Material instability with stress localization

J. Non-Newtonian Fluid Mech. 102 (2002) 251–261

Material instability with stress localization J.D. Goddard Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA

Abstract This paper is concerned with a possible form of material instability arising from a non-unique dependence of stress on deformation and leading to heterogeneous states of stress in otherwise homogeneous deformations. Following a discussion of certain theories and experiments that suggest this type of stress behavior, an analysis is given of some possible quasi-static bifurcations of steady-state stress fields for extensional motions of isotropic elastic solids and viscoelastic fluids. These involve extended lamellar and axisymmetric phase structures having common extension. For certain other restricted forms of non-linear elasticity, the classical Eshelby [Proc. R. Soc. London A 241 (1957) 376] theory is employed to describe dilute ellipsoidal phases having a deformation different from the globally imposed mean. © 2002 Elsevier Science B.V. All rights reserved. PACS: 01.30.-y Keywords: Material instability; Stress localization; Non-linear rheology; Buckling instability; Granular media; Polymeric drag reduction; Eshelby theory

1. Introduction The concept of material (constitutive) instability is now firmly established in continuum mechanics, particularly in solid mechanics, where it is manifest experimentally in various modes of localized deformation (strain localization) and failure, such as necks, shear bands and damage zones [1–4]. Such instability, the analog of phase transition in thermostatics, is mathematically associated with “loss of ellipticity” in the underlying field equations and often traceable to non-convexity [5] in the form of strain softening or other non-monotone stress-deformation behavior. This is illustrated paradigmatically Fig. 1, where Fig. 1(b) and (c) allow for coexistence of different states or “phases”, with the possibility of hysteretic transitions. Fig. 1(c), which is similar to the behavior associated with “disproportionation” and plastic flow in “dry” liquid foams [6], is qualitatively different from Fig. 1(a) and (b), in that it allows for the possibility of stress localization, the subject of the present paper. E-mail address: [email protected] (J.D. Goddard). Dedicated to Professor Acrivos on the occasion of his retirement from the Levich Institute and the CCNY. 0377-0257/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 0 1 ) 0 0 1 8 1 - 1

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Fig. 1. Non-monotone stress: (a) ductile failure; (b) gradual strain softening (or shear thinning) with coexistent strains; (c) catastrophic stress jumps.

Although axiomatic in continuum mechanics that the local stress in a material body is determined uniquely by the past history of deformation, reflecting the so-called “Principle of Determinism” [7], there are prominent examples to the contrary in structural mechanics, such as the (Euler) buckling of elastic rods [8]. Here, as in more complex structures, one has bifurcations between states with quite different axial load-displacement behavior. Therefore, to the extent that real materials may possess mechanically unstable microstructures, for example foamed and fiber-reinforced solids [9–14]and liquid crystalline and micellar systems [15–19], such instability should also be manifest in their continuum-level behavior. Fig. 2 provides a schematic illustration of the kind instability envisaged in the present work. In particular, Fig. 2(a) illustrates stress “blowup” in steady elongational flow, of a type found theoretically for suspensions of elastic spheres in viscous liquids [20], which resembles that found for elementary linear “bead-spring” models of polymers [21], or for the bursting of liquid droplets [22,23]. By contrast, Fig. 2(b) illustrates the steady elongational behavior of the “coil-stretch” model of de Gennes [24] (and see also [25–28]), as well as the shear behavior predicted in [29] for gas–solid dispersions in simple shear. Finally, Fig. 2(c) is intended to suggest the possibility of distinct phases, with differences in both stress and strain, a possibility discussed further below. The schematic diagrams in Figs. 1 and 2 cannot of course represent the full complexity of tensorial stress–strain relations, for which the basic issue is one of convexity in a higher dimensional space [5,30]. As another example, we note that the discontinuous shear thickening in colloidal dispersions, observed experimentally by Hoffman [31] and attributed to microscopic instability, may be subject to similar phenomenological interpretation. However, the author is unaware of any hysteresis observed in these

Fig. 2. (a) Stress blowup, and stress jumps with coexistent states having (b) the same, and (c) different states of deformation.

