F.E.M.for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes Antonio J.Gil Civil and Computational Engineering Centre,School of Engineering,University of Swansea,Singleton Park,SA2 8PP,United Kingdom a.j.gil@swansea.ac.uk Summary.This chapter presents a complete numerical formulation for the nonlin- ear structural analysis of prestressed membranes with applications in Civil Engineer- ing.These sort of membranes can be considered to undergo large deformations but moderate strains,consequently nonlinear continuum mechanics principles for large deformation of prestressed bodies will be employed in order to proceed with the anal- ysis.The constitutive law adopted for the material will be the one corresponding to a prestressed hyperelastic Saint Venant-Kirchhoff model.To carry out the computa- tional resolution of the structural problem,the Finite Element Method(FEM)will be implemented according to a Total Lagrangian Formulation (TLF),by means of the Direct Core Congruential Formulation (DCCF).Eventually,some numerical exam- ples will be introduced to verify the accuracy and robustness of the aforementioned formulation. Key words:Tension membrane structures,Total Lagrangian Formulation,Di- rect Core Congruential Formulation,Hyperelastic Saint Venant-Kirchhoff material, Newton-Raphson method 1 Introduction Tension structures constitute a structural form providing remarkable opportunities in the fields of architecture and civil engineering.Nowadays,numerous practical examples can be located throughout the entire world because of the acceptance by designers and their upward trend in popularity -see [1],[2],[3],[4]-.The increasing necessity of creating large enclosed areas,unobstructed by intermediate supports, has allowed the introduction of limitless possibilities for doubly curved surface forms -see [5]-.Although there are several categories that fall into the general term of tension structures,this paper will focus on the well known prestressed membranes, from which prestressed cables can be analyzed by extension.As an instance of these models,Fig.1 reflects several views of a prestressed cable reinforced membrane. We will focus on those particular membranes where strains can be assumed moderate,despite having large deformations.Two different and successive loading cases may be distinguished according to their effects on the stabilization of the 123 E.Onate and B.Kroplin (eds.).Textile Composites and Inflatable Structures,123-142. 2005 Springer.Printed in the Netherlands
F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes Antonio J. Gil Civil and Computational Engineering Centre, School of Engineering, University of Swansea, Singleton Park, SA2 8PP, United Kingdom a.j.gil@swansea.ac.uk Summary. This chapter presents a complete numerical formulation for the nonlinear structural analysis of prestressed membranes with applications in Civil Engineering. These sort of membranes can be considered to undergo large deformations but moderate strains, consequently nonlinear continuum mechanics principles for large deformation of prestressed bodies will be employed in order to proceed with the analysis. The constitutive law adopted for the material will be the one corresponding to a prestressed hyperelastic Saint Venant-Kirchhoff model. To carry out the computational resolution of the structural problem, the Finite Element Method (FEM) will be implemented according to a Total Lagrangian Formulation (TLF), by means of the Direct Core Congruential Formulation (DCCF). Eventually, some numerical examples will be introduced to verify the accuracy and robustness of the aforementioned formulation. Key words: Tension membrane structures, Total Lagrangian Formulation, Direct Core Congruential Formulation, Hyperelastic Saint Venant-Kirchhoff material, Newton-Raphson method 1 Introduction Tension structures constitute a structural form providing remarkable opportunities in the fields of architecture and civil engineering. Nowadays, numerous practical examples can be located throughout the entire world because of the acceptance by designers and their upward trend in popularity -see [1], [2], [3], [4]-. The increasing necessity of creating large enclosed areas, unobstructed by intermediate supports, has allowed the introduction of limitless possibilities for doubly curved surface forms -see [5]-. Although there are several categories that fall into the general term of tension structures, this paper will focus on the well known prestressed membranes, from which prestressed cables can be analyzed by extension. As an instance of these models, Fig. 1 reflects several views of a prestressed cable reinforced membrane. We will focus on those particular membranes where strains can be assumed moderate, despite having large deformations. Two different and successive loading cases may be distinguished according to their effects on the stabilization of the 123 E. Oñate and B. Kröplin (eds.), Textile Composites and Inflatable Structures, 123–142. © 2005 Springer. Printed in the Netherlands
