The emergence of auxetic material as a result of optimal isotropic design

Abstract

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.

Optimal internal architectures of femoral bone based on relaxation by homogenization and isotropic material design

“…One can circumvent the mentioned singularity problem by imposing the isotropy of the design from the very beginning, as proposed recently in [9,10]. The isotropic tensor is determined by the bulk modulus k and the shear modulus  ; the isoperimetric condition can be still expressed by the trace of the Hooke tensor: the trace takes the value 3 10 k   in 3D case, and 2 4 k   in a 2D context.…”

“…Recently, an isotropic version of FMD, called IMD, has been proposed in [9,10]; the unknowns are the distributions of the bulk k and shear  moduli within an isotropic body of minimal compliance. In the 2D case, the representation of the Hooke tensor will be used in the form …”

“…A second class of topology optimization methods, called IMD (Isotropic Material Design, a variant of the Free Material Design strategy) approach will be further applied to determine the optimal elasticity tensor in terms of the bulk and shear moduli for the present loading applied to the femoral bone sample. The IMD method has been recently developed in [9,10]; it is the isotropic version of the Free Material Design (FMD), originally proposed in [11], and then developed for various branches of optimization problems, see the review papers [12,13].…”

Negative Poisson's ratio in non‐porous smooth curve sheet

“…As the first re-entrant auxetic form was produced by Lakes [3], a variety of auxetic materials and structures has been fabricated, experimented, and analyzed based on their important property like enhanced shear modulus [4], higher resistance to indentation [5], larger fracture toughness [6], and enhanced vibration absorption properties [7,8]. After the first mechanical model with negative Poisson's ratio presented by R. F. Almgren [9], the scholars developed both analytical [10,11] and numerical [12][13][14][15] methods to investigate and utilize the auxeticity.…”

Material orientation design of planar structures with prescribed anisotropy classes. Study of rhombic systems

“…Due to the similarity of CMD and IMD methods a stress field which minimizes (2) can be found by using the numerical algorithm described in Czarnecki [2] and Czarnecki and Wawruch [5]. The minimal compliance is expressed with the formula: Y = Z 2 /Λ in which Z is a number expressed by the minimization problem (2) and Λ is the integral of the trace of the Hooke tensor.…”