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measurements. Finally, we should mention buckling instabilities in granular assemblies, as a plausible explanation of stress inhomogeneity in the form of “force chains” [32–34], which may reflect a bifurcation into “strong” and “weak” phases [35], but perhaps of the buckling type depicted in Fig. 1(c). The purpose of the present work is to provide a reasonably elementary, yet general rheological theory of bifurcation and spatial heterogeneity of steady-state stress in isotropic elastic solids and viscoelastic fluids. In Section 2, we focus attention on steady extensional motion, which appears more conducive to coexistent states having common strain but different stress than, say, simple shear, where the opposite is true [19]. Then, we consider briefly the possible effect of more complex deformations for a restricted class of elastic materials. We note that all the examples illustrated in Fig. 2 involve “critical” points with infinite derivatives of steady-state stress. While this suggests a strong instability of perturbed stress fields, a complete stability analysis would require more complete rheological models than those considered here. The aim here is not to work out detailed solutions for specific rheological models but rather to indicate the possibility of such solutions for a fairly broad class of constitutive equations. As with countless other viscoelastic phenomena, non-linear elasticity provides a useful point of departure in the investigation of non-linear effects. 2. Quasi-static and piecewise-homogeneous extension At this point, we restrict the analysis to isotropic materials, recalling that the constitutive equation for anisotropic elastic solid can be written in the implicit form Hij (Tjk , Blm ) = 0,

i, j, . . . , m = 1, 2, 3,

or

H(T, B) = 0

(1)

where T = TT is Cauchy stress, B = BT the (right Cauchy–Green or Finger) strain, and H = HT is a symmetric isotropic tensor-valued map (technically speaking, R6 × R6 → R6 ), such that H(QTQT , QBQT ) = QH(T, B)QT

(2)

for arbitrary real orthogonal tensors QT = Q−1 . We assume that H has a finite number of piecewise differentiable branches on which T can expressed as an explicit function of B and vice versa. We recall that B can be replaced by any isotropic function of B, e.g. E = 21 (B − 1)

(3)

which reduces to the standard infinitesimal strain measure in the small-strain limit. On the other hand, if E in (3) is replaced in (1) by the stretching (strain rate) D = 21 {∇ v + ∇ vT }

(4)

where ∇ v is the velocity gradient, a representation of the form (1) also applies to the steady pure stretching of simple fluids, where by definition D can be taken as independent of time at a given material point, in a frame where the vorticity W = 21 {∇ v − ∇ vT }

(5)

vanishes identically. It is worthwhile to regard (1) as the steady-state limit of a viscoelastic or hypoelastic model of the form T˙ = H(T, ∇ v) (6)

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where the dot denotes the material derivative. A dynamic model such as (6) would of course be required for a stability analysis and, hence, a comprehensive bifurcation analysis of the various steady-state solutions of (1) to be identified in the following sections. We focus attention mainly on two special deformations: (1) homogeneous extension or pure stretch, where the displacement or velocity fields are given on a cartesian system xyz, respectively by u = x x,

v = y y,

w = z z

(7)

where the principal strains or strain rates  are constant, and (2) axisymmetric extension, where these quantities are given on a cylindrical polar system rθ z, respectively by u = u(r),

v = 0,

w = z z

(8)

with z constant. In both the above cases we can replace B or D and T in (1), respectively by diagonal forms, in the first case by E = diag(x , y , z ),

T = diag(σx , σy , σz )

and in the second by
 du u E = diag , , z , dr r

T = diag(σr , σθ , σz )

(9)

