted as either kinking or splitting. Attention is restricted to (plane) deformations in which, on one side of the surface of discontinuity, the load axis is aligned with the fibre axis. Then the fibre stretch on this side of the discontinuity is a natural load parameter. The ellipticity status of the two-phase piecewise homogeneous plane deformations is shown to span all four possible ellipticity/non-ellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be non-elliptic. Moreover, it is shown that the mechanism is dissipative, and maximally dissipative quasi-static failure motion is examined in respect of both kinking and splitting. It follows that, firstly, surfaces of discontinuity perpendicular to the fibre direction, associated with fibre kinking, are nucleated followed by surfaces of discontinuity parallel to the fibre direction, associated with fibre splitting. With respect to kinking, such maximally dissipative kinks nucleate only in compression as weak surfaces of discontinuity, with the subsequent motion converting non-elliptic deformation to elliptic deformation.

Introduction

Based on the implications of the loss of ellipticity of the governing equations of equilibrium for fibre-reinforced incompressible nonlinearly elastic solids, the purpose of the present work is to examine the initiation of kink band instabilities and the possible coincidence of kinking and splitting in such materials. To set the scene we first consider some background on shear band initiation and related instabilities.

A general theoretical framework for shear bands was provided by Hill (1962), while, with a view to establishing a realistic model capable of predicting the critical bifurcation stress (and/or strain) leading to formation of a shear band, different constitutive models were analysed in a variety of contexts. For example, Rice (1976) showed that some isotropic elastic solids, and elastic-plastic solids with a smooth yield surface, do not develop shear band instabilities, while important theoretical contributions were made by Hill and Hutchinson (1975), Anand and Spitzig (1980) and Hutchinson and Tvergaard (1981) within the theory of plasticity under several loading conditions. In the context of the inelastic behaviour of over-consolidated clay soils shear band formation was investigated by Rice (1973) and Rudnicki and Rice (1975).

In connection with asymptotic studies of crack problems, Knowles, Sternberg, 1978, Knowles, Sternberg, 1980 showed that loss of ellipticity of the field equations of nonlinear compressible hyperelastic materials under plane deformation is a necessary condition for the emergence of solutions lacking the standard smoothness properties required by the governing differential equations of equilibrium. Identification of the surfaces on which ellipticity fails formed the subject of an earlier paper (Knowles and Sternberg, 1976), while Knowles and Sternberg (1978) were concerned with the emergence of surfaces of discontinuity and piecewise homogeneous deformations and energy considerations associated with the loss of ellipticity. and dissipation (see also Abeyaratne, 1980, Abeyaratne, Knowles, 1989, Knowles, 1979).

Budiansky and Fleck (1993) provided one of the most widely used models that incorporates the effect of combined stress loading for predicting realistic ranges of kink band angles in composites. Similar analyses were conducted by Sutcliffe and Fleck (1994) and Moran et al. (1995) for carbon fibre epoxy composites, by Moran and Shih (1998) for ductile matrix fibre composites, by Poulsen et al. (1997) for wood, and for an advanced fibre-reinforced composite by Kyriakides et al. (1995) and Vogler and Kyriakides (1997). These works focus not only on predicting the compressive strength but also on the propagation of the kink band. It has been assumed in the literature that a kink band may start from either a well-defined initial band of wavy fibres (for example, Kyriakides, Arseculeratne, Perry, Liechti, 1995, Kyriakides, Ruff, 1997) or from a deformation induced band, as in, for example, Christoffersen and Jensen (1996), Jensen and Christoffersen (1997) and Christensen and DeTeresa (1997).

Analysis of discontinuous deformation gradients, forming two-phases of the same material, provides both qualitative and quantitative information on the formation and broadening of (shear) kink bands (Merodio, Pence, 2001a, Merodio, Pence, 2001b). Two different phases of a given material formed by two joined homogeneously deformed half-spaces have been studied by Fu and Freidin (2004) with respect to (quasi-static) bifurcation and stability using a kinetic stability criterion and an energy analysis. Furthermore, Fu and Zhang (2006) analysed kink band formation with respect to the choice of the strain-energy function and the stability of the solutions based on the so-called Maxwell relation. Considering uni-directionally fibre-reinforced materials, they obtained the kink propagation stress, the kink orientation angle and the fibre direction within the kink band. More recently, Baek and Pence (2010) studied the effect of shearing deformation in fibre-reinforced materials on the emergence and disappearance of kink surfaces for several different boundary-value problems.

