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Computational Mechanics of Composite Materials part 8

Tham khảo tài liệu 'computational mechanics of composite materials part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 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 j-1 n n eff n e n e Il Al e e .e e .e I de i 1 i j 1 i I i 1 i i 12 i-1 i 1 n - I j IL Alee .e e .e I i 1 i i 12 i-1 i 1 n I n 1 j-1 n A ne LAlee.e e .e LAlee.e e .e i 1 i e 1 1 i i 12 i-1 i 1 n j 1 i i 12 i-1 i 1 n j V n A2 Il Alee .e e .e I 1 i i 12 i-1 i 1 n I 4.10 The geometrical sensitivity with respect to the cross-sectional area Aj is determined as n deleft n ei Ịje1e2.ej-1ej 1.en dA f 4.11 I LAilie1e2.ei-1ei 1.en I i 1 I Analogously geometrical sensitivity with respect to the member length lj is calculated from the following formula de eff dlj ei Aje1e2.ej-1ej 1.-en i 1__ n . . A2 I LAilie1e2.ei-1ei 1.en I i 1 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 .