128 Antonio J.Gil Pislat+pbi=0 in with fi=Pielatnjdr (14) The formula(14)along with the boundary conditions and continuity conditions -see [17]-,represents the strong formulation of the structural problem according to a Lagrangian description with respect to a reference stressed configuration.This is the equation that will be used from now on. Many engineering applications,particularly the one which concerns us,involve moderate strains and large rotations.Therefore,in these kind of problems the effects of large deformation are primarily due to rotations.The response of the material may then be modeled as an extension of the well known linear elastic law by replacing the Cauchy stress tensor by the second Piola-Kirchhoff stress one and the small strain tensor by the Green-Lagrange strain one. This material behaviour is named Saint Venant-Kirchhoff hyperelastic or simply Kirchhoff material.By accounting for the hyperelastic pattern of this constitutive model,the second Piola-Kirchhof stress tensor may be formulated in an elegant way by means of the Helmholtz free energy-also known as internal strain energy-. Thus,the second Piola-Kirchhoff stress tensor in the current configuration may be formulated by means of a Taylor series expansion truncated after the first order as follows: 。0wimt Owint+ SAB DEAB-EAB unt-(E吃D-EcD】 (15) OEABOECD The accuracy of this Taylor series depends directly on the smallness of the step Ecp-EcD.For tension membrane structures,as it was previously mentioned,this is a valid assumption.Thus: SAB -SAB+CABCDXICXDEeLat (16) By recalling (11)and (16): Saelat-JXSAB+JB CABCDXICXDEifelat (17) The fourth order tensor of elastic moduli can be referred to the prestressed configuration as follows: CXXCABCDXECX (18) Eventually,equation(17)may be reformulated to give the final expression: Sifelat=g+Cuk Eifelat (19) This final formula is set up to show the constitutive law for a prestressed Saint Venant-Kirchhoff hyperelastic material.The second Piola-Kirchhoff stress tensor is expressed in terms of an easy linear relationship which depends on three tensorial entities:Cauchy stress tensor in the primary state,fourth order tensor of elastic moduli and the Green-Lagrange strain tensor of the secondary state referred to the primary state. Recall -see [18]-that in a Saint Venant-Kirchhoff hyperelastic material,the con- stitutive tensor can be formulated as follows: Cikl=入ddkl+2μdkdl (20)
128 Antonio J. Gil P∗relat ji,j + ρbi = 0 in t, with f ∗ i = P∗relat ji njdΓ (14) The formula (14) along with the boundary conditions and continuity conditions -see [17]-, represents the strong formulation of the structural problem according to a Lagrangian description with respect to a reference stressed configuration. This is the equation that will be used from now on. Many engineering applications, particularly the one which concerns us, involve moderate strains and large rotations. Therefore, in these kind of problems the effects of large deformation are primarily due to rotations. The response of the material may then be modeled as an extension of the well known linear elastic law by replacing the Cauchy stress tensor by the second Piola-Kirchhoff stress one and the small strain tensor by the Green-Lagrange strain one. This material behaviour is named Saint Venant-Kirchhoff hyperelastic or simply Kirchhoff material. By accounting for the hyperelastic pattern of this constitutive model, the second Piola-Kirchhof stress tensor may be formulated in an elegant way by means of the Helmholtz free energy -also known as internal strain energy-. Thus, the second Piola-Kirchhoff stress tensor in the current configuration may be formulated by means of a Taylor series expansion truncated after the first order as follows: S∗ AB = ∂wint ∂E∗ AB = ∂wint ∂EAB + ∂2wint ∂EAB ∂ECD (E∗ CD − ECD) (15) The accuracy of this Taylor series depends directly on the smallness of the step E∗ CD −ECD. For tension membrane structures, as it was previously mentioned, this is a valid assumption. Thus: S∗ AB = SAB + CABCDXpret i,C Xpret j,D E∗relat ij (16) By recalling (11) and (16): S∗relat st = J−1Xpret s,A Xpret t,B SAB + J−1 Xpret s,A Xpret t,B CABCDXpret i,C Xpret j,D E∗relat ij (17) The fourth order tensor of elastic moduli can be referred to the prestressed configuration as follows: Cijkl = J−1 Xpret i,A Xpret j,B CABCDXpret k,C Xpret l,D (18) Eventually, equation (17) may be reformulated to give the final expression: S∗relat ij = σij + CijklE∗relat kl (19) This final formula is set up to show the constitutive law for a prestressed Saint Venant-Kirchhoff hyperelastic material. The second Piola-Kirchhoff stress tensor is expressed in terms of an easy linear relationship which depends on three tensorial entities: Cauchy stress tensor in the primary state, fourth order tensor of elastic moduli and the Green-Lagrange strain tensor of the secondary state referred to the primary state. Recall -see [18]- that in a Saint Venant-Kirchhoff hyperelastic material, the constitutive tensor can be formulated as follows: Cijkl = λδij δkl + 2µδikδjl (20)
F.E.M.for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 129 where A and u are known as the Lame constants.These two constants can be related to the classical Young modulus E and Poisson's ratio as follows: vE E 入=0+0-2“=201+可 (21) Another important feature which needs to be obtained is the incremental strain energy accumulated into the structure along the deformation path from the primary to the secondary states.By performing again a Taylor series expansion truncated after the second order,the internal strain energy functional per unit of nominally stressed volume may be developed as: (Eia-Ba)+5 EAROEC--Eu)--BcD）网 wint=wimt十8EAB This above formula can be rewritten as: Wint Wint++r (23) where the terms and T can be depicted as: =SABXX Eiyelat Jg Eirelat (24) T-0 ADGDX9E对X8-号C,uE写E 1 (25) By substituting(24)and (25)back into (22): a-0a=pgt+号0C写aE的= (26) By integrating over the initial undeformed volume Vo corresponding to the con- figuration Ro,and by applying the mass conservation principle from this volume Vo to the prestressed volume Vpret corresponding to the configuration we obtain the incremental Helmholtz's free energy functional,which is given as: △Wmt= (wint-wint)dV J亚dW= 亚dW= wnladV (27 Vpret Therefore,the internal strain energy functional per unit of volume of the primary state takes the final form: watatCkEiat eat (28) 3 Finite Element Discretization 3.1 From the Strong to the Weak Formulation The above mentioned primary and secondary states can be understood as an initial prestressed state Rpret and a final in service loading state R due to the consider- ation of live and dead load.Henceforth,the coordinates of any body's particle,in
F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 129 where λ and µ are known as the Lam´e constants. These two constants can be related to the classical Young modulus E and Poisson’s ratio ν as follows: λ = νE (1 + ν)(1 − 2ν) µ = E 2(1 + ν) (21) Another important feature which needs to be obtained is the incremental strain energy accumulated into the structure along the deformation path from the primary to the secondary states. By performing again a Taylor series expansion truncated after the second order, the internal strain energy functional per unit of nominally stressed volume may be developed as: w∗ int = wint+ ∂wint ∂EAB (E∗ AB −EAB )+ 1 2 ∂2wint ∂EAB ∂ECD (E∗ AB −EAB )(E∗ CD −ECD) (22) This above formula can be rewritten as: w∗ int = wint + Ω + Υ (23) where the terms Ω and Υ can be depicted as: Ω = SABXpret i,A Xpret j,B E∗relat ij = JσijE∗relat ij (24) Υ = 1 2 CABCDXpret i,A Xpret j,B E∗relat ij Xpret k,C Xpret l,D E∗relat kl = 1 2 JCijklE∗relat ij E∗relat kl (25) By substituting (24) and (25) back into (22): w∗ int − wint = J[σijE∗relat ij + 1 2 CijklE∗relat ij E∗relat kl ] = JΨ (26) By integrating over the initial undeformed volume V 0 corresponding to the con- figuration 0, and by applying the mass conservation principle from this volume V 0 to the prestressed volume V pret corresponding to the configuration t, we obtain the incremental Helmholtz’s free energy functional, which is given as: ∆Wint = V 0 (w∗ int−wint)dV = V 0 JΨdV = V pret ΨdV = V pret w∗relat int dV (27) Therefore, the internal strain energy functional per unit of volume of the primary state takes the final form: w∗relat int = σijE∗relat ij + 1 2 CijklE∗relat ij E∗relat kl (28) 3 Finite Element Discretization 3.1 From the Strong to the Weak Formulation The above mentioned primary and secondary states can be understood as an initial prestressed state pret and a final in service loading state due to the consideration of live and dead load. Henceforth, the coordinates of any body’s particle, in