Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1987 On analysis of viscoelastic structures. Kim, Ju-Eon. http://hdl.handle.net/10945/22393 OHOOL Mo 93943-8002 NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS ON ANALYSIS OF VISCOELASTIC STRUCTURES by Kim , Ju-Eon September 1987 The; sis Advisor: Ramesh Kolar Approved for public release; distribution is unlimited T 234281 UNCLASSIFIED SECuR^Y CLASSIFICATION OF PaGP TniS REPORT DOCUMENTATION PAGE lj REPORT SECURITY CLASSIFICATION 'b 2a Un clas s i f ied SECURITY CLASSIFICATION AUTHORITY 3 2b OEClASSiFiCATiON i 6* Approved for public release; distribution is unlimited PERFORMING ORGANI2ATION REPORT NUM8ER(S) NAME OF PERFORMING ORGANIZATION ADDRESS (Cry and State, /if> MONITORING ORGANISATION REPORT NUMBER(S) S 60 OFFICE SYMBOL (If 4ppl*4blt) 1* Code 6 7 Codt) ?b 8c NAME OF FUNDING /SPONSORING ORGANIZATION ADDRESS (Ofy Stitt ADDRESS *nd HP Cod*) Scjf*. (C/fy. Monterey, California 93943-5000 8b OFFICE (if NAME OF MONITORING ORGANIZATION Naval Postgraduate School Monterey, California 9.3943-5000 3a MARKINGS DISTRIBUTION /AVAILABILITY OF REPORT /DOWNGRADING SCHEDULE Naval Postgraduate School 6< RESTRICTIVE SYMBOL PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER 9 applicable) SOURCE OF FuNOlNG NUMBERS TASK PROJECT ELEMENT NO NO NO and ZiPCode) 10 PROGRAM TiTlE WORK UNIT ACCESSION NO (include Security CUmticttion) ON ANALYSIS OF VISCOELASTIC STRUCTURES PERSONA,. auTmOR(S) !»' Kim, Ju-Eon rypf of report 3j i3b COVERED T'ME FROM Master's Thesis DATE OF REPORT 14 TO 1987, (Yesr Month Day) IS PAGE COuM 60 September Supplementary notation 6 COSATi CODES F GROUP EiD 18 SUBJECT TERMS (Continue on reverie if neceuary and identify by block number) GROUP SUB Viscoelastic Structures Problems ABSTRACT (Continue on 9 reverie if neceuary and identify by bicxk num^r) The thesis begins with a comprehensive formulation of continuum-based finite element theory. The theoretical portion concludes with the details of both spatial and temporal discretization, including a discussion of nonlinearity This study is to understand the effects of nonlinear behavior of structures using a finite element method. The nonlinear behavior equations are derived from equations of motion and constitutive equations. The basic theory is the principle of virtual work. in particular, large displacement problems and visco-elastic problems remain a challenging engineering problems today. The viscoelastic problems depend on the relaxation function which is the source of material nonlinearities S"R'3UTiON' AVAILABILITY OF ABSTRACT !0 GSuNCLASSiFiEQ/UNL'MiTED 22a NAME OF RESPONSIBLE Q 1473. 84 mar O DTiC USERS ABSTRACT SECURITY CLASSIFICATION Unclassified 22b TELEPHONE (include Are* Code) NDiViOUAL Professor Ramesh Kolar DO FORM 21 SAME AS RPT (408) 83 APR edition ™*y b« uted until cihauited ah other •dt.om *>* obtoi«t* 546-2066 22c OFFICE SYMBOL Code 6 7 kj SECURITY CLASSIFICATION OF ThiS PAGE UNCLASSIFIED Approved for public release; distribution is unlimited On Analysis of Viscoelastic Structure by /-Kim, Ju-Eon Captain, Republic of Korea Air Force B.S., Korea Air Force Academy, Seoul, 1980 B.S., Seoul University, Seoul, 1983 Submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL » September 1987 . . ABSTRACT This study is to understand the effects of nonlinear behavior of structures using a finite element method. The nonlinear equations of behavior equations derived are constitutive equations. motion and from The basic theory is the principle of virtual work. The thesis begins with continuum-based finite with portion concludes temporal a comprehensive element the discretization, The theoretical theory. details both of including formulation of a spatial and discussion of nonlinearity In particular, elastic problems today. large displacement remain a The viscoelastic problems and visco- challenging engineering problems problems depend on the relaxation function which is the source of material nonlinearities ... . . TABLE OF CONTENTS I II . . INTRODUCTION A. ANALYSIS OF STRUCTURES 7 B. LITERATURE REVIEW 7 C. THESIS OVERVIEW OVERVIEW 12 B. NONLINEAR CONTINUUM EQUATIONS 12 . . Principle of Virtual Work 12 One-Dimensional Finite Element 15 C. CONSISTENT NODAL FORCES 17 D SOLUTION METHODS 21 E. IV 12 A. 2. . 