ANALYSIS OF ELASTO-PLASTIC STRESS WAVES BY A TIME-DISCONTINUOUS VARIATIONAL INTEGRATOR OF HAMILTONIAN

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1 Modern Physics Letter B World Scientific Publishing Company ANALYSIS OF ELASO-PLASIC SRESS WAVES BY A IME-DISCONINUOUS VARIAIONAL INEGRAOR OF HAMILONIAN SANG-SOON CHO, HOON HUH School of Mechanical, Aerospace & System Engineering, KAIS,5,Gwahangno, Deadoek Science own, Daejeon, KOREA, ss00@kaist.ac.kr, hhuh@kaist.ac.kr KWANG-CHUN PARK Department of Aerospace Engineering Sciences and Center for Aerospace Structures, College of Engineering and Applied Science, University of Colorado, Campus Box 49, Boulder, CO 8009, USA, kcpark@colorado.edu Received 5 June 008 Revised June 008 his paper proposes a numerical algorithm of a time-discontinuous variational integrator based on the Hamiltonian in order to obtain more accurate results in the analysis of elasto-plastic stress wave. he algorithm proposed adopts both a time-discontinuous variational integrator and spacecontinuous Hamiltonian so as to capture discontinuities of stress waves. he algorithm also adopts the limited kinetic energy to enhance the stability of the numerical algorithm so as to solve the discontinuities such as elastic unloading and internal reflection in plastic deformation. Finite element analysis of one dimensional elasto-plastic stress waves is carried out in order to demonstrate the accuracy of the algorithm proposed. Keywords: elasto-plastic stress waves; time-discontinuous variational integrator; Hamiltonian, limited kinetic energy. Introduction A study about the dynamic response of materials under impact or blast load is a crucial research topic in many engineering applications such as crashworthiness of vehicles, explosive devices used in aerospace and military industry. When impact or blast load is imposed to a machine, stress waves are propagated through a medium and fracture or damage occurs at a structural weak point such as crack. herefore, understanding of the stress wave propagation is very important for the reliable and safe design of machines. Stress waves propagate through a medium rapidly when dynamic forces are applied for very short period of time. he analytical prediction of the propagation of stress waves is very difficult in practical problems since the exact solution of the governing equation Corresponding Author.

2 S. S. Cho, H. Huh and K.C. Park becomes complicated to consider characteristic phenomena such as discontinuities. In spite of such difficulties, many researchers have progressed valuable studies by various methods in recent years. One of remarkable efforts is the numerical simulation using the finite element method, which faces two typical difficulties currently during the numerical procedure. he first difficulty is to control the dispersive and dissipative errors induced at the discontinuous or the singular domain. And the second difficulty is to preserve the shape of stress waves with the small wave length. his paper is concerned with the analysis of elasto-plastic stress waves by a timediscontinuous variational integrator based on Hamiltonian in order to obtain more accurate computational solution by reducing the dispersive and dissipative errors. he algorithm proposed adopts both time-discontinuous variational integrator and spacecontinuous Hamiltonian so as to capture discontinuities of stress waves. his algorithm adopts the limited kinetic energy to enhance the stability of the numerical algorithm so as to solve the discontinuities such as elastic unloading and internal reflection in plastic deformation. Finite element simulation of one dimensional elasto-plastic stress waves is carried out in order to demonstrate the accuracy of the algorithm proposed.. Formulation for ime-discontinuous Variational Integrator of Hamiltonian.. Hamilton s principle Hamilton presented in 84 that the following action integral t S= N L qt (, qt (, t dt, L = ( V ( t0 ( whose functional variation, δ S = 0, yields the equation of motion, where q denotes the generalized coordinates. In the Eq. ( is the kinetic energy, V is the potential energy and L is the Lagrangian. he Hamiltonian is expressed via Legendre s transformations from the Lagrangian as H ( q, p; t = pq L( q, q, t ( where p is the generalized momentum which can be expressed in Eq. ( as L p = ( q From which, one obtains the calculated Hamilton s equation as H H q =, p = (4 A fundamental property of the Hamiltonian, H, can be observed from Eq. (5. H H H = p q = 0 (5

3 Analysis of Elasto-Plastic Stress Waves by a ime-discontinuous Variational Integrator of Hamiltonian he equation indicates that the Hamiltonian, H, is constant for all t, which must hold for the long-time responses. 5 he Hamilton's principle shown in Eq. ( and (4 is an alternative formulation of the differential equations of motion for a physical system as an equivalent integral equation, using the calculus of variations. he principle is also called the principle of stationary action. Although formulated originally for classical mechanics, the Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields and has even been extended to quantum mechanics, quantum field theory and criticality theories... Finite element formulation of the Hamiltonian In order to solve the stress wave propagation in one-dimensional space, the Hamiltonian is discretized with the assumption that the generalized coordinates, q, and generalized momentum, p, are continuous in the space domain. he displacement and velocity fields can be express as Eq. (6 and (7, respectively. ux ( = Nq = N N q q (6 vx ( = NV = N N v v (7 Discretized kinetic and potential energy in the continuous domain, Ω, can be defined as follows: = Ω v ρv dω (8 ( ( V = U W = Ω u C u dω( Ωu fdω Γu t dγ (9 Substituting Eq. (8 and (9 into Eq. (, the Hamiltonian can be discretized as: H( qp, ; t = V = ( U W ( d d = pm p qkq Ω q ΩNfΩ Γ NtΓ h (0 where p, M and K Ω denote the momentum, the mass and the stiffness matrix, respectively. Each matrix is defined as follows: p = MV M = Ω N ρndω ( K = N C NdΩ Ω Ω.. ime-discontinuous Variational Integrator of Hamiltonian A variational integrator is applied in this paper based on the Hamiltonian shown in Eq. (0 in order to derive the algebraic finite element equation in the discretized time domain. It is assumed that the displacement and the velocity are discontinuous at the

