Impact response of flying objects modeled by positional finite eement method

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Impact Response of Flying Objects Modeled by Positional

Finite Element Method

João Paulo de Barros Cavalcante*_{, Daniel Nelson Maciel}†,§ and Marcelo Greco‡

*Civil Engineering Department Federal University of Rio Grande do Norte

Lagoa Nova, Natal 59078-900, Brazil †_{School of Sciences and Technology} Federal University of Rio Grande do Norte

Lagoa Nova, Natal 59078-900, Brazil ‡_{Structural Engineering Department}

Federal University of Minas Gerais

Av. Presidente Carlos, Belo Horizonte 31270-901, Brazil §_{dnmaciel@ect.ufrn.br}

Received 1 December 2016 Accepted 20 September 2017

Published 25 October 2017

This paper analyzes the dynamic response of space and plane trusses with geometrical and material nonlinear behaviors using di®erent time integration algorithms, considering an alternative Finite Element Method (FEM) formulation called positional FEM. Each algorithm is distinguished from each other by its speci¯c form of position, velocity, acceleration and equilibrium equation concerning the stability, consistency, accuracy and e±ciency of solution. Particularly, the impact problems against rigid walls are analyzed considering Null-Penetration Condition. This formulation is based on the minimum potential energy theorem written according to the nodal positions, instead of the structural displacements. It has the advantage of simplicity when compared with the classical counterparts, since it does not necessarily reply on the corotational axes. Moreover, the performance of each temporal integration algorithm is evaluated by numerical simulations.

Keywords: Contact; °ying object; impact; nonlinearity; positional formulation; stability; temporal integration.

1. Introduction

The main purpose of this paper is to evaluate the performance of di®erent algorithms of time integration (explicit and implicit) for nonlinear problems through an alter-native Finite Element Method (FEM) formulation. This formulation, called positional FEM, considers nodal positions (coordinates) as variables instead of displacements. Vol. 18, No. 6 (2018) 1850076 (18 pages)

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c World Scienti¯c Publishing Company DOI:10.1142/S0219455418500761

Int. J. Str. Stab. Dyn. 2018.18. Downloaded from www.worldscientific.com

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In nonlinear formulations for contact/impact, the aim is to determine the con-tact/impact forces as well as the ¯nal body con¯guration. The materials usually have relatively complex behavior, leading to the adoption of simpli¯ed models which results in approximated solutions. The nonlinearities associated with dynamic im-pact problems are also complex, even considering the simplest mathematical de-scription. Therefore, the numerical simulation of contact/impact in the study of deformable bodies has attracted the attention of researchers and remarkable progress in the computer ¯eld has been achieved during recent years, particularly using the FEM, and this progress implies increasing e±ciency and accuracy of the numerical solutions.

In terms of numerical methods applied to problems involving impact, the study by Hughes et al.1showed a signi¯cant contribution regarding approximations by ¯nite elements in the application of bars and shells considering elastic problems without friction. Also, Kuo et al.2 _{presented a time integration by using the momentum} equation where it showed good response for discontinuous applied loads for nonlinear problems with large displacements and rotations. Li et al.3_{highlight the importance} of dynamic response and failure process in a structural system, caused by discon-tinuous initial conditions.

The dynamic equilibrium equation can be conveniently modi¯ed, where the impact problems are approached through the combination of temporal integra-tion algorithms with the Lagrange multiplier in order to obtain better accuracy for the estimation of contact forces.4,5_{It is also possible to describe the problem} through variational formulations, expressing the Lagrange multipliers in terms of velocity and acceleration, combined with an e±cient contact algorithm. Hu6 proposed a formulation based on Lagrange multipliers ensuring the null-pene-tration condition, considering a modi¯ed Newmark algorithm. In most cases, the constraint condition is imposed by using Lagrange multipliers7–9 _{or penalty} function.10–12

The Lagrange multiplier technique is widely used in frictionless problems.13,14 Landenberg and Elzafrany15_{apply penalty functions using discrete elements on the} contact surfaces. The study of Laursen and Love16 _{presents an energy restoring} scheme, by enforcing the impenetrability condition and conserving the total body's energy along the contact instant. In both cases, restrictions have been modi¯ed with the purpose of achieving total energy conservation of the mechanical system.

