Michell trusses in the presence of microscale material randomness ...

10.1098/rspa.2001.0777

Michell trusses in the presence of microscale material randomness: limitation of optimality By M. O s t o j a-Starzewski† Institute of Paper Science and Technology and Georgia Institute of Technology, 500 10th St. NW, Atlanta, GA 30318-5794, USA ([email protected]) Received 24 January 2000; accepted 2 January 2001

The classical problem of a Michell (optimal) truss concerns a minimum-weight design of a planar truss that transmits a given load to a given rigid foundation with the requirement that the axial stresses in the bars of the truss stay within an allowable range σ0 σ σ0 . The present study considers this problem when the truss is made of a material with random microstructure, that is, when σ0 is a random field. The trusses tending to the optimal state can be determined through a net of characteristics generalized to a stochastic setting. While in the classical case of a homogeneous material this net gives the minimum weight as its spacing tends to zero, the presence of a random microstructure prevents the attainment of this state. Basically, the finer the net, the stronger the scatter of characteristics, which forces one to use more structural material to compensate for these fluctuations. In effect, there is a limitation to the attainment of the optimality of the Michell truss made of a hypothetical perfectly homogeneous material. Keywords: Michell truss; optimal structure; random media; microstructural disorder; biomaterials; nanotechnology

1. Introduction The classical problem of an optimal truss concerns a minimum-weight design of a planar truss T that transmits a given load to a given rigid foundation with the requirement that the axial stresses in the bars of the truss stay within an allowable range σ0 σ σ0 (Michell 1904). This forms the basis for a problem of layout of a truss whose locally orthogonal members are of a rigid perfectly plastic material with tensile and compressive stresses ±σ0 (see, for example, Save & Prager 1985; Rozvany et al . 1995; and references therein). The solution to this problem is provided by a so-called Michell truss-like continuum, whose members form a dense structure having the geometry of an orthogonal net of characteristics. That is, as the mesh spacing becomes infintesimally fine, the volume (and hence the weight) of the material reaches a minimum. The present study considers the effect of random material microstructure of which the Michell truss is manufactured on the ability to attain this minimum: the denser † Present address: Department of Mechanical Engineering, McGill University, Montréal, Quebec, Canada H3A 2K6. c 2001 The Royal Society

Proc. R. Soc. Lond. A (2001) 457, 1787–1797

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the truss—or, the finer the mesh spacing—the more significant is the effect of microstructural fluctuations on the plastic limit σ0 . Thus arises a question: is it sensible to refine the mesh ad infinitum, given the fact that the volume of material used to make the truss, beyond a certain point of refinement, would need to increase to compensate for the microscale fluctuations? Living plants built by nature provide beautiful examples of optimal structures. Upon a closer examination, however, one observes that no two plants of the same species, even if cloned from the same parent, possess a perfectly identical microstructure. This is due to unavoidable differences in environment (e.g. fluctuations in temperature, humidity, nutrients) and the microheterogeneity that is genetically controlled (plant structures are optimized for survival over a large range of conditions which have an effect on each plant’s growth). As a result, we deal with a statistical ensemble of nominally same, optimally built specimens, which nevertheless exhibit small microstructural differences. A perfect minimum of a deterministic homogeneous material is not really attained. We return to this issue at the end of the paper. In our analysis, we consider the manufacturing of ever finer Michell trusses, which are made by cutting them from a material with microstructural imperfections, such as a polycrystal. The material’s plastic limit is therefore described by a planar random field k, which depends on a non-dimensional scale parameter δ = L/d > 1, relating the mesh spacing L to the size of microheterogeneity d (e.g. crystal diameter). Due to the randomness of k on scales δ < ∞, the net of characteristics displays a statistical scatter that increases with decreasing δ, which translates into a trend to use more structural material to carry the prescribed loading. Since this is an opposing tendency to the convergence to the Michell truss with a mesh refinement, there is a limitation of this ideal optimality. It is well known that the bars of the optimal truss-like continuum follow the principal lines of the strain field, which are identical to the characteristics of the associated hyperbolic problem. Due to the randomness of k, we now have a stochastic hyperbolic problem—akin to the recently studied stochastic elliptic problem (Ostoja-Starzewski 1999) and a stochastic wavefront propagation problem (Ostoja-Starzewski & Tr¸ebicki 1999). In these problems, L cannot be made arbitrarily smaller than the macroscopic dimensions Lmacro , yet sufficiently larger than the microscale d, as is dictated by the representative volume element (RVE) prescription of deterministic continuum mechanics: d < L Lmacro or d L Lmacro . Rather, the RVE is replaced by a statistical mesoscale window.

