50 Computational Mechanics of Composite Materials Horizontal displacement fields and the shear stress fields for particular experiments are presented in Figure 2.15 and 2.19(test no 1),Figure 2.16 and 2.20 for test no 2,Figure 2.17,2.21 for test no 3 and Figure 2.18 for test no 4. Comparing these results,it is seen that a decrease of the Young modulus value lower than its expected value results in a jump of the horizontal displacements field within and around the interphase.This effect can be interpreted as the possibility of debonding of the composite components caused by the worsening of the interphase elastic parameters,which confirms the usefulness of the presented mathematical- numerical model in the interphase phenomena analysis.It should be underlined that in other models of interphase defects and contact effects at the interface,the horizontal displacements have discontinuous character too.On the other hand, increasing the Young modulus above its expected value does not introduce any sensible differences in comparison with the traditional deterministic model for the perfect interface. Analysing the shear stresses fields)collected in Figures 2.19 and 2.21 a jump of the respective values of stresses at the boundary between the fibre and the interphase region is observed in all cases.In the case of tests no.1,2 and 4 the shear stress fields have quite similar characters differing one from another in the interface regions placed near the horizontal and vertical edges of the periodicity cell quarter.The o2(x,)field obtained for test no.3 has decisively different character:for almost the entire interface the jump of stresses between the matrix, interphase and fibre regions is visible.It may confirm the previous thesis based on the displacement results dealing with the usefulness of the model proposed for the analysis of the interface phenomena. +3.48E-35 +5.64E-05 +1.69E-04 +2.82E-04 +3.94E-04 +5.07E-04 +6.20E-04 +6.77E-04 +7.33E-04 Figure 2.15.Horizontal displacements for test no.I
50 Computational Mechanics of Composite Materials Horizontal displacement fields and the shear stress fields for particular experiments are presented in Figure 2.15 and 2.19 (test no 1), Figure 2.16 and 2.20 for test no 2, Figure 2.17, 2.21 for test no 3 and Figure 2.18 for test no 4. Comparing these results, it is seen that a decrease of the Young modulus value lower than its expected value results in a jump of the horizontal displacements field within and around the interphase. This effect can be interpreted as the possibility of debonding of the composite components caused by the worsening of the interphase elastic parameters, which confirms the usefulness of the presented mathematicalnumerical model in the interphase phenomena analysis. It should be underlined that in other models of interphase defects and contact effects at the interface, the horizontal displacements have discontinuous character too. On the other hand, increasing the Young modulus above its expected value does not introduce any sensible differences in comparison with the traditional deterministic model for the perfect interface. Analysing the shear stresses fields ( )i x σ 12 collected in Figures 2.19 and 2.21 a jump of the respective values of stresses at the boundary between the fibre and the interphase region is observed in all cases. In the case of tests no. 1, 2 and 4 the shear stress fields have quite similar characters differing one from another in the interface regions placed near the horizontal and vertical edges of the periodicity cell quarter. The ( )i x σ 12 field obtained for test no. 3 has decisively different character: for almost the entire interface the jump of stresses between the matrix, interphase and fibre regions is visible. It may confirm the previous thesis based on the displacement results dealing with the usefulness of the model proposed for the analysis of the interface phenomena. +3.48E-35 +5.64E-05 +1.69E-04 +2.82E-04 +3.94E-04 +5.07E-04 +6.20E-04 +6.77E-04 +7.33E-04 Figure 2.15. Horizontal displacements for test no. 1
Elasticity problems 51 +3.49E.35 +5.64E-04 +1.69E-04 +2.82E-04 +3.95E-04 +5.08E-04 +6.21E-04 +6.77E-04 +7.34E-04 Figure 2.16.Horizontal displacements for test no.2 +8.63E-35 +7.88E-05 +2.33E-04 +3.89E-04 +5.44E-04 +7.00E-04 +8.56E-04 +9.34E-04 +1.01E-03 Figure 2.17.Horizontal displacements for test no.3
Elasticity problems 51 +3.49E-35 +5.64E-04 +1.69E-04 +2.82E-04 +3.95E-04 +5.08E-04 +6.21E-04 +6.77E-04 +7.34E-04 Figure 2.16. Horizontal displacements for test no. 2 +8.63E-35 +7.88E-05 +2.33E-04 +3.89E-04 +5.44E-04 +7.00E-04 +8.56E-04 +9.34E-04 +1.01E-03 Figure 2.17. Horizontal displacements for test no. 3
52 Computational Mechanics of Composite Materials +3.28E-35 +5.85E-05 +1.67E-04 +2.79E-04 +3.91E-04 +5.02E-04 +6.14E-04 +6.70E-04 +7.26E-04 Figure 2.18.Horizontal displacements for test no.4 -1.74E+02 +1.06E+02 +6.67E+02 +1.22E+03 +1.78E+03 +2.35E+03 +2.91E+03 +3.19E+03 +3.47E+03 Figure 2.19.The shear stresses for test no.1
52 Computational Mechanics of Composite Materials +3.28E-35 +5.85E-05 +1.67E-04 +2.79E-04 +3.91E-04 +5.02E-04 +6.14E-04 +6.70E-04 +7.26E-04 Figure 2.18. Horizontal displacements for test no. 4 -1.74E+02 +1.06E+02 +6.67E+02 +1.22E+03 +1.78E+03 +2.35E+03 +2.91E+03 +3.19E+03 +3.47E+03 Figure 2.19. The shear stresses for test no. 1
