Finite Element Formulation for Large Deformation Problem

Transcription

1 Finie Elemen Formulaion for Large Deformaion Problem Sushan Verma 1*, Sandeep Kumar Singh 1, Anuj Gupa 1, Anwer Ahmed 1 1 Mechanical Engineering Deparmen, G.L Bajaj Insiue of Technology and Managemen, India ABSTRACT In his paper finie elemen procedure has been applied o solve he large deformaion and large srain problems. Linear and nonlinear mechanics followed by finie elemen formulaion has been used o solve he russ problem. The aim of his research paper is o give an overview abou he algorihm used o solve he complex problem. For ha he behaviour of he hyperelasic maerial is sudied subjeced o large srain by using he code FLagSHyP (Finie Elemen Large Srain Hyperelasic Program) wrien by Javier Bone and Richard D. Wood. The resul obained from he code FLagSHyP is also compared from he resul obained by he abaqus sofware. In his Work hree hyperelasic consiuive models i.e. Odgen model, Incompressible neo-hookean model and Mooney-Rivlin model have been used o solve he russ problem subjeced o large srain. Keywords: Finie Elemen,Hyperelasic program,large deformaion I INTRODUCTION In linear and non-linear finie elemen analysis involving large displacemens, large srains and maerial nonlineariies, i is necessary o resor o an incremenal formulaion of he equaions of moion. Various formulaions are used in pracice. Some procedures are general and ohers are resriced o accoun for maerial non-lineariies only, or for large displacemens bu no for large srains, or he formulaion may only be applicable o cerain ypes of elemens. Limied resuls have been obained in dynamic non-linear analysis involving large displacemens and large srains. The earlies non-linear finie elemen analysis were essenially based on exensions of linear analysis and have been developed for specific applicaions. The procedures were primarily developed on an inuiive basis in order o obain soluions o he specific problems considered. However, o provide general analysis capabiliies using isoparameric (and relaed) elemens a general formulaion need o be used. The isoparameric finie elemen discreizaion procedure has proved o be very effecive in many applicaions, and laely i has been shown ha general nonlinear formulaions based on principles of coninuum mechanics can be efficienly implemened. Now a day, analysis of non-linear behavior of componens is mos likely o involve a simulaion, because of he easy availabiliy of finie elemen sofware s o provide good resuls wih accuracy i is necessary o undersand he 617 P a g e

2 fundamenal of linear and non-linear coninuum mechanics, non-linear finie elemen formulaion and soluion procedure. A number of books are available o provide background on his subjec, for example he book of JN Reddy [1], Klaus Hackl and Mehndi goodarazi [2] explain he basic conceps of linear and non-linear coninuum mechanics (sress and ensor, elasiciy, Hooke s law, displacemen funcions. The book of Javier Bone and Richard D. Wood [3], Moron E. Gurin [4], J. Tensley Oden [5], explain he conceps dealing wih differen ypes of nonlineariies viz geomerical non-lineariy, maerial non-lineariy and conac non-lineariy; kinemaics conceps viz deformaion ensors, srains; kineics conceps viz Cauchy sress, firs and second piola kirchoff sress and differen governing equaions involved in i. Klaus-Jurgen Bahe e al. [6] derived and compared he wo-finie elemen incremenal formulaions viz Updaed lagrangian formulaion and oal lagrangian formulaion for nonlinear saic and dynamic analysis. Concerning he applicaion of finie elemen mehods o he soluion of problem involving large displacemen and roaion we noe ha here are basically wo differen approaches used in FEM firs is he oal lagrangian formulaion and he second is updaed lagrangian formulaion. The formulaion includes large displacemen, large srain and maerial non lineariies. The formulaions are lised below Toal lagrangian formulaion: C ijrs v Ɛ rs δ Ɛ ij dv + S Updaed lagrangian formulaion: v + C ijrs ij δ η v ij dv + + Ɛ rs δ Ɛ ij dv + ζ + = R S + + ij δ ηij dv ij δ e v ij dv (1) + = R ζ + + ij δ e ij dv v (2) Where, and + are ime sep/load level showing he reference configuraion, new reference configuraion and curren configuraion respecively; C + ijrs, he configuraion a ime and respecively; Ɛ Cijrs are he componens of consiuive ensor a ime referred o + rs, + Ɛrs componens of srain incremen ensor referred o configuraion a ime and respecively; e ij, e ij are linear par of srain incremen a ime and respecively; η + ij, ηij are non linear par of srain imcremen a ime and respecively; componens of second piola kirchoff sress ensor a ime referred o configuraion a ime ; componens of Cauchy sress encor a ime and configuraion a ime +. S ij are he ζ ij are he + R is he exernal virual work expression corresponding o 1.1Finie elemen procedure In his procedure he problem is solved in incremenal seps. I uses he ieraive mehods o give he resuls. The seps involved in he procedure. 618 P a g e

