Computational Mechanics of Composite Materials part 8 pps

Sensitivity Analysis for Some Composites 195 conductivity coefficient k 1 is almost the same for all composites. However in the case of sensitivity to v f the 2D and 3D models are similar, while the 1D case is essentially different it results from the relevant equations forms. 4.1.3 Sensitivity of Homogenised Young Modulus for Periodic Composite Bars Let us consider periodic composite bar applied to the compressive/tensile stresses and the homogenised Young modulus of such a structure. For such a unidirectional n component composite structure, one can readily obtain the sensitivity gradients of the effective parameter e (eff) with respect to the modulus of its jth component e j as 2 1 1121 1 1121 1 1 1121 1 2 1 1121 1 1121 1 1 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∂ ∂ ∑ ∑∑∏ ∑ ∑ ∏∏ = +− + +− − = +− = = +− = +− + − = n i niiii n j niiii j i niiii j n i i n i niiii n i niiii n j i j i i j )eff( e ee eelA e ee eelAe ee eelA e e e ee eelA e ee eelAee e e (4.10) The geometrical sensitivity with respect to the cross-sectional area A j is determined as () 2 1 1121 1121 1 )( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= ∑ ∏ = +− +− = n i niiii njjj n i i j eff eeeeelA eeeeele A e ∂ ∂ (4.11) Analogously, geometrical sensitivity with respect to the member length l j is calculated from the following formula: () 2 1 1121 1121 1 )( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= ∑ ∏ = +− +− = n i niiii njjj n i i j eff eeeeelA eeeeeAe l e ∂ ∂ (4.12) 196 Computational Mechanics of Composite Materials It should be underlined that the equations obtained above can be relatively easily inserted in the 1D implementations of the FEM formulation for elastostatics as well as heat conduction problems, both in deterministic and stochastic computation. Now, the sensitivity gradients are derived first for a 1D two component composite with the RVE presented in Figure 2.42. Considering the fact that composite materials are characterised by numerous parameters, it is essential to reduce this number by introduction of non dimensional normalised parameters between the corresponding material and geometric characteristics of a composite. It is recommended to make the sensitivity analysis more focused with opportunity to compare the sensitivity gradients with each other. Determination of the first sensitivity gradient, cf. (4.11), makes it possible to verify how the interrelation between cross sectional area α of both components influences the final effective Young modulus of the composite. The next gradient is responsible for the sensitivity of the composite to the length of both components ratio γ, while the last one gives information about the influence of interrelation β of the Young moduli for composite components. The general observation in this analysis is that an increase in analysed structural geometrical parameters results in a decrease of the effective parameter value (negative derivative sign) and vice versa. Analogously, it is observed that increasing any Young modulus of composite components, the increase of the effective homogenised parameter is obtained. Quantitative verification of the most decisive parameter depends on the interrelations between particular material and geometrical characteristics and should be analysed in detail in further studies. In case of the unidirectional composite, the shape sensitivity studies with respect to the interface location can be done analytically. All the sensitivities calculated above enable us to design, during engineering studies, the most suitable