Axisymmetrical finite element for membrane large strains and displacements

Transcription

1 Axisymmetrical finite element for membrane large strains and displacements R. Bouzidi', Y. Ravaut' % C C.Wielgosz' 'Laboratoire de Ge'nie Civil de Nantes- St-Nazaire, 2, Rue de la Houssiniere B.P , Nantes University, F44322 France Abstract This paper presents numerical developments for calculating membrane deflections under lateral pressure. The computation is made by construction of an axisymmetrical finite element in the case of finite strains and large displacements. The numerical procedure is based on the minimization of the total potential energy. The choice of this numerical scheme is justified by the fact that both numerical and some physical instabilities are correctly treated. This numerical approach is applied to the case of membranes with no flexural stiffness like fabrics. For this kind of problems, the use of classical finite element scheme based on equilibrium equations, needs to add to the rigdity matrix the non-linear terms as geometric stiffness and those arising from the follower forces [l]. In the case of pressurized membranes we show that it is more interesting to solve the stationarity of the potential energy by direct numerical minimization. This numerical scheme presents more stability and is easiest to be constructed. Moreover we can take into account all non-linear terms due to the large strains and displacements and to the follower forces. 1 Introduction The finite element equilibrium equations in structure computation is obtained either by the virtual work principle or the minimum of the total potential energy. A

2 5 12 ISBN High PerjomanceStru~uresand Composites simplified form is obtained when the problem is linearized by neglecting large strains and displacements terms. Its presentation leads to solve a system of equations for which finding the unknowns constitute the solution of the problem. Some structures, like membranes based ones, exhibit a great difference between their in-plane rigidity and the orthogonal direction rigidity. These structures present singular classical stiffness matrix due to weak rigidity value in some directions. Solving such systems need hardness numerical algorithm witch take into account the additional components into the rigidity matrix. We consider here the problem of a circular plane membrane clamped at the rim and described in cylindrical coordinate system. An experimental survey was conducted in order to investigate the rheological properties of polymers [6]. In this paper we focus only on the construction of numerical solution when the membrane is submitted to lateral pressure and exhibits large deflections. Figure 1 shows the initial and the currentpositions of the membrane and the displacement vector. A Z Initial configuration Figure 1: Radial cross section of the membrane in cylindrical coordinate system. The numerical solutions are compared with two analytical solutions for isotropic elasticity case. We also consider the stability of the numerical scheme for large deflections. 2 Bibliography The first analyt~cal solution to the pressurized membrane problem is given by Hencky [4]. He consider a circular membraneclamped at the rim and subjected to lateral pressure i.e. the pressure action remain parallel z direction. Solutions for the radial stress resultant and the lateral deflection are expanded as: b A$+ Lp)2'35)2n p2" 4 Eh

3 High PeyfomanceS'trucruresand Composites 5 13 W(p)= E h 0 (4 The coefficients of these summations are obtained analflcally by the resolution of the continuum elastic equations with the boundary conditions. Fichter [3] supposes that the pressure action follows the membrane and remain orthogonal to its shape. He gives a similar description to the Hencky's one and propose to write these two quantities as:.(p)=-&,, (1++2) (4) 0 The Fichter's coefficients description depend on the material properties since the last ones do not appear explicitly in the proposed expressions. The reader can find full details of these two analflcal solution in [3]. Figure 2 shows the difference for these two deflections. We can see the effect of the uniform pressure given by Fichter's formulation for which the pressure remain normal to the membrane while loading. The radial effect of the pressure seems greater than that given by Hencky's formulation witch consider the pressure always parallel to the z direction Radius (cm) 15 Figure 2: Deflections of the analflcal models