3 1.1.1 Deformaion Gradien ensor The deformaion gradien ensor is he second order ensor which maps he line elemens in he reference configuraion o he line elemens in he curren configuraion F=dx/dX (3) WheredX is he line elemen emanaing from posiion X in he reference configuraion and dx is he line elemen in he curren configuraion Srain Srain (Ɛ) is a measure of deformaion represening he displacemen beween paricles in he body relaive o a reference lengh. A general deformaion of a body can be expressed in he form x=f(x) where X is he reference posiion of maerial poins in he body and I is he ideniy ensor Ɛ = Cauchy sress ensor θ x X θx = F I (4) The Cauchy sress ensor (σ) is a second order ensor ha compleely defines he sae of sress a a poin inside a maerial in he deformed sae. The ensor relaes a uni-lengh direcion vecor n o he sress vecor T n across an imaginary surface perpendicular on and described as Principle of Virual Work T j n = σ ij i, j = 1, 2, 3(5) Consider he moion of a body, he principle of virual work is used o express he equilibrium of he body δu T Tds + δu T fdv = δɛ T σdv(6) s v Whereare inernal virual work due o racion and body force andis an exernal virual work 1.1.5For linear and nonlinearproblem The soluion canno be obaineddirecly as above equaion give he soluion in discree ime sep so we use he lagrangian formulaion [1] o solve he problem which inegrae he unknown variables (displacemen and srain) wih reference o iniial posiion or firs posiion. If here is nonlinearproblem, hen i became necessary o linearize i afer he lagrangian formulaion 1.2 FORMULATION OF CONTINUUM MECHANICS INCREMENTAL EQUATION Consider he moion of a body in a Caresian co-ordinae sysem. The aim is o evaluae he equilibrium posiions of he body a he discree ime poins,,2,3 where is an incremen in ime. Assume ha he soluion for he kinemaic and saic variables for all ime seps from ime o ime, inclusive, have been solved, and ha he 619 P a g e

4 soluion for ime + is required nex. I is noed ha he soluion process for he nex required equilibrium posiion is ypical and would be applied repeiively unil he complee soluion pah has been solved. Since he soluion is known a all discree poins,, 2.., he basic aim of he formulaion is o generae an equaion of virual work from which he unknown saic and kinemaic variables in he configuraion a ime + can be solved. The principle of virual work is used o express he equilibrium of he body in he configuraion a he ime +. I is saed as + + σ ij δ e + v + ij dv = R + Where R + (7) is he exernal virual work expression and shown as, R = + f k δu k dv + + v + k δu k dv k = 1, 2, 3 (8) + a + Where f k +, + k are he body force vecor and surface racion vecor in he configuraion a he ime + respecively,δu k is he (virual) variaion in he curren displacemen componen. The difficuly here being ha configuraion of he body a ime + Δ is unknown so he above equaion canno be solved direcly. A soluion can be obained by referring all he variables o a known previously calculaed equilibrium configuraion for ha wo approaches have been used i.e. Toal lagrangian formulaion and Updaed lagrangian formulaion Truss Problem A four noded russ of dimension 14mm 1.414mm.77mm has been chosen. The iniial orienaion angle is 45 degree. There is a hinge suppor on node 1 and roller suppor on node 2 which consrained he moion in x direcion and allows o move in y direcion as shown in Fig.1(all dimension in mm). The displacemen of he op node is prescribed downward in incremen of 1mm. Fig.1 Truss wih prescribed displacemen in downward direcion 62 P a g e

5 The problem is solved using 198 incremens (1 mm incremen in every incremen sep). A russ is divided ino 7 elemens as shown in Fig.2 The convergence crierion is se o 1.e-1 in he code wih a maximum 25 ieraions in every sep. Fig2. Meshed Geomery wih node where analysis is performed II RESULT The verical Reacion vs. displacemen graph of node 2 obained from FLagSHyP code is shown in Fig.3 Fig.3 Verical Reacion v/s Displacemen of node 2 In Fig.3, x is he verical posiion of node 2 in every incremen sep and L is he lengh of he russ. I can be observed ha he russ exhibi non-linear snap hrough behaviour (also see Fig.4). In his non-linear insabiliy region he equilibrium pah goes from one sable poin (a which force is.1322kn) o anoher sable poin(a which force is.1329 KN). 621 P a g e

6 Fig.4 Non-Linear Snap hrough behaviour of Truss [7] III CONCLUSION In he work, he behaviour of hyperelasic maerial (incompressible elasomer) has been sudied for large srains hrough he Finie elemen large srain hyper elasiciy program (FLagSHyP) wrien by Javier Bone and Richard D. Wood. The resuls have been scruinized o describe he behaviour of he maerial in differen boundary and load condiions. The following conclusions are illusraed in his work. 1. The hree hyperelasic consiuive model odgen, neo-hookean and mooney-rivlin are used for he analysis of russ problem, plae wih hole problem and pressurized cylinder problem. 2. Only maerial non-lineariy is considered as he program is srucured using Updaed lagrangian formulaion. 3. The resuls depic ha he maerial is highly nonlinear elasic. 4. The non-linear snap hrough behaviour is obained in russ problem 5. In plae wih hole problem i is ha inensiy of sresses are high near he hole because of he sress concenraion in his region whereas he sresses are minimum on he edge of he plae because forces are evenly disribued here. This work can be exended o viscoelasic maerials and elaso-plasic maerials. The sofware on oal lagrangian mehod can be developed and resuls compared wih he updaed lagrangean mehod. REFRENCES [1] J.N. Reddy., (28). An Inroducion o Coninuum Mechanics, firs-ediion, Cambridge universiy press, New York. [2] Klaus Hackl., Mehdi Goodarzi., (21). An Inroducion o linear Coninuum Mechanics, lecure noes, Ruhr Universiy Press, Bochum (Germany). 622 P a g e

7 [3] Javier Bone., Richard D. Wood., (1997). Non-Linear Coninuum Mechanics for Finie Elemen Analysis, Cambridge Universiy Press, New York. [4] Moron E. Gurin., (1982). An Inroducion o Coninuum Mechanics, volume 158 firs- Ediion, Academic Press Inc, London. [5] J. Tinsley Oden., (26). Finie Elemen of Non-Linear Coninua, Dover Publicaion Inc, New York [6] Klaus-Jurgen Bahe., Edward L. Wilson., (1975). Finie Elemen Formulaions for Large Srain Deformaion Dynamic Analysis, Inernaional Journal for Numerical Mehod in Engineering, Vol:9, [7] Rubber Indusry Repor (214), hp:// 623 P a g e