2. Michell truss vis-` a-vis random microstructure Allowable stresses in all the truss members are in the range σ0 σ σ0 . Given a modulus E, we have a range for strains −k ε k, where k = σ0 /E. As pointed out by Rozvany (1996), equal permissible stresses in tension and compression are necessary for Michell’s (1904) criteria to hold. With length li and cross-sectional area Ai of bar i, the design variables are the yield forces Yi = σ0 Ai , and the design objective is the minimization of the cost Yi li , (2.1) Γ = i

which is proportional to the total volume of the bars V = Proc. R. Soc. Lond. A (2001)

i

Ai l i .

Michell trusses with microscale material randomness P

P

A

F

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A

F

(a) Eff = 0.939

(b) Eff = 0.965

P

F (c) Eff = 0.981

P

A

A

F (d ) Eff = 0.991

Figure 1. Successive approximations to the Michell truss, all governed by (3.6) for a homogeneous material Bdet , according to meshes based on, respectively, 2n + 1 (n = 2, 3, 4 and 5) boundary points on the rigid foundation F.

We now consider a problem of optimal layout of a truss set up on a rigid circular foundation F, which can support a force P acting at a point A (figure 1). The members are rigid/perfectly plastic. The solution is provided by a so-called field of type T (Save & Prager 1985), for which the principal strains have equal absolute values k, but carry opposite signs, and the principal lines are logarithmic spirals. Indeed, plots (a)–(d) of this figure show a sequence of four ever finer trusses providing Proc. R. Soc. Lond. A (2001)

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supports under the same global conditions (the foundation F and the force P ). These trusses are based on, respectively, 2n + 1 (n = 2, 3, 4 and 5) boundary nodes. With n growing, their geometries tend to an optimal truss-like continuum whose principal strain/stress directions are mutually orthogonal characteristics of a quasilinear hyperbolic system (cf. § 3). This convergence of trusses (of volume V ∗ = V (n), n finite) to the optimal truss-like continuum (of volume V = V (n)|n→∞ ) can be quantified by the efficiency, (2.2) Eff = V /V ∗ . We indicate Eff in all four cases in figure 1. Let us now suppose that we are going to use the same type of material (a polycrystal plate, say) to manufacture the trusses of sequence shown in figure 1, and continuing for higher n. As illustrated in figure 2, the finer the truss, the shorter and narrower the truss members. Associated with this there must be a growing dependence of the yield limit k on the chosen mesh spacing L. Henceforth, it will be convenient to introduce a non-dimensional parameter δ = L/d,

(2.3)

where d is the typical size of a microscale imperfection, such as a grain in a polycrystal. Evidently, a finite δ forces us to work with a statistical mesoscale window in place of a classical RVE. We note two effects: (i) the scatter in k grows as δ decreases; and (ii) the statistical average k changes as δ decreases. The theories of plasticity of random heterogeneous media are not sufficiently well advanced to provide an explicit rule for this dependence (see, for example, Jiang et al . 2001, and references therein), and we shall therefore assume that the mean of k is constant, while its standard deviation, σk , is inversely proportional to δ, (2.4) k = const., σk (δ) ∼ 1/δ. Strictly speaking, k is a random field, parametrized by location (x, y) in the truss plane. Equation (2.4) describes the dependence of k on the mesoscale δ. That is, k describes the effective plastic limit, as a result of local smoothing of the random microstructure (one realization of which being shown in figure 2), of a given truss member according to the given mesh spacing L = δd. This leads us to a concept of a random medium, i.e. a set B = {B(ω); ω ∈ Ω} of deterministic, inhomogeneous media B(ω) = B(k(ω)), where ω is an indicator of a given realization, and Ω is a sample space of inhomogeneous continua locally smoothed by a given scale δ. We can write k (ω) = 0, (2.5) k(ω) = k + k (ω), where k is the zero-mean noise in k. In the following we assume (i) the truss spacings L of interest to us are greater than the grain size d, so that k may be treated as a field of independent random variables when entering the finite difference formulation (§ 4 below); (ii) the underlying material microstructure is space-homogeneous and ergodic. Such assumptions are valid for, say, a Poisson–Voronoi polycrystalline mosaic (Ostoja-Starzewski & Wang 1989), which has been used here in figure 2. Note, Proc. R. Soc. Lond. A (2001)