124 Antonio J.Gil prestressed membrane.The first one or prestressed loading is developed to provide the necessary in-surface rigidity to the membrane in order to support the second loading step.The latter,also named in service loading step is comprised of a wide group of loads:snow,wind or live loads among others. Perspective view Plan view 10 0 0 -5 10 0 5 0 -10 OY axis (m) -10 -5 OX axis (m) -5 0 OX axis (m) Lateral view Lateral view (w)spxe ZO (w)sixe 3 3 2 0 -10 -5 0 5 10 0 -5 OY axis (m) OX axis (m) Fig.1.Prestressed cable reinforced membrane The theory of hyperelastic membranes,as for instance,propounded by 6,[7] and [8 treats the problem from an analytical point of view.Some simplicity may be accomplished if the Von Karman compatibility equations are used-see [9]and [10]-,whereby rotations are considered to be moderate.Regardless of the important implications of this approach for the theoretical understanding of these structures, a main disadvantage is that it results in a nonlinear partial differential system of equations with impossible analytical resolution. Because of this lack of numerical results,variational approaches ought to be taken into consideration as the best means to provide feasible solutions.Some au- thors have treated the problem of finite hyperelasticity set on rubberlike membrane materials by means of the Finite Element Method (FEM).By following this ap- proach,interesting papers are those due to [11],[12],[13],[14],[15]and [16].For these cases,the Updated Lagrangian Formulation (ULF)is considered to be the most suitable for the derivation of the tangent stiffness matrix.This matrix is re-
124 Antonio J. Gil prestressed membrane. The first one or prestressed loading is developed to provide the necessary in-surface rigidity to the membrane in order to support the second loading step. The latter, also named in service loading step is comprised of a wide group of loads: snow, wind or live loads among others. −5 0 5 −10 0 10 0 2 4 OX axis (m) Perspective view OY axis (m) OZ axis (m) −5 0 5 −10 −5 0 5 10 Plan view OX axis (m) OY axis (m) −10 −5 0 5 10 0 1 2 3 4 5 Lateral view OY axis (m) OZ axis (m) 5 0 −5 0 1 2 3 4 5 Lateral view OX axis (m) OZ axis (m) Fig. 1. Prestressed cable reinforced membrane The theory of hyperelastic membranes, as for instance, propounded by [6], [7] and [8] treats the problem from an analytical point of view. Some simplicity may be accomplished if the Von Karman compatibility equations are used -see [9] and [10]-, whereby rotations are considered to be moderate. Regardless of the important implications of this approach for the theoretical understanding of these structures, a main disadvantage is that it results in a nonlinear partial differential system of equations with impossible analytical resolution. Because of this lack of numerical results, variational approaches ought to be taken into consideration as the best means to provide feasible solutions. Some authors have treated the problem of finite hyperelasticity set on rubberlike membrane materials by means of the Finite Element Method (FEM). By following this approach, interesting papers are those due to [11], [12], [13], [14], [15] and [16]. For these cases, the Updated Lagrangian Formulation (ULF) is considered to be the most suitable for the derivation of the tangent stiffness matrix. This matrix is re-
F.E.M.for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 125 quired by the Newton-Raphson iterative scheme for the solution of the nonlinear equilibrium equations. The discussion to follow is divided into six parts.The second section reviews the classical nonlinear strong form equations.The consideration of the Saint Venant- Kirchhoff material as the adopted model will very conveniently provide a linear constitutive relationship of easy implementation.The third section entails a com- prehensive explanation of the Finite Element semidiscretization of the previously obtained strong form.After the weak form is derived in a straightforward man- ner,the displacement field is interpolated by means of shape functions based on a Lagrangian mesh geometry.The resulting formulation will be the so called To- tal Lagrangian Formulation (TLF).Afterwards,the exact linearization of the Total Lagrangian weak form of the momentum balance is carried out in detail.For the sake of further computing implementation reasons,the Direct Core Congruential Formulation (DCCF)is reviewed as the most appropriate formulation. Eventually,based on the aforementioned formulation,two numerical examples for both a cable network