An impact event is characterized by instantaneous velocity changes at the contact surface. Numerous authors have used velocity changes induced by impact to improve the results at the contact instant.1,17,18_{Some contact descriptions}19–21_{consider the} dynamic impact without friction, under the hypothesis of a conservative mechanical system, expanding the framework, to incorporate the conserving mathematical idealization of a frictionless surface interaction.

In respect to frictional contact/impact problems, several studies have presented formulations and solution algorithms.22–28 _{Complex models of friction presenting}

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nonlinear behavior at the contact surfaces are found in Oden and Martins29 _and Wriggers et al.30

Carrazedo and Coda31 _{studied the thermomechanical coupling in nonlinear} pro-blems for trusses impact with positional FEM. The thermomechanical behavior is also extensively studied on impact problems.32–34

The FEM based on nodal positions is developed using Total Lagrangian de-scription and the principle of minimum potential energy. In this paper, the impact without friction is considered between lattice structures and rigid wall.37,38_The so-called scheme of null-penetration condition is applied to restrict the impact surface.31 The performance of the following algorithms is also analyzed in this work: Hou-bolt, Newmark modi¯ed, Wilson-, Central Di®erences, Chung and Lee,37_{Souza and} Moura.38 Each algorithm di®ers from each other by its speci¯c way of updating equations of positions, velocities and accelerations and the balance equation. The Newmark modi¯ed method proposed by Hu6_{is the classical method with}_{¼ 1:5 and} ¼ 1:0. All algorithms are evaluated by numerical applications.

2. Adopted Kinematics and Strain Calculation

Equations (2.1), (2.2) and (2.3) describe the kinematics of the lattice element in the space, as shown in Fig.1. The element geometry is mapped as a function of a dimensionless variable (ranging from 0 to 1). As illustrated in Fig.1, the pair of coordinates (X0₁; Y₁0; Z₁0) and (X₂0; Y₂0; Z₂0) is the initial con¯guration of the element and the pair (Xt1; Y1t; Z1t) and (Xt2; Y2t; Zt2) de¯nes the current con¯guration of the element.

Fig. 1. Space truss element.

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x¼ X1þ ðX2 X1Þ; ð2:1Þ

y¼ Y1þ ðY2 Y1Þ; ð2:2Þ

z¼ Z1þ ðZ2 Z1Þ: ð2:3Þ

The nonlinear engineering strain is herein applied. In truss kinematics, only axial strains are considered. By considering an in¯nitesimal axial ¯ber, ds₀ and ds are its initial and ¯nal length respectively, then one has

" ¼ds ds0 ds0

: ð2:4Þ

Also, in terms of dimensionless variableðÞ, it is possible to write Eq. (2.4) as " ¼ds=d ds0=d

ds₀=d : ð2:5Þ

The terms ds=d and ds0=d described as a function of variable are de¯ned by the following equations: ds0 d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx d 2 þ dy d 2 þ dz d 2 s ! 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX2 X1Þ2þ ðY2 Y1Þ2þ ðZ2 Z1Þ2 q ¼ l0; ð2:6Þ ds d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx d ₂ þ dy d ₂ þ dz d ₂ s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðX2 X1Þ2þ ðY2 Y1Þ2þ ðZ2 Z1Þ2 q ¼ l; ð2:7Þ

where l0 and l are the initial and ¯nal lengths, respectively. 3. Nonlinear Positional Formulation

For the static problems in a conservative system, the total potential energy function (_{) is described by two terms of energy, which is the total strain energy (U}e) and the potential energy of the applied force (P ) according to the Eq. (3.1).

¼ Ue P: ð3:1Þ

As shown in expression (3.2), the total mechanical energy functional for dynamic problems is de¯ned by the addition of kinetic energy (KC) and energy dissipation (KA) in Eq. (3.1). KA is due to mechanical system damping.