4 4 S. S. Cho, H. Huh and K.C. Park discretized time domain. At first, the action integral of the k th discretized time domain can be defined theoretically from the integration of the Lagrangian as shown in Eq. (. 4 t k Δt Sk = lim t ( ( 0 k ε V dt ε pq pm p q t k ε lim t ( V( dt 0 k ε ε pq pm p q Δt pq Δ pm pδ q p q t tv( lim t 0 k ε ε tk he action integral shown in Eq. ( provides the first order accuracy. For the k th discretized time domain, the action integral can be approximated as ( ( ( Sˆ k p p q q Δ t V p q q k k k k Λ ( k k k where and V are the kinetic and potential energy which is expressed as Eq. (4 and Eq. (5, respectively. ( k k ( k k = ( α α ( α α = p p M p p (4 V( q V q ( q V ( q q ± α = : Gaussianintegration point 6 k k k k k Internal force at k th discretized time domain can be derived from the potential energy shown in Eq. (5 as follows: V F K q q K q q int q = = ( ( k k k 6 k k k V F K q q K q q int q = = ( ( k 6 k k k k k A limited kinetic energy, which is denoted as Λ, is newly introduced in this paper to the action integral as shown in Eq. ( in order to suppress the abrupt oscillation of the stress waves at discontinuity. he limited kinetic energy is defined as φ Λ function ( p p M ( p p k k k k S = φm ( p p k k (7 k S = φm ( p p k k k ( (5 (6

5 Analysis of Elasto-Plastic Stress Waves by a ime-discontinuous Variational Integrator of Hamiltonian 5 0 v L Velocity(m/s -5-0 λ=0.5xl (a (b Fig. One-dimensional stress wave propagation: (a schematic description; (b profile of the imposed velocity. Finally, the algebraic finite element equation is obtained in the discretized time domain as expressed in Eq. (8 by taking the variation of the action integral shown in Eq. (. 4 ime N ˆ δ Sk = 0 k = 0 F MV MV MV F int ext F MV MV = F U = U int ext = 0 0 Δt Δt U = U ( 4φ V ( 4 φ V,0 φ < 4 (8. Numerical Analysis of Elasto-Plastic Stress Wave Propagation A time-discontinuous variational integrator of Hamiltonian is applied to the numerical simulation of the elasto-plastic stress wave propagation. Wave propagation in onedimensional bar is considered as shown in Fig.. he length of the bar is 00 mm and is discretized with 600 elements. he Young s modulus and the density of a bar are assigned as 00 GPa and 8000 kg / m, respectively. A linear hardening model is used to p describe the flow stress of the bar as σ = 0. 0 ε [GPa]. he right end side of the bar is entirely fixed and a velocity boundary condition depicted in Fig. (b is imposed on the left end side. A constant velocity of 0 m/s is imposed on the left end side and the velocity boundary condition is suddenly eliminated at the time of 5μsec to describe the loading unloading condition. 6 In order to evaluate the efficiency of the method proposed, the results obtained from the proposed time-discontinuous variational integrator of Hamiltonian are compared with those obtained from the conventional finite element method of the continuous Galerkin method with the same finite element discretization. Fig. shows the stress waves configurations at two different times: t =5 and t =0μs. After an elastic wave followed by a plastic wave is generated, an unloading elastic wave is propagated if the loading is eliminated abruptly. Because elastic wave speed is larger than the plastic wave speed, the unloading elastic wave catches the loading plastic wave. hen two new elastic waves are

6 6 S. S. Cho, H. Huh and K.C. Park 5 μs 0 μs 5 μs 0 μs (a (b Fig. Simulation result for one-dimensional stress wave propagation: (a conventional finite element method; (b time-discontinuous variational integrator of the Hamiltonian. generated and propagated to the both sides. his is called by the internal reflection. 6 Fig. (a shows the stress profiles obtained from the conventional finite element method. Stress waves are severely oscillating due to the dispersive and dissipative errors induced at the discontinuous region such as the elastic unloading and the internal reflection. he reason is that the conventional finite element method cannot eliminate those two errors arise from the local truncation error of approximation. he oscillation is remarkably reduced as shown in Fig. (b when the time-discontinuous variational integrator of the Hamiltonian is utilized in the numerical simulation. he comparison indicates that the time-discontinuous variational integrator of the Hamiltonian is appropriate in the numerical simulation of the stress wave propagation. 4. Conclusion A time-discontinuous variational integrator of the Hamiltonian is newly proposed in order to obtain more accurate results in the analysis of stress wave propagation. he proposed algorithm adopts both a time-discontinuous variational integrator and a spacecontinuous Hamiltonian so as to describe discontinuities of stress waves in the numerical simulation with reduced dispersive and dissipative errors. Elasto-plastic numerical simulations for one-dimensional wave propagation are carried out in order to evaluate the efficiency of the proposed algorithm. he results indicate that dispersive and dissipative errors, which appear in the continuous Galerkin method, are remarkably reduced in the proposed algorithm using a time-discontinuous variational integrator of the Hamiltonian. References. W. J. Kang and H. Huh, Int. J. Automotive echnology, ( M. A. Meyers, Dynamic Behavior of Materials (John Wiley & Sons, Inc., New York, I. Harari,Wave Motion 9 ( B. Cockburn, Z. Angew. Math. Mech. 8 ( J. M. Wendlandt, J. E. Marsden, Physica D, 06 ( H. Kolsky, Stress Waves in Solids (Dover Publications, Inc., New York, 96.