392 Random Composites >kk:=1/k:kkk:=int(kk,x=0..1):khom:=eval(1/kkk); >plot3d(khom,contrast=5..100,g=0.1..0.9,title='khom'); Parametric sensitivity of the Mexican hat-Haar basis effective heat conductivity with respect to the contrast and volumetric ratio restart;sig:=-.5;k1:=contrast*k2;k2:=0.05; kmexh:=2+1/(sqrt(2*Pi)*sig 3)*((exp(-x2/(2*sig 2)))*(x 2/sig 2-1)); khaar:=piecewise(x<g,k1,x>g,k2); >k:=khaar+0.005*kmexh; >kk:=1/k:kkk:=int(kk,x=0..1):khom:=eval(1/kkk); dkhomdcontrast:=diff(khom,contrast)*contrast/khom; plot3d(dkhomdcontrast,contrast=5..100,g=0.1..0.9,title='dkhomdcontrast'); dkhomdg:=diff(khom,g)*g/khom; >plot3d(dkhomdg,contrast=5..100,g=0.1..0.9,title='dkhomdg'); Parametric variability of the multiresolutional homogenisation of the Mexican hat-Haar basis heat conductivity with respect to the contrast and volumetric ratio restart;sig:=-.5;k1:=contrast*k2;k2:=0.05; kmexh:=2+1/(sqrt(2*Pi)*sig 3)*((exp(-x^2/(2*sig 2)))*(x^2/sig 2-1)); khaar:=piecewise(x<=g,k1,x>=g,k2); >k:=khaar+0.005*kmexh; >kk:=1/k;kkk:=x/k;kkkk:=1/(2*k); kkk1:=eval(int(kk,x=0..1));kkk2:=eval(int(kkk,x=0..1)); kkk3:=eval(int(kkkk,x=0..1)); khomwav:=eval(1/(kkk1-2*kkk2+2*kkk3)); >plot3d(khomwav,contrast=5..100,g=0.1..0.9,title='khomwav'); Parametric sensitivity of the Mexican hat-Haar basis homogenised heat conductivity with respect to the contrast and volumetric ratio restart;sig:=-.5;k1:=contrast*k2;k2:=0.05; kmexh:=2+1/(sqrt(2*Pi)*sig 3)*((exp(-x^2/(2*sig2)))*(x 2/sig 2-1)); khaar:=piecewise(x<=g,k1,x>=g,k2); >k:=khaar+0.005*kmexh; kk:=1/k;kkk:=x/k;kkkk:=1/(2*k); kkk1:=eval(int(kk,x=0..1));kkk2:=eval(int(kkk,x=0..1)); kkk3:=eval(int(kkkk,x=0..1)); khomwav:=eval(1/(kkk1-2*kkk2+2*kkk3)); dkhomwavdcontrast:=diff(khomwav,contrast)*contrast/khomwav; >plot3d(dkhomwavdcontrast,contrast=5..100,g=0.1..0.9,title='dkhomwavdcontrast); dkhomwavdg:=diff(khomwav,g)*g/khomwav; >plot3d(dkhomwavdg,contrast=5..100,g=0.1..0.9,title='dkhomwavdg');
392 Random Composites > kk:=1/k: kkk:=int(kk,x=0..1): khom:=eval(1/kkk); > plot3d(khom,contrast=5..100,g=0.1..0.9,title='khom'); Parametric sensitivity of the Mexican hat - Haar basis effective heat conductivity with respect to the contrast and volumetric ratio > restart; sig:=-.5; k1:=contrast*k2; k2:=0.05; > kmexh:=2+1/(sqrt(2*Pi)*sig^3)*((exp(-x^2/(2*sig^2)))*(x^2/sig^2-1)); > khaar:=piecewise(x<g,k1,x>g,k2); > k:=khaar+0.005*kmexh; > kk:=1/k: kkk:=int(kk,x=0..1): khom:=eval(1/kkk); > dkhomdcontrast:=diff(khom,contrast)*contrast/khom; > plot3d(dkhomdcontrast,contrast=5..100,g=0.1..0.9, title='dkhomdcontrast'); > dkhomdg:=diff(khom,g)*g/khom; > plot3d(dkhomdg,contrast=5..100,g=0.1..0.9,title='dkhomdg'); Parametric variability of the multiresolutional homogenisation of the Mexican hat - Haar basis heat conductivity with respect to the contrast and volumetric ratio > restart; sig:=-.5; k1:=contrast*k2; k2:=0.05; > kmexh:=2+1/(sqrt(2*Pi)*sig^3)*((exp(-x^2/(2*sig^2)))*(x^2/sig^2-1)); > khaar:=piecewise(x<=g,k1,x>=g,k2); > k:=khaar+0.005*kmexh; > kk:=1/k; kkk:=x/k; kkkk:=1/(2*k); > kkk1:=eval(int(kk,x=0..1)); kkk2:=eval(int(kkk,x=0..1)); kkk3:=eval(int(kkkk,x=0..1)); > khomwav:=eval(1/(kkk1-2*kkk2+2*kkk3)); > plot3d(khomwav,contrast=5..100,g=0.1..0.9,title='khomwav'); Parametric sensitivity of the Mexican hat - Haar basis homogenised heat conductivity with respect to the contrast and volumetric ratio > restart; sig:=-.5; k1:=contrast*k2; k2:=0.05; > kmexh:=2+1/(sqrt(2*Pi)*sig^3)*((exp(-x^2/(2*sig^2)))*(x^2/sig^2-1)); > khaar:=piecewise(x<=g,k1,x>=g,k2); > k:=khaar+0.005*kmexh; > kk:=1/k; kkk:=x/k; kkkk:=1/(2*k); > kkk1:=eval(int(kk,x=0..1)); kkk2:=eval(int(kkk,x=0..1)); kkk3:=eval(int(kkkk,x=0..1)); > khomwav:=eval(1/(kkk1-2*kkk2+2*kkk3)); > dkhomwavdcontrast:=diff(khomwav,contrast)*contrast/khomwav; > plot3d(dkhomwavdcontrast,contrast=5..100,g=0.1..0.9,title='dkhomwavdcontrast'); > dkhomwavdg:=diff(khomwav,g)*g/khomwav; > plot3d(dkhomwavdg,contrast=5..100,g=0.1..0.9,title='dkhomwavdg');
References 1.ABAOUS,v.5.8,(1999)User's Manual.Hibbitt,Karlsson Sorensen, Pawtucket. ANSYS v.5.5.User's Manual (1999),Swanson,Houston,PA. 3. Abdelal GF,Caceres A,Barbero EJ,(2002)A micro-mechanics damage approach for fatigue of composite materials.Comp.Struct.56:413-422 4. Aboudi J,Elastoplasticity theory for composite materials,(1986)Solid Mech.Arch.11:141-183. 5. Achenbach JD,Zhu H,(1989)Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites.J.Mech.Phys.Sol. 37(3):381-393. 6. Adrianov IV et al,(1999)Homogenization procedure and Pade approximants for effective heat conductivity of composite materials with cylindrical inclusions having square cross-section.Proc.Roy.Soc.London A455:3401-3413. 7. Akin JE (1982)Application and the implementation of finite element methods,Academic Press,New York 8. Algam M,Bennett RM,Zureick AH,(2002)Three-parameter vs.two- parameter Weibull distribution for pultruded composite material properties. Comp.Struct.58:497-503. 9. Alzebdeh K et al.,(1998)Fracture of random matrix-inclusion composites: scale effects and statistics.Int.J.Sol.Struct.35(19):2537-2566. 10. Amaniampong G,Burgoyne CJ.(1996)Monte-Carlo simulations of the time dependent failure of bundles of parallel fibres.Eur.J.Mech.A/Solids 15(2):243-266. 11.Anthoine A.(1995)Derivation of the in-plane elastic characteristics of masonry through homogenization theory.Int.J.Sol.Struct.32(2):137-163. 12.Aravas N,(1987)On the numerical integration of a class of pressure- dependent plasticity models.Int.J.Num.Meth.Engrg.24:1395-1416. 13.Argyris J,Mlejnek HP(1991)Dynamics of Structures,North Holland 14. Augusti G,Baratta A,Casciati F(1984)Probabilistic Methods in Structural Engineering,Chapman and Hall 15.Auriault JL,(1983)Effective macroscopic description for heat conduction in periodic composites.Int.J.Heat and Mass Transfer 26:861-869. 