Michell trusses with microscale material randomness

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(a)

(b)

Figure 2. Manufacturing of a truss from one mosaic of polycrystal using either a coarse (a) or a twice finer (b) refinement of the mesh—it is understood that the material in the interior of squares is removed, thus leaving an orthogonal grid of bars. Clearly, the scatter of the effective plastic limit of a bar on the scale of single cell (mesoscale) increases as we go from (a) to (b). And, simultaneously, the thickness of the bars decreases with the mesh refinement. The dash-dot lines are the axes of bars of the truss (i.e. characteristics of the hyperbolic system).

however, that the k field is ergodic but generally space-inhomogeneous, given the non-uniformity of the truss. For the sake of comparison, we will also need a hypothetical deterministic homogeneous medium Bdet = B(k ), which corresponds to the absence of noise. Proc. R. Soc. Lond. A (2001)

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The effective properties display a statistical scatter which decreases to 0 as δ tends to ∞, and this is the classical deterministic continuum limit assumed in the theory of optimal trusses. However, as noted earlier, the latter theory postulates the existence of a truss-like continuum—a continuum with an infinitesimally fine spacing of truss connections. Thus, we observe the impossibility to physically attain this limit due to a competition of two opposing effects: (i) noise in k decreases to zero as the truss spacing L grows; (ii) classical deterministic solutions tend to hold as the spacing L decreases.

3. Field equations of the random medium Following Hegemier & Prager (1969), we let u and v be the displacement components with respect to the x, y coordinates in the plane of the truss. Then 1 ∂v ∂u ∂v 1 ∂v ∂u ∂u , W = (3.1) εx = , εy = , γ= + − ∂x ∂y 2 ∂x ∂y 2 ∂x ∂y are the strain components and the rotation. If by θ we denote the angle between the negative y-direction and arbitrarily assigned positive direction along the line with the unit extension k, then εx = −k cos 2θ,

εy = k cos 2θ,

γ = −k sin 2θ.

(3.2)

All the developments of this section are understood for a single realization k(ω) of the random medium B. It now follows from (3.1) and (3.2) that  ∂u ∂u  = −k cos 2θ, = −k(2w + sin 2θ),  ∂x ∂y (3.3) ∂v ∂v    = k(2w − sin 2θ), = k cos 2θ, ∂x ∂y where w = W/k. Eliminating u and v from (3.3) by cross-differentiation, we obtain

 ∂w ∂θ ∂θ 2w + sin 2θ ∂k cos 2θ ∂k  + k cos 2θ + k sin 2θ =− + ,  ∂x ∂x ∂y 2 ∂x 2 ∂y ∂θ ∂θ cos 2θ ∂k 2w − sin 2θ ∂k  ∂w  + k sin 2θ − k cos 2θ = + .  k ∂y ∂x ∂y 2 ∂x 2 ∂y k

(3.4)

Setting θ = π/2, we obtain ∂w w ∂k 1 ∂k ∂θ =− − , − ∂x ∂x k ∂x 2k ∂y

∂w ∂θ 1 ∂k w ∂k + =− − . ∂y ∂y 2k ∂x k ∂y

(3.5)

If k = const., we recover the equations of a Michell truss made of a homogeneous material, d d (w − θ) = 0, (w + θ) = 0, (3.6) ds2 ds1 which hold along two characteristics s1 and s2 , at angles specified, respectively, by θ = α + π/4 Proc. R. Soc. Lond. A (2001)

and θ = α + 3π/4.

(3.7)


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where α is defined as the angle formed by the positive direction along the foundation F with the positive x-direction. On this boundary, w = −1.

