From lattices and composites to micropolar continua

From lattices and composites to micropolar continua Iwona Jasiuk GWW School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA Martin Ostoja-Starzewski Department of Mechanical Engineering, McGill University, Montréal, Qué., Canada May 15, 2003

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ABSTRACT This paper starts out by recalling the basic assumptions required for the classical (Cauchy type) continuum mechanics to hold. It turns out that the micro- and nano-scale structures of many materials - such as lattice-type porous solids and two-phase composites - may not satisfy the requirement of separation of scales and/or the postulate of negligible internal couple force fields. If that is the case, the mechanics of material cannot be predicted correctly by a classical theory of a Cauchy continuum. We give a sketch of history and achievements of non-classical (Cosserat type) elastic continua, and then focus on micropolar models, their planar cases, and the somewhat enigmatic characteristic lengths. This is followed by an outline of formulation of micropolar elasticity models for three beam-type lattices of classical geometries: triangular, square, and honeycomb. Next, we review very recent studies on determination of such models for two-phase (inclusion-matrix) periodic composites, which are locally of Cauchy type. It is shown that, in cases of both lattices and composites, the micropolar moduli and the characteristic lengths can be determined from first principles. The paper also provides a summary of the pros and cons of Cauchy vis-à-vis Cosserat models, and of the outstanding challenges in this field.

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1 1.1

Introduction Motivation

In the newly emerging area of nanotechnology nanoscale material structures hold the potential to revolutionize the current technology. There arise, however, fundamental questions regarding the appropriate mechanical models and constitutive properties of such materials at the nanoscale? It is known that material properties depend on the actual micro-/nanoscale geometry, and, consequently, various size effects are observed. In general, properties such as stiffness and strength change at lower scales. The issue of size effects also arises in materials that have microstructure details that are not much smaller than specimen size. These can occur at larger scales, in heterogeneous materials such as foams (e.g., bone) or concrete. Size effects cannot be fully addressed by classical (i.e., Cauchy type) continuum mechanics theories. One can begin by considering atomic and/or molecular level but this brings forth great complexity and computational mechanics challenges. Another possibility is to employ a generalized continuum mechanics approach in which more material constants allow one to capture some microstructural features not detectable by classical theories. This indeed is the viewpoint taken in the present paper. When turning to generalized continuum mechanics, one recognizes that there is a certain hierarchy of higher-order continuum theories. Roughly speaking, in the order of increasing complexity, such theories include poroelasticity, Cosserat/micropolar theories, strain gradient theories, multipolar theories, nonlocal theories, and so on. However, as the complexity increases, so does the number of material coefficients that need to be determined. Thus, a compromise between the realism of the model and its usefulness has to be made. In this paper we focus on the Cosserat/micropolar theories for elastic materials in small strain. The basic distinction of any such theory from the classical one consists in endowing a continuum point with its local orientation relative to the surrounding material, and then with its deformation. The Cosserat brothers (1896, 1909) did so by placing a triad of vectors, or directors (trièdre cachè), at each continuum point, and thus introducing a means to account for three rotations besides three displacements. When these directors are rigid, such a model is called a micropolar continuum; when the directors are stretchable it is a microstretch continuum, and when they are deformable it is a micromorphic continuum. However, the terminology is not unified - sometimes the term Cosserat (or micropolar, couple-stress) continuum is simply used to make a 2

distinction from a Cauchy continuum.

1.2

History and comparisons of micropolar to classical elastic materials

The theory of Cosserat brothers (1896, 1909) remained dormant for half a century, apparently the only exceptions being the works of Somigliana (1910) and Sudria (1935); see also Ball & James (2002). The situation was likely due to the theory’s generality (as a nonlinear theory with finite motions and inelastic interactions) and its presentation as a unified theory incorporating mechanics, optics, magnetism and electrodynamics. The dynamic growth of continuum mechanics and thermodynamics (e.g. Ericksen & Truesdell, 1958; Truesdell & Toupin, 1960) begun in the fifties and sixties brought the work of Cosserat brothers back into focus. Fundamentals of a general linear Cosserat continuum were given by Günther (1958), who discussed in detail the 1-, 2-, 3-D Cosserat models, as well as their significance in the dislocation theory, and Schäfer (1962), who focused on the planar case. From that period one should also mention several other works. Thus, Grioli (1960) established the constitutive relations for finite deformations of perfectly elastic solids. Aero & Kuvshinskii (1960) independently derived the equilibrium equations and constitutive relations for anisotropic solids in the linearized theory. Mindlin & Tiersten (1962) established the boundary conditions; see also (Kröner, 1963; Koiter, 1963; Eringen, 1968). An expression of the widespreading interest in Cosserat theory was soon found in symposia (e.g. Kröner 1968) and monographs on the subject (e.g., Nowacki, 1970, 1986; Stojanovic, 1970; Brulin & Hsieh, 1982). Mention should also be made of a poroelasticity theory (Cowin & Nunziato, 1983). Building on the shoulders of Cosserats, and to account for increasing levels of complexity, other, more general theories accounting for higher order interactions such as monopolar, multipolar, and strain-gradient were introduced, see e.g. (Green & Rivlin, 1964; Toupin, 1962, 1964; Jaunzemis, 1967; Tiersten & Bleustein, 1974). There are also ‘micropolar’, ‘microstretch’ and ‘micromorphic’ continua (Eringen, 1999, 2001; Mariano, 2001). In a certain sense, all of these theories can be considered as simpler cases of ‘nonlocal continuum theories’ (Eringen & Hanson, 2002), which, according to these authors, “are concerned with material bodies whose behavior at any interior point depends on the state of all other points in the body — rather than only on an effective field resulting from these points — in addition to its own state and the state of some calculable external field.”

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Focusing more on micropolar theories, we would like to note their extensions beyond purely elastic material behaviors. An extension pertaining to thermoelasticity was already given by Nowacki (1966) and Tauchert et al. (1968), see (Dhaliwal & Singh, 1987) for a review. A micropolar generalization of viscoelasticity was presented by Maugin (1974). Beginning with Green & Naghdi (1965), Mi¸sicu (1964) and Sawczuk (1967), there has also been research on (elastic-)plastic continua with microstructure, e.g. Fleck et al., (1994), Hutchinson (2000). This has then led to strain-gradient models (Aifantis, 1987; Zbib & Aifantis, 1989; Fleck & Hutchinson, 1997). An extensive research has also been done on micropolar fluid mechanics (e.g., Cowin, 1974; Eringen, 2001).