¼ Ueþ KCþ KA P: ð3:2Þ

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The strain energy is de¯ned by the integral of the speci¯c strain energy (ue) along the initial volume:

U_e¼ Z

V0

u_edV₀: ð3:3Þ

As shown in Eq. (3.4), the potential energy of the applied forces is expressed as a function of the applied external forces (Fi) and respective positions (Xi).

P ¼ F_i_X_i: ð3:4Þ

Also, the kinetic energy is given by KC ¼

Z V0

x:ix:i

2 dV0: ð3:5Þ

Substituting Eqs. (3.3), (3.4) and (3.5) into (3.2), one has ¼ Z V0 uedV0þ Z V0 x:ix:i 2 dV0þ KA FiXi; ð3:6Þ where @KA @Xi ¼ Z V0 @KA @Xi dV₀¼ Z V0 c_mx:_idV₀; ð3:7Þ

where cm is the damping coe±cient.

This energy function is rewritten substituting the exact position ¯eld by its approximation described in Sec.2, i.e.:

¼ Z V0 u_eð; X_iÞdV₀þ1 2 Z V0 x:2 ið; XiÞdV0þ KAð; XiÞ FiXi: ð3:8Þ

Thus, the position of dynamic equilibrium is de¯ned using the minimum potential energy principle, by di®erentiating Eq. (3.8) regarding the generic nodal position XS, resulting in @ @xS ¼ Z V0 @ueð; XiÞ @xS dV₀þ Z V0 x:ið; XiÞ @x:ið; XiÞ @xS dV₀þ@KAð; XiÞ @xS FS ¼ 0: ð3:9Þ Substituting Eq. (3.7) into (3.9), we have

@ @xS ¼ Z V₀ @ueð; XiÞ @xS dV0þ Z V₀ x:ið; XiÞ @x:ið; XiÞ @xS dV0 þ Z V0 cmx:ið; XiÞdV0 FS¼ 0: ð3:10Þ

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In vector form, Eq. (3.10) can be written as follows: g¼@Ue

@xS

þ Finertþ Fdamp Fext¼ 0: ð3:11Þ In Eq. (3.11), the vector of inertial forces is F_inert, the force due to damping is F_damp and the external force is F_ext, while the term @U_e=@x_S characterizes the nonlinear material term.

Due to the nonlinear expression (3.10) or (3.11), it is di±cult to obtain a precise solution directly, hence the dissipation of the residual forces requires a numerical strategy. A convenient way of solving nonlinear problems is to linearize the equation and then to apply an iterative correction algorithm to ensure the solution convergence. For this purpose, the well-known Newton–Raphson procedure is used in this work.

While the external loads are independent of residual forces, the equation of equilibrium con¯guration (3.10) or (3.11) is obtained by Taylor expansion regarding Xi, and it is used as follows:

gðXkþ XkÞ ¼ 0 ﬃ gðXkÞ þ @gðXkÞ

@Xk

þ O2_: _ð3:12Þ

Neglecting higher-order terms O2_{, one writes} Xk¼ @gðXkÞ @Xk ₁ gðXkÞ; ð3:13Þ where @g @Xk ¼ Z V0 @2_u eð; XiÞ @Xk@xS dV0þ Z V0 @x:ið; XiÞ @Xk @x:ið; XiÞ @XS þ x:ið; XiÞ @2_x: ið; XiÞ @Xk@XS dV0þ Z V0 cm @x:ið; XiÞ @Xk dV0 FS ¼ 0: ð3:14Þ Expression (3.14) is so-called Hessian Matrix. The strategy to solve the Eq. (3.11) for a speci¯c time step is ¯rst to determine gðXkÞ to an arbitrary Xk through Eq. (3.11). Secondly,@g=@Xk is obtained through Eq. (3.14) and ¯nallyXk using Eq. (3.13), updates Xk. Then, it feeds Eq. (3.11) and repeats the iterative process until a given value of_Xk is su±ciently small, within a given required tolerance. Moreover, the dynamic nonlinear problem is solved through the combination of the iterative Newton–Raphson procedure with temporal integration schemes. In this study, it is considered uniform mass distribution along elements. Nonuniform mass distribution and mass eccentricity can be found in Lei et al.39