and for a prestressed membrane,are presented.These cases will show adequate performance as the required quadratically convergence of the Newton-Raphson method is obtained.Some conclusions are presented at the end. 2 Strong Formulation:General Structural Principles Before establishing the formulation in terms of particular finite elements,that is,ca- ble or membrane elements,we will develop in this section the general equations that govern the behaviour of prestressed membrane structures.For a complete under- standing,it is necessary to consider three successive configurations of the material body:an initial nominally stressed state Ro,a primary state and a secondary stateR,for the time instants t and t",respectively.It is important to point out that the term nominally stressed state is employed to describe a self-equilibrated configuration where the internal stresses are as small as required by the designer. Usually,o represents the nominally stressed state found at a form finding state. symbolizes the actual in service prestressed state prior to the live loading,which may be different to Ro and due to constructions prestresses.Finally,stands for the live loading in service state. Between these latter two stages,a displacement field u=(u1,u2,u3)may be defined in R3.To differentiate the coordinates of a body particle along the defor- mation path,the following convention will be employed throughout the remainder of this paper: XA,(A =1,2,3)for the initial nominally stressed configuration Ro. .,(j=1,2,3)for the initial prestressed configuration or primary state. 7 .i,(i=1,2,3)for the current spatial configuration or secondary state. Henceforth,we will consider as incremental those quantities which proceed from the movement from the primary to the secondary state.The spatial coordinates for the time t*of a particle can be related to its material coordinates in the initial nom- inally stressed configuration Ro according to the classical mapping equation x: ri(XA,t).By recalling the chain rule,relations among deformation gradient tensors
F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 125 quired by the Newton-Raphson iterative scheme for the solution of the nonlinear equilibrium equations. The discussion to follow is divided into six parts. The second section reviews the classical nonlinear strong form equations. The consideration of the Saint VenantKirchhoff material as the adopted model will very conveniently provide a linear constitutive relationship of easy implementation. The third section entails a comprehensive explanation of the Finite Element semidiscretization of the previously obtained strong form. After the weak form is derived in a straightforward manner, the displacement field is interpolated by means of shape functions based on a Lagrangian mesh geometry. The resulting formulation will be the so called Total Lagrangian Formulation (TLF). Afterwards, the exact linearization of the Total Lagrangian weak form of the momentum balance is carried out in detail. For the sake of further computing implementation reasons, the Direct Core Congruential Formulation (DCCF) is reviewed as the most appropriate formulation. Eventually, based on the aforementioned formulation, two numerical examples for both a cable network and for a prestressed membrane, are presented. These cases will show adequate performance as the required quadratically convergence of the Newton-Raphson method is obtained. Some conclusions are presented at the end. 2 Strong Formulation: General Structural Principles Before establishing the formulation in terms of particular finite elements, that is, cable or membrane elements, we will develop in this section the general equations that govern the behaviour of prestressed membrane structures. For a complete understanding, it is necessary to consider three successive configurations of the material body: an initial nominally stressed state 0, a primary state t and a secondary state t∗ , for the time instants t and t ∗, respectively. It is important to point out that the term nominally stressed state is employed to describe a self-equilibrated configuration where the internal stresses are as small as required by the designer. Usually, 0 represents the nominally stressed state found at a form finding state. t symbolizes the actual in service prestressed state prior to the live loading, which may be different to 0 and due to constructions prestresses. Finally, t∗ stands for the live loading in service state. Between these latter two stages, a displacement field u = (u1, u2, u3) may be defined in R3. To differentiate the coordinates of a body particle along the deformation path, the following convention will be employed throughout the remainder of this paper: • XA, (A = 1, 2, 3) for the initial nominally stressed configuration 0. • Xpret j , (j = 1, 2, 3) for the initial prestressed configuration t or primary state. • xi, (i = 1, 2, 3) for the current spatial configuration t∗ or secondary state. Henceforth, we will consider as incremental those quantities which proceed from the movement from the primary to the secondary state. The spatial coordinates for the time t ∗ of a particle can be related to its material coordinates in the initial nominally stressed configuration 0 according to the classical mapping equation xi = xi(XA,t ∗). By recalling the chain rule, relations among deformation gradient tensors
