An Investigation Of Antiplane Problem With Doubly Periodic Cylindrical Inclusions Based On The Virtual Stress

Composite material is a non-homogeneous medium, the effective properties of itsmacroscopic and microscopic stress field has important implications in the design andsafety assessment of composite structures. This paper employs an analytically method tostudy the local field of periodically arranged cylindrical inclusion in infinite matrixsubjected to far field uniform anti-plane stress and the far field uniform anti-plane strain,the analytic expression of local elastic field is given and combined mean field theory themacroscopic effective properties of the material can be accurately predicted.The concept of virtual stress is given based on the concept of eigenstrain andEshelby’s equivalent inclusion ideas. A problem of doubly-periodic virtual stress problem,which is equivalent to the originally heterogeneous materials problem (an infinite matrixwith doubly periodic arrangement cylindrical inclusions), is constructed by theintroduction of non-uniform distribution of virtual stress in the corresponding regions ofdoubly-periodic cylindrical in the originally problem. Equivalent equations to the originalproblem are derived. Taking into account the leap of virtual stress in the contact surfaceand displacement compatibility in the contact surface, and integrated with Laurent seriesexpansion and doubly quasi-periodic Riemann boundary value problem, perturbation fieldinduced by virtual stress can be obtained in series expression. The analytical solution ofthe elastic field inside the material is obtained subjected to two boundary conditions ofinfinity uniform longitudinal shear stress and longitudinal shear strain.Programming the stress and strain fields inside the material by using symboliccomputation software Mathematica, the elastic field within the basic unit cell can beanalyzed, effective elastic modulus of the composites can be accurately predicted usingthe mean-field theory, and comparison with existing results show the correctness of thismethod. Letting the two periods tending to infinite the solution in this article can bereduced to the classic solution of single inclusion. In the end, the conversion issues of thetwo boundary conditions of the far-field stress and far-field strain is discussed, and theconclusions are numerically verified.