1.3

Micropolar vis-à-vis classical elasticity models

While the non-classical theories have become very advanced mathematically and explained effects which could not be brought out by classical theories, they usually lacked the input of physically based constitutive coefficients. The progress that has been made over the last few decades is reviewed below in four categories of solid mechanics. Elastostatics and experiments Mindlin (1963) found that stress concentrations in the presence of holes are lowered in Cosserat-type versus those in Cauchy-type solids. This was followed by studies due to (Neuber, 1966; Kaloni & Ariman, 1967; Cowin, 1970a,b; Itou, 1973). On the other hand, an increase of stress concentrations in the vicinity of rigid inclusions was established by (Hartranft & Sih, 1965; Weitsman, 1964). The case of holes motivated one of the earliest experimental studies of couple-stress effects by Schijve (1966), who actually found that effect to be insignificant. However, given the fact that he used aluminum sheets - a macroscopically homogeneous material without, say, reinforcing inclusions his investigation pertained to couple-stress effects due to the atomic lattice of aluminum. This is not surprising in view of the fact that couple stress effects vanish on scales much larger than the microscale. Indeed, the situation is much different in, say, a lattice of beams (which may be interpreted as a material with large holes), if one looks at dependent fields on scales comparable to the lattice spacing, see Section 4 below. Discrete systems - lattices, porous materials, composites Several workers, already in the sixties, derived micropolar models explicitly from the

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microstructure. The work of theoreticians started from lattice-type models enriched with flexural - in addition to central - interactions (e.g. Askar, 1968; Banks & Sokolowski, 1968; Wo´zniak, 1970; Bazant & Christensen, 1972; Holnicki-Szulc & Rogula, 1979a,b; Bardenhagen & Triantafyllidis, 1994). From the outset, these models adopted Cosserat type continua in analyses of large engineering structures such as perforated plates and shells, or latticed roofs. There, the presence of beam type connections automatically led to micropolar interactions and defined the constitutive coefficients. In principle, such models have their origin in atomic lattice theories, e.g. (Berglund, 1982); see (Friesecke & James, 2000) for the latest work in that direction. Several workers (e.g. Perkins & Thompson, 1973; Gauthier & Jahsman, 1975; Yang & Lakes, 1982; Lakes, 1983, 1986) have provided experimental evidence of micropolar effects in porous materials such as foams and bones. In particular, Lakes (1995) was able to infer micropolar constants from his experiments, both for centrosymmetric and chiral materials. Another interesting application in the context of biomechanics was due to Shahinpoor (1978). It is also to be noted that composite materials may naturally lead to Cosserat models where the nonclassical material constants can directly be calculated from the microstructure, this was done in 1-D by Herrmann & Achenbach (1968). But, a similar task in 2- and 3-D has only been undertaken recently, and this is described in Section 4. In more recent years, progress has been made on derivation of effective (homogeneous) Cosserat models for heterogeneous composite materials of either Cauchy or Cosserat type. Here we point out that a central-force lattice (truss of two-force members) is an example of the former material, while a lattice of beams is an example of the latter one. Fracture mechanics All the studies in the area of stress singularities due to cracks were preceded by (Muki & Sternberg, 1965), who studied stress concentrations caused by concentrated surface loads or discontinuously distributed surface shear tractions. Next, Sternberg & Muki (1967) and Bogy & Sternberg (1967) studied the implications of the couple-stress theory on unbounded concentrations of stress and on locally infinite deformation gradients. Basically, it was found that, depending on a given situation, where the classical elasticity would predict infinite (singular) stresses, the couple-stress theory may give either finite stresses or weaker singularities, or have an opposite tendency (see also Cowin, 1969; Atkinson & Leppington, 1977). This involves a proper generalization of conservation integrals, which has recently been given in the setting of couple-stress elasticity (Lubarda & Markenscoff, 2000).

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Recently, the Griffith’s fracture theory has been generalized to rectilinear and fractal cracks in micropolar solids (Yavari et al., 2002). In particular, two cases of the Griffith criterion were considered, depending on whether the effects of stresses and couple-stresses are coupled or uncoupled, the key finding being that both cases give equal orders of stress and couple-stress singularities, which is the same result as that in a classical continuum. Also, the effect of fractality of fracture surfaces on the powers of stress and couple-stress singularity was studied. Elastodynamics Many studies of wave propagation in the context of harmonic disturbances were conducted. First, in addition to classical dilatational and shear waves in an unbounded medium, there also exist rotational waves. Next, it turns out that only the dilatational waves propagate non-dispersively (Nowacki, 1986; Eringen, 1999). In general, this is indicative of various new dispersion effects in other wave problems, which are not present in classical continua. In some cases of Cosserat continua, entirely new phenomena arise such as, for instance, that a layer on top of an elastic half-plane is not necessary for the propagation of Love waves - in the classical case, a layer is necessary. Many results on periodic and aperiodic waves were collected by Nowacki (1986), see also (Eringen, 1999).

1.4

Outline of the article

In the next section we discuss three ‘paths’ to non-classical models: (i) a heterogeneous microstructure which intrinsically carries moments besides forces, (ii) a random medium lacking the separation of scales, (iii) a homogenization technique for a random local medium which results in a deterministic nonlocal model. We next focus on the micropolar elastic continuum, and in Section 3 we discuss its basic relations. The fourth section, on the other hand, outlines lattice models which, with reference to Fig. 1, serve to explicitly derive constitutive micropolar laws, including the characteristic lengths. The fifth section shows that this can also be done for two-phase composite materials, and it involves inhomogeneous boundary conditions. The sixth and last section of this review lists some outstanding issues and challenges.

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2 2.1

Why Cosserat models? Force-stress, couple-stress, and kinematics

Every course on solid mechanics starts out with an introduction of the Cauchy stress concept. This first involves identification of a finite surface area ∆A (= L2 ) - either in the interior of the body or on its external surface - defined by an outer unit normal n, and a force ∆F acting on ∆A, Fig. 1(a). [In the following, we shall interchangeably use the vector notation (F) and the subscript notation (Fi ).] Next, one considers a ratio of ∆F to ∆A, and takes the limit ∆F(n) (2.1) = t(n) . lim ∆A→0 ∆A It is a basic postulate of conventional solid mechanics that such a limit is well defined, i.e. that it is finite except the singularity points in the body, such as crack tips. In a third step, following Cauchy himself, one introduces his force-stress tensor σ as a linear mapping from n into t(n) t(n) = σ · n.

(2.2)

Let us note, however, that any consideration of a finite area ∆A should involve a surface couple ∆M accompanying ∆F. Thus, in analogy to (2.1), we must consider ∆M(n) = m(n) , (2.3) lim ∆A→0 ∆A and, following Voigt (1887) and the brothers Eugène and François Cosserat (1909), should introduce a couple-stress tensor µ as a linear mapping from n into m(n) m(n) = µ · n. (2.4) If the microstructure is disregarded, we are dealing with an idealized, homogeneous continuum in which ∆M must vanish in the limit L → 0. To see this, take n to be aligned with n1 , and consider shear stresses σ 12 and σ 13 . The torque caused by them, proportional to L3 (σ 12 − σ 13 ), must disappear as L → 0, because the cube’s volume scales as L3 . Otherwise, we would be left with a non-zero angular acceleration of a continuum point. This, in fact, is the case with classical/conventional solid mechanics of Cauchy-type continua. One then only has displacement u at a point, and assumes that m(n) = 0. But, if the material intrinsically carries couples, we cannot disregard ∆M. Such a situation occurs when the material has a discrete-type microstructure such as a beam-lattice shown in Fig. 1(b), which simply precludes one from taking 7

Figure 1: (a) Force ∆F and couple ∆M acting on an internal (or external) surface area ∆A (= L2 ) in a continuum; ∆A is the area of any face of a cubic element of side L. (b) A porous medium in 2-D, viewed as a beam lattice, with each beam carrying a force and a couple locally. A unit cell of size L is indicated with dashed lines. ∆A to 0. Here one needs to take ∆A equal to the area of the elementary cell’s cross-section, and the moment traction m(n) in (2.3) is defined at ∆A finite. For this model of force distribution in a continuous body to be fully consistent with kinematics, each continuum point is endowed with the degrees of freedom of a rigid body: displacement u and rotation ϕ, which are, in general, independent functions of position and time. In particular, this implies that ϕ is not the same as the macro-rotation given by the gradient of u, such as in Bernoulli-Euler beams which are known as kinematically constrained 1-D continua of Koiter type, a special case of Cosserat continua to be discussed in Section 2.1 below. Now, whether we deal with a Cauchy or a Cosserat continuum, the small ∆V volume of side area ∆A indicated in Fig. 1(a) is called a Representative Volume Element (RVE) of continuum mechanics. Therefore, before proceeding to the discussion of Cosserat-type continua, it is imperative to recall the RVE concept and the closely related separation of scales.