126 Antonio J.Gil Re 孔o Re e X e Fig.2.Deformation path and their respective jacobians are summarized as: 工，A=X→J产=J'J (1) where the implied summation convention for repeated indices as well as the comma differentiation symbol=r/have been introduced to simplify the alge- bra.The termJ above represents the jacobian at the primary state,Jstands for the jacobian at the end of the secondary state and J symbolizes the jacobian as a consequence of the incremental deformation. The description of the deformation and the measure of strain are essential parts of nonlinear continuum mechanics.From a kinematically point of view,a material particle position in the primary and secondary states may be expressed in terms of the incremental displacement field.Analogously,the deformation gradient tensor may be introduced to characterize adequately the deformation path as: 五=Xgr+→x,A=X+山，A=X+4X (2) where: Oui Ui.= oxprei (3) In contrast to linear elasticity,many different measures of strain may be used in nonlinear continuum mechanics.Nevertheless,the Green-Lagrange strain tensor is considered to be the most appropriate measure specially when dealing with moderate strains.The Green-Lagrange strain tensor for the secondary state with respect to the initial nominally stressed configuration is defined in terms of the deformation gradient tensor and the Delta-Kronecker tensor as: 1 EAB=(Ei.AZi.B -6AB) (4) By substituting equation (2)into equation (4): EAB-6AB)+A) (5)
126 Antonio J. Gil e1 e2 e3 X x u pret X R0 Rt Rt* Deformation path and their respective jacobians are summarized as: xi,A = xi,jXpret j,A ⇒ J∗ = J J (1) where the implied summation convention for repeated indices as well as the comma differentiation symbol xi,j = ∂xi/∂Xpret j have been introduced to simplify the algebra. The term J above represents the jacobian at the primary state, J ∗ stands for the jacobian at the end of the secondary state and J symbolizes the jacobian as a consequence of the incremental deformation. The description of the deformation and the measure of strain are essential parts of nonlinear continuum mechanics. From a kinematically point of view, a material particle position in the primary and secondary states may be expressed in terms of the incremental displacement field. Analogously, the deformation gradient tensor may be introduced to characterize adequately the deformation path as: xi = Xpret i + ui ⇒ xi,A = Xpret i,A + ui,A = Xpret i,A + ui,jXpret j,A (2) where: ui,j = ∂ui ∂Xpret j (3) In contrast to linear elasticity, many different measures of strain may be used in nonlinear continuum mechanics. Nevertheless, the Green-Lagrange strain tensor is considered to be the most appropriate measure specially when dealing with moderate strains. The Green-Lagrange strain tensor for the secondary state with respect to the initial nominally stressed configuration is defined in terms of the deformation gradient tensor and the Delta-Kronecker tensor as: E∗ AB = 1 2 (xi,Axi,B − δAB) (4) By substituting equation (2) into equation (4): E∗ AB = 1 2 (Xpret i,A Xpret i,B − δAB) + 1 2 (2Xpret i,A Xpret j,B eij + ui,Aui,B) (5)
F.E.M.for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 127 with: 1 e=2(ui+西，) (6) The difference between the Green-Lagrange strain tensor for the primary and secondary states can be carried out in a straightforward manner as: EAB-EAB=XXEe+2山，Au,B (7) By applying the chain rule: EAn-EABeg+)a (8) where the tensor Efelat has been introduced for the sake of convenience and it represents a relative measure of the strain at the secondary state by taking the primary one as an adequate reference. In nonlinear problems,various stress measures can be defined.In this paper, in addition to the Cauchy or real stress tensor,two tensorial entities referred to as the second Piola-Kirchhoff and the nominal stress tensors are to be used.The latter is known as well as the transpose of the first Piola-Kirchhoff stress tensor.By considering as initial configuration the initial nominally stressed one,the Cauchy stress tensor oi;may be related to the nominal stress tensor PA,and the second Piola-Kirchhoff stress tensor SAB as: J产南=x4,APA=x4,A,BSAB,A=XA 0r: (9) The same relationship