Computational Mechanics of Composite Materials part 5 ppt

Elasticity problems 105 Figure 2.70. Horizontal stresses in the homogenisation problem 11 Figure 2.71. Vertical stresses in the homogenisation problem 11 Figure 2.72. Shear stresses in the homogenisation problem 11 106 Computational Mechanics of Composite Materials Figure 2.73. Vortex visualization of the homogenisation function 11 Figure 2.74. Relative error of the stresses determination in the problem 11 Figure 2.75. Relative error for strain determination in the homogenisation problem 11 Elasticity problems 107 Figure 2.76. Relative error of the strain energy determination 11 Figure 2.77. Horizontal components of the homogenisation function 12 Figure 2.78. Vertical components of the homogenisation function 12 108 Computational Mechanics of Composite Materials Figure 2.79. Total values of the homogenisation function 12 Figure 2.80. Horizontal stresses in the homogenisation problem 12 Figure 2.81. Vertical stresses in the homogenisation problem 12 Elasticity problems 109 Figure 2.82. Shear stresses in the homogenisation problem 12 Figure 2.83. Equivalent von Mises stresses in the homogenisation problem 12 Figure 2.84. Vortex visualization of the homogenisation function 12 110 Computational Mechanics of Composite Materials Figure 2.85. Relative error of the stresses determination in the problem 12 Figure 2.86. Relative error of the strain determination in the problem 12 Figure 2.87. Relative error of the strain energy determination 12 Elasticity problems 111 Figure 2.88. Horizontal components of the homogenisation function 22 Figure 2.89. Vertical components of the homogenisation function 22 Figure 2.90. Total values of the homogenisation function 22 112 Computational Mechanics of Composite Materials Figure 2.91. Horizontal stresses in the homogenisation problem 22 Figure 2.92. Vertical stresses in the homogenisation problem 22 Figure 2.93. Shear stresses in the homogenisation problem 22 Elasticity problems 113 Figure 2.94. Vortex visualization of the homogenisation function 22 Figure 2.95. Relative error of the stresses determination in the problem 22 Figure 2.96. Relative error of the strain determination in the problem 22 114 Computational Mechanics of Composite Materials Figure 2.97. Relative error of the strain energy determination 22 The results of the computational analysis carried out in this section show that the effective properties of the composite and, at the same time, the overall behaviour of the composite, in the context of the homogenisation method, are sensitive to the interphase between the constituents and its material parameters. It should be underlined that the interphase, improved in the example presented above, has small total area in the comparison to the fibre and matrix. It can be expected that the previous, simplified approach (upper and lower bounds or direct approximations of effective properties cited above) do not enable us to arrive at such effects. Considering the assumption that the scale factor between the RVE and the whole composite structure tends to 0 in our analysis and, on the other hand, that this quantity in real composites is small but differs from 0, the sensitivity of the effective characteristics to this parameter are to be calculated in the next analyses based on this approach. To carry out such studies, the scale parameter has to be introduced in the equations describing effective properties and next, due to the well-known sensitivity analysis methods, the influence of the scale parameter ε relating composite micro and macrostructure may be shown. In the analogous way we can study the sensitivity of the effective characteristics of the composite to the component material parameters but there is no need in this case to introduce any extra components into the equations cited above. Further mathematical and computational extensions of the model presented should be provided to include in the constitutive tensor the components responsible for the thermal expansion [228,311]. Having computed the effective characteristics on the basis of Young moduli, Poisson ratios, coefficient of thermal expansion and heat conduction coefficient [106,163,347] it will be possible to provide the coupled temperature displacement FE analyses of periodic composite materials. At the same time it will be valuable to work out the problem presented in the context of viscoelastic or elastoviscoplastic material models of the composite constituents [74,368]. It will enable us to approximate computationally the fracture and failure phenomena in composites resulting from the interface defects or partial debonding using the homogenisation approach. [...]