(3.8)

If k is a random field, the characteristics’ directions are still given by (3.7) and w = −1 on the foundation F, but the evolution of w and θ along the characteristics is governed by (3.5). Now, if the derivatives ∂/∂x and ∂/∂y are replaced by the tangential derivatives ∂/∂s1 and ∂/∂s2 along the s1 and s2 characteristics, equations (3.5) become independent of the orientation of the axes, and result in dw − dθ = −

w 1 ∂k ds1 , dk − k 2k ∂s2

dw − dθ = −

w 1 ∂k ds2 . dk − k 2k ∂s1

(3.9)

Summarizing, with k being a random function of x and y, we have a stochastic quasi-linear hyperbolic system governing the field.

4. Solution by finite differences At the typical point Q of the foundation F, let the positive direction along this boundary form the angle α with the positive x-direction. Since the rigid foundation is inextensible, its tangent and normal at Q bisect the right angles formed by the principal axes of strain at Q. Since u vanishes along F, cos α

∂u ∂u + sin α =0 ∂x ∂y

at Q.

From this, along with (3.3) and (3.7) there follows w = ∓1

(4.1)

along F; in figures 1 and 3 the upper sign in the above is appropriate. Thus, we have an inverse Cauchy problem: ‘find the net of characteristics supporting the given load P at point A, which emanates from the foundation F with conditions (3.7) and (3.9) specified on it.’ Figure 1a–d displays four deterministic solutions, all governed by (3.9) according to meshes based on, respectively, 2n + 1 (n = 2, 3, 4 and 5) boundary points. If k is a random field, we have a stochastic inverse Cauchy problem, that is, a set of deterministic inverse Cauchy problems, each one corresponding to one realization k(ω) of k. A solution to any one of these Cauchy problems—denoted by a truss T (ω)—represents some departure (or a perturbation) from that of the deterministic homogeneous medium Bdet . Since all solutions converge at A, we deal with an ensemble T = {T (ω); ω ∈ Ω} of nets of backward dependence wedges, which contains all T (ω)s; see figure 3a–d. In these figures we show the wedges in black and the solutions of Bdet in white; the latter are repeated from figure 1a–d for a reference. Analogous scatter of Hencky–Prandtl nets of characteristics, also due to the resolution by finite mesoscales, was observed in plasticity problems (Ostoja-Starzewski & Ilies 1996). In figure 3d, where truss spacings on the foundation arc F are about 10% of the radius, we take the noise k to be a zero-mean uniform random variable having standard deviation 0.015; the mean of k is k = 1.0. All noises k away from F in figure 3d, as well as in figure 3a–c, are scaled according to (2.4)2 , that is, in inverse Proc. R. Soc. Lond. A (2001)

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M. Ostoja-Starzewski P

P

A

F

A

F

(a) Eff = 0.93

(b) Eff = 0.935

P

F (c) Eff = 0.942

P

A

A

F (d ) Eff = 0.95

Figure 3. Shown in black are successive sets of trusses, all governed by (3.9), according to meshes based on, respectively, 2n + 1 (n = 2, 3, 4 and 5) boundary points on the rigid foundation F. Shown in white are the trusses of Bdet , same as those of figure 1.

proportion to the local truss spacing L. For the small noise-to-signal ratio adopted here, taking Gaussian as opposed to uniform noise, while keeping the same variance, made hardly any difference on the final results. The efficiencies Eff of four random truss systems are also given in figure 3. We observe at once that they are lower than those obtained under the deterministic homogeneous medium assumption. Moreover, a finer random truss than that shown Proc. R. Soc. Lond. A (2001)


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in figure 3d is not possible since the noise is too strong for the net of characteristics to continue in a stable manner up to the point A—the characteristics tend to intersect prior to A. This suggests that, should a larger number than 25 + 1 of boundary nodes on F be desired, a different truss topology—one of a disordered type rather than that of an orthogonal net—might be needed to solve the problem. Nets of characteristics for any field k(ω) were determined from equations (3.7) and (3.9) by finite differences. Here let 1 and 2 be the points on characteristics s1 and s2 , respectively, and let N stand for the point of their intersection. Next, by wi and θi (i = 1, 2) we denote values on these characteristics at the two points, while by wN and θN we denote those at N . Thus, we have  y2 − y1 + x1 tan(α + π/4) − x2 tan(α + 3π/4)  xN = ,  tan(α + π/4) − tan(α + 3π/4) (4.2) x2 − x1 + y1 cot(α + π/4) − y2 cot(α + 3π/4)   yN = , tan(α + π/4) − tan(α + 3π/4) and

wN = 12 (w1 + w2 − θ1 + θ2 + r1 − r2 ),

θN = 12 (−w1 + w2 + θ1 + θ2 − r1 + r2 ),

whereby r1 r2 ds1 ds2

 kN − k1 kN − k2 ds1  = −w1 , −  (kN − k1 )/2 (kN + k1 )/2 ds2      kN − k2 kN − k1 ds2  = −w2 , − (kN + k2 )/2 (kN + k2 )/2 ds1  