may be developed when the initial prestressed configura- tion is adopted to be the reference state.The super index relat is added to distinguish the new nonlinear stress tensors with respect to those shown in the above formula: ig=iPirelat =Selat Ti.s= 0x1 oxpret (10) Formulae (9)and(10)can be modified to set up expressions(11)which summa- rize the relationship among the nominal and second Piola-Kirchhoff stress tensors obtained in both the initial undeformed configuration Ro and the primary state Re. Pafelat-JXXSAB (11) Sarelat =J-XA XPE SAB The local equilibrium equations in the secondary state may be expressed with respect to three possible descriptions:Ro,R and,these being a Lagrangian formulation for the first two configurations and an Eulerian formulation for the final configuration.These expressions may be gathered as follows: oji.j+p'bi=0 in,with f=ajndr" (12) PAi.A pobi =0 in o,with fi=PAinAdTo (13)
F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 127 with: eij = 1 2 (ui,j + uj,i) (6) The difference between the Green-Lagrange strain tensor for the primary and secondary states can be carried out in a straightforward manner as: E∗ AB − EAB = Xpret i,A Xpret j,B eij + 1 2 ui,Aui,B (7) By applying the chain rule: E∗ AB − EAB = Xpret i,A Xpret j,B (eij + 1 2 us,ius,j ) = Xpret i,A Xpret j,B E∗relat ij (8) where the tensor E∗relat ij has been introduced for the sake of convenience and it represents a relative measure of the strain at the secondary state by taking the primary one as an adequate reference. In nonlinear problems, various stress measures can be defined. In this paper, in addition to the Cauchy or real stress tensor, two tensorial entities referred to as the second Piola-Kirchhoff and the nominal stress tensors are to be used. The latter is known as well as the transpose of the first Piola-Kirchhoff stress tensor. By considering as initial configuration the initial nominally stressed one, the Cauchy stress tensor σ∗ ij may be related to the nominal stress tensor P∗ Aj and the second Piola-Kirchhoff stress tensor S∗ AB as: J∗σ∗ ij = xi,AP∗ Aj = xi,Axj,BS∗ AB xi,A = ∂xi ∂XA (9) The same relationship may be developed when the initial prestressed configuration is adopted to be the reference state. The super index relat is added to distinguish the new nonlinear stress tensors with respect to those shown in the above formula: J σ∗ ij = xi,sP∗relat sj = xi,sxj,tS∗relat st xi,s = ∂xi ∂Xpret s (10) Formulae (9) and (10) can be modified to set up expressions (11) which summarize the relationship among the nominal and second Piola-Kirchhoff stress tensors obtained in both the initial undeformed configuration 0 and the primary state t. P∗relat sj = J−1Xpret s,A Xpret t,B xj,tS∗ AB S∗relat st = J−1Xpret s,A Xpret t,B S∗ AB (11) The local equilibrium equations in the secondary state may be expressed with respect to three possible descriptions: 0, t and t∗ , these being a Lagrangian formulation for the first two configurations and an Eulerian formulation for the final configuration. These expressions may be gathered as follows: σ∗ ji,j + ρ∗ bi = 0 in t∗ , with f ∗ i = σ∗ jin∗ j dΓ ∗ (12) P∗ Ai,A + ρ0bi =0 in 0, with f ∗ i = P∗ AinAdΓ0 (13)
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
130 Antonio J.Gil both prestressed and final loaded states,are related by means of the incremental displacement field u as follows: x Xpret +u,ti=Xpret ui (29) According to this nomenclature,the strong formulation of the problem in a Lagrangian description with respect to the prestressed configuration is summarized in Fig.3.The configurations Rpret represents a material body of domain Vpret with frontier re.As can be observed,the super indexrelat has been suppressed for the sake of simplicity. 1.Balance of linear momentum Xpre=0 in vpret OPj (30) 2.Transformation of stress tensors. Oxi Oxprer P.y= Oxi J0对= 8xi 0xprei axprer Sat with J=det( (31) 3.Green-Lagrange strain tensor. 1 OTkOtk Bxx5u) (32) 4.Constitutive law. Sy=e+CykEk (33) 5.Internal strain energy functional per unit volume of prestressed config- uration. t=aE+号CkHE,E1 (34) 2 6.Boundary conditions. ta=Pn喝ret=on rpret u=is on Iapet (35) Fig.3.Strong formulation for a Lagrangian description. Thus,the weak form may be developed in a Total Lagrangian Format (TLF)by means of the so called Principle of Virtual Work.By neglecting inertia forces: 6Wint(oui,ui)=SWert(oui,ui) (36) 6E Sdv Vpret Vpret Vpret VPret (37) 6uT.idr (38) Vpret Tpret
