The paper deals with optimal distribution of the bulk and shear moduli minimizing the compliance of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support. The isoperimetric condition is expressed by the integral of the trace of the Hooke tensor being a linear combination of both moduli. The problem thus formulated is reduced to an auxiliary 3D problem of minimization of a certain stress functional over the stresses being statically admissible. The integrand of the auxiliary functional is a linear combination of the absolute value of the trace and norm of the deviator of the stress field. Thus the integrand is of linear growth. The auxiliary problem is solved numerically by introducing element-wise polynomial approximations of the components of the trial stress fields and imposing satisfaction of the variational equilibrium equations. The under-determinate system of these equations is solved numerically thus reducing the auxiliary problem to an unconstrained problem of nonlinear programming.
Using both mathematical and numerical methods, the optimal distributions of material characterized by the Young modulus and Poisson ratio (as well as other moduli of isotropy) maximizing the overall stiffness of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support are constructed. The overall stiffness of the body is defined as the inverse of the work of external forces on displacements, called here the compliance of the structure. The isoperimetric condition bounds the integral of the trace of the Hooke tensor. It is proved that isotropic composite materials forming the bodies of extremely high stiffness exhibit negative Poisson ratio in large subdomains, which points at the significance of the auxetic material in modern structural design. The obtained results show that the whole range of possible variation of the Poisson ratio is used, from −1 to 1/2, which proves usefulness of the auxetic materials.
The paper deals with minimization of the weighted sum of compliances related to the load cases applied non-simultaneously. The design variables are all components of the Hooke tensor, subject to the isoperimetric condition bounding the integral of the sum of the Kelvin moduli. This free material design problem is reduced to an equilibrium problem -in two formulations -of an effective body with locking. The stress-based formulation reduces to minimization of an integral of a certain norm of stress fields over the stress fields which equilibrate the given loads. The equivalent displacement-based formulation involves a locking locus defined by using a norm being dual to the previous one. The optimal Hooke tensor is determined by using the stress fields solving the auxiliary locking problem. To make the optimal Hooke tensor positive definite one should consider at least 3 load conditions in the 2D case and not less than 6 load conditions in the 3D case.
Abstract. The problem to find an optimal distribution of elastic moduli within a given plane domain to make the compliance minimal under the condition of a prescribed value of the integral of the trace of the elastic moduli tensor is called the free material design with the trace constraint. The present paper shows that this problem can be reduced to a new problem of minimization of the integral of the stress tensor norm over stresses being statically admissible. The eigenstates and Kelvin’s moduli of the optimal Hooke tensor are determined by the stress state being the minimizer of this problem. This new problem can be directly treated numerically by using the Singular Value Decomposition (SVD) method to represent the statically admissible stress fields, along with any unconstrained optimization tool, e.g.: Conjugate Gradient (CG) or Variable Metric (VM) method in multidimensions.
The Free Material Design (FMD) is a branch of topology optimization. In the present article the FMD formulation is confined to the minimum compliance problem within the linear elasticity setting. The design variables are all elastic moduli, forming a Hooke tensor C at each point of the design domain. The isoperimetric condition concerns the integral of the p-norm of the vector of the eigenvalues of the tensor C. The most important version refers to p = 1, imposing the condition on the integral of the trace of C. The paper delivers explicit stress-based formulations and numerical solutions of the FMD problems in the case of a single load case as well as for a general case of a finite number of load conditions.
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