Elasticity problems 45 4.5 3.5 2.5 一4"bubbles'" 1.5 。-----10"bubbles" -·--20"bubbles --40 "bubbles" 0.5 0 4.00E-036.00E-038.00E-031.00E-021.20E-021.40E-021.60E-021.80E-022.00E-02 Figure 2.9.Expected values of probabilistically averaged Young modulus in matrix 0.222 0.22 0.218 0.216 一4"teeh" 0.214 -·。-·-l0"teeth" 一.20"1egeh 0.212 -·一·40"1eeth" 0.21 4.00E036.00E038.00E-031.00E-021.20E-021.40E-021.60E-021.80E-022.00E-02| Figure 2.10.Probabilistically averaged Poisson ratio in fibre
Elasticity problems 45 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02 1.40E-02 1.60E-02 1.80E-02 2.00E-02 4 "bubbles" 10 "bubbles" 20 "bubbles" 40 "bubbles" Figure 2.9. Expected values of probabilistically averaged Young modulus in matrix 0.21 0.212 0.214 0.216 0.218 0.22 0.222 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02 1.40E-02 1.60E-02 1.80E-02 2.00E-02 4 "teeth" 10 "teeth" 20 "teeth" 40 "teeth" Figure 2.10. Probabilistically averaged Poisson ratio in fibre
46 Computational Mechanics of Composite Materials 0.35 0.3 0.25 02 0.15 4 "bubbles" ·-·-I0"bubbles'" 0.1 20"bubbles" ·一40."bubbles” 0.05 0 4.00E-036.00E-038.00E-031.00E-021.20E-021.40E021.60E-021.80E-022.00E-02 Figure 2.11.Probabilistically averaged Poisson ratio in matrix As is expected,the resulting expected values of the homogenised Young modulus both in the matrix and the fibre regions,and similarly the Poisson ratio, are linear functions of the contact zone widths.The variances of the averaged Young modulus are second or higher order functions of this variable and this order is directly dependent on the number of interface defects. Comparing Figures 2.8 with 2.9 and 2.12 with 2.13 it can be seen that the Young modulus in the matrix contact zone is,for the present problem,much more sensitive to variation of its parameters than the same modulus in the fibre interphase.Larger coefficient of variation of this modulus is obtained in the matrix interface region rather than in the fibre contact zone.On the other hand,the homogenised elastic properties are derived by averaging their values in both regions.Thus,greater changes in these properties can be expected in the matrix because of the larger volume of bubbles related to the fibre teeth. Another interesting effect (cf.Figures 2.12 and 2.13)is the increase of variances of the homogenised Young modulus in the matrix contact zone for increasing width of this zone and the number of bubbles.The reverse effect occurs for the fibre side of the interface and its teeth.This is due to the fact that bubbles occupy more than half of a volume of the corresponding contact zone,and teeth less than a half
46 Computational Mechanics of Composite Materials 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02 1.40E-02 1.60E-02 1.80E-02 2.00E-02 4 "bubbles" 10 "bubbles" 20 "bubbles" 40 "bubbles" Figure 2.11. Probabilistically averaged Poisson ratio in matrix As is expected, the resulting expected values of the homogenised Young modulus both in the matrix and the fibre regions, and similarly the Poisson ratio, are linear functions of the contact zone widths. The variances of the averaged Young modulus are second or higher order functions of this variable and this order is directly dependent on the number of interface defects. Comparing Figures 2.8 with 2.9 and 2.12 with 2.13 it can be seen that the Young modulus in the matrix contact zone is, for the present problem, much more sensitive to variation of its parameters than the same modulus in the fibre interphase. Larger coefficient of variation of this modulus is obtained in the matrix interface region rather than in the fibre contact zone. On the other hand, the homogenised elastic properties are derived by averaging their values in both regions. Thus, greater changes in these properties can be expected in the matrix because of the larger volume of bubbles related to the fibre teeth. Another interesting effect (cf. Figures 2.12 and 2.13) is the increase of variances of the homogenised Young modulus in the matrix contact zone for increasing width of this zone and the number of bubbles. The reverse effect occurs for the fibre side of the interface and its teeth. This is due to the fact that bubbles occupy more than half of a volume of the corresponding contact zone, and teeth less than a half