130 Antonio J. Gil both prestressed and final loaded states, are related by means of the incremental displacement field u as follows: x = Xpret + u, xi = Xpret i + ui (29) According to this nomenclature, the strong formulation of the problem in a Lagrangian description with respect to the prestressed configuration is summarized in Fig. 3. The configurations pret represents a material body of domain V pret with frontier Γ pret . As can be observed, the super index ∗relat has been suppressed for the sake of simplicity. 1. Balance of linear momentum. ∂Pji ∂Xpret j + ρpret bi = 0 in V pret (30) 2. Transformation of stress tensors. Jσij = ∂xi ∂Xpret s Psj = ∂xi ∂Xpret s ∂xj ∂Xpret t Sst with J = det( ∂xi ∂Xpret j ) (31) 3. Green-Lagrange strain tensor. Eij = 1 2 ( ∂xk ∂Xpret i ∂xk ∂Xpret j − δij ) (32) 4. Constitutive law. Sij = σpret ij + CijklEkl (33) 5. Internal strain energy functional per unit volume of prestressed configuration. wint = σpret ij Eij + 1 2 CijklEijEkl (34) 6. Boundary conditions. ti = Pjinpret j = ¯ti on Γ pret ti ui = ¯ui on Γ pret ui (35) Fig. 3. Strong formulation for a Lagrangian description. Thus, the weak form may be developed in a Total Lagrangian Format (TLF) by means of the so called Principle of Virtual Work. By neglecting inertia forces: δWint(δui, ui) = δWext(δui, ui) (36) δWint = V pret δFijPjidV = V pret δFT : PdV = V pret δEijSijdV = V pret δE : SdV (37) δWext = V pret δuibidV + Γpret δuit ¯idΓ = V pret δuT · bdV + Γpret δuT · ¯tdΓ (38)
F.E.M.for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 131 where the work conjugacy property of the tensors S and P with E and FT respec- tively,has been employed for the equalities in(37);b is the body force vector and t are the surface tractions. 3.2 Semidiscretization of the Weak Form The weak form equations obtained formerly may be combined with a finite element discretization of the displacement field in terms of the nodal values u'and shape functions N!as: i=u:'Ni=1,2,3;I=1..N (39) enables the nodal equivalent internal and external vector forces,fint and fert,re- spectively,to be obtained in a straightforward manner for a given node I as: f品nta= ONI ON! P Oxprerdv (40) Vpret Vpret firt.= biN'dV+ EN'dr (41) Vpret Tpret Assembling these forces for all the nodes of the Lagrangian mesh gives the global equilibrium equations: fint =fert fres fint -fert =0 (42) where fint is the global vector of internal forces,frt is the global vector of external forces and fres is the global vector of residual forces.This last vector represents clearly the out of balance forces as a result of the strong nonlinearity contained into the structural problem. 3.3 Linearization of the Global Equilibrium Equations The set of equations depicted by (42)presents a geometrically nonlinear feature,so an iterative solution scheme will be required.Among all the available methods,the second-order Newton-Raphson one accomplishes the best convergence properties. The total tangent stiffness matrix required by the later one is formed by linearizing the global equilibrium equations (42)in the direction of the incremental displace- ment u. By carrying out the linearization of the global vector of internal forces,it turns out to be: dfint=dfimgdf(K)du (43) ONI ON (44) Vpret =δ切 aN! ONJ (45) 8XL Vpret
F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes 131 where the work conjugacy property of the tensors S and P with E and FT respectively, has been employed for the equalities in (37); b is the body force vector and t are the surface tractions. 3.2 Semidiscretization of the Weak Form The weak form equations obtained formerly may be combined with a finite element discretization of the displacement field in terms of the nodal values uI and shape functions NI as: ui = u I i NI i = 1, 2, 3; I = 1 ...N (39) enables the nodal equivalent internal and external vector forces, fint and fext, respectively, to be obtained in a straightforward manner for a given node I as: f I int,i = V pret Pji ∂NI ∂Xpret j dV = V pret FikSkj ∂NI ∂Xpret j dV (40) f I ext,i = V pret biNI dV + Γpret t ¯iNI dΓ (41) Assembling these forces for all the nodes of the Lagrangian mesh gives the global equilibrium equations: fint = fext =⇒ fres = fint − fext = 0 (42) where fint is the global vector of internal forces, fext is the global vector of external forces and fres is the global vector of residual forces. This last vector represents clearly the out of balance forces as a result of the strong nonlinearity contained into the structural problem. 3.3 Linearization of the Global Equilibrium Equations The set of equations depicted by (42) presents a geometrically nonlinear feature, so an iterative solution scheme will be required. Among all the available methods, the second-order Newton-Raphson one accomplishes the best convergence properties. The total tangent stiffness matrix required by the later one is formed by linearizing the global equilibrium equations (42) in the direction of the incremental displacement u. By carrying out the linearization of the global vector of internal forces, it turns out to be: df I int = df matI int + df geoI int f = (KmatIJ + KgeoIJ )duJ (43) KmatIJ ij = V pret Fik ∂NI ∂Xp CpklmFjl ∂NJ ∂Xm dV (44) KgeoIJ ij = δij V pret ∂NI ∂Xl Slk ∂NJ ∂Xk dV (45)