4. Necessary Algebraic Step

The total strain is given by the sum of the elastic and plastic terms according to the following equation:

" ¼ "e_{þ "}p_: _ð4:1Þ

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The strain energy considering the nonlinear material behavior is written as follows: Ue¼ Z V₀ Z "d"dV0¼ Z V₀ Z " E"d" Z " E"pd" dV0 ¼ Z V0 1 2E" 2_E""p dV0: ð4:2Þ

Considering Eqs. (2.6) and (2.7), the total strain energy expression (Eq. (4.2)) is divided into an integral of length and area, as shown below:

Ue¼ l0 Z 1 0 1 2EA" 2_E Z A ""p_dA d ¼ l0 Z 1 0 ued: ð4:3Þ

Using index notation, the ¯rst- and second-order derivatives of the total strain energy are de¯ned, respectively, by

@Ue @Xi ¼ l0 Z 1 0 ue;id; ð4:4Þ @2_U e @Xi@XK ¼ l0 Z 1 0 ue_;ikd: ð4:5Þ

According to Greco et al.,40 _{the explicit determination of the derivatives of} ¯rst and second orders of the strain energy is necessary to calculate the residual force vector as well as the Hessian matrix. Recalling Eq. (4.3) and performing the appropriate considerations, one has

l0ue¼ EAl0 2 ffiffiffiffi B p l0 1 !₂ l0E Z A ffiffiffiffi B p l0 1 ! "p_dA_l 0E Z A ffiffiffiffi B p l0 1 ! "_dA_; ð4:6Þ where B¼ ðX2 X1Þ2þ ðY2 Y1Þ2þ ðZ2 Z1Þ2: ð4:7Þ The ¯rst derivative of Eq. (4.6) relative to the nodal parameter i (ranging from 1 to 6) is de¯ned by l0ue;i¼ EA 2l0 1 l0ffiffiffiffi B p B_;i l0E Z A ";i"pdA l0E Z A ";i"dA; ð4:8Þ ";i¼ B_;i 2l0 ffiffiffiffi B p : ð4:9Þ

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The expression (4.10) denotes the derivative of Eq. (4.8) relative to the nodal parameter k (ranging from 1 to 6), given by

l0ue;ik¼ EA 2l0 l0B_;iB_;k 2ðpffiffiffiffiBÞ3 þ 1 l0_ffiffiffiffi B p B_;ik " # l0E Z A ";ik"pdA l0E Z A ";ik"dA; ð4:10Þ ";ik¼ 1 2l0 B_;ik ffiffiffiffi B p B;iB;k 2ðpffiffiffiffiBÞ3 ! : ð4:11Þ

The terms involving "p _and " _{are calculated only when the element exhibits} plastic and thermal behavior, respectively.

As shown in Greco et al.,40_Table₁_{presents the derived B}

;i; B;k and B;ik. 5. Numerical Results and Discussion

5.1. Impact of a rod on a rigid wall

This example, presented by Armero and Petocz,14 _{consists of the uniaxial impact} of a bar (with a constant velocity) against a rigid wall, as shown in Fig.2. The geometrical and material characteristics are dimensionless and given by: E¼ 1; L ¼ 1; A¼ 1; ¼ 1 and ¼ 0:05. The time discretization is 500 time steps of t ¼ 0:01. The bar is discretized by 20 ¯nite elements.

Table 1. Derivatives of variable B. B_;1¼ 2ðX2 X1Þ B;11¼ 2 B;41¼ 2 B_;2¼ 2ðY2 Y1Þ B;14¼ 2 B;44¼ 2 B;3¼ 2ðZ2 Z1Þ B;22¼ 2 B;52¼ 2 B;4¼ 2ðX2 X1Þ B;25¼ 2 B;55¼ 2 B;5¼ 2ðY2 Y1Þ B;33¼ 2 B;63¼ 2 B_;6¼ 2ðZ2 Z1Þ B;36¼ 2 B;66¼ 2

Fig. 2. Impact of a bar against a rigid wall: Problem de¯nition.

