Computer Methods in Applied Mechanics and Engineering
The stiffest designs of elastic plates: Vector optimization for two loading conditions
Introduction
The problem of maximization of the overall stiffness of a structure can be formulated in several manners. One can assume that two or three materials are at our disposal, the amount of both being given. The aim is to find the optimal placement of these materials, see e.g. Cherkaev [11] and Allaire [1]. The first paper on two-material plates is due to Gibiansky and Cherkaev [15], where the problem was formulated in a proper setting. Majority of papers in this vein are targeted at the shape design, see Bendsøe and Sigmund [6] and Allaire [1]. The papers mentioned are based on the relaxation by homogenization method, with using a specific energy estimate, called translation method, cf. Chapter VI in Ref. [21].
The alternative method is FMD (free material design approach) where all the anisotropic properties are treated as design variables of non-uniform distribution, cf. Ringertz [29], Bendsøe et al. [4], [5], see chapter 3 in the book by Bendsøe [3], cf. Banichuk [2]; an alternative approach can be found in Taylor [38] and Du and Taylor [13], Gaile et al. [14]. The Hooke tensor encompasses all these properties. In FMD no point-wise conditions are imposed on this tensor; instead an integral isoperimetric condition is assumed involving e.g. the Frobenius norm as the integrand. This approach is still developed in papers by Hörnlein et al. [16] and Kočvara and Stingl [17], see the web page of the PLATO-N Project coordinated by Bendsøe.
The essential drawback of the FMD approach as outlined above is the unclear physical meaning of the isoperimetric condition. The relaxation by homogenization approach was free from this drawback, since the isoperimetric condition is there imposed on the volume of the materials being at our disposal. The mass density was also linked with Hooke’s tensor. The FMD does not use the density of mass, since the Hooke tensor itself is viewed as design variable, with no link to the underlying microstructure. The other drawback of FMD is the very result which leads to one non-zero elastic modulus. This degeneracy is a direct consequence of the formulation in which the moduli must follow the one stress field.
The present paper puts forward a new version of the FMD in which the Kelvin moduli are viewed as distributed according to a given pattern, as announced in Lewiński [20] and in Czarnecki and Lewiński [12]. The other elastic characteristics ωK, K = 1, 2, 3, the so called eigen-states, are design variables. We refer here to the theory of spectral decomposition of Hooke’s tensor developed in: Rychlewski [31], [32], [33], [34], Blinowski and co-workers [7], [8], [9], Theocaris and Sokolis [39], Sutcliffe [37], Moakher [24], Norris [26], Moakher and Norris [23]. No integral type isoperimetric condition is imposed. Two loading conditions are considered. The merit function is the weighted sum of the total compliances corresponding to two loading conditions. Introduction of two loading states will make the distribution of Hooke tensor unique, of lesser degeneracy than in the one loading case. Just this is the main reason to consider two types of loads. As in early paper by Gibiansky and Cherkaev [15] the departure point is stress-based. This makes the formulation a minimum problem with respect to: statically admissible stress fields and design variables.
The problem discussed brings about local minimization problems which turn out to be solvable by relatively elementary methods. By solving the local problems explicitly, the main problem is rearranged to the form of an equilibrium problem of a hypothetic hyperelastic body composed of two constituents, within a mixture theory. The constitutive equations couple two strain fields with two stress fields in a nonlinear manner. These equations are inverted to the primal form, which makes it possible to recover the strain-based potential. This effective potential turns out to be convex thus corresponding to monotonicity of the stress–strain equations and to solvability of the optimization problem considered. The FEM is applied using quadrilateral elements with bilinear shape functions. We show that a special Newton type algorithm is a right tool to solve a rich family of optimization problems for various loading cases and contrasts between the Kelvin moduli.
The following conventions are adopted: the space of symmetric tensors of degree 2 is denoted by . The summation convention applies to repeated indices at all levels. Small Latin indices run over 1, 2.