132 Antonio J.Gil where Kand Kstand for the elemental material or constitutive stiffness matrix and the elemental geometrical or initial stress stiffness matrix,respectively. By assuming that the body forces b and external surface tractions t not associ- ated to pressure forces remain constant and by taking into account that the pressure component is dependent upon the geometry due to changing orientation and surface area of the structure,the linearization of the global vector of external forces is given through the following derivation: =厂w'an=-paw'ar (46) By applying the Nanson rule for the unit normal n-see for instance [17]or [19] for details-and particularizing for an isoparametric three-node linear finite element: =-p J-TnpreN!dr=-prpret x 3( (47) ONJ 3 amafm2+adn ONJ eprer) (48) where pis the pressure scalar acting on the considered finite element,and are the local plane coordinates and e is the so called alternating third order tensor. 3.4 Direct Core Congruential Formulation (DCCF) From the computational viewpoint,a very elegant procedure termed the Direct Core Congruential Formulation(DCCF)may be applied to perform the implementation stage of the formulation developed above.This methodology,which is hardly used in the existing literature is due to pioneer studies in [20]and [21].The main ideas behind this formulation can be discovered in the notable paper due to [22]. The scope of the DCCF is establishing the set of global equilibrium equations whose unknowns are the components of the displacement gradient tensor G which is given as: G,=0 oxpret (49) Therefore,this new set of equations is completely independent on the geometry of the structure and on the adopted discretization properties.Afterwards,every single component of the displacement gradient tensor may be easily expressed in terms of the nodal displacements of the Lagrangian mesh.Naturally,it is right then when properties concerning geometry and discretization are brought to light.The consideration of only traslational degrees of freedom for the nodes of the Lagrangian mesh makes the DCCF specially simple and easy of being implemented.The Fig.4 shows a summary of this formulation: Gradient Congruential Equations in equations transformation DOFs Fig.4.DCCF scheme
132 Antonio J. Gil where KmatIJ and KgeoIJ stand for the elemental material or constitutive stiffness matrix and the elemental geometrical or initial stress stiffness matrix, respectively. By assuming that the body forces b and external surface tractions ¯t not associated to pressure forces remain constant and by taking into account that the pressure component is dependent upon the geometry due to changing orientation and surface area of the structure, the linearization of the global vector of external forces is given through the following derivation: f I ext = Γ tNI dΓ = −p Γ nNI dΓ (46) By applying the Nanson rule for the unit normal n -see for instance [17] or [19] for details- and particularizing for an isoparametric three-node linear finite element: f I ext = −p Γpret JF−T npret NI dΓ = −pΓ pret 3 ( ∂x ∂ξpret 1 × ∂x ∂ξpret 2 ) (47) KpIJ ij = −pΓ pretilm 3 (δlj ∂NJ ∂ξpret 1 Fm2 + Fl1δmj ∂NJ ∂ξpret 2 ) (48) where p is the pressure scalar acting on the considered finite element, ξpret 1 and ξpret 2 are the local plane coordinates and is the so called alternating third order tensor. 3.4 Direct Core Congruential Formulation (DCCF) From the computational viewpoint, a very elegant procedure termed the Direct Core Congruential Formulation (DCCF) may be applied to perform the implementation stage of the formulation developed above. This methodology, which is hardly used in the existing literature is due to pioneer studies in [20] and [21]. The main ideas behind this formulation can be discovered in the notable paper due to [22]. The scope of the DCCF is establishing the set of global equilibrium equations whose unknowns are the components of the displacement gradient tensor G which is given as: Gij = ∂ui ∂Xpret j (49) Therefore, this new set of equations is completely independent on the geometry of the structure and on the adopted discretization properties. Afterwards, every single component of the displacement gradient tensor may be easily expressed in terms of the nodal displacements of the Lagrangian mesh. Naturally, it is right then when properties concerning geometry and discretization are brought to light. The consideration of only traslational degrees of freedom for the nodes of the Lagrangian mesh makes the DCCF specially simple and easy of being implemented. The Fig. 4 shows a summary of this formulation: Gradient equations =⇒ Congruential transformation =⇒ Equations in DOFs Fig. 4. DCCF scheme