Elasticity problems 53 -1.45E+02 +1.37E+02 +7.02E+02 +1.26E+03 +1.83E+03 +2.39E+03 +2.96E+03 +3.24E+03 +3.52E+03 Figure 2.20.The shear stresses for test no.2 -8.25E+01 +1.38E+02 +5.81E+02 +1.02E+03 +1.46E+03 +1.91E+03 +2.35E+03 +2.57E+03 +2.79E+03 Figure 2.21.The shear stresses for test no.3 The general purpose of the computational experiments performed is to verify the stochastic elastic behaviour of the composite materials with respect to probabilistic moments of the input random variables:both the Young moduli of the constituents as well as the stochastic interface defects parameters.The starting point for such analyses is a verification of the probabilistically averaged Young modulus in the interphase(example 1).This has been done by the use of the special FORTRAN subroutine,while the next tests have been carried out using the 4-node isoparametric rectangular plane strain element of the system POLSAP.Material parameters of the composite constituents are taken in examples 1 to 3 as
Elasticity problems 53 -1.45E+02 +1.37E+02 +7.02E+02 +1.26E+03 +1.83E+03 +2.39E+03 +2.96E+03 +3.24E+03 +3.52E+03 Figure 2.20. The shear stresses for test no. 2 -8.25E+01 +1.38E+02 +5.81E+02 +1.02E+03 +1.46E+03 +1.91E+03 +2.35E+03 +2.57E+03 +2.79E+03 Figure 2.21. The shear stresses for test no. 3 The general purpose of the computational experiments performed is to verify the stochastic elastic behaviour of the composite materials with respect to probabilistic moments of the input random variables: both the Young moduli of the constituents as well as the stochastic interface defects parameters. The starting point for such analyses is a verification of the probabilistically averaged Young modulus in the interphase (example 1). This has been done by the use of the special FORTRAN subroutine, while the next tests have been carried out using the 4-node isoparametric rectangular plane strain element of the system POLSAP. Material parameters of the composite constituents are taken in examples 1 to 3 as
54 Computational Mechanics of Composite Materials E(e1)=84.0GPa,V1=0.22,o(e1)=4.2GPa,E(e2)=4.0GPa,o(e2)=0.4GPa, v,=0.34 (expected values and standard deviations of the Young modulus and Poisson ratio,respectively). 2.2.2 Random Composite without Interface Defects The main aim of the numerical analysis is to verify numerically the elastic behaviour of a fibre composite when the Young modulus of composite components is Gaussian random variable.Moreover,numerical tests are carried out to state in what way,for various contents of fibre (with round section)in a periodicity cell, the random material properties of reinforcement and matrix influence the displacement and stress distribution in the cell.A quarter of a fibre composite periodicity cell is tested in numerical analysis and its discretisation is shown in Figure 2.22. Figure 2.22.Discretisation of the periodicity cell quarter Numerical implementation enabling the computations is made using a 4-node rectangular plane element of the program POLSAP (Plane Strain/Stress and Membrane Element).The composite structure is subjected to the uniform tension (100 kN/m)on a vertical cell boundary (60 finite elements with 176 degrees of freedom).Vertical displacements are fixed on the remaining cell external boundaries and the plane strain analysis is performed.Twelve numerical tests are carried out assuming the fibre contents of 30,40 and 50 and the resulting coefficients of variation are taken from Table 2.2. Table 2.2.Coefficients of variation for different numerical tests Test number a(e) a(e2) 0.10 0.10 2 0.10 0.05 3 0.05 0.10 4 0.05 0.05
54 Computational Mechanics of Composite Materials E(e1 ) = 84.0 GPa, ν1 =0.22, 2 ( ) 4. σ e1 = GPa, E(e2 ) = 4.0 GPa, σ() . e2 = 0 4 GPa, ν 2 =0.34 (expected values and standard deviations of the Young modulus and Poisson ratio, respectively). 2.2.2 Random Composite without Interface Defects The main aim of the numerical analysis is to verify numerically the elastic behaviour of a fibre composite when the Young modulus of composite components is Gaussian random variable. Moreover, numerical tests are carried out to state in what way, for various contents of fibre (with round section) in a periodicity cell, the random material properties of reinforcement and matrix influence the displacement and stress distribution in the cell. A quarter of a fibre composite periodicity cell is tested in numerical analysis and its discretisation is shown in Figure 2.22. Figure 2.22. Discretisation of the periodicity cell quarter Numerical implementation enabling the computations is made using a 4-node rectangular plane element of the program POLSAP (Plane Strain/Stress and Membrane Element). The composite structure is subjected to the uniform tension (100 kN/m) on a vertical cell boundary (60 finite elements with 176 degrees of freedom). Vertical displacements are fixed on the remaining cell external boundaries and the plane strain analysis is performed. Twelve numerical tests are carried out assuming the fibre contents of 30, 40 and 50 % and the resulting coefficients of variation are taken from Table 2.2. Table 2.2. Coefficients of variation for different numerical tests Test number ( )1 α e ( ) 2 α e 1 0.10 0.10 2 0.10 0.05 3 0.05 0.10 4 0.05 0.05