F. Erchiqui∗ and A. Bendada∗∗


  1. [1] G. Xiaoping, Kinematic modelling of finite axisymmetric in-flation for an arbitrary of polymeric of membrane inflation,Polymer Plastics Technology and Engineering, 40(3), 2001,341–361. doi:10.1081/PPT-100000253
  2. [2] H.G. DeLorenzi & H.F. Nied, Finite element simulation of ther-moforming and blow molding. Progress in polymer processing(Hanser Verlag, 1991), 117–171.
  3. [3] D. Laroche & F. Erchiqui, Experimental and theoretical studyof the thermoformability of industrial polymers, Journal ofReinforced Plastics and Composites, 19(3), 2000, 231–239. doi:10.1106/6MKH-YFJR-MNEE-796G
  4. [4] D. Laroche & F. Erchiqui, 3D modelling of the blow mouldingprocess, in Hu´etink & Baaijens (Eds.), Simulation of materialsprocessing: theory, methods and applications (Rotterdam:Balkema, 1998), 483–488 (ISBN 90 5410 970 X).
  5. [5] F. Erchiqui & A. Gakwaya, Mod´elisation du comportementvisco´elastique d’une membrane thermoplastique par la m´ethodedes ´el´ements finis, Revue Europ´eenne des ´El´ements Finis,12(1), 2003, 43–58. doi:10.3166/reef.12.43-58
  6. [6] A. Derdouri, F. Erchiqui, A. Bendada, & E. Verron, Vis-coelastic behaviour of polymer membranes under inflation,XIII International Congress on Rheology, Cambridge, 2000,394–396.
  7. [7] E. Verron, R.E. Khayat, A. Derdouri, & B. Peseux, Dynamicinflation of hyperelastic spherical membranes, Journal of Rhe-ology, 43(5), 1999, 1083–1097. doi:10.1122/1.551017
  8. [8] N. Chevaugeon et al. Instabilit´e et bifurcation du soufflage demembranes hyper´elastiques, Revue Europ´eenne des ´El´ementsFinis, 11, 2002, 479–492.
  9. [9] E. Verron, G. Marckmann, & B. Peseux, Dynamic inflationof nonlinear elastic and viscoelastic rubberlike membranes,International Journal for Numerical Methods in Engineering,50, 2001, 1233–1251. doi:10.1002/1097-0207(20010220)50:5<1233::AID-NME77>3.0.CO;2-W
  10. [10] R.E. Khayat, A. Derdouri, & A. Garcia-R´ejon, Inflation ofhyperelastic cylindrical membranes as applied to blow molding,International Journal of Solids and Structures, 29(1), 1992,69–87. doi:10.1016/0020-7683(92)90096-C
  11. [11] M.A. Dokainish & K. Subbaraj, A survey of direct time-integration methods in computational structural dynamics.Computational Structure, 32(6), 1989, 1371–1386. doi:10.1016/0045-7949(89)90314-3
  12. [12] N. Chevaugeon, Contribution `a l’´etude des membranes hy-per´elastiques en grandes d´eformations, PhD Thesis, Ecole Cen-trale of Nantes and of University of Nantes, 14th January2002.
  13. [13] E. Verron, Experimental and numerical contribution to blow-molding and thermoforming processes, PhD Thesis, EcoleCentrale de Nantes, France, 1997.
  14. [14] F. Erchiqui & A. Gakwaya, Analysis of gaz pressure effectduring the thermoplastic membrane forming using the dynamicfinite element method, PPS-19, Annual Polymer ProcessingSociety, Session 7: Blow Moulding and Thermoforming, Mel-bourne, Australia, 7–10 July, 2003.
  15. [15] O.C. Zienkiewicz & R.L. Taylor, The finite element method,Fourth Edition, Vols. 1 and 2 (McGraw-Hill, 1991).
  16. [16] R.S. Rivlin, Large elastic deformation of isotropic materials-IV. Further developments of the general theory, PhilosophicalTransaction of the Royal Society, A241, 1948, 379–397. doi:10.1098/rsta.1948.0024
  17. [17] D.D. Joye, G.W. Poehlein, & C.D. Denson, A bubble inflationtechnique for the measurement of viscoelastic properties inequal biaxial extension flow. Transactions of the Society ofRheology, 16, 1972, 421–445. doi:10.1122/1.549259
  18. [18] F. Erchiqui, A. Derdouri, A. Gakwaya, & E. Verron, Analyseexp´erimentale et num´erique en soufflage libre d’une membranethermoplastique, Entropie, 235 /236, 2001, 118–125.
  19. [19] K. Levenberg, A method for the solution of certain non-linearproblems in least squares, Quarterly of Applied Mathematics,2(2), 1944, 164–168.
  20. [20] D. Marquardt, An algorithm for least squares estimation ofnonlinear parameters, SIAM Journal of Applied Mathematics,11(2), 1963, 431–441. doi:10.1137/0111030
  21. [21] J.E. Denis, Schnabel (Englewood Cliffs, NJ: Prentice-Hall,1983).Appendix A: External ForcesThe element external forces are given by:Fext,e= Δp∂Vg∂un= (p(Vg) − pa)∂Veg∂un(A.1)The calculation of this vector thus requires the calcu-lation of single element volume Veg and the total volumeenclosed by hyperelastic structure Vg.The total volume enclosed by the hyperelastic struc-ture relatively to an unspecified regular surface (real orimaginary) has been obtained with fixed reference point,p, pertaining to the one of these surfaces. If we subdividethe total surface into triangular finite element, the total en-closed volume, Vg, of a closed structure can be determinedby:Vg =Nee=1Veg (A.2)where Veg is the tetraedrical volume element contribution,defined by the reference point p and the corner nodes ofthe element. Ne is the total number of elements. Thevolume Veg is given by the formula:Veg =16x1 · (x2 × x3) (A.3)and its variational form is given by:δVeg =16δx1 ·(x2 ×x3)+16x1 ·(δx2 ×x3)+16x1 ·(δx2 ×x3)(A.4)By taking account of:δxi = δui (A.5)we obtain:δVeg =16δu1 ·(x2 ×x3)+16x1 ·(δu2 ×x3)+16x1 ·(x2 ×δu3)(A.6)264which is written:δVeg =16⎧⎪⎪⎪⎨⎪⎪⎪⎩δu1δu2δu3⎫⎪⎪⎪⎬⎪⎪⎪⎭T·⎧⎪⎪⎪⎨⎪⎪⎪⎩x2 × x3x3 × x1x1 × x2⎫⎪⎪⎪⎬⎪⎪⎪⎭(A.7)One deduces then:∂Veg∂un=16⎧⎪⎪⎪⎨⎪⎪⎪⎩x2 × x3x3 × x1x1 × x2⎫⎪⎪⎪⎬⎪⎪⎪⎭(A.8)Finally, the elementary vector of the external forces,given by (A.1), becomes:Fext,e=16Δp ·⎧⎪⎪⎪⎨⎪⎪⎪⎩x2 × x3x3 × x1x1 × x2⎫⎪⎪⎪⎬⎪⎪⎪⎭(A.9)Appendix B: External Element Tangent StiffnessMatrixThe external element tangent stiffness matrix is given by:Kext,e=∂∂uenΔp∂Veg∂uen=dpdV∂Veg∂uen∂Veg∂uen+Δp∂2Veg∂uen∂uen(B.1)Let us pose the following definitions:κ =dp(Vg)dV, a =∂Veg∂uen, Λe=∂2Veg∂uen ∂uen(B.2)In this case:Ke,ext= κaeae,T+ [p(Vg) − pa]Λe(B.3)

Important Links:

Go Back