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Figure 2: (a) Boolean model of Poisson polygons (e.g., modeling tungstencarbide (black) and cobalt (white)); (b) dead leaves random tessellation of Poisson polygons (e.g., modeling randomly microlayered systems); (c) macroscopic body - like a part of a MEMS device - whose length scale Lmacro is not necessarily much larger than L. Figures (a) and (b) are generated by MicroMorph (1999).

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2.2

Separation of scales

The assumption of separation of scales in classical continuum mechanics states that d < L ¿ Lmacro or d ¿ L ¿ Lmacro , (2.5) where Lmacro is the length scale of the macroscopic domain Bmacro of the body generally having some random microstructure; L is the length scale of the intermediate (mesoscopic) domain BL of random microstructure over which smoothing (or homogenization) is being done relative to the microscale (e.g. heterogeneity size) d, which may be required to be smaller or much smaller than L, depending on the actual mismatch of properties and the random microgeometry of the composite, Fig. 2. Hashin (1983) calls (2.5)2 a MMM principle MICRO ¿ MINI ¿ MACRO

(2.6)

where MICRO, MINI, and MACRO stand, respectively, for the microscale, the RVE scale, and the macroscopic dimensions of the body. As discussed below, the RVE may be attained on scales L (or MINI) not necessarily much larger than d (or MICRO), and thus we have two possibilities in (2.5). Let us next discuss the fundamental issues involved in (2.5) in the setting of materials made of linear elastic phases (governed by Hooke’s law) σ = C · ε,

(2.7)

where C (≡ Cijkl ) is stiffness tensor of a given phase and ε is infinitesimal strain tensor. The material is homogenized according to the Hill condition (Hill, 1963) σ · ε = σ·ε, (2.8) which ensures that the mechanical and energetic definitions of the apparent stiffness tensor C(L/d) are equivalent (Huet, 1990, 1999). Following the latter reference, we use the adjective apparent to distinguish the properties which apply on mesoscale L/d from the effective ones, which apply on infinite scale L/d → ∞. Now, C(L/d) is determined from Z 1 V (L/d) εij Cijkl εkl dV = ε0ij Cijkl ε0kl . (2.9) 2 V 2 This is an equality of the total energy stored in the mesoscopic domain with the energy of an approximating continuum subjected to a uniform displacement 10

(also called kinematic, essential, or Dirichlet) boundary condition on surface ∂BL of BL u(x) = ε0 · x ∀x ∈ ∂BL . (2.10) On the other hand, we can determine an apparent compliance tensor S(L/d) from an analogous equality of complementary energies Z 1 V (L/d) σ ij Sijkl σ kl dV = σ 0ij Sijkl σ 0kl , (2.11) 2 V 2

where the mesoscopic domain is subjected to a uniform traction (also called static, natural, or Neumann) boundary condition t(x) = σ 0 · n ∀x ∈ ∂BL .

(2.12)

In (2.10) and (2.12) ε0 and σ 0 are uniform strain and stress tensors, respectively. Note that equations (2.9) and (2.11) employ average strain (ε0 = ε) and av£ ¤−1 erage stress (σ 0 = σ) theorems, respectively. In general, C(L/d) 6= S(L/d) , £ ¤−1 tend to converge to Cef f , and how but, with L/d → ∞, C(L/d) and S(L/d) fast an acceptable approximation is£ reached ¤ depends on a given microstruc(L/d) (L/d) −1 ture. In fact, C (respectively, S ) provides a stiffer (softer) L/def f dependent bound on C . For reviews of studies on the resulting coupled scale and boundary conditions effects see (Huet, 1999; Ostoja-Starzewski, 2001). When there is no material randomness, i.e. when the microstructure is periodic, d is comparable to L, and we can use a homogeneous periodic boundary condition u(x) = ε0 · x + u(x + L) ∀x ∈ ∂BL ; (2.13)

to determine C(L/d) ≡ Cef f from (2.13). In that case, we clearly have (2.5)1 . The homogenization of periodic media (e.g., Hornung, 1997), just like the homogenization of random media outlined above, requires that the inequality L ¿ Lmacro in (2.5) holds, and this is the range over which classical continuum mechanics of homogeneous media applies. It follows from studies on the coupled scale £ (L/d) ¤−1 and boundary conditions effects (L/d) that the convergence of C and S to within a few percent of the stiffness tensor characterizing the RVE is strongly dependent on the actual microstructure. Two general conclusions that can currently be made are: (i) mesoscale responses of composites with soft inclusions in a stiff matrix converge (sometimes much) more slowly with increasing size to RVE than those of composites with stiff inclusions in a soft matrix; (ii) anti-plane elasticity is slowest, in-plane elasticity is faster, and 3-D elasticity is fastest. 11

Now, it may turn out that, in a particular situation, L required by homogenization of a random medium may be so large (e.g., L = 103 d) that we will only have L < Lmacro or not even that. In such a case, also inhomogeneous boundary conditions need to be used besides either (2.10) and (2.12), or (2.13) above. One way to see the reason for introducing inhomogeneous loadings on BL is to note that, roughly speaking, Lmacro is the wave length of applied stress/strain gradients in the macroscopic body domain. Evidently, if L is not much smaller than Lmacro , the effective Hooke’s law should reflect the dependence on such gradients, and this is where non-classical (Cosserat-type) theories also arise.

2.3

Homogenization of a random Cauchy continuum

Suppose we consider statics of a random medium of linear elastic type. Thus, the material spatial randomness is described by a random stiffness tensor field in some domain D, which is written as an ensemble of realizations ω ∈ Ω varying smoothly over x ∈ D Cijkl = {Cijkl (ω, x); ω ∈ Ω, x ∈ D}

(2.14)

It follows that the governing equations for any realization ω ∈ Ω are σ ij,j = 0, σ ij = Cijkl (ω, x)εkl , εkl = u(k,l) , ω ∈ Ω, x ∈ D.

