《纺织复合材料》课程参考文献（Textile Composites and Inflatable Structures）08 F.E.M. for Prestressed Saint Venant-Kirchhoff Hyperelastic Membranes

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 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. 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 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-

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 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 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 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 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 nom￾inally 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 alge￾bra. 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 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: 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 summa￾rize 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)