Material And Structural Analysis And Design Of Cellular Solids Based On The Couple-stress Theory

It is an interesting issue for the community of cellular solids to discover the relationship between the effective macroscopic properties and the microstructures as well as to design the materials and structures through the effective model. The cellular material is usually homogenized to a continuum, based on the classical continuum theory. Since this kind of theory contains no intrinsic scale parameter and is only the first order approximation of the microstructures, it is merely able to describe the phenomenon where the size of macrostructures is much larger than that of microstructures. On the contrary, if the macro structures and the microstructures have the same or comparable size order, the classical effective continuum model cannot provide the real response of the structures. In other words, there are size effects in the analysis of the effective properties of cellular solids. Moreover, these size effects inevitably affect the designing results of cellular solids. It is a pronounced feature that the macro structural size often has the similar order to that of macrostructure. As a result, the effective classical continuum model cannot explain the size effects in the deformation of cellular solids. Furthermore, the optimization model, based on the conventional theory, is also defective.The purpose of this paper is to discuss the size effects during the procedure of the analysis and optimization of materials and structures of cellular solids. Based on the couple-stress theory, the effective continuum model of cellular solids is established. Moreover, the thesis constructed the microstructural optimization model, the macrostructural optimization model and the concurrent optimization model of materials and structures of cellular solids, respectively. The size effects are also studied through the continuum model. The main works of this thesis are as follows:1) Taking the lattice truss material plate for example, the dissertation investigated the size effects problems in the analysis of bending deformation of thin-wall structures of cellular solids. The out-plane size of the truss material plate are in the same order with that of the microstructures. Hence, remarkable size effects exist in the analysis of the effective bending rigidity of the truss material plate. Thus, the conventional integral method through the effective in-plane properties of truss materials, based on the classical continuum theory, cannot achieve the accurate effective bending stiffness of the plate model since it contains no size effects. This thesis constructed a Representative Volume Element (RVE) model to determine the effective stiffness and recommended the corresponding displacement boundary conditions. We compared the results of bending deformation of typical truss material plates through the effective models, the experiments and discrete finite element models. Good agreements show that our model is reasonable.2) The author constructed two effective finite element (FE) formulations of couple-stress theory. Since the couple-stress theory is more complicated than the classical theory, the FE method is still the useful technique. The constraint of rotations leads to the requirement of C1 continuity of the couple-stress FE formulation. This dissertation developed two kinds of FE formulations:the 8-node quadrilateral Serendipity element, based on the penalty functions and the 4-node quadrilateral discrete couple-stress element, based on the discrete constraints. The comparison of the numerical results with the analytical solutions for some typical questions shows that the two presented formulations have high precise and can satisfy the requirement of numerical accuracy of this thesis.3) The dissertation studied the effective couple-stress continuum model of cellular solids. A RVE model for the effective continuum constitutive constants as well as the corresponding boundary conditions was established in terms of equivalence strain energy. The author derived the definitions of the characteristic lengths as material engineering constants for the orthotropic continuum. The results comparison showed that the present effective constitutive constants and the characteristic lengths agree well with those analytical solutions in literature. Besides, the influences of microstructural features (the topology, the size and the relative density) on the effective continuum model were discussed. Furthermore, the phenomenon and the nature of size effects in bending deformation of cellular solids were researched using the effective couple-stress continuum model.4) This dissertation proposed a microstructural topology optimization formulation of cellular solids, based on the effective couple-stress continuum model. This optimization formulation was established on the basis of the developed effective couple-stress continuum model. In order to improve the computational efficiency, the sensitivity analysis formulation was derived from the conclusions of the optimization formulation of structures under displacement loading. The present optimization model can exhibit the evolution of the micro structural topology and the objective function of cellular solids versus the size ratio of the macrostructures to microstructures. Thus, this model overcomes the shortcomings of the inverse-homogenization method that is based on the conventional continuum theory that cannot mimic the size effects.5) This thesis investigated the formulation for topology optimization of couple-stress material structures. In the optimization of cellular material structures, the optimal topology is usually size-dependent when the macrostructural size has a comparable order with that of microstructures. However, the conventional formulation of topology optimization, based on the classical continuum mechanics theory, containing no intrinsic scale parameter of material, cannot show any size effects. For this reason, we proposed the topology optimization formulation of couple-stress continuum and generalized the classic SIMP scheme to the couple-stress continuum. Moreover, we discussed the numerical instabilities and recommended the corresponding prevention schemes. The results showed that the present optimization model can reveal the size effects and that this model presented high accuracy on the optimization of cellular material structures.6) This dissertation proposed a concurrent optimization model of materials and structures of cellular solids, based on the couple-stress continuum model. The conventionally separate optimization model of materials and structures contains on coupling effects of materials and structures. At the same time, the structures often have a comparable size order with the material for cellular solids while the classical theory contains no higher order information of material microstructures. Owing to the above reasons, the author established the concurrent optimization model of materials and structures, based on the couple-stress theory. In this model, the topology of macrostructures is generated by the SIMP model while the topology of materials is produced by the couple-stress based optimization model of microstructures of cellular solids. The effective constitutive constants of macroscopic continuum that are determined by the effective couple-stress continuum model actually play the role of bridge between the macro-design and micro-design. The results show that the present simultaneous optimization model contains the coupling effects of materials and structures and can describe the size effects precisely.The research of this dissertation is supported by National Natural Science Foundation of China (No.10572030,10721062,10332010), National Basic Research Program of China (No. 2006CB601205), and Program for New Century Excellent Talents in University of the State Ministry of Education of China (No. NCET-04-0272). The financial contributions are gratefully acknowledged.