Two-scale FEM Analysis For Equivalent Elasticity Displacement Field Of Composite Material With Random Grains

In this paper,a statistical two-scale finite element method(STSFEM)is used to compute the equivalent displacement field of composite materials in order to analyze the influence of grains on elastic properties.The error of STSFEM is given with some assumptions and the numerical example is given to illustrate the validity.The details are shown as follows.Firstly,based on the properties of composite materials with random grains,grains are described simply,and two probability spaces on composites with random grains are introduced: finite-dimension probability space and infinite countable probability space.Secondly,the partial differential equation of elastic mechanics is described.Based on the partial differential equations,the two-scale formula of elastic displacement field is analyzed to prove the existence and uniqueness of the solution of homogenization equation.And then the error between the expected displacement field and the homogenization displacement field is computed.The finite element approximation error produced by the two-scale method is analyzed based on the algorithm of equivalent displacement field.Finally,a numerical example is given to verify the feasibility of the two-scale method predicting the mechanical performance of the composite material with random grains and the accuracy of the error analysis.Numerical simulation model is established.Comparing the computed results of the finite element method with the two-scale method,we will find that the expected value of the displacement fields will be closer to the equivalent displacement field with the increase of the number of samples.So it is verified the effectiveness of the two-scale method to compute the equivalent displacement field.