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Only the right free-edge of the bar su®ers impact (see Fig.2). Figures3 and4 present the numerical solution of velocity and contact force of the impacted node.

As expected, the impact causes a velocity jump in the impacted node as observed in Fig.3. In the ¯rst segment of this ¯gure, it is also observed for explicit algorithms, especially the method of Central Di®erences, much lower oscillations on impact when compared with the implicit ones. As a consequence of being a multistep method, the velocities obtained by Houbolt method provide high perturbations near the impact instant. The method of Chung and Lee and the Newmark method present similar behavior, with a short jump of velocities during the impact instant and quickly converging to the analytical response. The method of Moura and Souza presents lower oscillations, however, it takes more time than the Newmark method to dissipate these oscillations.

Fig. 3. Impacted node velocity response.

Fig. 4. Contact force response.

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The Wilson- method presented the worst performance, and although it is not visible after the impact, the numerical response has small oscillations around the analytical response, decreasing over time. With the exception of the Wilson- method, the results of all methods coincide in the second segment. The Houbolt and Wilson- algorithms have better performance in the third segment of Fig.3 (when compared with the analytical response).

In Fig.4, very high spurious oscillations are shown for Houbolt and Wilson- responses, emphasizing their ine±ciency related to solving impact problems. Based on Figs.3 and4, it is possible to conclude that explicit methods and the Newmark method are more convenient for solving impact problems. The analytical solution, based on conservation of momentum and re°ection velocity, is the average velocity of the body instead of the right free-edge one.

5.2. Unidirectional impact between two bars

In this example, which is studied in Carpenter et al.,3_{the case of impact between two} identical bars with the same initial velocity (Fig.5) is analyzed, however, moving in opposite directions. The geometrical and material characteristics of both elements are given by: E¼ 207 GPa, A ¼ 645 mm2and ¼ 7844 N s2/m4.

The problem analysis is carry on taking advantage of its symmetry, according to Fig.6. The dynamic response of the contact forces and velocities of the impacted nodes are investigated.

The dynamic response is obtained for two di®erent time steps_{t ¼ 2:010}6s and t ¼ 0:5106s. The bar is discretized by 20 ¯nite elements and the initial distance between the bars is equal to ¼ 0:0508 mm. Plasticity is considered in the analysis. The parameter of isotropic hardening is K¼ 6; 9 GPa with Y ¼ 69 MPa is adopted.

Fig. 5. Problem description for one-dimensional impact example.

Fig. 6. Impact of a rod on a rigid wall. Problem de¯nition.

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In Figs.7and8, the velocity of the impact point for the two di®erent time steps oscillates around5,13 m/s after the separation of the bars. This occurs due to the oscillatory motion presented by the bar after the impact, which is not represented by the simpli¯ed analytical solution. It is also observed in Figs.7 and8 that the impact causes a velocity jump at the contact node. Moreover, the explicit algorithms, especially the method of Central Di®erences, eliminate oscillations at the instant of impact when compared with the implicit algorithms, therefore representing results closer to the analytical response.

(a) ∆t = 0.5 10–6s (b) ∆t = 2.0 10–6s Fig. 7. Horizontal velocity of the impacted node (geometric nonlinearity only).

(a) ∆t = 0.5 10–6_s _(b) _∆_{t = 2.0 10}–6_s Fig. 8. Horizontal velocity of the impacted node (geometrical and material nonlinearities).

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The Newmark method presents similar behavior to explicit ones, while spurious oscillations occur in Houbolt and Wilson- algorithms, especially after the re°ection of the bars, and the response comes with high numerical oscillations compared with the analytical one.

When comparing the results of Figs.7 and8, it is observed that the methods of Houbolt and Wilson- have high perturbations at the moment of impact due to the e®ects of the material nonlinearity. Therefore, due to energy loss, the bar contact time is larger than the elastic case and its velocity return (see Fig.8). In addition, after their separation, the bars move in opposite directions, in relation to the initial movement, with oscillations of the velocity amplitudes.