Experiments by Lee and Anthony (1999) examined the effect of fibre diameter and initial misalignment angle on the compressive behaviour of glass fibre unidirectionally-reinforced composites and the prediction of their compressive strength for a wide range of fibre volume fractions. The initial fibre misalignment angle and the axial propagation of kink bands have also been studied experimentally by Kyriakides et al. (1995) and Kyriakides and Ruff (1997). Lee et al. (2000) observed experimentally that glass fibre epoxy composites fail predominantly by a splitting failure mode at lower fibre volume fractions and by a combination of splitting and kinking at higher fibre volume fractions, while, by contrast, carbon fibre epoxy composites were found to fail only by kinking.

With this background in mind, the motivation for the present work is to provide qualitative understanding of kink band phenomena in fibre-reinforced materials, with special attention to the simultaneous existence of fibre kinking and fibre splitting observed by Prabhakar and Waas (2013a). Various efforts have been aimed at developing models that can capture and predict fibre failure (Yerramalli, Waas, 2004, Prabhakar, Waas, 2013a, Prabhakar, Waas, 2013b, El Hamdaoui, Merodio, Ogden, 2015, El Hamdaoui, Merodio, Ogden, 2018, Hasanyan, Waas, 2018) since, to some extent, the use of fibre-reinforced materials has been constrained by lack of understanding of the failure mechanisms.

To analyze and predict failure mechanisms, micromechanics approaches that consider the geometry of the microstructure and the imperfections associated with them have been used to obtain the load bearing capacity of the material (see Prabhakar and Waas, 2013b and references therein). It is therefore essential to consider models that account for fibre and matrix constituents of the composite since such models provide physical insight into the failure of both the fibre and the matrix, as well as the load transfer between them. However, micromechanics approaches are often less practical than macromechanical models since, for example, they are more costly in computational time due to the large number of degrees-of-freedom involved. Our preferred approach is therefore to adopt a macroscopic continuum model that embodies information about the microstructure, such as fibre orientation.

Within a macroscopic continuum framework, initiation of material failure is often associated with the loss in ellipticity of the governing equations. Analysis of ellipticity is therefore important, not only for characterizing macromechanical behaviour and failure but also, for instance, in computational mechanics, where loss of ellipticity leads to mesh-dependent results that depend on the mesh size, and this needs to be well understood (Hasanyan and Waas, 2018).

A continuum-mechanical model in the setting of nonlinear elasticity theory that predicts the onset of material instabilities for fibre-reinforced materials has been established in a series of paper by Merodio, Ogden, 2002, Merodio, Ogden, 2003, Merodio, Ogden, 2005a, Merodio, Ogden, 2005b, Merodio, Ogden, 2005c and references therein. The loss of stability and the onset of fibre failure in fibre-reinforced materials were related to the ellipticity status of the governing equation of equilibrium, a status which changes locally in type as a result of deformation. This change is referred to as loss of (ordinary) ellipticity, which can be interpreted as the onset of a failure mechanism.

For a given strain-energy function the loss of ellipticity condition determines both the deformation associated with the existence of surfaces of weak discontinuity and the direction of the normal to that surface. Surfaces of weak discontinuity (or weak surfaces) are surfaces across which the second derivative of the deformation field is discontinuous. For some historical background related to this type of discontinuity it is of interest to consult the book by Hadamard (1903). It has been found in Merodio and Ogden (2002) that for reinforcing models that depend only on the fibre stretch ellipticity is lost under fibre contraction, and the associate failure mode is fibre kinking. On the other hand, for reinforcing models that depend on the shear as well as stretch the failure mechanism can involve combination of fibre kinking and fibre splitting.

These failure modes are characterised by the emergence of weak surfaces of discontinuity that are normal to the fibre direction (fibre kinking) or parallel to the fibre direction (fibre splitting). Weak surfaces do not provide information about either the (quasi-static) development of the discontinuities or the different kinematic and stress variables across the surfaces of discontinuity. These data are provided by surfaces of strong discontinuity. In a fully developed or strong surface of discontinuity the first derivative of the deformation field (i.e. the deformation gradient) suffers a finite jump.