Section snippets
The plane problem of linear elasticity. Equilibrium for two loading cases
Consider a thin elastic plate loaded in plane. Assume that the plane stress assumptions are fulfilled. The plate is subject to a boundary loading acting on a part Γ1 of the boundary of a given plane domain Ω. Two kinds of the loading will be considered, indexed by α = 1 or 2. The plate is fixed along Γ2 being a part of ∂Ω. Assume that Ω is parameterized by Cartesian coordinates (x1, x2) with the orthonormal basis (e1, e2). Let the virtual displacement field v = (v1, v2), referred to Ω, be
Formulation of an optimal design problem for two loading conditions
In the present section the case of two kinds of loads acting independently will be discussed. The aim is to find optimal distribution of the eigenstates ω1(x), ω2(x), ω3(x) to make the plate as stiff as possible with respect to both the load cases. The distribution of the Kelvin moduli λ1(x), λ2(x), λ3(x) within Ω is viewed as prescribed. These moduli do not undergo optimization. Since ω1(x) is determined by ω2(x) and ω3(x), only the latter two fields will be design variables. The fields ω2(x)
Construction of potential Wη
Let and choose (a, b) ∈ Q. Let us define the auxiliary functionwhich can be put in the formwithwhereThe problem (3.5) can be rewritten as follows:whereWe see thatwithThe latter function will be found explicitly. To this end it is
Construction of the optimal Hooke tensor C
Let us come back to the result (4.40). The minimizing value of y = y∗ is such thatwhereand y0 = 2α. HenceMinimum in (4.37) is attained for with y0 = 2α, being determined by (4.44). Thus the angle x in Fig. 3 is given as above. Our task is now to express ω1 and ω3 in terms of and being non-colinear. Let us redraw Fig. 3 as in Fig. 4.
We introduce the unit vector τ⊥, orthogonal to .We shall find z1, z2 from the
Question of well-posedness of problem (P)
Let us focus our attention on problem (P) cf. (3.7). The relevant Lagrangian has the formwithHere are kinematically admissible; they vanish on Γ2. These fields are also Lagrangian multipliers for the statical admissibility conditions of and . The variation δLη = 0 for all , . Let . ThenThe above variational equation implies
Displacement-based formulation of the optimal problem considered
The optimal design problem (3.2) has been rearranged to the form (3.7) or (P). Problem (P) is expressed in terms of the stress fields. Since the available numerical methods of nonlinear mechanics are based on displacement fields as the primal unknowns, it is expedient to work with the problem dual to (P). The aim of this section is to put forward this formulation and deliver its equations explicitly. To this end we invert the nonlinear constitutive equations (6.20) arriving at
Numerical treatment of the problem (P∗)
The problem (P∗) formulated in Section 7 has not its counterpart in the literature. All what is known has been said above, the main feature being monotonicity of the constitutive equations (7.17). The degree of complexity of this problem resembles that of the equilibrium of strongly curved thin hyperelastic shells, where two stress resultant fields and two strain fields are involved. The constitutive equations are simultaneously coupled and nonlinear. The hitherto existing papers on numerical
Case studies
The aim of this section is to construct the optimal layouts of elastic moduli within rectangular plates of various kinematic boundary conditions subject to two kinds of in-plane surface loads: contributing to the compliance functional (3.22) with the weight factors and , respectively; η ∈ [0, 1]. The optimal moduli Cijkl are chosen among all possible moduli referring to a given distribution of the Kelvin moduli λ1, λ2, λ3 within the plate domain. In the examples
Final remarks
The main problem of the present paper, e.g. problem (3.2) of minimization of the weighted sum of compliances (corresponding to two kinds of loads applied non-simultaneously) over all fields of projectors P1, P2, P3 which determine Hooke’s tensor (2.4) in each point of Ω has been reduced to problem (P), see (3.7) which can be viewed as an equilibrium problem of an effective two- component mixture of specific non-linear and anisotropic properties. This result makes it possible to solve the
Acknowledgements
The paper was prepared within the Research Grant no N506 071338, financed by the Polish Ministry of Science and Higher Education, entitled: Topology Optimization of Engineering Structures. Simultaneous shaping and local material properties determination.
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