Similar to the case of velocity, the methods of Newmark, Central Di®erences and Chung and Lee presented practically identical contact forces, whereas the response of methods Houbolt and Wilson- diverged from the analytical response, as shown in Figs.9and10.

In addition, for the bar with elastoplastic behavior, the contact forces are smaller than in the elastic case as well as the acceleration levels and the contact time.

5.3. Impact between truss and rigid wall

In this example, the impact of a circular truss against a rigid wall (Fig.11) is ana-lyzed. The distance between the structure and the rigid wall is 0:10 m. Plasticity is considered in the analysis. The material yield stress is Y ¼ 100 MPa and the hardening kinematic model parameters are H¼ 500 MPa and K ¼ 0. The other

geometrical and material characteristics of both elements are given by:

E¼ 210 GPa, A ¼ 0:0036 m2 and ¼ 7850 kg/m3. All elements of the structure move initially at a constant velocity of 20 m/s.

(a) ∆t = 0.5 10–6_s _(b) _∆_{t = 2.0 10}–6_s Fig. 9. Contact force of the impacted node (geometric nonlinearity only).

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Responses are obtained for two nodes, as highlighted in Fig.11. The numerical response of displacements is shown for two di®erent time steps (_{t ¼ 0:0005 s and} t ¼ 0:00005 s).

Comparing Figs. 12and13, it is noticeable that the results of the temporal in-tegration methods tend to converge for a smaller time step. Here, again, the Houbolt

(a) ∆t = 0.5 10–6_s _(b)_∆_{t = 2.0 10}–6_s Fig. 10. Contact force of the impacted node (geometrical and material nonlinearities).

Fig. 11. Impact of the truss against a rigid wall: Problem de¯nition.

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and Wilson- diverge when compared with other methods. In both cases, the results of the methods Central Di®erences and Newmark are practically identical, while the method of Chung and Lee has small numerical damping controlled by a free damping parameter.

The explicit methods and the Newmark one exhibit re°ection at time 0.035 s and re-impacted near the instant 0.12 s. The Houbolt and Wilson- methods are impacted only once. The strain levels of explicit methods and Newmark are larger, therefore,

(a) Node 1 (b) Node 2

Fig. 12. Horizontal displacement (t ¼ 0:0005 s).

(a) Node 1 (b) Node 2

Fig. 13. Horizontal displacement (t ¼ 0:00005 s).

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the energy stored and dissipated is also larger. Moreover, a smaller contact time is observed.

Figure14shows the deformation of the structure in di®erent instants of time. Due to the accumulation of permanent deformation, the structure does not return to its initial con¯guration.

6. Conclusions

A simple and e®ective alternative formulation to deal with the dynamic nonlinear truss structures based on the minimum potential energy written regarding nodal positions has been successfully applied to impact problems. The formulation includes both geometric and material nonlinearities and a comparison between six di®erent time integration algorithms has been performed, that is, Houbolt, Newmark modi-¯ed, Wilson-, Central Di®erences, Chung and Lee,37_{Souza and Moura.}38

The obtained results are considered satisfactory for most of the implemented algorithms. The numerical algorithms used by positional formulation have appro-priate characteristics of consistency, convergence and stability according to their peculiarities. Nevertheless, the use of methods Chung and Lee, Souza and Moura,

(a) t=0.00005s (b) t=0.04s (c) t=0.08s

(d) t=0.12s (e) t=0.16s (f) t=0.2s Fig. 14. Deformed con¯guration.

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Newmark and Central Di®erences is highly recommended for impact analysis, while Houbolt and Wilson- presented unsatisfactory responses for such problems.

An important detail related to the impact scheme herein applied is the necessity, in some cases, of considering a small_{t for avoiding undesirable numerical damping} which can result in phase error. Numerical simulations have demonstrated the merits of the proposed treatment.

Acknowledgments

The authors would like to thank the Brazilian Research Funding Agencies CAPES (Coordination of Improvement of Higher Education Personnel), CNPq (National Council for Scienti¯c and Technological Development) and FAPEMIG (Foundation for Research Support of Minas Gerais State) for the ¯nancial support.

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