The Two Phase Hybrid Stress Element Method For Composite Materials

Compared with the traditional single-phase materials,particle reinforced composites have many superior mechanical properties,which is of great importance in the industrial field and daily life.The addition of reinforcements improves the properties of the composites,at the same time,it brings some negative effects on the materials.The mechanical properties of particle reinforced composites are related to the volume ratio,microstructure size,shape,spatial distribution,material type and interface characteristics.Therefore,it is necessary to construct accurate models and fully consider the microstructure characteristics to analyze the composites.There are two main methods to analyze composite materials.One is homogenization method,which is easy to obtain the macroscopic mechanical properties of materials;the second is numerical simulation method,including unit cell method and finite element method.The unit cell method considers periodic distribution of multiphase materials,and the finite element method is based on the real structural model of materials.In this paper,a new finite element method is proposed.Based on the principle of minimum complementary energy,an element with two material phases is constructed,which is called two phase hybrid stress element method(TPHSE).The main contents of this paper are as follows:1.The importance of analyzing composite materials is reviewed.Several representative analysis methods including their advantages and disadvantages are introduced.2.Based on the principle of minimum complementary energy,and using Lagrange multiplier to implement several constraints,the modified complementary energy functional of two-phase hybrid stress element is derived.Solving the function and discretize to obtain the matrix form of the corresponding variables.The program is written in FORTRAN.Two examples are given to verify the effectiveness and advantages of the two phase hybrid stress element method,and comparing two results calculated with or without the stress interaction term.The results show that it is more accurate to consider the stress interaction term.3.Two improvements have been made to the implementation of the two-phase hybrid stress element method.The first part is to optimize the division of the integral region of the element.The division of the integration region before optimization is very unfriendly to the concave polygon,and it is very easy for an integral region to span two material phases.After optimization,Delaunay triangulation method is used to divide the concave polygon area(mostly the matrix area)of the element to avoid the above situation.The second is that when the number of elements is large,singlephase material element is easy to appear in the process of element division.Treating this kind of element as single-phase hybrid stress element,and considering the interaction effect of adjacent inclusion interface of the element.The above two improvements make the calculation results more accurate.4.The plastic theory is applied to the two-phase hybrid stress element method.The incremental form of the two-phase hybrid stress element is given and the program is compiled.Using MARC and the two phase hybrid stress element method considering plasticity to simulation the same model.By comparing and analyzing the stress results obtained by these two methods,it is verified that the two phase hybrid stress element method considering plasticity is feasible and accurate.5.Introducing several methods for predicting equivalent modulus of composite materials by micromechanics are introduced,and applying the direct homogenization technique to the two-phase hybrid stress element method.Setting the material parameters of the matrix and the inclusion as the same value,obtaining the average stress-strain curve and the equivalent elastic modulus of the model,and the equivalent elastic modulus value is same with the set value,which verifies the effectiveness of the method.The direct homogenization method and the two phase hybrid stress element method are combined to further analyze the influence of the topological structure of inclusions on the equivalent modulus of composite materials.The effects of the angle between the long axis of inclusion and the direction of tensile deformation,the ratio of height to width and the volume fraction of inclusions on the mechanical properties of the composites were considered respectively.