11 THEORETICAL FORMULATION 1 III 7 1 Incremental Method 21 2 Newton-Raphson Method 22 3. Modified Newton-Raphson Method 24 ANALYTIC SOLUTIONS 24 1 Elastic Materials 26 2. Visco-Elastic Materials 30 PROGRAM IMPLEMENTATION 35 A. OVERVIEW 35 B. SOLUTION PROCEDURES 36 1 Algorithm Implementation 36 2. Flow Chart of Finite Element Program 38 NUMERICAL EXAMPLES 39 4 V. A. ELASTIC MATERIALS 39 B. VISCO-ELASTIC MATERIALS 45 CONCLUSION 48 LIST OF REFERENCES 49 INITIAL DISTRIBUTION LIST 59 ACKNOWLEDGEMENTS I am appreciative provided to me by faculty members I would of Professor the assistance Ramesh Kolar and and guidance the other at the Naval Postgraduate School. also like to thank the R.O.K. Air Force for allowing the opportunity to study. I am greatful to my wife, Mi-Seon and my daughters, Myeon-Joa and Nam-Lyang with whom God has blessed me. INTRODUCTION I. A. ANALYSIS OF STRUCTURES structures subjected to large displacements Analysis of requires the inclusion nonlinear of terms the strain in In the formulation of the problem, displacement relations. either total Lagrangian or Eulerian approach is adopted. In order to assure structural integrity for loads beyond linear range, nonlinear analysis is very much mandated. Another class of problems of structures problems - vary with time at where stresses in the components deformation state constant a viscoelastic behavior is the or strains vary with time at constant stress states. problems These as applied simple one-dimensional to structures will be addressed in this research work. follows is end the a brief survey of literature in this area. this of In what chapter, overview of At thesis the is described. B. LITERATURE REVIEW A method for nonlinear static and dynamic analysis using cable and beam Formulation large elements allows for deformations. comparison of finite is given material Gadala element and by Bergan et formulations [2] [1]. and very nonlinearities Oravas al present based a on the Lagrangian, updated Lagrangian and Eulerian formulations for nonlinear mechanics problems. Henriksen [3] applies to a single integral constitutive law viscoelastic describe Ghoneim et al [4] treat the viscoelastic and behavior his in strain total formulation. rate Pinsky and use variational approach for studying the nonlinear Kim [5] viscoelastic shell as behavior shell representing by multi-director field. a within individual layers, the transverse strain normal Saigel and Yang present describe the layered The formulation includes, effects layer thickness coordinate [6, and sum of viscoplastic contributions in implementing the constitutive equations in their formulation, and as to transverse shear of arbitrary orders in the 7]. quadrilateral a implementation shell element elastic-viscoplatic of constitutive law for large deformation. A large collection of papers on the application of ADINA may be found in [8]. Response nonlinear of mechanical excitation is given by To [9]. with thermoviscoplastic discussed Allen contacting methods bodies are given deformations of on to random Thermomechanical response of uniaxial bars by systems [10]. Large the contact consideration composite beams constitutive laws is deformations [11, the and solution surfaces in of 12]. Large subjected to axial dynamic load is reported in [13]. Based on Eulerian-Lagrangian formulation, frictional contact analysis is discussed by Haber et al [14]. Halleux and Casadei [6] analysis structures of Polar-decomposition derived tangent based used is biquadratic on arriving in large strain at elements. explicitly stiffness matrices for a frame element by method based on co-rotational Kondoh and Atluri [15]. coordinate present transient where system, system attached to A element each cartesian coordinate local the translate and rotate is described by Mattiasson [16]. based problems Inelastic equations is given by Nagtegaal rate during assumed is an incremental virtual work on A [17]. increment constant strain in integrating the constitutive equations. curved Planar beams undergoing rotations large is analyzed using Helinger-Reissner mixed variational principle by Noor Steele [18]. discusses several nonlinearities including buckling, creep, and nonlinear elastic problems in [19]. plastic The use model Hirashima et al of Lagrangian is presented [21] formulation using an elasto- by Voyiadjis and Navaee [20]. present an approach for the snap- through analysis of curved shell panels and determination of initial stress matrix. Importance pellet are of viscoelastic investigated by relaxation Basombrio [22], observed that the duration of the ramp input response considerably. effects on a where it is influences the Viscoplastic constitutive models are obtained explicitly, without resorting directly implemented to inversion, to be finite element formulation [23]. in a An intrinsic time scale that can vary with time and position together with means to control integration in the approximation of constitutive