Elasticity problems 47 17.8 17.6 17.4 17.2 17 4 "teeth" ····-l0"ieeh” ·-·20"teeth 16.8 一一一40"teeth' 16.6 4.00E-036.00E-038.00E-031.00E-021.20E-021.40E-021.60E-021.80E-022.00E-02 Figure 2.12.Variances of probabilistically averaged Young modulus in fibre 1.6 1.4 一4"bubbles" 1.2 ····-lO"bubbles" ·一·20"bubbles" ---40 "bubbles" 0.8 0.6 0.4 ”-‘ 0.2 04 4.00E-036.00E-038.00E-031.00E-021.20E-021.40E-021.60E-021.80E-022.00E-02 Figure 2.13.Variances of probabilistically averaged Young modulus in matrix
Elasticity problems 47 16.6 16.8 17 17.2 17.4 17.6 17.8 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02 1.40E-02 1.60E-02 1.80E-02 2.00E-02 4 "teeth" 10 "teeth" 20 "teeth" 40 "teeth" Figure 2.12. Variances of probabilistically averaged Young modulus in fibre 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 4.00E-03 6.00E-03 8.00E-03 1.00E-02 1.20E-02 1.40E-02 1.60E-02 1.80E-02 2.00E-02 4 "bubbles" 10 "bubbles" 20 "bubbles" 40 "bubbles" Figure 2.13. Variances of probabilistically averaged Young modulus in matrix
48 Computational Mechanics of Composite Materials 2.2 Elastostatics of Some Composites Elastic properties and geometry of so defined result in the random displacement field u,(x;o)and random stress tensor o(x,@)satisfying the classical boundary-value problem typical for the linear elastostatics problem.Let us assume that there are the stress and displacement boundary conditions,on,and respectively,defined on Let C be a random function of C class defined on the entire region.Let p denote the mass density of a material contained in and pf denote the vector of body forces per a unit volume.The boundary-differential equation system describing this equilibrium problem can be written as follows ij(x:)=Cik(x:0)ER(x:0) (2.48) cGo)=为 u,x）uj(xω) (2.49) dxj x 0i,(x四)+p(@)f=0 (2.50) Eu,co)]=Ei,(xo)小x∈。 (2.51) Var(u,(xo)=0;x∈a2. (2.52) Elog(x:)j=E(x)]:xed, (2.53) Varli(x:@))nj=0;xEd (2.54) for a=1,2.....n and ijj,k,l=1,2. Generally,the equation system posed above is solved using the well- established numerical methods.However it should first be transformed to the variational formulation.Such a formulation,based on the Hamilton principle,is presented in the next section.To have the formulation better illustrated,an example of the periodic superconducting coil structure is employed.The stochastic non- homogeneities simulate the technological innacuracies of placing the superconducting cable in the RVE.Its periodicity cell in that case is subjected to horizontal uniform tension on its vertical boundaries to analyse the influence of the stochastic defects on the probabilistic moments of horizontal displacements.The stochastic variations of these displacements with respect to the thickness of the interphase constructed are verified numerically.Stochastic computational experiments are performed using the ABAQUS system and the program POLSAP specially adapted for this purpose
48 Computational Mechanics of Composite Materials 2.2 Elastostatics of Some Composites Elastic properties and geometry of Ω so defined result in the random displacement field ) ui (x;ω and random stress tensor ) σ ij (x;ω satisfying the classical boundary-value problem typical for the linear elastostatics problem. Let us assume that there are the stress and displacement boundary conditions, Ωt ∂ and Ωu ∂ respectively, defined on Ω . Let Cijkl be a random function of 1 C class defined on the entire Ω region. Let ρ denote the mass density of a material contained in Ω and i ρf denote the vector of body forces per a unit volume. The boundary-differential equation system describing this equilibrium problem can be written as follows σ (x;ω) C (x;ω) ε (x;ω) ij = ijkl kl (2.48) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + i j j i ij x u x x u x x ∂ ∂ ω ∂ ∂ ω ε ω ( ; ) ( ; ) ; 2 1 (2.49) σ ij, j(x;ω) + ρ(ω) fi = 0 (2.50) E[ ] u (x;ω) E[ ] uˆ (x;ω) i = i ; u x∈∂Ω (2.51) Var( ) u (x;ω) = 0 i ; u x∈∂Ω (2.52) E[σ (x;ω)]n E[ ] t (x;ω) ij j = i ; t x∈∂Ω (2.53) Var(σ ij (x;ω)) n j = 0 ; t x∈∂Ω (2.54) for a=1,2,...,n and i,j,k,l=1,2. Generally, the equation system posed above is solved using