AN EXPLICIT COMPUTATIONAL FORMULATION FORFRICTIONAL CONTACT/IMPACT PROBLEMS

X. Zhou∗, D. Sha† and K. K. Tamma‡

Abstract

An efficient formulation for frictional contact/impact problems based on a newly developed decoupledpoint to segment gap vector projection model and employing the so-called arbitrary reference configurationformulation for finite elastic/inelastic deformation and the forward incremental displacement central differ-ence time integration scheme for the equation of motion is developed. The resulting normal and tangentialcontact boundary formulations are easy to solve, and the contact constraints are satisfied exactly. Theoverall formulation is robust and efficient for the numerical simulation of finite deformation elastic/inelasticfrictional contact/impact problems.

1 Introduction

The numerical simulation of frictional con-tact/impact problems involve three major compu-tational aspects: (i) time integration algorithms fornonlinear dynamic system of equations; (ii) stressupdate formulation for the constitutive equations;and (iii) the frictional contact formulation for thefrictional contact boundary conditions. Althoughthe development of this exposition primarily focuseson the frictional contact formulation, the aforemen-tioned three computational aspects closely rely oneach other.

The time integration algorithms for the dynamicsemi-discretized equation can be implicit or explicit.The implicit time integration algorithms such as theNewmark average acceleration method [1] and itsspectrally identical method of mid-point rule methodand the velocity based implicit scheme [2], theHilber-Hughes-Taylor-α method [3], the generalized-α method [4], and the GSSSS family of methods [5]are unconditionally stable and more suitable for thestructural dynamic problems in contrast to the ex-plicit algorithms. The explicit schemes such as thecentral difference method [6], the velocity based ex-plicit method [7], the forward incremental displace-ment central difference (FIDCD) method [8], and thenonlinearly explicit second-order accurate L-stablealgorithm [9] are more suitable for the class of wavepropagation problems in contrast to the implicit al-gorithms. For the situation of impact problemswhere the shock waves induced by the impact areparticipating in the solution, the explicit schemesare favored over the implicit schemes [10]. In thepresent study, we choose the FIDCD method as thetime integration algorithm for integrating the dy-namic equation of motion.

In regards to the computational finite elas-tic/inelastic deformation dynamic problems, the up-dated Lagrangian formulation is a standard and the

mainstream approach [11, 12]. The updated La-grangian formulation is a three configurations formu-lation, and it involves the initial configuration, thecurrent configuration, and the incremental configu-ration. Since the Eulerian description of the con-stitutive model is more appropriate for the finitedeformation problem in contrast to the Lagrangiandescription, more recent efforts [13] are directed toan arbitrary reference configuration (ARC) formu-lation which is degenerated from the Eulerian for-mulation and involve only the current configurationand the corresponding incremental configuration toaccount for the deformation path. A stress updateformulation for hypoelasto-plasticity with the Trues-dell stress rate and the additive decomposition ofthe plastic strain rate based on the ARC formula-tion which is also developed in [13] is incorporatedin the present study as the stress update formulation.

Regarding the contact boundary conditions, var-ious contact algorithms have been developed, suchas the Lagrangian multiplier method [11, 14–16],the penalty method [10, 17–19], the augmented La-grangian method [20, 21], the perturbed Lagrangianmethod [22,23], the mixed method [24], and the com-plementary equation method [8]. Most of the ex-isting contact algorithms are based on the implicittime integration schemes and not suitable for theimplementation of the explicit schemes except thepenalty method [12] and the complementary equa-tion method. When the penalty method is employedalong with the central difference scheme, the impen-etrability condition in the contact normal directionis most likely violated [10], while the complementaryequation method incorporating with FIDCD schemeindeed satisfies the impenetrability condition in thenormal contact direction exactly [8]. Nevertheless,when the widely used point to segment (PTS) gapvector model is employed, the resulting coupled nor-mal contact constraint and the tangential contact

∗Doctoral student, xiangmin@me.umn.edu†Researcher, Army High Performance Computing Research Center (AHPCRC), sha@msi.umn.edu‡Professor and Technical Director, to receive correspondence, Department of Mechanical Engineering/AHPCRC, University of

Minnesota, 111 Church St. SE, Minneapolis, MN 55455, ktamma@tc.umn.edu‡Copyright c©2003 by Xiangmin Zhou, Published by the American Institute of Aeronautics and Astronautics, Inc. with

permission

1American Institute of Aeronautics and Astronautics

44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Confere7-10 April 2003, Norfolk, Virginia

