On minimum compliance problems of thin elastic pla

Authors: S Czarnecki T Lewiński

Publish Date: 2013/02/26

Volume: 48, Issue: 1, Pages: 17-31

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Abstract

The paper deals with two minimum compliance problems of variable thickness plates subject to an inplane loading or to a transverse loading The first of this problem called also the variable thickness sheet problem is reduced to the locking material problem in its stressbased setting thus interrelating the stressbased formulation by Allaire 2002 with the kinematic formulation of Golay and Seppecher Eur J Mech A Solids 20631–644 2001 The second problem concerning the Kirchhoff plates of varying thickness is reduced to a nonconvex problem in which the integrand of the minimized functional is the square root of the norm of the density energy expressed in terms of the bending moments This proves that the problem cannot be interpreted as a problem of equilibrium of a locking material Both formulations discussed need the numerical treatment in which stresses bending moments are the main unknownsThe problem of optimum design of the thickness h of a linearly elastic anisotropic plate loaded inplane to minimize its compliance is well posed provided that the thickness variation is subjected to the conditions h ge h textrm min 0 as proved in Cea and Malanowski 1970 see also Litvinov and Panteleev 1980 Bendsøe 1995 Sec 151 and Petersson 1999 This optimization problem is equivalent to the problem of the optimal transversely homogeneous distribution of one material within a plate loaded inplane in its convexified version see Sec 525 in Allaire 2002 By virtue of this analogy one can note that the question of correctness of the formulation of the minimum compliance problem of plates loaded in plane of varying thickness with the condition h textrm min 0 can be concluded from the Th 528 in Allaire 2002 The main aim of the present paper is the discussion of the problem of optimal distribution of the plate thickness under the condition h textrm min = 0 To be specific let us set the problemFind optimal distribution of the thickness h of a plate made of an elastic material of the inplane reduced moduli C ijkl subject to the inplane loading and fixed on a part Gamma 2 of the contour of Omega to minimize the compliance of the plate under the condition of the plate volume being given The inplane stiffnesses A ijkl=hxC ijkl are involved in the formulation they are referred to a Cartesian frame x 1 x 2 x=left x 1 x 2 right in Omega The assumption h textrm min = 0 is essential since it makes it possible to determine the subdomains of the design domain where hast = 0 or the appearance of openings as well as the unnecessary segments close to the edges The solution boldsymbol tau to the problem 11 can vanish on a subdomain Omega 0 of Omega There the thickness hast of the optimal plate vanishes Thus the solution of the problem 11 cuts off the part of the plate which is unnecessary In this manner specific mathematical difficulties are circumvented linked with admitting very small values for h textrm min in the original setting of the optimization problem discussedIn the present paper it is shown that the problem 11 can be the point of departure for the numerical approach developed in Czarnecki and Lewiński 2012 to solve the free material design problem of planar elasticity Due to some mathematical similarities this numerical approach with slight adjustment applies here The stress fields tau are interpolated by polynomials over the polygons forming the mesh—see Czarnecki and Lewiński 2012The very idea of using a stressbased numerical scheme to solve the minimum compliance problem is not new see the paper by Allaire and Kohn 1992 where the Airy stress function method had been used to set the numerical method based on the FE approach The novelty of the present paper is that the subject of the numerical analysis is the problem 11 in which the thickness h is absent hence it does not need to be bounded from below while in the shape optimization algorithm proposed by Allaire and Kohn 1992 the volume fraction had to be bounded by a small value to assure the numerical stability A price to pay is to deal with a non typical problem 11 in which the functional involves an integrand of linear growth Thus this problem should be numerically solved in its original setting by appropriate interpolating the statically admissible stress fields and by performing the minimization over the stress representations The numerical approach does not lead to a set of linear equations with a nonsingular square matrix which is typical while using the finite element method In particular no stiffness matrices occur The method proposed exceeds the FEM framework we have to develop a new numerical method in which only the meshing of the domain is a step common with any FE approaches The algorithm put forward has been by no means supported by the experience we have from solving problems of solid body mechanicsIf the plate is transversely loaded its bending stiffnesses equal h3x/12C ijkl while the isoperimetric condition has the same form as in the inplane loaded case The problem of the compliance minimization of the plate in bending of thickness h ge h textrm min 0 is badly posed which had been the subject of discussions in the numerous papers starting from Kozłowski and Mróz 1970 Cheng and Olhoff 1981 Rozvany 1989 Lur’e and Cherkaev 1986 cf the papers published in the volume Cherkaev and Kohn 1997 Krog and Olhoff 1997 Bendsøe 1995 Cherkaev 2000 and Lewiński and Telega 2000 Sec 27 In the paper by Muñoz and Pedregal 2007 a review of the relaxation methods of this problem can be found the aim of the relaxation is to make the problem well posed without losing its original setting

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