16. Babuska I,(1976)Homogenization and its application.Mathematical and computational problems.In:Numerical solution of partial differential equations III,Academic Press,pp.89-116. 17. Bacry E,Mallat S,Papanicolaou G,(1990)A wavelet based space-time adaptive numerical method for partial differential equations.Rep.AFOSR- 90-0040,New York 18.Bahei-El-Din YA,(1996)Finite element analysis of viscoplastic composite materials and structures.Int.J.Comp.Mat.Struct.3:1-28
References 1. ABAQUS, v. 5.8, (1999) User’s Manual. Hibbitt, Karlsson & Sorensen, Pawtucket. 2. ANSYS v.5.5. User’s Manual (1999), Swanson, Houston, PA. 3. Abdelal GF, Caceres A, Barbero EJ, (2002) A micro-mechanics damage approach for fatigue of composite materials. Comp. Struct. 56:413-422. 4. Aboudi J, Elastoplasticity theory for composite materials, (1986) Solid Mech. Arch. 11:141-183. 5. Achenbach JD, Zhu H, (1989) Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites. J. Mech. Phys. Sol. 37(3):381-393. 6. Adrianov IV et al, (1999) Homogenization procedure and Pade approximants for effective heat conductivity of composite materials with cylindrical inclusions having square cross-section. Proc. Roy. Soc. London A 455:3401-3413. 7. Akin JE (1982) Application and the implementation of finite element methods, Academic Press, New York 8. Algam M, Bennett RM, Zureick AH, (2002) Three-parameter vs. twoparameter Weibull distribution for pultruded composite material properties. Comp. Struct. 58:497-503. 9. Alzebdeh K et al., (1998) Fracture of random matrix-inclusion composites: scale effects and statistics. Int. J. Sol. Struct. 35(19):2537-2566. 10. Amaniampong G, Burgoyne CJ, (1996) Monte-Carlo simulations of the time dependent failure of bundles of parallel fibres. Eur. J. Mech. A/Solids 15(2):243-266. 11. Anthoine A, (1995) Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int. J. Sol. Struct. 32(2):137-163. 12. Aravas N, (1987) On the numerical integration of a class of pressuredependent plasticity models. Int. J. Num. Meth. Engrg. 24:1395-1416. 13. Argyris J, Mlejnek HP (1991) Dynamics of Structures, North Holland 14. Augusti G, Baratta A, Casciati F (1984) Probabilistic Methods in Structural Engineering, Chapman and Hall 15. Auriault JL, (1983) Effective macroscopic description for heat conduction in periodic composites. Int. J. Heat and Mass Transfer 26:861-869. 16. Babuska I, (1976) Homogenization and its application. Mathematical and computational problems. In: Numerical solution of partial differential equations III, Academic Press, pp. 89-116. 17. Bacry E, Mallat S, Papanicolaou G, (1990) A wavelet based space-time adaptive numerical method for partial differential equations. Rep. AFOSR- 90-0040, New York 18. Bahei-El-Din YA, (1996) Finite element analysis of viscoplastic composite materials and structures. Int. J. Comp. Mat. Struct. 3:1-28