Homogenizated Numeration Of Effective Properties For Composite Material With Periodicity

The effective properties of composites with periodicity were studied by the two-scaleasymptotic homogenization method. In the process, the performance of the componentmaterials, the geometry of the fibres and the stochastic effects were considered. Basedon the displacement expansion of the small parameter epsilon, a series of perturbationequations were obtained by applying the perturbation method to the control equationsof linear elasticity. And the basic theoretical formulas were achieved by introducing theeigendisplacement and the homogenization coeffcient.The finite element solution of the characteristic function was derived from the controlformulas of the homogenization equations with the principle of the minimum potentialenergy. Through the further derivation, the distribution force between the interface ofthe hetergenous materials was obtained, and the boundary force method was proposed tosolve the characteristic function. To evaluate the invalidity of the method, simulationswere carried out by ABAQUS, of which the results were coincident with the ones ofthe thermal-stress method. Besides, the boundary force method is helpful to the greatercomprehension of the homogenization coeffcient and the process of homogenization.The effective properties of fibre reinforced composites, which were divided into uni-directional fiber reinforced composites and three dimension four direction braided com-posites, were discussed in the paper. And the inffuences of the boundary conditions, thechoice of unit cell, the fibre volume fraction, the arrangement of fibres and the shape offibres on the effective properties were investigated. For the three dimension four directionbraided composites, the homogenization process was applied twice to calculate the ef-fective properties, considering the inffuences of the fibre volume fraction and the braidedangle.The sensitivity formula of the homogenization coeffcient was derived from thegeneral homogenization-oriented sensitivity analysis with the aid of the finite elementmethod. For the unidirectional fibre reinforced composites, the sensitivity of effectiveproperties to the component modulus, the Poisson's ratio, the fibre volume fraction andthe shape of fibres was calculated. The inffuences of the design parameters and the meshsize on the accuracy of the sensitivity were also discussed. Furthermore, the effects of the same design parameters on the different effective properties were compared by introduc-ing the scaled sensitivity coeffcient.In the last section, the spatial variability theory in geostatistics, such as the vari-ogram and the kriging method, was introduced to the homogenization method to forecastthe effective properties of the composites in consideration of the stochastic microstruc-ture of the composites. Then the stochastic analysis method was proposed, including thekriging-based approximation of density function and kriging-based integral approxima-tion. Based on the method, the inffuences of the random variation of the elastic modulus,the Poisson's ratio and the shape of fibres on the effective properties of the unidirec-tional fibre reinforced composites were discussed. The average and the variance of theeffective properties were also calculated, which were compared with the results of MonteCarlo simulations.