Sensitivity Analysis for Some Composites 195 conductivity coefficient ki is almost the same for all composites.However in the case of sensitivity to vr the 2D and 3D models are similar,while the ID 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 parameterm with respect to the modulus of its jth component ej as de'e) de (4.10) The geometrical sensitivity with respect to the cross-sectional area A,is determined as ie,l,ee2e-e4e） (4.11) Analogously,geometrical sensitivity with respect to the member length l,is calculated from the following formula: ee)1 eee,） al; (4.12)
Sensitivity Analysis for Some Composites 195 conductivity coefficient k1 is almost the same for all composites. However in the case of sensitivity to vf 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 ej as 2 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 1 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∂ ∂ ∑ ∏ ∑ ∑ ∑ ∏ ∏ ∑ = − + + − + − = − + = = − + = − + + − = n i i i i i n n j i i i i n j i i i i i n j n i i n i i i i i n n i i i i i n n j i j i i j ( eff ) Al e e ...e e ...e Al e e ...e e ...e Al e e ...e e ...e e e Al e e ...e e ...e e e Al e e ...e e ...e e e (4.10) The geometrical sensitivity with respect to the cross-sectional area Aj is determined as ( )2 1 1 2 1 1 1 2 1 1 1 ( ) ... ... ... ... ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − ∑ ∏ = − + − + = n i i i i i n j j j n n i i j eff Al e e e e e e l e e e e e A e ∂ ∂ (4.11) Analogously, geometrical sensitivity with respect to the member length lj is calculated from the following formula: ( ) 2 1 1 2 1 1 1 2 1 1 1 ( ) ... ... ... ... ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − ∑ ∏ = − + − + = n i i i i i n j j j n n i i j eff Al e e e e e e A e e e e e 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 ID 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 a.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 y,while the last one gives information about the influence of interrelation B 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:e2=2.0E9,the coefficient y relating the lengths of composite components is arbitrarily taken as equal to 1.Other parameters are adopted in the following form:A2=0.2 and 12=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
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: e2=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: A2=0.2 and l2=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 1e+010 8e+09 6e+019 4◆+009 2e+0094 10 10 6 b时a Figure 4.10.Parameter variability of the effective Young modulus 1.4g+00g 12e+0091 1+00 8e+008 68+008 48+008 2a008 0 2 8 10 alfa 10 beta Figure 4.11.Parameter variability of sensitivity gradient wrt parameter a
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