is used, the numerical model by Chambers and Becker [24]. discuss applications of visco- plastic models in the metal-forming. Elasto-plastic models Chandra et are used al 26] [25, analyzing in unilateral problems by Cheng et al [12]. contact Chung et al [27] suggest varying parameters in guiding solutions for the methods for simulation of plastic forming processes. propose friction and formulation for et al [28] path-dependent materials based on arbitrary Lagrangian Euler method plastic wave Liu and include an elastic- propagation problem. Rheinboldt and Riks describe several algorithms based on continuation method and mesh refinements for the solution of nonlinear finite element propose an approach equations. to solve Ryu nonlinear and Arora [29] element finite equations based on substructures concept. Several plastic [10, 7, papers behavior 4, 30, pertaining and their 31]. 10 to visco-elastic , visco- implementation is reported in C. THESIS OVERVIEW The objective of this thesis is to review some of the literature available in the analysis of large deformation of structures, develop to displacements and structures, the the of and incremental load method. large of one dimensional to understand the physics of nonlinearities theoretical formulation based on include to viscoelastic behavior and in the process, effects theory Chapter . principle of II presents virtual work Descretization based on a simplex element is given. Chapter III procedure. gives the algorithms used in the solution Full Newton-Raphson and Modified Newton-Raphson methods are described. Chapter IV describes some test cases and numerical results obtained. Finally, Chapter V provides conclusions and some remarks for future work in this area. 11 THEORETICAL FORMULATION II. OVERVIEW A. chapter this In we consider a nonlinear continuum This approach is based equations for a rod element. on the The resulting equations are principle of virtual work [48]. in convenient form for numerical solution via finite element methods [49]. course, of discretized-solid the J T Oden et . This, al . [50] the conceptual basis of is finite introduced elements, by in the elastic structures and nonlinear continua. B. NONLINEAR CONTINUUM EQUATIONS Discuss a rod element with the It may proved be that external virtual work. equilibrium if is equal and only the to total total Lagrangian Method. internal virtual work is equal to Namely a system deformable is in if the total external virtual work internal virtual work for every virtual displacement consistent with the constraints. 1. Principl e of Virtual Work In this section we prove that total internal virtual work is equal to total external virtual work. concept of is minimum potential energy [51]. satisfied potential with to equilibrium state and respect to the displacement 12 Consider the The condition the change of must remain stationary. And consider the load- displacement and stress-strain relationship of figures shown below. -^sukFigure 2.1 Load-Displacement Curve 6- Figure 2.2 Stress-Strain Curve In General, the total potential energy is equal to the summation of strain energy and potential energy. ie, where, = 7T P Kp w U + W (2.1) Total potential energy Strain energy (internal work) Potential energy of load system (external work) 13 , From Figure (2.1), W = - = - where, p 6u J 2 Pi (2.2) ui i th load deflection under Pi ui i th load From Figure (2.2), \ U (\ o 6€ d(vol) = vol o 5 a € d(vol) J (2.3) vol Substituting the equation (2.2) and (2.3) in equation (2.1), we get a € d(vol) 5 vol p Equation can (2.4) - be 2 p. (2.4) u. represented by matrix form, the following is the matrix formula. tup = -5 J v where, {€} d(vol) {a} - { u. } { p. } 2.5 x a vector of stresses a corresponding vector of strains a vector of elemental nodal displacements a vector of elemental nodel forces {a} {€} (ui } {Pi} Let us define [B] , a matrix relating strains and displacements le {€} = and define [D] le, {a} = [B] , a [D] (2.6) {u} matrix relating stresses and strains, (2.7) {€} 14 } Substituting equation (2.6) & (2.7) equation (2.5), we in get T Kp {ui - T T {B} } [D] d(vol) [B] {ui - } {ui } {pi } (2.8) vol Now we consider the minimum potential energy, the condition is as follows: dTTp (2.9 = d{ui } with respect Taking derivatives to {ui } in equation (2.8) and applying to equation (2.9), we get {pi } :J CD] [B] d(vol) {ui [B] (2.10) } vol relating From loads displacements, and elemental stiffness matrix [k] le, [K] = J [B] [D] [B] we can get the . d(vol) (2. 