(2.15)

Using a smoothing procedure, Beran & McCoy (1970a) have shown that the field equations for an ensemble averaged medium are ∂ hσ ij i + fi = 0 ∂xj

(2.16)

whereby the ensemble averaged constitutive law is given by a nonlocal equation

hσ ij (x)i = hCijkl (x)i hεij (x)i +

Z

Λijkl (x, x0 ) hεij (x0 )i dx0

(2.17)

wherein Λijkl (x, x0 ) = Dijkl (x0 ) δ (x − x0 ) + Eijkl (x, x0 )

(2.18)

and the ensemble averaged strain tensor is related to the ensemble averaged displacement field by 12

1 hεij (x)i = 2

µ

∂ hui i ∂ huj i + ∂xj ∂xi

¶

(2.19)

There are two things to note here: (i) Dijkl (x0 ) and Eijkl (x, x0 ) in (2.18) are functions determined by the statistical properties of Cijkl and the free space Green’s function of the homogeneous medium (deterministic) problem. In essence, Λijkl (x, x0 ) is an infinite sum of integro-differential operators, involving, in principle, moments of all orders for the random field Cijkl . (ii) The ensemble average response of a random linearly elastic (local) solid is that of a deterministic nonlocal elastic solid. When the fluctuations about the mean are small, Dijkl (x0 ) and Eijkl (x, x0 ) can be evaluated explicitly. This has been done in the special case when Cijkl (x) can be written as a locally isotropic tensor, that is, in terms of two Lamé constants Cijkl (ω, x)=λ(ω, x)δ ij δ kl + µ(ω, x)(δ ik δjl + δ il δ jk ),

(2.20)

Furthermore, considering the contribution of the field within the correlation radius of Cijkl (x), and writing the strain εkl (x0 ) as Taylor series

εkl (x0 ) = εkl (x) + (x0m − xm ) εkl,m (x) +

(x0m − xm ) (x0n − xn ) εkl,mn (x) + ... 2 (2.21)

we obtain ∗ (x) hεij (x)i+Dijklm hεkl,m (x)i+Eijklmn hεkl,mn (x)i+... (2.22) hσ ij (x)i = Cijkl

This constitutive law shows that the average stress is related to the average strain and its first, second, and higher gradients. This subject was pursued in a subsequent paper (Beran & McCoy, 1970b), where, among others, it was found that a first strain gradient model is not well posed, and higher order terms need to be retained. See also McCoy (1991) for an extension of this approach to two-phase composites, and (Drugan & Willis, 1996; Buryachenko, 2001) for related work.

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3 3.1

Elements of micropolar elasticity Basic relations

It is well known that the requirement for a body B of bounding surface area ∂B to be in a static equilibrium is Z Z ti dA = 0 (eijk xj tk + mi ) dA = 0, (3.1) ∂B

∂B

which implies that the local equilibrium equations, in the absence of body and inertia forces and couples, are σ ji,j = 0

eijk σ kj + µji,j = 0.

(3.2)

It is apparent that σ and µ are generally asymmetric tensors. Here eijk is the Levi-Civita permutation tensor. Kinematics of the micropolar continuum is described by two tensors: strain εji and torsion κij . They are related to u and ϕ as follows εji = ui,j − ekji ϕk

κij = ϕj,i .

(3.3)

Evidently, εji and torsion κij are generally asymmetric; κij is also called curvature, or torsion-curvature tensor. Just like in a classical continuum, we need compatibility equations, and these are εli,h − εhi,l − ekhi κik + ekli κhk = 0 κli,h = κhi,l . (3.4)

Assuming a micropolar material of a linear elastic type, its energy density is given by a scalar product

1 1 (3.5) U = σ ij εij + µij κij , 2 2 ¡ ¢ where we see that (σ ij , εij ) and µij , κij are conjugate pairs. Now, the Hooke’s law generalized to such a material is (1)

(2)

σ ij = Cijkl εkl

µij = Cijkl κkl ,

(3.6) (1)

whereby (3.5) becomes a quadratic form in strains and curvatures. Here Cijkl (2) and Cijkl are two micropolar stiffness tensors. Note that, due to the existence of U, we have basic symmetry of stiffness (and hence, compliance) tensors (1)

(1)

(2)

Cijkl = Cklij

(2)

Cijkl = Cklij , 14

(3.7)

¡ ¢ but not two other symmetries since (σ ij , εij ) and µij , κij are, in general asymmetric. Indeed, this is the reason for calling this theory an asymmetric elasticity by Nowacki (1970, 1986). The inverse of (3.6) is written as (1)

(2)

εij = Sijkl σ kl

(3.8)

κij = Sijkl µkl .

In general, there is a possibility of a direct coupling between force-type and couple-type effects in the constitutive model, whereby σij would also be a function of κkl , and µij would be a function of εkl (Nowacki, 1986). This tends to be called non-centrosymmetry, or chirality (e.g., Lakes & Benedict, 1982). A simple one-dimensional (1-D) case of chirality is a helix, see Section 1.3 in (Ostoja-Starzewski, 2002). Focusing henceforth on the centrosymmetric case, for an isotropic material, (1) (2) tensors Cijkl and Cijkl of (3.6) become (Nowacki, 1986) (1)

Cijkl = (µ − α) δ jk δ il + (µ + α) δ jl δ ik + λδ ij δ kl

(3.9)

(2)

Cijkl = (γ − ε) δ jk δ il + (γ + ε) δjl δ ik + βδ ij δ kl , where λ and µ are the Lamé constants of classical elasticity, while α, γ, ε, and β are the micropolar constants. Of use will also be the inverse forms of the constitutive law, namely γ ij = 2µ0 σ (ij) + 2α0 σ [ij] + γ 0 δ ij σ kk κij = 2γ 0 µ(ij) + 2ε0 µ[ij] + β 0 δ ij µkk ,

(3.10)

in which 1 1 1 2α0 = 2γ 0 = 2µ 2α 2γ −λ −β = β0 = K =λ+ 6µK 6γΩ

2µ0 = λ0

2ε0 = 2 µ 3

1 2ε

(3.11)

2 Ω = β + γ. 3

Here we recognize the familiar bulk modulus K, and its mathematically analogous micropolar quantity Ω. Round brackets and square brackets indicate symmetric and antisymmetric parts of the tensors, respectively. Our constitutive tensors may alternatively be expressed in the notation of Eringen (1966, 1999) (1)

Cijkl = µE δ jk δ il + (µE + αE ) δ jl δ ik + λE δ ij δ kl (2)

Cijkl = β E δ jk δ il + γ E δ jl δ ik + αE δ ij δ kl , 15

(3.12)

where, using the subscript E to denote quantities in Eringen’s notation, we have µE = µ − α

κE = 2α

λE = λ

γE = γ + ε

β E = γ − ε.

(3.13)

We end this section by noting that, just like in the classical elasticity, we can express a micropolar field problem in displacements and rotations, or in stresses. In the first case, six such equations (for ui and rotation ϕi , i = 1, ..., 3) are obtained by substituting (3.6) with (3.9) into the equilibrium equations (3.2) and using (3.3) (µ + α) ui,jj + (λ + µ − α) uj,ji + 2µεijk ϕk = 0 (γ + ε) ϕi,jj − 4αϕi + (β + γ − ε) ϕj,ji + 2αεijk uk = 0.

(3.14)

This generalization of the Navier equations of classical elasticity is to be supplemented by the kinematic boundary conditions on ∂Bk and traction conditions on ∂Bt ; here ∂Bk ∪ ∂Bt = ∂B. The field equations in stresses are a generalization of the Beltrami-Michell equations. Here we make a reference to a study of Schäfer (1967) who generalized the functions of Morrey and Maxwell & Kessel. We return to this topic in the 2-D setting of Section 3.3 below. As pointed out in Section 1.1, ϕ is an independent kinematic quantity. However, a special model assuming the equality 1 ϕi = eijk uk,j 2

(3.15)

is sometimes used, and this is the same definition as in classical elasticity. It is called a pseudo-continuum, a restricted, a couple-stress, or a Koiter model, Koiter (1963), Truesdell & Toupin (1960), Grioli (1960), Toupin (1962), and Mindlin & Tiersten (1962), and Mindlin (1963). In view of (3.15), the strains εij are symmetric and simply defined as εij = u(i,j) .

