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 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)

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