11 vol Summing all the elements of the entire structure, equation (2.10) becomes {P} = where, 2 . [K] {P} - [K] [u] = = (2.12) {u} 1 {Pi 2 [K] 2 {ui } One Dimensional Finite Element Consider a shape function which is linearly function with geometric boundary condition 15 with one dimensional We define that the shape function, finite element. \J/M (x) aN = bN + tn, is X In order to derive the constants in the shape function for a with element rod displacement linear field, use the geometric boundary condition. We may obtain the constants as follows: X2 Lo i b (2.13) = Ln 1 XI a 2 Lo 1 b 2 The relationship between displacement and shape function is given below M u = N uN ^(x) = Another T |\{/N (N=l; u x (t) ^.(x) + consideration M=2, ....) (2.14) u 2 (t) the viscoelastic materials. is viscoelastic materials, the stresses may terms of relaxation functions as given by 't s(t)= J G(t - £ ) s (£) 16 d£ be For expressed in ] = S I(t) (t) e e where, (2.15) S (t) S = [ r G I(t) ( t t - ( elastic response : ) £ ) = S* ( £ ) d£ ae relaxation function. G(t): Assume the element is in equilibrium and quasistatic state. The motion equation for a one dimensional finite given by J T . . ODEN [ref.6], S\^ N Vo where, Vo P element is : ) dV = M M, x Ao Lo = u *£/ 1 ( x , P (u N M ,t undeformed volume total generalized force at node N : N On substituting for above functions shape and their derivatives, we may obtain fN (u M t, , I) = Vob N Se [ - I] (1 + b k ) - P N (u M ,t) (2.16) = C. uk CONSISTENT NODAL FORCES Consider a rod element in two configurations: undeformed and deformed. Let the distributed forces configurations be q and q* load vector both the depends on per unit length in these respectively. the deformed state. 17 The consistent magnitude of the load and . One dimensional rod element under distributed load is showed as following figures. (x,V> -*. %*C*>*^ * *. V 2- i 2,-- i a L. where, Lo L undeformed rod element length : deformed rod element length : q(x,t) is distributed load per unit undeformed length q*(y,t) is distributed load per unit deformed length Figure One-Dimensional 2.3 Rod Element under Distributed Load Consider the relationship undeformed coordinate. between the deformed It may be written as y(x,t)-x+u(x,t) where and (2.17) x: undeformed coordinate of a y: deformed coordinate of point u: displacement of a a point point within element The consistent nodal forces vector is given by (2.18) In index nutation, P = [ nodal forces may.be written as Pp%dV - fjh* dVo (2.19) with N where F = body forces per unit mass deformed : Fo body forces per unit mass undeformed : P deformed density : Po 2 1, undeformed density : Relating q*(y,t) and q(x,t) is given by q(x,t) where, (2.20) dy/dx = A. q*(y,t) \ = stretch : undeformed body Consider the force per unit mass, then we may write (2.21) r°" dm P Ao - " P. A Using equation (2.17) y x + \Jf = (2.22) u M M dy = (2.23) u + \|/ 1 dx M M, x The undeformed body force per unit mass is then A P q* = ( o ' +^M,x 1 um) (2.24) Ao The consistent nodal forces take the following form, L f PN = J q* L ° t Vf?x f + J o q*>^ MiX u M\^ N dx (2.25) Substituting the shape function in equation (2.25), we may get 19 PLo = 5 N + b u M M) (1 q*(a N J + bN x) dx (2.26) Now we consider some special cases of equation (2.26) One case is the uniformly distributed loads. V / *>* S* Figure 2.4 Rod Element under Uniformly Distributed Loads The uniformly distributed loads are given q*(y,t) = h(t) qo Substituting the loads in equation (2.26), we may get the consistent nodal forces. qo p /I { P„ > = > = h(t) [Lo X J + *- u ui] U (2.27) du where, h(t dx Another case is the linearly distributed loads Figure 2.5 Rod Element under Linearly Distributed Loads 20 . Linearly distributed loads are given q(x,t) q (t) + [q (t) = Substituting loads the - (t)] q ey equation (2.26) we may get the in consistent nodal forces. (Lo P N < D. - + u u a l" ) < > 13 2 M (2.28) » ,<\ SOLUTION METHODS In this section we incremental method, derive the equilibrium equations of Newton-Raphson, Full and Modified Newton-Raphson Method. It is based on Lagrangian total Method use the and Taylor Serier at a given equilibrium point. element The stiffness matrices vectors are assembled to form global prescribed The manner. and consistent load matrices in the usual displacements are implemented by replacing the diagonal element of the corresponding row by unity and the rest of the element of the rows and columns by zeros [49] 1 Incremental Method (INC) . Consider the incremental process, we can rewrite the equation of load-displacement relationship, [K]i (A>i where, { i APc }i + : : i r (APMi + {APc}i the current configuration coordinates. the force unbalance 21 (2.29) {/\P}i+i an increment in the external load : The method is described by the Figure 2.6 c c -A , H Pb ....v* /f \ ' "ICj A 1 1 As p. 1 *s> J 1 1 1 ' k where, -p. « 1 1 1 JL The algorithm of equation (2.29) AEC AB' —^—4 \ * C : The algorithm of purely incremental process Figure 2.6. The Algorithm of Incremental process The approximate values of A, To improve E, is computed from p -f(u). C, these values, take the iteration process at each load level with {AP}i+i= 0, as is Figure 2.7. The iteration processes are load levels with many corrective another is many load levels This is sometimes called two with ways, one is few iterations in each, or few iterations in each. 