AIAA 2003-1599

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

constraint equations are difficult to solve, since re-sulting coupled contact constraint equations are anunsymmetry operator. Via the point to point (PTP)gap vector model, the normal contact constraint andthe tangential contact constrain can be naturally de-coupled, and the resulting contact boundary con-straint equations are easy to be solved. In the presentstudy, we employ a variational inequality formula-tion for the PTS gap vector and obtain a PTS gapvector projection operator. This is positive definitefor the normal contact forces and also has the pri-mary orthogonality property for any given normaland tangential contact force pair associated with thePTS gap vector. Based on the physical argumentof the PTP gap vector model that the effect of thetangential force away from a given contact point af-fecting the normal contact force of the given contactpoint is negligible, we assume that the resulting PTSgap vector projection operator to be fully orthogo-nal. Thus the contact boundary constraint equationsresulting from PTS gap vector model becomes decou-pled and is easy to be solved as with that from thePTP gap vector model. Thus the complementaryequation method and the return mapping algorithm(or the augmented Lagrangian method) can be read-ily employed for the normal contact constraint equa-tion and the tangential contact constraint equation,respectively.

This exposition is presented as follows. First,the Eulerian governing equation for the finite defor-mation hypoelastic/hypoelasto-plastic problem withthe frictional contact boundary condition are pre-sented and incorporated with the ARC-end stressupdate formulation. Then the FIDCD schemes isbriefly described, and is followed by the developmentof the PTS gap vector projection operator. The re-sults of an explicit formulation for finite deforma-tion frictional contact/impact problems are then out-lined. The numerical example of two identical 3-Dbar impacts by the proposed formulation is comparedwith the degenerated one dimensional exact solu-tion to demonstrate that the impenetrability condi-tions is satisfied. Also the numerical simulation of ahypoelasto-plastic bar impact on rigid base is finallypresented.

2 Governing Equations

The governing equations for the finite deforma-tion frictional contact/impact problem include threecomponents: (i) the dynamic equations of motion,(ii) the constitutive equation, and (iii) the contactboundary conditions. The governing equations areformulated as follows.

2.1 Equation of Motion

Let the open set Ωt ⊂ <3 be the domain of inter-est at the configuration at time t, with the bound-ary ∂Ωt = Γf

t ∪ Γdt , Γf

t ∩ Γdt = ∅, and the closure

Ωt = Ωt ∪ ∂Ωt. For finite strain dynamic problems,the dynamic equilibrium equation of configurationat time t in Eulerian form with Cauchy stress is de-scribed by

∂(ρtvt)

∂t+ ρtηvt −∇ · t = bt in Ωt (1)

tn = f on Γft (2)

xt = xd on Γdt (3)

where ρt is the material density at configuration attime t, vt is the material particle velocity at time t,t is the Cauchy stress at configuration at time t, ηis the viscous damping ratio in the sense of Rayleighdamping, bt is the body force, n is the boundarysurface outward unit vector, xt is material particlecoordinate, Γf

t is the boundary subject to tractionboundary condition, and Γd

t is the boundary subjectto prescribe displacement boundary condition.

2.1.1 Frictional Contact Boundary Con-ditions

When the frictional contact boundary conditions areconsidered along with the dynamic equilibrium equa-tion, the frictional contact boundary conditions canbe treated as traction boundary conditions, i.e., Γf

t =Γf (t) ∪ Γc(t) and ∅ = Γf (t) ∩ Γc(t). The frictionalcontact boundary condition can be decomposed into the normal contact direction and the tangential(frictional) direction, respectively. The impenetra-bility condition and a friction law are employed alongthe normal contact direction and the tangential di-rection, respectively.

2.1.2 Impenetrability condition in thenormal direction

Let n = n = σnk = −niσijnjnk ∈ <3 bethe normal stress on the contact surface, and n bethe normal direction vector on the contact surface,g ∈ <3 be the relative gap vector between point pairson the contact surface. Also let n(x) be the normaldirection at point x ∈ <3 of the surface Γc ⊂ Γc, andn(x) be the normal direction at point x ∈ <3 of thesurface Γc ⊂ Γc. The initial gap vector g0 satisfies,

g0 :=g0 ∈ <3 | g0 = x− x = minz

(z − x),

n(z) · n(x) = −1,

x ∈ <3 on Γc, x ∈ <3, z ∈ <3 on Γc(4)

Let gn and gτ be the normal and tangential direc-tions of the gap vector g,

g = gn + gτ (5)

gn · gτ = 0 (6)

Then, the impenetrable constraint in the normal di-rection on the contact surface is given as,

(n, gn − λcnn)Γc(t) = 0

λcn ≥ 0, ψn(n) = · n ≥ 0

λcnψn(n) = 0, ∀ n ∈ ψn(n) ≥ 0

(7)

2American Institute of Aeronautics and Astronautics

where λcn ≥ 0 represents the impenetrability condi-

tion on the contact surface.