(10)

representing principal values j and σj , j = 1, 2, 3. Then, (1) reduces to three generally non-linear equations of the form hi (σj , k ) = 0,

i, j, k = 1, 2, 3

(11)

with 1,2,3 representing x, y, z or r, θ, z, respectively. In the special case of incompressible materials, the three relations (11) reduce to two relations involving only two each of the deviatoric quantities, say, ˆj and σˆ j which satisfy 3 

σˆ i = 0,

i=1

3 

ˆi = 0

(12)

i=1

For the simple deformations (9) and (10), we now investigate the multiplicity of solutions of (11) and the quasi-static equation of equilibrium: ∇ ·T=0

(13)

in spatially unbounded regions. 2.1. Lamellar bifurcation It is clear that (13) admits piecewise homogeneous solutions of the type (9) provided the tractions and displacements or velocities can be matched at the boundaries of the various homogeneous regions

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Fig. 3. Lamellar bifurcation.

or “phases”. One class of such solutions is obtained with parallel lamellar or “smectic” phases, lying perpendicular to the x-direction, say, as illustrated in Fig. 3, and distinguished here by primed and unprimed quantities. For example, with all the quantities y = y ,

z = z ,

σx = σx

(14)

specified, (11) provides three equations for x , σy , σz and three identical equations for x , σy , σz . Of course, if the solution to the above set of simultaneous equations is unique, then the phases are identical and the corresponding globally homogeneous state is unique. Otherwise, one has the possibility of a space-filling “stack” of distinct lamellae, each of arbitrary thickness and each representing a distinct solution to the equations in question. Here, as with thermostatic phase transitions, higher gradient or “non-local” effects are required to set length scales and regularize field equations, and such effects may also influence material instability [4,42,43]. In the case of incompressible materials, we have x + y + z = 0, with only two equations for the deviatoric stresses σy , σz and their primed counterparts (Fig. 3). 2.2. Axisymmetric bifurcation One class of axisymmetric solutions is represented by (8) and (10) with all relevant variables being independent of θ and z. Then, (13) reduces to a single differential equation r

dσr + σr − σθ = 0 dr

(15)

For the sake of simplicity, we assume that the relations (11) can be solved explicitly for the ’s, with du = E1 (σr , σθ , σz ) dr u = E2 (σr , σθ , σz ) r z = E3 (σr , σθ , σz )

(16) (17) (18)

Obviously, (15)–(18) represent a set of two ODEs subject to two algebraic constraints. We note that certain special integrals are given by Green and Adkins [36] for the case of purely elastic (hyperelastic) solids

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endowed with strain–energy function, with considerable simplification arising from the assumption of incompressibility. For given, constant z the essential problem is to integrate these equations, subject to continuity of σr (r) and u(r) and to certain regularity conditions at r = 0. To this end, we further assume that (18) possesses a solution, possibly non-unique, for σz in terms of σr , σθ and z , so that σz can be eliminated from (16)–(18), after which z can be treated as a constant parameter. Then, letting u ξ := log r (19) ω := , r (15)–(18) can be replaced by the autonomous set dσr = σθ − σ r dξ

(20)

dω = γ (σr , σθ ) dξ

(21)

ω = Ω(σr , σθ )

(22)

where Ω(σr , σθ ) := E2 {σr , σθ , σz (σr , σθ )}

(23)

γ (σr , σθ ) := E1 {σr , σθ , σz (σr , σθ )} − E2 {σr , σθ , σz (σr , σθ )}

(24)

and z is an implicit parameter. Once again, we have simultaneous ODEs (20) and (21) with algebraic constraint (21), and once again we could employ one of the possibly multiple solutions of the latter to eliminate σθ and, hence, to reduce (20) and (21) to simultaneous ODEs for σr and ω. Without making this explicit, we see that the non-uniqueness associated with Ω leads to the possibility of solution branches exhibiting any number of radial discontinuities in azimuthal (hoop) stress σθ , depicted in Fig. 4, and shear strain γ , while maintaining continuity of radial stress σr and displacement u. In the case of an incompressible material, the continuity equation du u + + z = 0 dr r

Fig. 4. Axisymmetric bifurcation.