'incremental with one-step Newton- Raphson method" 2. Full Newton-Raphsor: Method CFNR) Consider the load versus f(u). Take the Taylor series of P 22 displacement function = f(u) P - , , . <#>. J(U^ JCUA + A,)= A, (2.30) dP where tangent stiffness at A K du f(u ) A f (u A + ) = P A.such that We look for k (A) A = p - f(u A P - ) , SO B (2.31) P B I + B The consequent steps are explained by the Figure 2.7 Figure 2.7. The Newton Raphson solution of the equation P f(u) = The above algorithms are below; update displacements use . ui : ui to get stiffness find next increment Eventually ua + A = l|A and load ki ^a from +• , A + ki A^ ( Ax Pi ) = + Pb - Pi = ub to a close approximation Then, the load can be process can be started increased again. 23 from Pb to Pc , and the This solution algorithm is known as the Newton-Raphson Method. 3. Modified Newton-Raphson Method (MNR) Newton-Raphson The [k] solving in every iteration An economic. that tangent alternative is process This cycle. modified tangent same iterations cycles. stiffness matrix is is not Newton-Raphson also called "constant stiffness iteration". iteration, the requires generated and reduced for equation stiffness matrix be method used Here for several The MNR is described by the Figure 2.8 Figure 2.8 The MNR Iteration but As compared with FNR, the MNR requires more iterations, each quickly because [K] is formed and is done more reduced in only the first iteration. E. ANALYTIC SOLUTIONS At first we define the concept of stresses [52] Eulerian stress, a , is defined as follows: 24 ) : N(y,t) (y,t) a (2.32) = A(y) and is called True stress. Lagrangian stress, T, is defined as follows: N(y,t) T(x,t) (2.33) - Ao (x) and is called Engineering stress. Kirchloff stress, S(x,t) defined as follows: N(y,t) — = is S, (2.34) \(x,t) Ao(x) The relationship between strain r , and stretch is as follows ( \* " 1 du \ = i (2.36) + dx where, N A For the Applied force : deformed area : special case in cross-sectional of small strain, area are small. both u, In this x and change case, we can write r * u = € a * T » S where, € is Thus in engineering strain. small deformation case, and engineering strain can € 25 be the Lagrangian stress T used to replace the ) and strain engineering can € be used Kirchhoff stress S and Lagrangian strain r to replace the . Elastic Materials 1. Consider cross- sectional one-dimensional a area, Ao Ao(x), = with rod varying in equilibrium under distributed external loading. (T+£*X) AoOO (A*+% *x) (a) where, (b) external distributed load per unit length f T : Lagrangian stress within rod Figure 2.9. One-Dimensional Rod under Axial Loading For static problems and elastic materials, both f and T are independent of time, £ — -C f From figure n«-J .. \ 2.9(b), 1* - rr' i • - x the static equilibrium of forces in the x-direction is £ F(x) = -TA + fax (T (dT/dx) ax Expanding in Taylor series, 26 ) (A (dA (2. 37 / dx) Ax ) = = ^c For a in the limit, continuous rod, i + i^TA^ (2.38) - (2.39) Substitution of equations (2.34), and (2.36) in the (2.35), above (2.40) where, u(x) displacement w.r.t. x : u, x first derivative of u(x) : u.xx second derivative of u(x) : Note that equation (2.40) force (per f unit displacement field values of distributed gives the length) u(x) of a required to one-dimensional produce the rod once the boundary conditions, the stress-strain law, and varying area function Ao ( x constant area f(x) = -A have been ) Ao , u,xx Consider the rod with specified. equation (2.40) becomes 1(1 Some special cases of + u, x ) **2 ( dS* /dr ) + S«] equation (2.41) case is the specified displacements. 