Prediction Of The Stress Field And Effective Stiffness Of Two-dimensional Elastomer Containing Periodic Nano-holes Or-Inclusions

Analysis of inhomogeneities(including holes or inclusions)is of great importance in solid mechanics.For macro-scale materials,the surface/interface effects imposed on the hole surface or matrix-inclusion interface are usually ignored due to the negligible amount of atoms on surface/interface compared to that of the entire materials.However,for micro/nano-scale,these effects are essential due to the high ratio of surface to bulk.A lot of studies have been done on this topic,and some significant theories and results are reported,one of which is the Gurtin-Murdoch model.The surface/interface effect and the linearized stress-strain relationship are described in this model,respectively.Most of the previous works are mainly for the cases of nano-scale materials containing single hole or inclusion.Meanwhile,the analyses of nano-scale materials containing multiple holes or inclusions are always limited to numerical simulation,and the shapes of these inhomogeneities are always restricted to circular.In fact,the holes or inclusions are always embedded in nanocomposites periodically with irregular shapes.To address these deficiencies and give a more accurate analysis for the real nano-scale materials,we devote to develop in this thesis a semi-analytical scheme to calculate the stress field and effective stiffness of two-dimensional elastomer containing periodic nano-holes or-inclusions based on the Gurtin-Murdoch model.The main contents are summarized as follows:In the first Chapter,a brief introduction for materials containing periodic nano-inhomogeneities(including holes or inclusions)is given and the problems needed to be solved are outlined.In the second Chapter,the complex variable method for solving two-dimensional elastic problems is presented,and the surface/interface effect and periodic boundary condition are introduced briefly.In the third Chapter,the stress field and anti-plane effective shear modulus of two-dimensional elastomer containing periodic nano-holes or-inclusions are calculated in anti-plane shear,respectively.The residual surface/interface tension-induced stress fields in two-dimensional elastomer containing periodic nano-holes or-inclusions are analyzed in the fourth Chapter.In the fifth Chapter,we predict the effective plane stiffness of two-dimensional elastomer containing periodic nano-holes or-inclusions under plane stain deformations.Finally,the present work is summarized and some future works are proposed on the topic in the last Chapter,and the main innovative points of the thesis are listed below:(1)Based on complex variable methods,an effective and efficient scheme is proposed in this thesis to calculate the stress field and effective stiffness of two-dimensional elastomer containing periodic nano-holes or-inclusions.Using this scheme we can build a theoretical model for the problem with periodic boundary condition to obtain the semi-analytical solution incorporating surface/interface effect.(2)The mechanical problems about two-dimensional elastomer containing periodic arbitrarily-shaped nano-holes or-inclusions are first discussed in anti-plane shear using semi-analytical method,and the stress field and effective shear modulus are obtained in the thesis.(3)The semi-analytical solutions for residual surface/interface tension-induced stress fields of two-dimensional elastomer containing periodic nano-holes or-inclusions with various shapes and volume fractions are studied and obtained in this thesis.(4)In the context of the Gurtin-Murdoch model,we study the effective elastic modulus of two-dimensional elastomer containing periodic arbitrarily-shaped nano-holes or-inclusions under plane stain deformations and predict its effective plane stiffness using semi-analytical method.