interrelations between particular components for unidirectional tensioned/compressed structural members. Considering the nature of the presented 1D homogenisation approach, it is clear that the sensitivity of the Young modulus holds true for the effective heat conductivity and other related coefficients. The first and second order sensitivity gradients together with the mean value of the homogenised Young modulus have been computed and collected in the figures below. The following input data are adopted: e 2 =2.0E9, the coefficient γ relating the lengths of composite components is arbitrarily taken as equal to 1. Other parameters are adopted in the following form: A 2 =0.2 and l 2 =10.0. The effective Young modulus is determined with respect to the reinforcement ratio as well as to the cross-sectional area ratio of the components and presented below. Sensitivity Analysis for Some Composites 197 Figure 4.10. Parameter variability of the effective Young modulus Figure 4.11. Parameter variability of e (eff) sensitivity gradient wrt parameter α 198 Computational Mechanics of Composite Materials Figure 4.12. Parameter variability of e (eff) sensitivity gradient wrt parameter β Figure 4.13. Parameter variability of e (eff) sensitivity gradient wrt parameter γ Sensitivity Analysis for Some Composites 199 Figure 4.14. Second order sensitivity gradient of e (eff) wrt parameter α Figure 4.15. Second order sensitivity gradient of e (eff) wrt parameter β 200 Computational Mechanics of Composite Materials Figure 4.16. Second order sensitivity gradient of e (eff) wrt parameter γ It is seen that in the case of both ratios equal to 1, the effective elasticity modulus is obtained as the value corresponding to a weaker material, which perfectly agrees with engineering intuition. Next, first and second order derivatives of the effective Young modulus of the composite with respect to the coefficients relating composite components are computed and analysed. It is typical that all the first order gradients are positive, while second order derivatives are less equal to 0. It reflects the fact that the overall effective Young modulus increase is obtained by the corresponding increase of any of these parameters. The second order sensitivity gradients computed and visualised above enable one to confirm the existence of an extremum of the first order derivatives presented before. 4.1.4 Material Sensitivity of Unidirectional Periodic Composites The formulas describing the effective elasticity tensor components for the periodic composite with unidirectional distribution of the heterogeneities (see (2.103) (2.107)) have been implemented in the symbolic computations package MAPLE to derive the appropriate sensitivity gradients [177]. The two component composite shown schematically in Figure 4.17 was examined with the following input data for (a) weaker material e 2 =4.0E9, ν 2 =0.34, c 2 =1-c 1 and (b) stronger material: e 1 =4.0 E9 α, ν 1 =0.34 β, c 1 =0.5. Sensitivity Analysis for Some Composites 201 αe 2 , βν 2 e 2 , ν 2 x 3 l l Figure 4.17. RVE of two component composite bar Design parameters α and β are introduced to make the visualisation of particular sensitivity gradients for some variations of the contrast between Young moduli and Poisson ratios of laminate layers. It will enable more successful optimisation of the composite in case of the homogenisation theory applications. The gradients collected on figures given below are normalised to make all the surfaces presented comparable to each other. First, quite obvious engineering interpretation of these results is that if particular