3.2

(3.16)

Planar problems

There are, in general, two planar problems of Cosserat elasticity (Nowacki, 1986): i) The so-called first planar problem with u = (u1 , u2 , 0) and ϕ = (0, 0, ϕ3 ), which is a generalization of the classical planar elasticity. 16

ii) The so-called second planar problem u = (0, 0, u3 ) and ϕ = (ϕ1 , ϕ2 , 0), which is a generalization of the classical anti-plane elasticity. With the exception of Section 4.2, in this paper we focus on the first planar problem. From (3.2), the equilibrium equations are σ 11,1 + σ 21,2 = 0

σ 12 − σ 21 + µ13,1 + µ23,2 = 0, (3.17)

σ 12,1 + σ 22,2 = 0

while the kinematic relations (3.3) are ε11 = u1,1 κ13 = ϕ3,1

ε22 = u2,2 κ23 = ϕ3,2 ,

ε12 = u2,1 − ϕ3

ε21 = u1,2 + ϕ3 (3.18)

and these satisfy the compatibility equations ε21,1 − ε11,2 = κ13

ε22,1 − ε12,2 = κ23

κ23,1 = κ13,2 .

(3.19)

In the isotropic planar Cosserat medium, compliances of equations (3.8) become 1 (1) Sijkl = [(S +P )δ ik δjl +(S −P )δ il δ jk +(A−S)δij δkl ] 4

(2)

Si3k3 = δ ik M, (3.20)

where A, S, P and M are four independent planar Cosserat constants defined in (Ostoja-Starzewski & Jasiuk, 1995) A=

1 1 = κ λ+µ

S=

1 µ

P =

1 α

M=

1 . γ+ε

(3.21)

Note that A and S define planar bulk and shear compliances of classical elasticity (Dundurs & Markenscoff 1993), while P and M are additional two Cosserat constants; in the couple-stress elasticity P = 0. The restriction that the strain energy be nonnegative implies the following inequalities 0≤A≤S

0≤P

0 ≤ M.

(3.22)

In the case of orthotropy for plane Cosserat elasticity, constitutive equations (3.8) become (1)

(1)

(1)

(1)

ε11 = S1111 σ 11 + S1122 σ 22 ε22 = S2211 σ 11 + S2222 σ 22 (1) (1) (1) (1) ε12 = S1212 σ 12 + S1221 σ 21 ε21 = S2112 σ 12 + S2121 σ 21 (2) (2) κ13 = S1313 µ13 κ23 = S2323 µ23 .

(3.23)

In the above, given (3.7), we have (1)

(1)

(1)

S1122 = S2211

(1)

S1221 = S2112 . 17

(3.24)

Since for the couple-stress formulation ε12 = ε21 (recall equation 3.16), we must have (1) (1) (1) (1) S1212 = S2112 S1221 = S2121 . (3.25) This, combined with (3.24)2 above implies (1)

(1)

(1)

(1)

(3.26)

S1212 = S2112 = S1221 = S2121 , so that the constitutive relations (3.23) take on a simpler form (1)

(1)

(1)

(1)

ε22 = S1122 σ 11 + S2222 σ 22 ε11 = S1111 σ 11 + S1122 σ 22 (1) ε12 = ε21 = S1212 (σ 12 + σ 21 ) (2) (2) κ23 = S2323 µ23 . κ13 = S1313 µ13

(3.27)

Finally, for the special type of orthotropy (symmetric) referred to in Section 5 we have two additional simplifications (1)

(1)

(2)

S1111 = S2222

(2)

S1313 = S2323 .

(3.28)

Thus, the constitutive law for such an orthotropic and symmetric planar (1) couple-stress model involves four independent compliance components: S1111 , (1) (1) (2) S1122 , S1212 and S1313 .

3.3

Characteristic lengths in an orthotropic medium

In the early sixties when the Cosserat models began to undergo a revival following half a century of dormancy after the invention by the Cosserats, several people realized that, contrary to classical elasticity, an intrinsic length scale was involved in the governing equations. It was denoted l, and called a characteristic length. Let us now see how this l can be arrived at. Following Mindlin (1963) and Schäfer (1962), we employ a stress function formulation, which for the planar Cosserat (as well as the couple-stress) elasticity involves two stress functions, φ and ψ σ 11 = φ,22 − ψ,12 σ 22 = φ,11 + ψ,12 σ 12 = −φ,12 − ψ,22 σ 21 = −φ,12 + ψ,11 µ13 = ψ,1 µ23 = ψ,2 .

(3.29)

Note that φ is the Airy’s stress function known from the classical elastostatics. Recall also that, for the isotropic plane Cosserat elasticity, the compatibility conditions in terms of φ and ψ are given by (e.g. Nowacki, 1986) [ψ −

P +S 2 A+S 2 ∇ ψ],1 = − ∇ φ,2 4M 4M

[ψ − 18

P +S 2 A+S 2 ∇ ψ],2 = ∇ φ,1 . 4M 4M (3.30)

These are the Cauchy-Riemann conditions for the functions P +S 2 ∇ ψ], so that we actually have two harmonic functions 4M

A+S 2 ∇φ 4M

and [ψ −

P +S 2 (3.31) ∇ ψ] = 0. 4M The coefficient (P + S)/4M appearing above has the dimension of length squared and has thus led to a definition of characteristic length l via ∇2 ∇2 φ = 0

∇2 [ψ −

(1)

S P +S l = ≡ 1212 . (2) 4M S1313 2

(3.32)

In the couple-stress theory P = 0 in equations (3.30-3.32). For the orthotropic Cosserat elasticity case, the compatibility conditions (3.19) result in (1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

[S1111 − S1221 − S1122 ]ψ,122 + S2121 ψ,111 − [S1122 + S2121 + S1221 ]φ,112 − S1111 φ,222 (2) = S1313 ψ,1 (1) (1) (1) (1) (1) (1) (1) (1) [S2222 − S1221 − S1122 ]ψ,112 + S1212 ψ,222 + [S1122 + S1212 + S1221 ]φ,122 + S2222 φ,111 (2) = S2323 ψ,2 , (3.33) which suggest the following definitions of four characteristic lengths (Bouyge et al., 2002) v v u (1) u (1) (1) (1) uS uS − S − S 1111 1221 1122 t 2121 l1 = t l = 2 (2) (2) S1313 S1313 v v u (1) u (1) (1) (1) uS uS − S − S 2222 1221 1122 t 1212 . l3 = t l = (3.34) 4 (2) (2) S2323 S2323

In the special case of plane isotropic Cosserat elasticity, the following relations hold (1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

S1111 − S1221 − S1122 = S2121 = S2222 − S1221 − S1122 = S1212 ,

(3.35)

and equations (3.34) reduce to a single characteristic length defined in (3.32). For the planar orthotropic couple-stress case the compatibility equations (3.19) yield (1)

(1)

(1)

(1)

(1)

(1)

(1)

[S1111 − S1212 − S1122 ]ψ,122 + S1212 ψ,111 − [S1122 + 2S1212 ]φ,112 − S1111 φ,222 (2) = S1313 ψ,1 (1) (1) (1) (1) (1) (1) (1) [S2222 − S1212 − S1122 ]ψ,112 + S1212 ψ,222 + [S1122 + 2S1212 ]φ,122 + S2222 φ,111 (2) = S2323 ψ,2 . (3.36) 19

Thus, the four characteristic lengths are

l1

l3

v u (1) (1) (1) uS − S1212 − S1122 = t 1111 (2) S1313 v u (1) (1) (1) uS 2222 − S1212 − S1122 t = (2) S2323

whereupon equations (3.36) can be rewritten as (1)

l12 ψ,122

+

l22 ψ,111

−

(1)

S1122 + 2S1212 (2)

S1313 (1)

l32 ψ,112 + l42 ψ,222 +

(2)

S2323

(3.37)

(1)

φ,112 −

(1)

S1122 + 2S1212

v u (1) uS l2 = t 1212 (2) S1313 v u (1) uS l4 = t 1212 , (2) S2323 S1111 (2)

S1313

φ,222 = ψ,1

(1)

φ,122 +

S2222 (2)

S2323

φ,111 = ψ,2 .