27 (2.41) is considered. One x \oo> /-=* //- where, X J A« u(x) " u, x + ao - a + 2 i = u, x x 2 + a, x aa x 2 aa x a» Figure 2.10 Test Model of "Inverse Solution" Apply the boundary contition u,x(x) a u x ( ) a = , x — =0 at x = Lo (2.42) x2 - 2Lo If specified, we may get the exact solution u(x) and is a t the corresponding required distributed load f(x). Linear displacements field: a. u(x) u, x U, xx = = a, a, = From equation (2.42), we may get f(x) Note that = a linear displacement with distributed load. 28 field cannot be produced r Quadratic displacement field: b. u(x) u, x u 11, - a,x = = a - - { - ( (ai/2Lo)x 2 - y ( a i Lo / a | / Lo ) x ) Consider the linear materials, Se E - r We may get the distributed forces from equation (2.41) f(x) - - EAuU.xx - [(1 + u, x ) + + (u, x % u2, x )] (2.43 Consider the nonlinear materials, - Se Ei r = E2 + (quadratic form) 2 From equation (2.41), we may get f(x) AoU.xx - = + (u, El x { *- (1 + u, x ^u2,x) ) + [Ei E2 E2 2 + (u,x + (u,x >2U 2 ,x)] % u2,x)] + } (2.44) Another case is the prescribed loads. / ^p /e A -* 3C Figure 2.11 Test Model of Rod with Prescribed End Element From equation (2.39), since no distributed loads exist, d ( TAo ) = with constant area. dx We may get G = constant (2.45) 9Q r ) s The applied load is given P - Ao\ - (2.46) S Linear material; ^e Er r E(u, = + x *i u^ , x ) From equation (2.46), we may get P - EAo = (u, + x u2, x + ^ u3, x (2.47) ) Nonlinear material; Se El r = E2 + 2 (quadratic form) From equation (2.46) we may get P Ao = + U, (1 X [El ) (U, x y2 U= ,x + ) + E2 (U, x + % U2 , x )2] (2.48) 2 . Quas i lin ear Viscoel a tic materials The Viscoelastic materials has than problems the elastic materials, the more complicated since the external loads and displacements are time-dependent. lU-t) u Lo c*> > Fu c*) < x where, Ao ^ Lo , X=Lo uo(t) —^1 u< : : uLo(t) Fo ( t FLo(t) Underf ormed Area and Length time-varying left-end displacement time-varying right-end displacement : time-varying left-end force time-varying right-end force Figure 2.12 One-Dimensional Rod Subjected to End Loads 30 Time-varying . For quasilinear viscoelastic materials, Sit) Jn The Lagrangian (2.15) &, constant throughout the rod since stress is the distributed loads is not applied. From equation (2.39), the stress is time variable only. T(x,t) g(t) - \(t) = (2.49) S(t) And the internal forces produced with the rod is F(t) = TAo Aog(t) = (2.50) Substituting equation (2.15) in equation (2.49) yields g(t) \(t) = (1 S(t) + u.) S - e (r) (2.51) tf o of equation hand side The right (2.51) is a time function only Therefore must also be a time function. g(t) du u,x(x,t) h(t) = (2.52) dx Integrating w.r.t. u= u(x,t) where, C at equation (2.52) x xh(t) - + C(t) integrating function to be determined by the : boundary condition at one end. Assume that u ( o t , ) (2.53) = uo ( t ) = C t ( ) , 31 ) ) therefore u(x,t)=xh(t)+uo(t), u,x (x, t) = (2.54) h(t) Substituting equation (2.54) in equation (2.51) yields ;] (2.55) Se Once the G(t) and ( r been specified, have ) the equation be used with the boundary conditions to find the (2.55) can Consider the exact solution in some cases. with linear material property. boundary conditions at 3 element solid The prescribed displacements ends both described is in Figure 2.13. / / / / > /£ x -J L, u(o, t) u ( Lo , uo = t ) = ( u, t) ( - (2.56) t Figure 2.13 Test Model of a Rod with Prescribed Time Varying Displacement at the End. The prescribed displacements are linear, u(x, t) U, X ( X t , ) (x/Lo u. I' ( o u. ) t ) (t) / Lo (2.57) = h(t 32 (2.58) . where, t^ (t) = Loh(t) Assume linear material property, S e And for a G(t) where, element rod, G(t)is defined below 3 = - (1 a Er = a P , = (2.59) constants for material properties. ; Note that G(o) e-Pt + ) 1 at t = G(t) , = - 1 a at (3-0 No relaxation occurs and the material exhibits elastic response Specifying the prescribed displacement function u r (t). L o Ramp Function a. At x = Lo C.'S:u(o,t)=0 B. , u(Lo,t)= u u r (t) o = at (a: constant) From equation (2.57) and 2.58), we may get u(x,t) u, x ( x t , = (a/Lo ) = ( ) a/Lo x t ) (2.60) t Substituting equations (2.57) - We may get the reaction force, EAo at [1 + F. a a a2 1 2 [{at+ ] Lo in equation (2.55) (2.59) 2Lo Lo a a a {!-( } -|3t )}e pLo (3 a {(Pt - 1) + (p 2 t 2 2 (3 - 2(3 1 + 2)}] (2.61) pLo Equation (2.61) describes the reaction force produced at the fixed end for the prescribed displacement u , ( t ) - at 33 ) Step Function b. At x Lo = (Relaxation Test) B.C.'S , u ( Lo where, 1 ( t ) , t) u o t ) ui^l t ( : = , ( = unit step function : From equation (2.57) and (2.58), we may get ULo u(x,t) x l(t) = (2.62) Lo ULo u, x i (t) (2.63) Lo Substituting equations (2.57) - (2.59) and (2.62) in equation (2.55), we may get the reaction force, EAo F(t) = ( ULo ULo ulo ) + (1 ) G(t) [1 + 2Lo Lo F. ) ( 1 (t)] Lo (2.64) Consider the Kirchhoff stresses. From equation (2.34) ULo S = E ( ULo ) ( + 1 ) ULo rL = prescribed E ru G(t) (2.65) ULo ( Lo a = 2Lo Lo where, G(t) 1 + ) Lagrangian strain due to 2Lo step displacement function end. 34 u, l(t) at the rod . . . III. PROGRAM IMPLEMENTATION OVERVIEW A. are We analysis, (P) : {u} constrained stiffness matrix know we For static nodal displacements vector {u} If analysis. external loads vector where {P} [k] static in load-displacement relationship is as following: [k] = interested problems by {P} or then {u}, using elementary we can analyze the static elastic theory. In general, the static analysis steps are shown as follows: Step Enter geometry of material properties, 1 loads, et al Step 2 Calculate element stiffness matrices. Step 3 Determine appropriate part of [k] and {P}. Step 4 : If [k] or {P} is appropriate, through step Step 5 : If [k] 3 or {P} is not appropriate, constraints Step 6 Step 7 Step 8 : : : then repeat step Determine {u}. Regenerate elements. Calculate element stresses. 35 then apply 2 B. SOLUTION PROCEDURES structural problems, to find the solutions In nonlinear we use the INC and FNR and MNR. in advance what increments In fact, of loads or iteration should be good approximation used to obtain a solution in program application. for elastic structural problems. solution. The strategy subroutines The or visco-elastic time-dependent Here the initial state for incremental was selected as a null displacement vector. loading process Next we consider finite viscoelastic and to the was used in determining the strategy for interactive method were developed it is hard to know element method (FEMEV) programs. for elastic (FEMEL) IN both programs, equilibrium was checked at each stage for solutions obtained from other INC or FNR and MNR. 1 . Algorithm Implementation Consider based on total Algorithm is Step 1. the algorithm Lagrangian of Method. Newton-Raphson Method Full Newton-Raphson as follows: Establish local coordinates X-Y for the element at hand. Step 2. Generate the element stiffness matrix [k] in local coordinates so that it operates on the local D.O.F. Step 3. Transform [k] to global coordinates so that it operates on global D.O.F. Step 4. Repeat step 1 through 36 3 until all elements have been treated. Step 5. Compute displacements. Step 6. Compute internal forces, Step 7. Solve [k] {Au} where, Step 8. uN + = 1 (A p > = uN + [k]{u}. A uN Test for convergence. If not satisfied, Where, convergence return to step = ((ART AR) 0.001 37 1. / (AprAp))**0.5 < 2. Flow Cha rt of the Finite In E lement Program put: geometry, mat. prop; loads 3L PROBLEMTYPE: ELASTA/ISCOELASTIC F" ELE. MATRICES STIFF/LOAD: GLOBASSEMBLY SOLN:FORWARD SOLUTION ELIMINATION MONITOR -r- SOLN:BACKWARD DIAGNOSTICS SUBSTITUTION £ UPDATE DISPS 4/ COMPUTE RESIDUAL FORCES <- NEW LOAD INC/UPDATE DISPS e PRINT RESULTS STOP [Flowchart of the Finite element Program I j NUMERICA L EX AM PLES IV. two consider We numerical examples for elastic and viscoelastic materials. A. ELASTIC MATERIAL 1 • Presc r ibed conce ntrated fo rces . Specify the rod element. / / » Pz / / M L Pii - 22,5000 lbs A = 0.25 [in: L =10 E - [in] 106 [lbs/in2] Consider the stiffness matrix of rod element, the [k] . For rod element, Kn EA r L |_k2i ki 2- [K] (4.1) kc 2 Consider the boundary conditions, ui = Pi = 39 P2 22,5000 at = From loads - x - L displacements relationship, Solve for the displacements, Hence, u2 , we may get LP2 9.0 U2 (in) EA 2 P res cr ibed di . splacem e nts / X / / -I L Consider the same rod the element before, quadratic displacements, u(x), u(x) = 0.02 x - 0.001 x The first derivative is du — 0.0 2 = - 0.002 x dx The second derivative is d'^u = 0.0U2 dx2 40 2 and specified From equation 2.41 ( ) with the constant area, we may getthe distributed loads for any du d2 u f ( X ) = - Ao [ ( 1 + 3.0 * 10-3 dx X2 ds e ) dx 2 = x. + 2 3.06 x - 41 S* ] dr + 530 (4.2) e a o ts -2w 6 fiu -o O 2 T 00009 T 00000 > I 000* O'OOOB saoaoi 42 I 00003 I 0001 00 s 8 O U CO > \ 1 lit oro tot to o too oo o ooo wo S1N3W33VTHSK1 43 coo coo too oot S3 | Q O O CO CO w on fiU o'er 09i 0*21 oe oo S3SS3H1S iJOHHOHIM 44 O'C 00 . B. ) . ) VISCOELASTIC MATERIAL Consider the rod element below, U.