4 5 14 ISBN f Iigh Perjortnance Structwes mdconzposites 3 Kinematics and energy assumptions We will use the total Lagrangian formulation to represent the kinematics of the membrane. We consider only the case of thin membranes without bending stifhess. For axisymmetrical problems, the kinematics can be described by the radial cross section in the (r,z) plane. The analytical description is followed by numerical discretization into finite elements. Let s recall the theoretical frame of finite element formulationin the case of finite strains and large displacements. For a continuum with volume V, the total potential energy n is written as: n={e,, dv v (5) The first term is the elastic strain energy and the second one represents the work of the external forces. The second Piola-Kirchhoff stress tensor is obtained from the elastic energy density ed : S, = de, / a,. The variational theorem in the reference configurationis written as:?!l, a4 This last equation is solved by a descent method like conjugate gradient or Newton method. From mathematicalpoint of view, this scheme leads to the same results to those that we obtain by the rigidity equilibrium equations. However, it s easiest to handle all non-lineanties by evaluating the scalar ri rather than constructing the non-linear rigdity matrices. 3.1 Total Lagrangian formulation We consider the position of the membrane at the initial position at the time to and the current position at time t. The initial position of the membrane is circular and flat,so that the initial position is defined by the radius value. The deformation state is then defined by the displacement vector between these two positions [Fig. l]. - d(r)= U(& +W@)< (7) Wecan note that the radial u(r) and lateral w(r) displacements are assumed to depend only on the radius r in the reference configuration.this is dueto the flatten shape of the membrane in the initial configuration. In the following, they will be

5 High Performance 3ructure.s and Composites 5 15 noted simply U and W. The displacement gradient tensor is then written in cylindrical coordinates as: Bj =[; The Green-Lagrange tensor in the case of large-strain is defined as: Only the following components of the strain tensor are not nil E, =U, +Ut/2+W:/2 >. Ee =u~r+(u/r)~~ &E = W,rl2 In these expressions, we can discern the following terms: the classical linear terms for small strains and deflections:,.=u,., e=uir. terms for large strains: ut /2 and (U /r)*12, and terms for large deflections: W,/2 and W: /2. The stress-strain relationship for linear elastic behavior in the plane of membrane can be written as: (13) In the linear elastic and isotropic case only the Young modulus and Poisson coefficient are needed. 3.2 Potential energy and pressure work The elastic strain energy density can be written as: ed =ei+e&+e,d+ei (14)

6 5 16 ffighpc~ortnancesii-uctwesandc omposites with the following expressions: e, =idrr&f =!-D ( u, ~ + u : / ~ + w : / ~ ) ~ 2 2 efo =+DR,& = ~ D,(u/r +(~/r)~/2)~ 2 (15) The external work considered provides only from to the inflation pressure and gives: W,= JpZds=p(~-Vo) (19) S Where V and V. are the actual and initial volume of the membrane. The summation of the strain energy density over the volume leads to the potential energy: n=er~+e~+erd++~-p(v-vo) with E,; ={e:odv (20) 4 Finite element approximation The evaluation of the potential energy functional (eq. 20) is done by the discretization of the integration domain. The radial and lateral displacements are approximated with the following shape functions: v Thus the derivative functions are constants within the finite element :

7 High Per.fimmnce Stwctures and Composites 5 17 The density of elastic energy is then related to the nodal displacement vector. By integrating the energy density over the element volume, we obtain the elastic energy. Their expressionsin the case of small strains are : E, = nhdm IrhD, (2(% -%)(% Erd:- -5>-(Y-%) ) 4 (1, -5) (24) Similar expressions in the case of finite strains are obtained with formal calculation tools and not presented here because of thei complexity. Since the initial volume is zero, the pressure work is calculated by evaluating the actual volume. This for one element is given by: where qd = q +U, and r: = r2+u, are the radii of the element nodes in the deformed configuration. At this stage, for a given finite element defined on the initial configuration by 5 and r,, we can evaluate the total potential energy l7 as a hnction of its nodal displacements. The minimization of I Irelatively to those displacements gives thesolution of the problem. Thenumerical results are given in the next section. 5 Numerical results in the case of large displacements and small strains The model developed above is tested for its numerical stability and compared with \~~~~&~sa,d&issns. AlLtherexult< tiven hereafter are comnuted with: Eh=311Wa and v=o.34