gradient is less than 0 – an increase of design parameter accompanies a decrease of particular effective characteristic value. Otherwise (gradient greater than 0), an increase of the design parameter results in the appropriate increase of the homogenised quantity, while gradient comparable to 0 means that the given design parameter almost does not influence the overall effective characteristic. The figures plotted from the specially implemented MAPLE script present the sensitivity gradients of the homogenised elasticity tensor components – for )eff( C 1111 (Figures 4.18 4.21), )eff( C 3333 (Figures 4.22 4.25), )eff( C 1133 (Figures 4.26 4.29), )eff( C 1122 (Figures 4.30 4.33) and )eff( C 1212 (Figures 4.34 4.37). Parameters α and β equivalent to the contrasts between stronger and weaker materials Young moduli and Poisson ratios are marked on the vertical axes of these figures, correspondingly. Figure 4.18. Sensitivity of )eff( C 1111 wrt e 1 Figure 4.19. Sensitivity of )eff( C 1111 wrt 1 ν 202 Computational Mechanics of Composite Materials Figure 4.20. Sensitivity of )eff( C 1111 wrt e 2 Figure 4.21. Sensitivity of )eff( C 1111 wrt 2 ν Figure 4.22. Sensitivity of )eff( C 3333 wrt e 1 Figure 4.23. Sensitivity of )eff( C 3333 wrt 1 ν Figure 4.24. Sensitivity of )eff( C 3333 wrt e 2 Figure 4.25. Sensitivity of )eff( C 3333 wrt 2 ν Sensitivity Analysis for Some Composites 203 Figure 4.26. Sensitivity of )eff( C 1133 wrt e 1 Figure 4.27. Sensitivity of )eff( C 1133 wrt 1 ν Figure 4.28. Sensitivity of )eff( C 1133 wrt e 2 Figure 4.29. Sensitivity of )eff( C 1133 wrt 2 ν Figure 4.30. Sensitivity of )eff( C 1122 wrt e 1 Figure 4.31. Sensitivity of )eff( C 1122 wrt 1 ν 204 Computational Mechanics of Composite Materials Figure 4.32. Sensitivity of )eff( C 1122 wrt e 2 Figure 4.33. Sensitivity of )eff( C 1122 wrt 2 ν Figure 4.34. Sensitivity of )eff( C 1212 wrt e 1 Figure 4.35. Sensitivity of )eff( C 1212 wrt 1 ν Figure 4.36. Sensitivity of )eff( C 1212 wrt e 2 Figure 4.37. Sensitivity of )eff( C 1212 wrt 2 ν How is demonstrated in all these figures, an increase of Young moduli of both stronger and weaker material result in the increase of all effective elasticity tensor components. Sensitivity gradients computed with respect to Poisson ratios of both composite components have mixed signs and all gradients essentially differ from 0. Taking into account particular variations and values of these results it can be observed that [...]... for the unidirectional periodic composite ∂C h ( eff ) 1111 ∂h 0.9041 0.0959 -0.0476 0.03 38 e1 e2 ν1 ν2 ∂C ( eff ) 3333 ∂h 0.11 38 0 .88 62 0.03 68 1.20 18 ∂C ( eff ) ∂C 1133 ( eff ) 1122 ∂h 0.0603 0.9397 -0. 281 1 0.6254 ∂h 0.7696 0.2304 0. 784 9 0. 189 1 ∂C ( eff ) G 1212 ∂h 0.9 584 0.0451 -0.17 28 -0.0105 h 3.9570 2.5430 -1.1099 1 .85 38 h Furthermore, the sensitivity gradients of G with respect to all design... increments of the design parameters These values are used to approximate the value of G h computed on the basis of (4.21), which are scaled over the RVE total area Using such a composite structure response functional, the Poisson ratio of a matrix and - 2 18 Computational Mechanics of Composite Materials the next Young modulus of the fibre are detected as the most decisive design material parameters of this composite. .. )1-(O 3 2 2 1 ∆ν 0 9 8 7 6 5 4 3 2 1 Figure 4.44 Sensitivity of C1111 wrt ν 2 (eff ) 1ed/2211Cd 52.0 )3-(O 2.0 )2-(O )1-(O 51.0 1.0 50.0 1e ∆ 0 9 8 7 6 5 4 3 2 1 ( eff Figure 4.45 Sensitivity of C1122) wrt e1 Cd 2ed/2211 69.0 49.0 29.0 9.0 88 .0 )3-(O 68. 0 )2-(O 48. 0 )1-(O 28. 0 8. 