(3.38)

For the special case of plane orthotropic couple-stress case with symmetry, in view of (3.28), there are only two characteristic lengths v v u (1) u (1) (1) (1) uS uS − S − S 1212 1122 t 1212 . l1 = t 1111 l = (3.39) 2 (2) (2) S1313 S1313 We return to this case further below in Sections 4 and 5.

4

Micropolar models of lattices

Lattice models have their roots in the solid state physics as well as in structural mechanics. But, what is of essence for nanoscale mechanics is that such lattices may transfer moments in addition to forces. Whether this is the case needs to be checked by inspection of actual physics in a given material. Here we outline the basic ideas in the planar setting of triangular, square, and honeycomb (or hexagonal) lattices, Fig. 3. A fuller review of this topic has been given in (Ostoja-Starzewski, 2002), with (Wo´zniak, 1970) forming the key reference.

4.1 4.1.1

In-plane elasticity Triangular lattice of beams

We now consider the triangular lattice of equilateral geometry, made of identical Bernoulli-Euler beams of length s, which is the spacing of the mesh, Fig. 20

Figure 3: Three periodic lattices: honeycomb, square, and triangular. In each case a possible periodic unit cell is shown. 3 and Fig. 4. Thus, each beam has a cross sectional area A = wh, w being the width and h the out-of-plane height; a centroidal moment of inertia with 1 3 respect to an axis normal to the plane of the network is I = 12 w h; and E (b) is the Young’s modulus of the beam’s material. We next focus on the deformations of a typical beam, its bending into a curved arch allowing the definition of its curvature, and a cut in a free body diagram specifying the normal force F (b) , the shear force F˜ (b) , and the in-plane bending moment M (b) . It follows that, for 2-D motions, the force field within the beam network is described by fields of force-stresses σ kl and moment-stresses mk , so that, we have a micropolar medium. The kinematics of the beam network is now described by two displacements and a rotation u1 (x) u2 (x) ϕ(x), (4.1) which coincide with the actual displacements (u1 , u2 ) and rotations ϕ at the network nodes. Within each triangular pore, these functions may be assumed to be linear, and hence, the local fields of strain εkl and curvature κi (≡ κ3i ) are related to u1 , u2 , and ϕ (≡ ϕ3 ) by εkl = ul,k + elk ϕ

κi = ϕ,i

i, k, l = 1, 2,

where elk is the Ricci symbol (a 2-D analogue of the Levi-Civita tensor). 21

(4.2)

Figure 4: Triangular lattice as a plate with axial force, shear force and in-plane bending moment in a typical beam connection. It may now be shown by geometric considerations that the beam’s average axial strain and its curvature are (b) (b)

ε(b) ≡ nk nl εkl

(b) (b)

e ε(b) ≡ nk n el u(l,k) − ϕ

(b)

κ(b) ≡ nk κk ,

(4.3)

where n(b) and n(b) are the unit vectors parallel and normal to the beam’s cross section. Furthermore, ε(b) , e ε(b) , and κ(b) are the average axial strain, shear strain and curvature, respectively. Next, from the elementary beam theory we have force-displacement and moment-rotation response laws F (b) = EAγ (b)

12E (b) I (b) e ε F˜ (b) = s2

M (b) = EIκ(b) .

(4.4)

Turning now to the continuum picture, the strain energy of the micropolar continuum is specialized from (3.5-3.7) as Ucontinuum =

V V (1) εij Cijkm εkm + κi Dij κj 2 2

i, j, k, m = 1, 2,

(4.5)

from which we determine (1) Cijkm

=

6 X b=1

(b) (b) ni nk

³ ´ (b) (b) (b) (b) (b) ˜ (b) nj nm R + nj nm R

where R=

2E (b) A √ s 3

(b) e = 24E√ I R s3 3

22

S=

Dij =

6 X

(b) (b)

ni nj S (b) ,

b=1

2E (b) I √ . s 3

(4.6) (4.7)

Written another way, we have an isotropic continuum (1)

Cijkm = δ ij δ km Ξ + δ ik δjm Λ + δ im δ jk Π

Dijef f = δ ij Γ,

(4.8)

3 Γ = S, 2

(4.9)

in which Ξ=Π=

´ 3³ ˜ R−R 8

Λ=

´ 3³ ˜ R + 3R 8

and these are related to our four compliances defined in (3.21) as A=

1 Ξ + Λ+Π 2

S=

2 Λ+Π

P =

2 Λ−Π

M=

2 . 3Γ

(4.10)

The effective bulk and shear moduli as well as Young’s modulus and Poisson’s ratio are now identified as 3 κ= R 4

´ 3³ ˜ µ= R+R 8

E = 3R

1+ 3+

˜ R R ˜ R R

ν=

1− 3+

˜ R R . ˜ R R

(4.11)

Note that with the increasing beam width, ν goes down, but, in view of the basic assumption of Bernoulli-Euler beams’ slenderness, this model ceases to be applicable for Poisson’s ratios below 0.2. This restriction can, to some extent, be removed by treating the lattice’s node-to-node connections as Timoshenko beams. In that case, only the relation between the shear force F˜ (b) and the displacement se ε(b) in (4.4) is modified 12E (b) I (b) (b) s˜ γ , F˜ (b) = 3 s (1 + β) where

(4.12)

E (b) ³ w ´2 12E (b) I (b) = (4.13) G(b) A(b) s2 G(b) s is the dimensionless ratio of bending to shear stiffness, with A(b) = ta being the beams’ cross-sectional area, and I (b) = ta w3 /12 its centroidal moment of inertia. Let us recall here from structural mechanics two limiting cases (e.g., Hjelmstad, 1997): β → 0, high shear stiffness and, hence, less deflection owing to shear; the Bernoulli-Euler slender beam is recovered; β > 1, low shear stiffness and, hence, deflection owing to shear dominates over that due to the Young’s modulus E; the Timoshenko beam applies. In fact, taking typical values E/G = 2 and w/s = 0.1 , we find that β is closer to 0 than 1. Thus, taking Timoshenko beams results in only a rather β=

23

small modification of the overall stiffnesses. This observation also holds with respect to the micropolar characteristic length, which for the Bernoulli-Euler beam lattice is computed as ¡ w ¢2 1 + 3 l (4.14) l2 = ¡ s¢ . 24 1 + w 2 s

See the aforementioned review for a study of this lattice when beams become so stubby that the material has to be viewed more appropriately as a plate perforated with small triangular holes. 4.1.2