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together with regularity (absence of a mass source) at r = 0, imply that ω = −z /2. Then (20)–(22) reduce to dσ 1 = σr − σθ = const., where σ := (σr + σθ ) (25) dξ 2 giving the mean in-plane mean stress σ in terms of the in-plane shear stress σr −σθ . The latter is piecewise constant by virtue of the deviatoric forms of (16)–(18), whose multiple solutions allow once more for contiguous regions with different (constant) values of σz and σr − σθ . Returning to the compressible material, we note that one possible solution of (20)–(22) is given by the transversely-isotropic, homogeneous state θ ≡ r ≡ ω ≡ const.,

σθ ≡ σr ≡ const.

(26)

Such a solution serves to represent a far-field state for r → ∞ and another homogeneous state for r ≤ R having different values for the constants in (26). The transition between these states is then governed by (20)–(22), subject to the relevant continuity conditions at r = R. The above type of solution, or its incompressible counterpart, represents an axisymmetric filamentary (or annular) structure embedded in a large body of material in a different state in the far field. Although this structure resembles the “birefringent strands” or “pipes” of [37,38], the latter appear to arise from a strongly inhomogeneous flow field. In any event, since the axial stress in a filament σz can be much higher than that in the surrounding material, as suggested by Fig. 2(b), a dilute array of non-interacting filaments could make a major contribution to the axial stress. If the two different stress states are assumed to represent the macromolecular coil-stretch transition envisaged in [24], the resulting model is relevant to phenomena such as the suppression of turbulence by high molecular-weight polymers [39] or, for that matter, by other orientable slender bodies [40,41]. However, this type of problem will generally involve a difference in axial extension z between filament and far-field and, hence, axial shears, rz , σrz , of a type found, e.g. in the analysis of linear-elastic fibrous composites [44]. This requires a modification of the above theory, and one possibility is considered next. 3. Differential extension In the preceding examples, at least one principal extension is identical in the two phases, and the question arises as to possibility of phases having distinct extensions, as depicted schematically in Fig. 2(c). For example, consider the axisymmetric extension treated in Section 2, with imposed global mean z : z  = ϕz + (1 − ϕ)z

(27)

where z and z denote volume averages over two distinct phases occupying respective volume fractions ϕ and 1 − ϕ. With a similar expression for mean stress σz , the mean state is then represented by a point lying on the line segment connecting points on the upper and lower branches in Fig. 2(c) and related by a well-known “lever rule”. For the general non-linear form (1) an example has not been found in the present work of a two-phase structure with completely distinct principal extensions. However, some progress can be made for the situation, illustrated schematically in Fig. 5, where the relation (1) involves at least one linear branch, on which T and E are linearly related, as we now show.

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Fig. 5. (a)–(c) Bifurcations with linear (L) and non-linear (N) branches.

3.1. Dilute ellipsoids in a linear matrix The classical theory of Eshelby [45] for the static stress field around an ellipsoidal inclusion in an infinite linear-elastic matrix has been applied to a wide range of problems involving various heterogenieties in elastic bodies [46], including thermostatic equilibrium between different phases of the same material. In a similar way, it can be applied to the two-phase structure postulated in the present work, provided the external or matrix phase is represented by a linear branch of (1), which we now assume to have the standard linear-elastic form Tij = Cijkl Ekl ,

or

T = CE

(28)

where C = [Cijkl ] represents the elastic constants. Here, as below, we employ upper case script to denote fourth-rank tensors regarded as linear transformations of second-rank tensors, with essentially all being symmetric. Furthermore, we relax our assumption of isotropy, supposing (1) to be replaced by a relation appropriate to anistropic materials, which generally involves a dependence on the finite rotation [7] . Following the analysis of Eshelby [45] and Mura [46], consider an infinite linear-elastic region subject to an (infinitesimal) homogeneous strain E∞ at infinity and a displacement u = E0 x at points x on the boundary x · Ax = Aij xi xj = 1,

with

A = AT = const.