0 9 8 7 6 5 4 3 ( eff ) 2 1 Figure 4.46 Sensitivity of C1122 wrt e2 2e ∆ Sensitivity Analysis for Some Composites 1 ν d/2211Cd... 2.0 1e 1.0 ∆ 0 9 8 7 6 5 4 3 2 1 (eff Figure 4.41 Sensitivity of C1111) wrt e1 Cd 2ed/1111 50.1 )3-(O 1 )2-(O 59.0 )1-(O 9.0 58. 0 8. 0 2e 57.0 ∆ 7.0 9 8 7 6 5 4 3 2 1 (eff ) Figure 4.42 Sensitivity of C1111 wrt e2 1 ν d/1111Cd 7.0 )3-(O 6.0 )2-(O 5.0 )1-(O 4.0 3.0 2.0 1.0 0 8 7 6 5 4 3 (eff ) 9 2 1 Figure 4.43 Sensitivity of C1111 wrt ν 1 1 ∆ν 213 Computational Mechanics of Composite Materials ν 2 214... Sensitivity of C1212 wrt ν 2 ν Figure 4.51 Sensitivity of C1212 wrt ν 1 81 .0571.071.0561.061.0- )1-(O 7 6 5 4 551.0- )2-(O 8 51.0- )3-(O 9 3 1 541.0- 2 41.0- ∆ν 1 ν d/2121Cd Figure 4.50 Sensitivity of C1212 wrt e2 ( eff ) 9 2e 8 7 6 5 4 3 2 1 20.0 ∆ 520.0 )1-(O 30.0 )2-(O 530.0 )3-(O 40.0 540.0 50.0 550.0 60.0 Cd 2ed/2121 216 Computational Mechanics of Composite Materials Sensitivity Analysis for Some Composites... fibre reinforced composite and collected in Tab 2 it is observed that quite similar values are obtained in both cases and, moreover, both composites show negative sensitivity to Poisson ratios of stronger material The fibre reinforced composite is however the most sensitive with respect to the Poisson ratio of a composite weaker component - - 206 Computational Mechanics of Composite Materials Finally,... is presented in Figure 4.40 Computational sensitivity studies are carried out to determine the sensitivity gradients of the effective elasticity tensor components with respect to material parameters of the constituents, i.e Young moduli and Poisson ratios of fibre and - - - 212 Computational Mechanics of Composite Materials matrix All computational tests are done by the use of the specially tailored... the perturbation of a given material parameter employed as the design parameter - - Table 4.2 Sensitivity gradients of the effective elasticity tensor ( eff ( eff ( eff h ∂C1111) ∂C1122) ∂C1212) ∂h e1 ν1 e2 ν2 ∂h ∂h 0.141 0.056 0 .86 7 1.205 0.072 0. 180 0.926 2 .81 4 0.9 58 -0.173 0.044 -0.011 G h 2.129 -0.090 1 .88 1 3. 987 As can be observed on all these graphs, the worst numerical stability of sensitivity... from the computational error of the homogenisation method itself This numerical phenomenon can be studied in terms of the discretisation density of the RVE in the homogenisation analysis and with respect to the reinforcement ratio of the entire composite Another phenomenon, resulting from physical aspects of the composite being visible especially in Figures 4.44, 4. 48 and 4.52 in the case of the sensitivities... moment ( eff ) 1111 E[C ] ( ) β (C ) ‘Bubbles’ 0. 081 9 ∂ ∂Var (e2 ) ‘No bubbles’ 0. 081 7 ‘Bubbles’ -0.0001 ‘No bubbles’ -0.0001 (eff ) 1111 -0.07 48 -0.0747 0.0405 0.0405 ( eff ) 1111 -0.0076 -0.0127 -0.0005 -0.0064 -0.0003 0.0005 -0.0004 0.0000 ( eff E[C1122) ] 0. 089 2 0. 089 3 -0.0001 -0.0001 (eff α C1122) -0. 081 5 -0. 081 5 0.04 38 0.04 38 ( eff ) 1122 0.0 082 0.0059 0.0012 0.0013 ( eff ) 1122 0.0003 0.0002 . 0.2304 0.0451 2.5430 ν 1 -0.0476 0.03 68 -0. 281 1 0. 784 9 -0.17 28 -1.1099 ν 2 0.03 38 1.20 18 0.6254 0. 189 1 -0.0105 1 .85 38 Furthermore, the sensitivity gradients of h. G with respect to all design. )eff( C 1111 wrt 1 ν 202 Computational Mechanics of Composite Materials Figure 4.20. Sensitivity of )eff( C 1111 wrt e 2 Figure 4.21. Sensitivity of )eff( C 1111 wrt 2 ν Figure 4.22. Sensitivity of )eff( C 3333 . )eff( C 1133 wrt 2 ν Figure 4.30. Sensitivity of )eff( C 1122 wrt e 1 Figure 4.31. Sensitivity of )eff( C 1122 wrt 1 ν 204 Computational Mechanics of Composite Materials Figure 4.32. Sensitivity of )eff( C 1122 wrt