Square lattice of beams

The same procedure as above may be used to derive a micropolar continuum model of a square lattice network, Fig. 3. Thus, assuming Bernoulli-Euler beams, we find a formula analogous to (4.6) (1) Cijkm

=

4 X b=1

(b) (b) ni nk

³ ´ (b) (b) (b) ˜ nj n(b) R + n n R m m j

Dij =

4 X

(b) (b)

ni nj S,

(4.15)

b=1

where

E (b) A ˜ 12E (b) I E (b) I S = R= . s s3 s When all the beams are identical, this leads to R=

(1) (1) (1) (1) ˜ D11 = D22 = S, C1111 = C2222 = R C1212 = C2121 = R

(4.16)

(4.17)

with all the other components of the stiffness tensors being zero. Clearly, this beam lattice results in a special case of an orthotropic continuum. In accordance with the definitions (3.34) and (3.39), we will now have two micropolar characteristic lengths r r r S S I s =r≡ , (4.18) l1 = l2 = = √ ˜ R A 2 3 R where the radius of gyration r is easily recognized. For beams of a rectangular √ cross-section, the second one of these becomes l2 = b/2 3. In the foregoing derivation, lattice nodes were taken as rigid objects. As Wo´zniak (1970) showed, this model may be generalized to a situation of de-

24

formable nodes, in which case we have " # 4 4 X X (1) (b) (b) (b) (b) ⊥ (b) (b) (b) (b) ˜ Cijkm = ni nj nk nm R + ni nj nk nm R b=1

Dij =

4 X

b⊥=1

(b) (b)

(4.19)

ni nj S,

b=1

where R

⊥

=

d 1 − ν (I) ν (II )

(b) ˜ = 24E√ I R s3 3

·

˜(I) ν (I) E E˜(I) ν (II) E˜(II) E˜(II)

S=

2E (b) I √ . s 3

¸ (4.20)

Recently, an extension of such micropolar models (needed for wave propagation and vibration phenomena) has been carried out through the introduction of internal variables (Wo´zniak, 1997; Cielecka et al., 1998). Such models, in contradistinction to the more classical homogenization methods, do more correctly account for the internal microstructure. 4.1.3

Honeycomb lattice of beams

One can also derive a micropolar model of a hexagonal lattice of beams (Chen et al., 1998; Wang & Stronge, 1999), recall Fig. 3. Assuming Timoshenko beams, the latter authors found √ · ¸ s 3 ³ s ´2 (1) (1) +3 C1111 = C2222 = 4wE (b) w √ · ³ s ´2 ¸ s 3 (1) (1) 1− C1122 = C2211 = 4wE (b) w √ · ³ ´ ¸ s 2 s 3 (1) (1) 3 +1 C1212 = C1212 = 4wE (b) w √ · ³ s ´2 ¸ s 3 (1) (1) 1− C1221 = C2112 = 4wE (b) w √ ³ ´ 12s 3 s 2 D11 = D22 = , (4.21) 4wE (b) w

where s, w, and E (b) have the same meaning as before. Clearly, this beam lattice is an isotropic continuum, with an effective Poisson’s ratio being approximately zero. Wang & Stronge (1999) also provide analyses of rate decay 25

of stresses and couple-stresses away from a force acting on a half-space, in function of slenderness of beam connections. We refer the reader to (Chen et al., 1998) for continuum-type fracture analyses of porous materials with hexagonal as well as square and triangular microgeometries. In another recent paper, Mora & Waas (2000) calculated the characteristic length l in a circular-cell honeycomb using experiments. Two configurations were used: a plate with a centrally located circular hole of diameter a, or a rigid inclusion, larger than the cell microstructure. The values ranged from 1.8 to 6.

4.2

Plate-bending response

When the motions are anti-plane (recall the second planar problem from Section 3.2), the planar lattice can be approximated as a plate. Here we sketch the basic ideas in terms of a triangular lattice, within the assumptions of a Kirchhoff (thin) plate model. The kinematics is therefore described by three functions: one out-of-plane displacement and two rotations (with respect to the x1 , and x2 axes) x) u3 (˜

ϕ1 (˜ x)

(4.22)

ϕ2 (˜ x) ,

which coincide with the actual displacement (u3 ) and rotations (ϕ1 , ϕ2 ) at the lattice vertices. Within each triangular pore these functions may be assumed to be linear. It follows then that the strain and curvature fields are related to u3 , ϕ1 , ϕ2 by κkl = ϕl,k

εk = u3,k +

kl ϕl

k, l = 1, 2.

(4.23)

With reference to Fig. 5, for a single beam b the mechanical (forcedisplacement and moment-rotation) response laws of each beam are given as M (b) = C (b) κ(b)

˜ (b) = E (b) I (b) κ ˜ (b) M

P (f ) =

12E (b) I (b) (b) e ε . s(b)

(4.24)

˜ (b) is an out-of-plane Thus, in a given beam b, M (b) is a twisting moment, M (b) bending moment, and P is a shear force. These are related to the beam’s angle of twist κ(b) , beam’s curvature e κ(b) , and beam’s shear deformation e ε(b) . The constitutive quantities involve the beam’s torsional stiffness C (b) and the Young’s modulus E (b) . The strain energy of the unit cell is Ucontinuum =

V V (2) f εi Aef κij Cijkl κkl , ij εj + 2 2 26

(4.25)

Figure 5: Triangular lattice as a plate with shear force P (b) , bending moment ˜ (b) , and twisting moment M (b) in a typical beam connection. M which is consistent with the Hooke’s law pk = Akl εl

(2)

(4.26)

µkl = Cijkl κkl ,

(1)

(1)

where Akl = Ci3k3 is recognized as the anti-plane part of Cijkl . Here pk is the vector of shear tractions and µkl is the tensor of couple-stresses. Proceeding in a fashion analogous to the in-plane problems, Wo´zniak (1970) found 6 6 ³ ´ X X (b) (b) (b) (2) (b) (b) (b) (b) (b) (b) (b) ˜(b) ef f Aij = ni nj R Cijkm = ni nk nj nl S + nj nl S , b=1

b=1

(4.27)

where S (b) =

C (b) s(b)

E (b) I (b) S˜(b) = s(b)

(b) (b) ˆ (b) = 12E I . R 2 s˜(b) (s(b) )

(4.28)

In the case of a triangular lattice made of identical beams (E (b) = E, etc.) we find (2) Cijkl = δ ij δ kl ∆ + δ ik δ jl Y + δil δjk Ω Aij = δ ij B, (4.29) in which ∆ = Ω= S =

2C √ s 3

3 8

³ ´ S − S˜

Y =

2EI S˜ = √ s 3

3 8

³ ´ S + 3S˜

ˆ = 24EI √ . R s3 3 27

ˆ B = 32 R (4.30)

Figure 6: (a) A periodic, globally orthotropic, matrix-inclusion composite, of period L, with inclusions of diameter d arranged in a square array; (b) a periodic unit cell with soft inclusions at corners; (c) a periodic unit cell with a stiff inclusion at the center. Same type of derivation may be conducted for a lattice of rectangular geometry.