Surfaces of strong discontinuity, also referred as elastostatic shocks, non-evolutionary jump discontinuities, phase boundaries, stationary kink surfaces, stationary kinks, etc., were considered in (Merodio, Pence, 2001a, Merodio, Pence, 2001b) for analyzing fibre kinking using a reinforcing model that depends on the fibre stretch. Their analysis focused on the (plane) deformation gradients across the shock, where, in addition, one of the (in-plane) principal stretches associated with one side of the shock was aligned with the fibre direction. Henceforth in this paper, for simplicity of terminology, we mainly refer to surfaces of discontinuity as (elastostatic) shocks, although they are not shocks in the conventional sense.

The kink band is given by a three-zone state with the kink zone separating the two sides of the band (designated the ‘ + ’ and ‘ - ’ sides). However, it should be emphasized that the actual width of a kink band and its development is not captured by the pure elasticity theory used here. This requires a more general constitutive theory such as a second-gradient theory.

Our aim is to exploit the elasticity theoretical framework further in order to capture the combination of fibre kinking and fibre splitting. This requires the emergence of equilibrated shocks that are able to describe kinking and splitting simultaneously to be accommodated. The onset of fibre splitting also has a three-zone state, where in this case there is a splitting zone separating the ‘ + ’ and ‘ - ’ sides of the discontinuity. It can be argued that fibre splitting is not related to continuous displacements, as it is in fact a catastrophic failure mechanism. However, analysis of piecewise homogeneous deformations provides useful input for understanding and describing the mechanisms involved. Indeed, connecting the fibre stretching and shearing with the formation of shocks, i.e. with the formation of different fibre failure mechanisms, can be exploited in the nonlinear constitutive modelling of fibre-reinforced materials.

The main goals of the present analysis are to predict the combined splitting–kinking failure mode observed experimentally by Lee et al. (2000) and to capture the evolution of the elastostatic shock direction and kinking angle. Two-phase deformations of a fibre-reinforced material that correspond to two different phases of the same elastic material are constructed and analyzed for this purpose. The material model adopted here consists of a neo-Hookean matrix in which are embedded aligned fibres so that the material response is transversely isotropic. The fibres are characterized by a so-called reinforcing model that penalizes deformation in the fibre direction, and the overall material strain-energy function is taken to be the sum of the neo-Hookean and reinforcing energy functions.

Sections 2–4 focus on purely mechanical aspects of shocks. In Section 2, the main equations are presented. These include the requirements of displacement continuity and traction continuity that have to be satisfied by the set of all considered piecewise homogeneous plane deformations. In Section 3 some special cases are studied, namely both weak and strong shocks orthogonal and parallel to the fibre direction, while more general shocks are analyzed in Section 4. The ellipticity status of the piecewise homogeneous plane deformations is also provided. The results show that the existence of a strong shock involves loss of ellipticity at a suitable intermediate deformation, in agreement with previous analyses.

In Section 5, quasi-static shock motion is considered based on a detailed energy analysis involving the use of the so-called driving traction, i.e. the magnitude of a fictitious nominal traction acting on the shock by the surrounding material. Following (Merodio, Pence, 2001a, Merodio, Pence, 2001b) two families of solutions are studied with the purpose of establishing a criterion for selecting shock evolution, both for non-dissipative solutions and maximally dissipative shocks, which can be considered as two extreme cases. For sufficiently small values of the reinforcing parameter, as fibre contraction is increased, non-dissipative shocks nucleate at the particular values that give the first modes associated with loss of ellipticity in, for example, the ‘ + ’ zone. Incipient loss of ellipticity may occur in two different modes, which are associated with fibre kinking and fibre splitting. It follows that, firstly, shocks perpendicular to the fibre direction, associated with fibre kinking, are nucleated followed by shocks parallel to the fibre direction, associated with fibre splitting, and the quasi-static evolution of both kinking and splitting is analyzed.

