APPLICATION OF THE THERMODYNAMICAL LAW OF GASES FOR MODELLING OF FREE INFLATION OF THE CLOSED THIN HYPERELASTIC STRUCTURE

F. Erchiqui∗ and A. Bendada∗∗

References

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  21. [21] J.E. Denis, Schnabel (Englewood Cliffs, NJ: Prentice-Hall, 1983). Appendix A: External Forces The element external forces are given by: Fext,e = Δp ∂Vg ∂un = (p(Vg) − pa) ∂Ve g ∂un (A.1) The calculation of this vector thus requires the calculation of single element volume Ve g and the total volume enclosed by hyperelastic structure Vg. The total volume enclosed by the hyperelastic structure relatively to an unspecified regular surface (real or imaginary) has been obtained with fixed reference point, p, pertaining to the one of these surfaces. If we subdivide the total surface into triangular finite element, the total enclosed volume, Vg, of a closed structure can be determined by: Vg = Nee =1 Ve g (A.2) where Ve g is the tetraedrical volume element contribution, defined by the reference point p and the corner nodes of the element. Ne is the total number of elements. The volume Ve g is given by the formula: Ve g = 1 6 x1 · (x2 × x3) (A.3) and its variational form is given by: δVe g = 1 6 δx1 ·(x2 ×x3)+ 1 6 x1 ·(δx2 ×x3)+ 1 6 x1 ·(δx2 ×x3) (A.4) By taking account of: δxi = δui (A.5) we obtain: δVe g = 1 6 δu1 ·(x2 ×x3)+ 1 6 x1 ·(δu2 ×x3)+ 1 6 x1 ·(x2 ×δu3) (A.6) 264 which is written: δVe g = 1 6 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ δu1 δu2 δu3 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ T · ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ x2 × x3 x3 × x1 x1 × x2 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ (A.7) One deduces then: ∂Ve g ∂un = 1 6 ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ x2 × x3 x3 × x1 x1 × x2 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ (A.8) Finally, the elementary vector of the external forces, given by (A.1), becomes: Fext,e = 1 6 Δp · ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ x2 × x3 x3 × x1 x1 × x2 ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ (A.9) Appendix B: External Element Tangent Stiffness Matrix The external element tangent stiffness matrix is given by: Kext,e = ∂ ∂ue n Δ p ∂Ve g ∂ue n = dp dV ∂Ve g ∂ue n ∂Ve g ∂ue n +Δp ∂2 Ve g ∂ue n∂ue n (B.1) Let us pose the following definitions: κ = dp(Vg) dV , a = ∂Ve g ∂ue n , Λe = ∂2 Ve g ∂ue n ∂ue n (B.2) In this case: Ke,ext = κae ae,T + [p(Vg) − pa]Λe (B.3)

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