   = (xN − x1 )2 + (yN − y1 )2 ,  

  2 2 = (xN − x2 ) + (yN − y2 ) .

(4.3)

(4.4)

The derivatives of k with respect to s1 and s2 are treated in the finite difference sense. With the above equations we solve our stochastic inverse Cauchy problem by generating a large number (10 000, say) of trusses emanating from F in a Monte Carlo sense, and accepting only those that result in the end vertex being very close (within some acceptable numerical accuracy; e.g. 0.05%) to the prescribed point A. For each and every ω we iterate equations (4.2)–(4.4) so as to attain convergence to a solution, and this requires some three or four loops. For the rather fine mesh resolution of figure 3d, this is done in just a matter of minutes on a conventional Unix workstation or a personal computer and yields practically the whole range of possible behaviours, that is, the probability distributions of nets and their strain fields. Thus, having determined the ensemble T = {T (ω); ω ∈ Ω}, where each T (ω) has volume V (ω), we compute the efficiency of an ensemble average truss, Eff = V /V ∗ .

(4.5)

Here V ∗ = V (n) (n finite) is the ensemble average volume, and V is the same volume as that used in the limit n → ∞ of equation (2.2) and already implied by figure 1. Proc. R. Soc. Lond. A (2001)

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5. Closure Save & Prager (1985) said, ‘Although a truss-like continuum is not practical, it uses the smallest possible volume V of structural material for the considered behavioral constraint and thus furnishes a useful basis for computing the efficiencies of practical structures.’ Similarly, the present study uses the Michell truss-like continuum to provide better guidance on optimal structures in the presence of practically unavoidable microscale material randomness. One can readily argue that Michell-type truss-like continua with infinitesimal spacing are rather theoretical concepts in conventional engineering structures where relatively few members can already ensure a high efficiency (Rozvany & Prager 1976). This, however, is not the case with (very) smallscale systems, such as are encountered in nanotechnology and biostructures. It is here that the microscale noise may significantly alter predictions of conventional continuum mechanics. While the literature on optimal structures is quite extensive, the aspect of randomness does not appear to have been studied hitherto. In particular, we arrive at the following conclusions: (i) The presence of random material microstructure prevents the realizability of the optimal, deterministic truss-like continuum. Such a continuum may only be approximated to a certain degree by a truss of finite spacing. (ii) We have assumed throughout that the truss spacing was larger than a single crystal size. One may, of course, consider making a truss on smaller scales, but then the scale dependence of the mean of k—as opposed to our equation (2.4)1 —would need to be considered. Such a modified (2.4)1 would most likely play a stronger role than (2.4)2 . While some guidance on the choice of scaling of k in single crystals may be offered (e.g. Nix & Gao 1998), we do not want to introduce the non-classical (e.g. gradient) models at this stage. (iii) The scatter of trusses displayed in the sequence of figure 3 may have bearing on the models of morphogenesis in living systems. For example, we may consider two leaves coming from the same branch of one tree: they are very similar but somewhat different in the local detail. Note here that an optimal truss resulting from a circular foundation arc of angle 2π resembles a leaf shape (Cox 1965). Now, the veination geometries of such two leaves are, to some degree, disordered rather than perfectly regular as would be implied by a deterministic model. While some stochastic geometric modelling strategies for this have been proposed (see Prusinkiewicz & Lindenmayer (1990) and references therein), the present study suggests the interplay of structural optimization and material randomness as a key element in that kind of a problem. We thank Professor G. Peter of IPST for a biologist’s critique. We benefited from comments of two reviewers. This research was made possible by support of the National Science Foundation (grant CMS-9713764) and the US Department of Agriculture (grant 99-35504-8672).

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