... Deterministic values 73 .50 4.02 30.47 4.64e-2 21 .51 E[C] 73.34 4.02 30.37 4.64e-2 21.49 1.03e-1 2.34e-3 1 .57 e-1 3.34e-2 6 .56 e-3 α(C) 2.42e-7 -5. 64e-7 4.30e-7 5. 57e-7 -7.35e-8 β(C) 3.182 3.730 3.398 3.704 3.049 γ(C) Voigt-Reuss bounds Deterministic values 94.84 2 .55 41.27 1.23e-2 26.79 E[C] 95. 23 2 .55 41.74 1.23e-2 26.74 1.22e-1 9.72e-4 1.81e-1 3 .55 e-2 7.67e-2 α(C) 3.23e-7 -5. 80e-7 5. 15e-7 5. 79e-7 -2.30e-8... 35. 21 22.74 32 .52 29. 75 E[C] 100.37 82. 05 35. 45 22.68 32.46 29.69 8.18e-2 4.11e-2 1.40e-1 5. 84e-2 4.99e-2 3.46e-2 α(C) 2.12e-7 -2.38e-7 4.16e-7 -1 .58 e-7 -9.73e-8 -2.89e-7 β(C) 3.16 3. 15 3.38 3.08 3.06 3.21 γ(C) Voigt-Reuss bounds Deterministic values 113.11 71.80 43.86 16.64 34.63 27 .58 E[C] 113 .50 71. 65 44.34 16.61 34 .58 27 .52 1.03e-2 2.48e-2 1.71e-1 2 .57 e-2 5. 94e-2 2.46e-2 α(C) 3.23e-7 -4.13e-7 5. 15e-7... 146.47 75. 56 63.27 43.97 41.60 E[C] 163.60 146.18 75. 81 63.16 43.89 41 .51 6.89e-2 5. 76e-2 9.78e-2 8.14e-2 4.42e-2 3.95e-2 α(C) 1.79e-7 -1.04e-7 3.32e-7 -1.12e-8 -1.15e-7 -2 .51 e-7 β(C) 3.09 3.06 3.20 3.02 3.07 3.17 γ(C) Voigt-Reuss bounds Deterministic values 171.49 130.33 80. 95 52.63 45. 27 38. 85 E[C] 171.88 129.97 81.43 52 .46 45. 23 38.76 6.78e-2 4.72e-2 9.29e-2 6.60e-2 4 .54 e-2 3.45e-2 α(C) 3.23e-7 -2 .51 e-7... elasticity tensor components [GPa] in test 1 ( eff ( eff Effective C1111) C1212) characteristics Deterministic values 154 .94 68. 85 E[C] 154 .27 68 .52 5. 56e-2 5. 44e-2 α(C) -2.06e-1 -2.41e-1 β(C) 3.27 3.29 γ(C) ( eff C1122) 43.67 43.94 5. 76e-2 9.98e-2 3. 15 132 Computational Mechanics of Composite Materials Table 2.16 Upper and lower bounds for effective elasticity tensor [GPa] in test 1 ( eff ( eff ( eff Effective... the real values of these parameters, the numerical sensitivity of these estimators to the number of iterations should be analysed Such an analysis is performed on the periodicity cell taking the total number of random trials as N =5, 10, 25, 50 , 100, 250 , 50 0, 1000, 250 0, 50 00 and 10000, respectively ( eff Only the probabilistic parameters of C1111) (ω ) are shown, because variations of ( ) ( eff the... test 1 ( eff Figure 2.117 Coefficients of variation α C1122) (ω ) in test 2 1 25 126 Computational Mechanics of Composite Materials ( ) ( ) ( eff Figure 2.118 Coefficients of variation α C1122) (ω ) in test 3 ( eff Figure 2.119 Coefficients of variation α C1122) (ω ) in test 4 ( ) ( eff Analysing the coefficients of variation α Cijkl ) (ω ) , a nonlinear increase of these coefficients with a DB increase... 124 Computational Mechanics of Composite Materials ( ) ( ) ( ) ( eff Figure 2.112 Coefficients of variation α C1111) (ω ) in test 1 ( eff Figure 2.113 Coefficients of variation α C1111) (ω ) in test 2 ( eff Figure 2.114 Coefficients of variation α C1111) (ω ) in test 3 Elasticity problems ( ) ( ) ( ) ( eff Figure 2.1 15 Coefficients of variation α C1111) (ω ) in test 4 ( eff Figure 2.116 Coefficients of. .. 3.45e-2 α(C) 3.23e-7 -2 .51 e-7 5. 15e-7 -1.75e-7 -2.30e-8 -2 .50 e-7 β(C) 3.26 3.17 3 .54 3.09 3.03 3.30 γ(C) Table 2.17 Effective elasticity tensor components [GPa] in test 2 ( eff ( eff Effective C1111) C1212) characteristics Deterministic values 102.33 36.47 E[C] 102 .50 36.69 5. 83e-2 5. 90e-2 α(C) -1.86e-1 -1.92e-1 β(C) 3.23 3. 25 γ(C) ( eff C1122) 33.69 33.49 6.38e-2 -9.96e-2 3. 15 Table 2.18 Upper and lower... optimum number of the samples for estimation of any ( eff probabilistic coefficient and/or moment for the tensor C ijkl ) (ω ) [ ( eff Figure 2.120 Statistical convergence of the expected value E C1111) (ω ) ] 128 Computational Mechanics of Composite Materials [ ] [ ] ( eff Figure 2.121 Statistical convergence of the expected value E C1122) (ω ) ( eff Figure 2.122 Statistical convergence of the expected... convergence of coefficient of variation α C1111) (ω ) ) Elasticity problems 129 ( ) ( ) ( eff Figure 2.124 Statistical convergence of coefficient of variation α C1122) (ω ) ( eff Figure 2.1 25 Statistical convergence of coefficient of variation α C1212) (ω ) ( eff It is seen from the analysis of the expected values of C ijkl ) (ω ) that the estimator convergence character is described by a curve of similar . components of the homogenisation function 12 Figure 2.78. Vertical components of the homogenisation function 12 108 Computational Mechanics of Composite Materials Figure 2.79. Total values of the. 12 Figure 2.84. Vortex visualization of the homogenisation function 12 110 Computational Mechanics of Composite Materials Figure 2. 85. Relative error of the stresses determination in the problem. over random spaces Do for iter=1,M Generation of () ω e , () ων 118 Computational Mechanics of Composite Materials Enddo Computations of PDFs of elasticity tensor components Upper and lower