8 5 18 f/i& I'er~j?mnanc.e Stmctures and C'omposites The different numerical results presented hereafter have been obtained with the finite element in Large Deflection and Small Strain with Follower Pressure (the pressure remain normal to the membrane when loading). This numerical model, denoted LDSSFP, are based upon the same hypothesis as those used by Fichter [3] ~ ~ Central deflection (m) for p = 250 kpa '! \ - FP - M i ' -, --LDSSFP \, 1. +~ - ' -_- I I number of finite elements Figure 3: Mesh sensitivity of the numerical solution. First, we study mesh refinement effects on the solution in order to evaluate how many finite elements are needed to obtain a good accuracy. Figure 3 shows the evolution of the central deflection as a fimction of the mesh density. This deflection is quite stable beyond 25 elements. We can also notice that the numerical central deflection is enclosed by the two analytical solutions. Hereafter we retain the mesh with 29 finite. The number 29 is chosen for a future accordance with experimental measurementpoints conducted in our laboratory [6]. U e e, 1.cm% 0.00% -1.00% % - '$ -3.00% -4.00% -5.00% 4 Figure 4: Relative error distribution between the finite element and Fichter's solution

9 High Perjhnance Stmctwesand Composites 5 19 Comparison is made between the numerical modelling and an analytical solution and expressed as the relative difference on the deflections between Fichter s solution and our finite element. The difference is relative to the central value: [W,,,,,(r)-W,,,,,,(r)]/WFichter(0). We can see that the differences increase with the pressure value. The maximum difference doesn t stand at the middle of the membrane but near the rim. The maximum relative enor doesn t exceed 5% near the rim, where the deflection value is small. 6 Effect of the finite strain terms Let s study now the effect of large strain terms. In this case, the finite element takes into account the Large Deflection, Finite Strain terms, and Follower Pressure. This numerical model is denoted LDFSFP. The only approximation made concerns the displacement field in each element which is assumed linear. This approximation vanishes with the mesh refinement. So we can say that the proposed numerical solution tends, with the mesh refinement, to the exact solution of the problem without any other approximation. We didn t find any references with a complete model, so the comparison is made with the large deflection finite element with uniform pressure. 0.50% 0.00% % E -1.00% % e, % c. d -2.50% % -3.50% Figure 5 : Relative error between small and large strains cases. For p =lookpa, the deflections of the two numerical solutions are nearly identical. At this load stage, the strains remain small. The deflection differences increase with the pressure level. For p = 4OOkPu, the maximum relative difference reach 4%. The finite strain terms cannot be neglected if we need an accurate modelling. The large strain causes a less spherical shape, but a higher central deflection. 5

10 520ISBN High Pe~omanceStrwctwes a r 7 dc onzposites 7 Conclusions In this paper, we have showed that the numerical solution of a discretized problem by finite elements could be obtained directly with the help of the minimization of the total potential energy. Minimization is achieved with descent methods as like the conjugate gradient either or Newton method. Handling the geometric non- linearities and those providing from the follower forces is easy to do, in spite of the complexity of the deformation energy expressions. Also, it is easier to either activate or not auxiliary terms in order to evaluate their weight. A complete axisymmetical finite element is proposed, including large deflection, large strain and uniform pressure loading. The results obtained are satisfactory in comparison with the analytical solutions gwen by Hencky [4] or Fichter [3] The iterative process of minimization has shown a good numerical stability and allows the extension of this approach to three-dimensional tiangular finite element. Building such element permits the computation of more general shape of membranes. References [l] Zienkiewicz, O.C. &L Taylor R.L., The finite element method, Chapter 11, Geometrically non-linear problems - finite deformation, ed. Butterworth Heinemann, Vol. 2, Solid mechanics,oxford, pp , [2] Campbell, J.D., On the theory of initially tensioned circular membranes subjected to uniform pressure, Quart J. Mech. C% Appl. Math., IC, Pt 1, pp ,1956. [3] Fichter, W.B., Some solutions for the large deflections of uniformly loaded circular membranes, NASA Technical Paper 3658,NASA Langley Research Center,Hampton, [4] Hencky, H., On the Stress State in Circular Plates With Vanishing Bending Stiffness. Zeitschnpfur Mathematik und Physik, 63, pp , [5]Marker, D.K., & Jenkins, C.H., Surfaceprecision of optical membranes with curvature, Optics Express, 11, [6] Ravaut, Y., & Bouzidi, R., CaractCrisation du comportement d une membrane circulaire bloquee sur sa circonference et soumise a wepression unifome, XV Corwx% Franqais de Me canique, CD-ROM support proceedings, Nancy-France, 3-7 sept