Non-dissipative motion or neutral stability (analogous to the Gibbs and Maxwell phase equilibrium conditions) from the values of the ‘ + ’ side gives rise to a simultaneous rapid (almost instantaneous) large increase of the fibre contraction together with a rapid escalation of shear deformation, subsequently referred to as ‘fibre shearing’, in turn associated with an increase in the transverse strain. This occurs at the point of loss of ellipticity for any value of the reinforcing parameter, and, for sufficiently large values of the fibre reinforcement, a snap-back mechanism is associated with the non-dissipative kinking solutions. This means that the considered failure mechanisms are clearly dissipative, i.e. energy is dissipated in the process of kink band and splitting formation (see Prabhakar and Waas, 2013b).

It has been noted that large strains within the kink band suggest fibre/matrix splitting, which has been observed in experiments in conjunction with the formation of kink banding for certain materials (Prabhakar and Waas, 2013a).

Maximally dissipative shocks are also studied. Then no snap-back is found and a unique solution is singled out in the ‘ - ’ zone for both kinking and splitting, as for the non-dissipative case, although in that case just for sufficiently small values of the reinforcing parameter. Therefore, two families (kinking and splitting) nucleate, each being associated with the two different failure modes given by the breakdown of ellipticity in the ‘ + ’ zone. During subsequent motions for the maximally dissipative kinking solutions, there is a rapid increase of fibre contraction and fibre shearing while non-elliptic deformations convert into elliptic deformations. With respect to fibre kinking angles, as well as kink orientation angles, it is shown that maximally dissipative solutions might capture experimental observations closely although we are dealing here with just a simple prototype model and our results have a qualitative rather than a quantitative meaning. In addition, with respect to fibre splitting, maximally dissipative shocks are associated with fibre angles in the splitting zone which are almost zero and there is also a rapid increase of fibre contraction and fibre shearing.

Section snippets

Material model under plane strain

We consider an incompressible elastic material under plane strain deformation. Let D be the (plane) domain occupied by a body in its undeformed configuration and D* be the (plane) domain occupied by the same body in its deformed configuration. Vectors and tensors are described with respect to the planar orthonormal bases {Ei},i=1,2, in D, and {ei},i=1,2, in D*. Let χ denote the smooth invertible mapping that deforms D onto D* according tox=χ(X)=X+u(X),where X=X1E1+X2E2 and x=x1e1+x2e2 are the

Weak elastostatic shocks

The governing differential equation changes locally in type from elliptic to hyperbolic as the deformation of the material proceeds. This change is referred to as loss of ordinary ellipticity and it leads to the emergence of weak surfaces of discontinuity (otherwise referred to as weak shocks or characteristic surfaces). Let ξ be a coordinate in the direction normal to the shock. The surfaces carrying the discontinuities of the deformation gradient F, the stress tensor S and the pressure p are

The general solution for elastostatic shocks

In this section, for a fixed reinforcing parameter ρ, a general three-dimensional solution in (λ, k, α) space of (25)2 is determined and examined. It is first illustrated for ρ=3 since for larger values of ρ, the main features of the solution have similar characteristics. The projection of the solution for ρ=3 on to the (λ, k) plane is shown in Fig. 9, which reflects the symmetry (λ,n1,n2,k)(λ,n1,n2,k) about the horizontal line at k=0.

The solutions S2 and S3 are associated with angles α ≈ 90

Shock driving traction

Various aspects of the theory of finite elastostatics for materials characterized by non-elliptic elastic potentials have been studied in a number of investigations. Under suitable conditions, such materials can sustain equilibrium deformations with jump discontinuities in the derivative of the displacement field and the stress field across the shocks.

A quasi-static motion involving equilibrium elastostatic shocks at each instant may be dissipative. Thus, the existence of an elastostatic shock

Conclusions

It can be argued that fibre splitting is not related to continuous displacements as it is a catastrophic failure mechanism. However, the approach that we have adopted here provides useful data for understanding and describing the failure mechanisms. Furthermore, we have shown that connecting the invariants I4 and I5 with the formation of shocks, i.e. with the formation of different fibre failure mechanisms, can be exploited in the nonlinear constitutive modelling of fibre reinforced materials.

Declaration of Competing Interest

There are no conflicts of interest.

Acknowledgement

The authors acknowledge support from the Ministerio de Ciencia, Spain, under the Project number DPI2011-26167. Mustapha El Hamdaoui also thanks the Ministerio de Economía y Competitividad, Spain, for funding under the Project number DPI2008-03769 and Grant number BES-2009-027812.

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