2.1.3 Coulomb friction law in tangentialdirection

Let R = V | V ∈ <3 × [0,∞); then a ”yield func-tion” of the frictional stress vector in the tangentialdirection on the contact surface can be defined by

ψτ (τ ) = µc | n | − | τ |≥ 0 on Γc × [0,∞) (8)

where τ = σtk = −niσik − niσkjnjnk ∈ <3 isthe tangential stress vector on the contact surface,and µc ∈ < is the Coulomb friction coefficient.

Let gτ ∈ <3 be the the relative tangential veloc-ity of the point pairs on the contact surface. Theclassical Coulomb friction law thus is represented by

( τ − τ , gτ ) ≤ 0

τ ∈ ψτ (τ ) ≥ 0

∀ τ ∈ ψτ ( τ ) ≥ 0

(9)

The corresponding weak form employing the La-grangian multiplier is given as

( τ , gτ − λcτ∇στ ψτ (τ ))Γc(t) = 0

ψτ (τ ) ≥ 0, λcτ ≥ 0

λcτψτ (τ ) = 0, ∀ τ ∈ ψτ ( τ ) ≥ 0

(10)

where

∇στ ψτ (τ ) =∂ψτ (τ )

∂τ(11)

2.2 Eulerian Weak Form of Gov-erning Equations

The Eulerian weak form of governing equations in-cluding the dynamic equilibrium equation, consti-tutive equation, boundary conditions, impenetrableand friction constraints are set up at the configu-ration at time t. Define the following set of vir-tual displacement field as, [H1

0 (Ωt)]3 := w | w ∈

[H1(Ω)]3, w = 0 on Γd and the set of virtual ve-locity field as, [H1

d(Ωt)] := v | v ∈ [H1(Ω)]3, v =xd

t on Γd, where H1(Ω) is the first order differen-tiable Hilbert space. Such that the Eulerian weakform of the governing equations are stated as the fol-lowing problem.

Problem 1Let ∆t = t − tα, tmid = 1

2(t + tα). Given the

configurations at time tα and time t as xtα andxt, and the Cauchy stress at time tα as tα , withthe one parameter family assumption for the arbi-trary reference configuration deformation gradient,

∀ w ∈ [H10 (Ωt)]

3, find v ∈ [H1d(Ωt)]

3, such that

(w, ∂(ρv)∂t

+ ρηv)Ωt + (D(w),)Ωt (12)

= (w, b)Ωt + (w,f)Γf (t) + (δg,n + τ )Γc(t)

(D(w),(t))Ωt = (D(w),tαt (13)

+J ttmid

Ftmidt

C : (Etmid)ttα

(F

tmidt )T

−λpC : ∇σF(t))Ωt

(n, gn − λcnn)Γc(t) = 0 (14)

( τ , gτ − λcτ∇στ ψτ (τ ))Γc(t) = 0 (15)

where

∇σF(t) =∂F(t)

∂t(16)

C = λI⊗ I + 2µI (17)

tαt = J tα

t Fttαtα(Ft

tα)T (18)

(Etmid)ttα

= (Ftαtmid

)T Ettα

Ftαtmid

(19)

Ettα

=1

2[(Ft

tα)T Ft

tα− I] (20)

Fttα

= (∇xtαxt)

T =∂xt

i

∂xtαj

(21)

Ftmidtα

= Fttmid

=1

2

I + Ft

tα

(22)

J ttα

= detFttα

(23)

J ttmid

= detFttmid

(24)

F(t) ≤ 0, λp ≥ 0, λpF(t) = 0 (25)

and

λcn ≥ 0, ψn(n) ≥ 0, λc

nψn(n) = 0) (26)

ψτ (τ ) ≥ 0, λcτ ≥ 0, λc

τψτ ( ) = 0 (27)

where λ and µ are the Lame constants, λp is the La-grangian multiplier which serves as the plastic flowvelocity, and ∇σF(t) is the plastic flow direction.