(*> F.C*) -> -ya Boundary conditions; u(0,t) u(Lo,t) Lagrangian Stress; T(x,t) Internal Force; F(t) Kirchhoff stress; = S = g(t) T Au g(t) = (t) = u 1 L o L o - A(t) S(t) ( t (4.3) = t) , Ac (x) element solids, the relaxation function is 3 G(t) u = X(x For (t) uo = = - (1 a) OL where, t a e-pt and (4.4) constants are |3 of the material properties Assume unit step function for prescribed displacements. the boundary condition is At that time, u(o, u ( Lo t) t) = , where, We may = u. uo = t) u. u o 1 ( t unit step function l(t): get ( displacements the function with unit step function u ( x , t ) = -- - ul o 1 ( (4.5) t Lo 4 5 . du — ULo l(t) = dx (4.6) Lo And the Kirchhoff stress is specified S = (4.7) E r Substituting equations (4.4) EAo F t ( ) - (4.7) UL UL o Ulo = 1 G t + ( ) 2Lo Lo in equation 1 o where, o + 1 ( t ) S, G(t) (4.9) ULo ULo rLo (4.8) Lo Hence, we may get the Kirchhoff stress, S = r rL (4.3), = 1 + : Lagrangian strain 2Lo Lo due to the prescribed step displacement function u 1 ( t ) at the rod end Equation (4.9) describes the relaxation cases. equation (4.8), we may get the internal forces. 46 From « •4 ! • e • © rr f% • — o II : j— i © a "S w 1 ; i ;' i o : • ! e | : 1 83• 1 1 j i i • • ! : : : : ; : : O \ \ j i »o «< !• I j /- -: o • ../... ': 1 i i i o a; • L^<<~^--- r 1 0' I \y^ ! 61 J r^r 90 I : | e • j- o I . 10 90 NOUOMOJ 47 91 9 fo co NO] [ivxviaa r 10 O'O b CONCLUSION V. study This directed is towards understanding the nonlinear behavior of structures. The formulation based is work. We consider the one based rod viscoelastic effects. incremental process dimensional nonlinear continuumincludes that element To with Newton-Raphson Method. principle of virtual on the the get large displacements and solution we use the full Newton-Raphson and modified Numerical solutions agree well with exact solutions. Further structures, Consideration panels at high studies such of may viscoelastic to two dimensional composite laminated as the extended be effects temperatures is another study to verify of possible plates. composite area of study. An experimental present study is also recommended. 48 the results of the , LIST OF REFERENCES Bergan, Pal "Nonlinear Floating Wales) . 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Thompson, B.S.; Sung, "Variational Formulation C.K., for the Dynamic Viscoelastic Finite Element Analysis of Robotic Constructed Manipulators Mechanisms. Materials," Journal of Automation in Design , v 106 190. 56 n 2, From Composite Transmissions, and Jun 1984, pp. 183- 47. Thompson, Sung, B.S.; Manipulators Intelligent "Design C.K., Using of Robots and Modern Composite Materials", Mechanism & Machine Theory Eighth OSU Conf Appl Mech, - 48. 21 1983, 471 pp. Cook, Oden, 52. Ross, Inc. Limited, 1984. Y.C., Bathe, Analysis 55. Huebner, Engineers 56. . . Ross/Ellis C.T.F. , Horwood in Continuum Mechanics , 1969. and Implementation Academic Press, Inc. , of Finite 1982. Finite Element Procedures in Engineering Inc., 1982. Element Finite The K.H., Methods for . Malvern, C ontinuous 57. Inc., Prentice-Hall, . 1981. Nonlinear Continua of Course J.E., Application K.J., 1974, 1972. First A Element Methods 54. Elements Engineering in Akin, , Inc., Finite Element Programs for Axisymmetric C.T.F., Prentice-Hall, 53. , 1975. Finite J.T., Problems Fung, Inc., John Wiley & Sons, . McGraw Hill, 51. 482. Concepts and Applications of Finite Element R.D., Analysis 50. Sep 19 USA, MO, 1985, , Gallagher, R.H., Finite Element Analysis: Fundamentals Prentice-Hall, 49. - Louis, St. , v 20 n 6 , Introduction L.E., Medium Novozhilov, V.V., of Elasticity , , to Prentice-Hall, Foundations of Gray lock Press, 57 the Inc., Mechanics of a 1969. the Nonlinear Theory 1953. 58. Oden, J.T., Mechanics of Publishing Corporation, 59. Segerlind, L.J., Wiley & Sons, 60. Yang, Prentice- Hall, 1981, , Hemishere 1967. Applied Finite Element Analysis Inc., T.Y., Elastic Structures . John 1976. Finite Inc., Element 1986. 58 Structural Analysis . ,, INITIAL DISTRIBUTION LIST No. 1 Copies Defense Technical Information Center Cameron Station Alexandria, Virginia 22304-6145 2 2. Library, Code 0142 Naval Postgraduate School Monterey, California 93943 2 3. Chairman, Dept of Aeronautics, Code 67 Naval Postgraduate School Monterey, California 93943-5000 4. Prof. Ramesh Kolar, Dept. of Aeronautics, Code 67 Kj Naval Postgraduate School Monterey, California 93943 1 5. Personnel Management Office Air Force Headquarters Sindaebang Dong, Kwanak Gu Seoul, Republic of Korea 2 6. Air force Central Library Sindaebang Dong, Kwanak Gu Seoul, Republic of Korea 2 7. 3rd Department of Air Force College Sindaebang Dong, Kwanak Gu Seoul, Republic of Korea 2 8. Library of Air Force Academy Chongwon Gun, Chung Cheong Bug Do, Republic of Korea 2 9. Kim, Ju-Eon 369-4, Yeon-Nam Dong Ma-Po Ku . . Seoul 10. Lim. , 1 12 Korea Jong-Chun 4 SMC 110 3 Naval Postgraduate School Monterey, California 93943-5000 59 ^h RY MO,: rai SCHOOL ORBTIA 93943-5008 Thesis K417253 c.l Thesis K417253 c.l Kim On analysis of viscoelastic structures. Kim On analysis of viscoelastic structures.

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