5

Micropolar models of composite materials

Any of the lattices considered in the previous section can be viewed as a planar, two-phase composite material: one made of the solid that goes into the beams and another, quite trivially, of the vacuum in the pores. Clearly then, the mismatch of elastic properties - i.e. the ratio of moduli E (1) /E (2) is infinite (Ostoja-Starzewski et al., 1999). Starting from this consideration, one may now consider ‘non-trivial’ two-phase composites made of two kinds of solids having a finite mismatch, and generalizing the previously established method for passage from heterogeneous Cauchy to a homogeneous Cosserat continuum. This passage is done according to the following equality Z 1 V (1) (2) εij Cijkl εkl dV = [ε0ij Cijkl ε0kl + κ0i3 Ci3k3 κ0k3 ], i, j, k, l = 1, 2, (5.1) 2 V 2 where the left-hand side is the total elastic strain energy stored in the unit cell of the matrix-inclusion composite (a function of Cauchy strain fields εij ), while the right-hand side is the energy of a couple-stress continuum (a function of volume-average strains ε0ij and curvatures κ0i3 of the unit cell). V is the volume of the unit cell B. Cijkl is the elastic stiffness of the composites’ constituents, (1) (2) while Cijkl and Ci3k3 are the sought (effective) micropolar stiffnesses. We have recently computed numerically those stiffnesses for planar matrixinclusion composites arranged in periodic arrays: triangular (Bouyge et al., 2001) and square (Bouyge et al., 2002), Fig. 6, using a finite element method. 28

Note that even though our microstructure was periodic we could not use periodic boundary conditions because we could not apply periodic bending boundary conditions. Thus, several different boundary conditions - ranging from displacement-type to traction-type, and various combinations thereof - were used. For example, using displacement boundary conditions, we determine (1) Cijkl from three tests: (i) Uniaxial extension: u1 (x) = 0

u2 (x) = ε22 x2

∀x ∈ ∂B

(1)

(5.2) (1)

(1)

gives C2222 (= 2U cell /V when we set ε22 = 1). For our composite, C1111 = C2222 due to symmetry of square arrangement. (ii) Biaxial extension: u1 (x) = ε11 x1 (1)

∀x ∈ ∂B

u2 (x) = ε22 x2

(5.3)

(1)

yields 2C1111 + C1122 (= 2U cell /V when we set ε11 = ε22 = 1). (iii) Shear test: u1 (x) = ε12 x2

u2 (x) = 0

∀x ∈ ∂B

(5.4)

(1)

yields C1212 (= 2U cell /V when we set ε12 = 1). (2) Finally, to determine Ci3k3 , we conduct the fourth test. (iv) Bending test: u1 (x) = −κ13 x1 x2 (2)

u2 (x) = κ13 (2)

x21 2 (2)

∀x ∈ ∂B

(5.5)

gives C1313 . Note that in our study C1313 = C2323 due to the symmetry of square arrangement. The resulting deformation modes for the above four tests under displacement boundary conditions are shown in Fig. 7. Two distinct situations are considered here depending on whether the inclusion is softer or stiffer than the matrix. In the first case, the inclusion is located at the corner, whereas in the second at the center. In the special case of no mismatch in the properties we recover a homogeneous medium of Cauchy type, whereby the composite microstructure disappears and no Cosserat continuum is to be set up. In the case of traction boundary conditions, we use Z 1 V (1) (2) σ ij Sijkl σ kl dV = [σ 0ij Sijkl σ 0kl + µ0i3 Si3k3 µ0k3 ] (5.6) 2 V 2 29

(1)

(1)

(1)

Figure 7: Tests for the determination of constants C2222 , C1122 , C1212 , and (2) C1313 of a periodic composite with circular inclusions in a square arrangement under displacement boundary conditions (Bouyge et al., 2002). Left (right) column corresponds to the inclusion at the corner (center).

30

where, on the left we have the total complementary strain energy in the unit cell (a function of Cauchy stresses σ ij ), while on the right we have the complementary strain energy of a couple-stress continuum (a function of volumeaverage stresses σ 0ij and couple-stresses µ0i3 of the unit cell). Here Sijkl (inverse (1) of Cijkl ) is the elastic compliance of the composites constituents, while Sijkl (2) and Si3k3 are the sought micropolar compliances. Summing up, for the Koiter model of the composite, the micropolar moduli are bounded from above and below, respectively, by displacement and traction boundary conditions. In fact, since these bounds are wide, we recommend three mixed types of loadings to get tighter results. On the other hand, the characteristic lengths are highly insensitive to the mismatch in moduli, especially in the case of stiff inclusions, and this must be contrasted with the sensitivity of moduli. The aim of this study is the replacement of a two-phase material, locally of Cauchy type, by a homogeneous one of couple-stress type. Such a replacement offers savings - by about one order of magnitude - with respect to the classical modeling; this type of a comparison was made by Forest and coworkers (Forest, 1998; Forest & Sab, 1998). While none of our boundary conditions were of periodic type, the situation changes when an unrestricted model is used. Indeed, such a derivation has been done in (Forest & Sab, 1998) by extending the homogenization method (e.g., Sanchez-Palencia & Zaoui, 1987). The loading on BL in 2-D is then effected by boundary conditions involving polynomials of the general form u1 (x) = B11 x1 + B12 x2 − C23 x22 + 2C13 x1 x2 + D12 (x32 − 3x21 x2 ) u2 (x) = B12 x1 + B22 x2 − C13 x21 + 2C23 x1 x2 − D12 (x31 − 3x1 x22 )

(5.7)

Upon a comparison of this with equations (4.2-4.5), we observe that the derivation of the Koiter model involves a second-order polynomial, while that of the unrestricted one requires a third-order polynomial. Work has also been done on homogenization of a heterogeneous Cosserat-type continuum by a homogeneous one (Forest, 1999; Forest et al., 1999); see also (Forest et al., 2000, 2001). Most recently, a related homogenization procedure was outlined by Onck (2002) for the derivation of the micropolar model. In particular, a loading via a skew-symmetric part of the strain tensor, applied in terms of boundary rotations, was proposed to grasp the effect of difference between the microrotation ϕi and the macro-rotation eijk uk,j ; recall eq. (3.15).

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6

Further Research Directions

As stated in the introduction, the Cosserat-type theories developed more fully in the sixties. However, after about two decades of active research, the interest in those theories almost died down due, primarily, to the challenge of obtaining higher-order material constants entering constitutive laws. With the exception of lattice-type models where such constants could be derived explicitly, modeling of locally Cauchy-type composite materials by Cosserat-type continua remained a challenge. The situation started to change in the second half of the nineties, driven, among others, by the interest in addressing phenomena on lower scales. More recently, this has been amplified by the emergence of nanotechnology. It is in the latter setting that the non-classical continua become a viable option for modeling of materials with the inclusion of mechanics on several scales. Indeed, as indicated here, composite materials can be homogenized by Cosserat-type models and their constitutive coefficients can be computed. However, there remains a number of challenges: 1. Determination of micropolar constants for various other periodic microstructures (lattices as well as composites), besides the simple ones discussed here. 2. Extension of such methods to spatially non-periodic, indeed random, microstructures. 3. Assessment of ranges of applicability of micropolar theories for given elastic/inelastic materials, and, if need be, the development of higher-order, generalized continuum theories. 4. Experimental verification of such micropolar, and higher-order, models, including the measurement of constitutive coefficients.

Acknowledgment I.J. acknowledges the support by the NSF (Grant CMS-0085137) and the AFOSR (Grant F49620-01-1-013; Dr. B.-L. Lee is the program monitor). M.O.-S. acknowledges the support by the NSERC, and the Canada Research Chairs Program.

References 32

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