Remark 2.11. The Eulerian weak form governing equations of

Problem 1 are set up at the configuration attime t.

2. Hypoelasto-plasticity in the sense of Truesdellstress rate constitutive equation is employed inProblem 1.

3. An ARC-end stress update formulation is em-ployed in Problem 1.

3 FIDCD Scheme for theEquation of Motion

Setting tα = tn and t = tn+1 = tn +∆t, Problem 1becomes the semi-discretized system of equations attime t, and the stress update part relates the Cauchystress at the configurations at time tn and time tn+1.Thus a time integration algorithm is necessary to re-late the configuration at time tn to the configurationat time tn+1. Here we choose to employ a so-called

3American Institute of Aeronautics and Astronautics

forward incremental displacement central difference(FIDCD) method for integrating the equation of mo-tion because of its inherent advantages [8].

The FIDCD method is a self-starting second-order accurate conditionally stable explicit schemebased on the Eulerian configuration at time tn +∆t [8]. The scheme is listed as follows.Initial phase:

(w, ρt0(2

∆t2∆xt0 − 2

∆tvt0) + ρt0ηvt0)Ωt0

+ (D(w),t0)Ωt0= (w, bt0)Ωt0

+ (w,f t0)Ωf (t0)

+ (δg,n(t0) + τ (t0))Ωc(t0)

(28)

vt0 =2

∆t2∆xt0 − 2

∆tvt0 (29)

Predictor phase:

xtn+∆t = xtn + ∆xtn+∆t (30)evtn+∆t =1

2vtn +

1

4∆tvtn (31)evtn+∆t

= − 1

∆tvtn − 1

2vtn (32)

Solver phase:

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆xtn+∆t)Ωtn+∆t

+ (w, ρtn+∆tevtn+∆t+ ρtn+∆tηevtn+∆t)Ωtn+∆t

+ (D(w),tn+∆t)Ωtn+∆t = (w, btn+∆t)Ωtn+∆t

+ (w,f tn+∆t)Ωf (tn+∆t)

+ (δg,tn+∆tn + tn+∆t

τ )Ωc(tn+∆t)

(33)

Corrector phase:

vtn+∆t = evtn+∆t +1

2∆t∆xtn+∆t (34)

vtn+∆t = evtn+∆t+

1

∆t2∆xtn+∆t (35)

Remark 3.11. The unknown term in the solver phase is

the forward incremental displacement ∆xtn+∆t

which is associated with the displacement attime tn + 2∆t; therefore the frictional contactconstraints are enforced at time tn +2∆t of thedisplacement.

2. The critical time step for the FIDCD methodis same as the central difference method.

4 Treatment of FrictionalContact Boundary Condi-tions

4.1 Forward Lagrangian Formula-tion for Normal Contact Direc-tion

Based on the FIDCD scheme, the normal impene-trability condition of the gap vector can be eitherenforced at time tn + ∆t or at time tn + 2∆t. Whenthe normal impenetrability condition of the gap vec-tor is enforced at time tn + 2∆t on the configurationat time tn+∆t, the resulting formulation is termed asthe forward Lagrangian formulation The forward La-grangian formulation enforces the forward gap vectorat time tn +2∆t for the normal impenetrability con-dition at configuration at time tn +∆t. This is givenby

(n, gtn+2∆tn − (λc

n)tn+2∆tn)Γc(tn+∆t) = 0 (36)

(λcn)tn+2∆t ≥ 0, ψn(tn+2∆t

n ) ≥ 0 (37)

(λcn)tn+2∆tψn(tn+2∆t

n ) = 0

where gtn+2∆tn is the projection of the gap vector in

the normal direction of the contact surface at timetn + 2∆t, and (λc

n)tn+2∆t is the normal gap distanceat time tn + 2∆t. The forward Lagrangian formula-tion represents the normal impenetrability conditionof the contact surface at the configuration at timetn + ∆t. The variational inequality of the normalimpenetrability condition of the forward Lagrangianformulation is computed implicitly via the comple-mentary equation in the solver phase, and the detailare discussed in a later section. In the forward La-grangian formulation, the impenetrability constraintin the normal direction of the contact surface is en-forced at time tn + 2∆; but the dynamic equation ofmotion and the frictional constraint is set up at timetn + ∆t. This feature ensures that the impenetrabil-ity constraint in the normal direction of the contactsurface is satisfied at every time step during the sim-ulation. Hence we have the following theorem:

Theorem 1If the impenetrability of the contact surface is satis-fied for the initial time step, the solution obtained byemploying the forward Lagrangian formulation sat-isfies the impenetrability of the contact surface forevery time step during the simulation time history.