(29)

of a solitary, couple-free ellipsoidal inclusion undergoing a hypothetical homogeneous strain E0 . It follows from Eshelby’s results that the matrix exerts a traction T0 n on the surface of the inclusion, where n = Ax/|Ax| is the unit outer normal, and T0 is a constant stress, corresponding to a hypothetical uniform stress field inside the inclusion, given by T0 = C {S −1 E∞ + (I − S −1 )E0 }

(30)

where I is the fourth-rank idemfactor and S −1 is the inverse of the so-called Eshelby tensor S = [Sijkl ], which depends on C and A. Mura [46] gives the Sijkl for an isotropic matrix and for various special cases of A (spheroids, elliptical cylinders, etc.), together with citations of literature dealing with other special forms of C . Since a homogeneous strain E0 of the inclusion induces the homogeneous stress T0 in (30), the pair E0 , T0 serves to represent at least one solution to (1) and (13) in the interior of the ellipsoidal inclusion.

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Hence, with a linear branch of (1) representing the infinite matrix, a second, possibly non-linear branch represents the ellipsoidal inclusion. This idea underlies the suspension model of Roscoe [20] and has also been invoked recently by Ledbetter and Dunn [47] in their treatment of the thermostatic phase transition between a non-linear elastic inclusion and a linear-elastic matrix. In the case of finite, non-interacting ellipsoids with volume fraction φ  1, it is, therefore, possible to have a strain E0 in the inclusion which differs from the mean strain E∞ in the matrix, neither of which is restricted to purely extensional deformations. Then, given an imposed global strain E = φE0 + (1 − φ)E∞

(31)

T0 , E0 , E∞ presumably can be expressed in terms of E, by means of (30) and (31), and the appropriate branch of (1). This is particularly obvious in the special case of a linear-elastic inclusion, represented schematically by Fig. 5(c), since the substitution of the form T0 = C ∗ E0 into (30) immediately yields a linear relation between E0 and E∞ involving the elasticities C , C ∗ of matrix and inclusion. The resulting linear-elastic formula can also be derived by somewhat less direct methods [46]. As a final issue, we note that a solitary cylindrical or lamellar structure of the type considered in Section 2 can be represented by an ellipsoid having one or two infinite axes, respectively. In this case, it follows from the formulae given by Mura [46] for linear isotropic elastic matrices that the Eshelby tensor S becomes singular, with unbounded inverse S −1 . This suggests that, barring some extreme rheological non-linearity giving rise to a virtual “slip” between phases, such as that envisaged in the non-linear fluid model of [48], it is generally not possible to sustain a differential extension between indefinitely extended phases, in directions lying parallel to the phases. While this conjecture is relevant to a number of fields, particularly composite materials, its resolution would take us well beyond the scope of the present article. In closing, it should be emphasized that, in contrast to Section 2, the analysis of the present section is generally restricted to elastic solids, since the representation (1) is valid at best as an approximation for restricted classes of viscoelastic fluids. 4. Conclusions The preceding analysis serves to establish the theoretical possibility of heterogeneous stress fields arising from rheological models with non-unique stress. If the admissible stress states have widely disparate magnitudes, this serves to define stress localization, which is to be distinguished the more familiar strain localization usually associated with material instability. However, the steady-state analysis of the present paper generally does not provide a description of the evolution and stability of various heterogeneous states. The possible importance to several phenomena, such as force transmission in particulate media or the suppression of fluid turbulence by polymer additives, would seem to warrant further theoretical and experimental investigation. Acknowledgements Partial support from the US National Aeronautics and Space Administration (Grants NAG3-1888 and NAG3-2465), and the National Science Foundation (Grant CTS-9510121) is gratefully acknowledged. Special thanks are due to Professor Andreas Acrivos, Mentor worthy of Odysseus.

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