the wellestablished numerical methods. However it should first be transformed to the variational formulation. Such a formulation, based on the Hamilton principle, is presented in the next section. To have the formulation better illustrated, an example of the periodic superconducting coil structure is employed. The stochastic nonhomogeneities simulate the technological innacuracies of placing the superconducting cable in the RVE. Its periodicity cell in that case is subjected to horizontal uniform tension on its vertical boundaries to analyse the influence of the stochastic defects on the probabilistic moments of horizontal displacements. The stochastic variations of these displacements with respect to the thickness of the interphase constructed are verified numerically. Stochastic computational experiments are performed using the ABAQUS system and the program POLSAP specially adapted for this purpose
Elasticity problems 49 2.2.1 Deterministic Computational Analysis The main idea of the numerical experiments provided in this section is to illustrate the horizontal displacements fields and the shear stresses obtained for the deterministic problem of uniform extension of the periodicity cell quarter.Both Young modulus and Poisson ratio are assumed here as deterministic functions;for the purpose of the tests,the program ABAQUS [1]is used together with its graphical postprocessor.The periodicity cell quarter has been discretised by 224 rectangular 4-node plane strain isoparametric finite elements according to Figure 2.14. Figure 2.14.Discretisation of the fibre-reinforced composite cell quarter The symmetry displacement boundary conditions are applied on the horizontal edges of the quarter as well as on the left horizontal edge,while the uniform extension is applied on the right vertical edge of the RVE.The standard deviations of the composite component Young moduli are taken as o(e )=4.2 GPa,o(e,)= 0.4 GPa and the stochastic interface defect data are approximated by the following values:Eml]=3.σ(n0.05En0.15,El0.02R,o(r)=0.1R=8.0E-4. Probabilistically averaged values of the interphase elastic characteristics are obtained from these parameters as follows Eleg 3.82 GPa,Var(e)=1.48 GPa, v.=0.324 with the interphase thickness equal to A =0.0104.Four numerical experiments have been carried out for these parameters taking the values collected in Table 2.1. Table 2.1.Young modulus values of the interphase for particular tests Test number 2 4 ek 93 Ee]-3.o(e) Eles]+3.G(ex)
Elasticity problems 49 2.2.1 Deterministic Computational Analysis The main idea of the numerical experiments provided in this section is to illustrate the horizontal displacements fields and the shear stresses obtained for the deterministic problem of uniform extension of the periodicity cell quarter. Both Young modulus and Poisson ratio are assumed here as deterministic functions; for the purpose of the tests, the program ABAQUS [1] is used together with its graphical postprocessor. The periodicity cell quarter has been discretised by 224 rectangular 4-node plane strain isoparametric finite elements according to Figure 2.14. Figure 2.14. Discretisation of the fibre-reinforced composite cell quarter The symmetry displacement boundary conditions are applied on the horizontal edges of the quarter as well as on the left horizontal edge, while the uniform extension is applied on the right vertical edge of the RVE. The standard deviations of the composite component Young moduli are taken as ) ( 1 σ e = 4.2 GPa, ) ( 2 σ e = 0.4 GPa and the stochastic interface defect data are approximated by the following values: E[ ] n =3, σ ( ) n =0.05 E[ ] n =0.15 , E[ ]r =0.02 R , σ ( )r =0.1 R=8.0E − 4 . Probabilistically averaged values of the interphase elastic characteristics are obtained from these parameters as follows E[ ] e GPa k =3.82 , Var( ) e GPa k =1.48 , =0.324 ν k with the interphase thickness equal to 0.0104 k ∆ = . Four numerical experiments have been carried out for these parameters taking the values collected in Table 2.1. Table 2.1. Young modulus values of the interphase for particular tests Test number 1 2 3 4 k e 2 e [ ] k E e [ ] ( ) k k E e − 3⋅σ e [ ] ( ) k k E e + 3⋅σ e