4.1.1 The Coulomb law along the tangen-tial direction

The constraint on the tangential direction only canbe formulated on the configuration at time tn + ∆tsince the resulting velocity needs to be updated tothe corrector phase. The Coulomb friction law andthe sliding velocity in the tangential direction at the

4American Institute of Aeronautics and Astronautics

configuration at time tn + ∆t is given by

( τ , gtn+∆tτ − (38)

(λcτ )tn+∆t∇στ ψτ (tn+∆t

τ ))Γc(tn+∆t) = 0

ψτ (tn+∆tτ ) ≥ 0, (λc

τ )tn+∆t ≥ 0 (39)

(λcτ )tn+∆tψτ (tn+∆t

τ ) = 0

where (λcτ )tn+∆t is the Lagrangian multiplier which

serves as the magnitude of the sliding velocity on thetangential direction of the contact surface based onthe Coulomb friction law. This tangential directionconstraint formulation is an Eulerian formulation.

5 Superposition of the Solu-tion for the Frictional Dy-namic Contact Problems

Consider the following frictional dynamic contactproblem.

Problem 2Find tn+2∆t

n , tn+∆tτ , and ∆xtn+∆t, such that

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆xtn+∆t)Ωtn+∆t

− (δg,tn+2∆tn + tn+∆t

τ )Γc(tn+∆t)

=(w, btn+∆t)Ωtn+∆t + (w,f tn+∆t)Γf (tn+∆t)

− (w, ρtn+∆tevtn+∆t+ ρtn+∆tηevtn+∆t)Ωtn+∆t

− (D(w),tn+∆t)Ωtn+∆t

(40)

and

(n, gtn+2∆tn − (λc

n)tn+2∆tn)Γc(tn+∆t) = 0 (41)

( τ , gtn+∆tτ (42)

−(λcτ )tn+∆tψτ (tn+∆t

τ ))Γc(tn+∆t) = 0

subjected to the constraints:

(λcn)tn+2∆t ≥ 0, ψn(tn+2∆t

n ) ≥ 0 (43)

(λcn)tn+2∆tψn(tn+2∆t

n ) = 0

ψτ (tn+∆tτ ) ≥ 0, (λc

τ )tn+∆t ≥ 0 (44)

(λcτ )tn+∆tψτ (tn+∆t

τ ) = 0

Due to the fact that the equation of motion whichis obtained by employing the FIDCD scheme is alinear equation in terms of the forward incremen-tal displacement ∆xtn+∆t, the principle of superpo-sition can be employed. Hence, we decompose theforward incremental displacement into the followingthree components,

∆xtn+∆t = ∆xtn+∆tf + ∆xtn+∆t

n + ∆xtn+∆tτ (45)

where ∆xtn+∆tf is the solution of the equation

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆xtn+∆t

f )Ωtn+∆t

= (w, btn+∆t)Ωtn+∆t + (w,f tn+∆t)Γf (tn+∆t)

− (w, ρtn+∆tevtn+∆t+ ρtn+∆tηevtn+∆t)Ωtn+∆t

− (D(w),tn+∆t)Ωtn+∆t

(46)

∆xtn+∆n is the solution of the equation

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆xtn+∆t

n )Ωtn+∆t

= (δg,tn+2∆tn )Γc(tn+∆t)

(47)

and ∆xtn+∆tτ is the solution of the equation

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆xtn+∆t

τ )Ωtn+∆t

= (δg,tn+∆tτ )Γc(tn+∆t)

(48)

To solve equations (47) and (48), let

gtn+2∆t = ∇g ·∆xtn+∆t + gtn+∆t0 (49)

where gtn+∆t0 is the initial gap vector for the time

step tn+∆t, ∇g is the projection operator to projectthe forward incremental displacement ∆xtn+∆t tothe Lagrangian gap vector gtn+2∆t which referenceto the configuration at time tn + ∆t. The variationof the gap vector gtn+2∆t is obtained as

δgtn+2∆t = ∇g · δxtn+∆t (50)

and the gap velocity gtn+∆t is given by

gtn+∆t = ∇g · vtn+∆t

= ∇g · ( 1

2∆t∆xtn+∆t + evtn+∆t)

(51)

Substituting the variation of the gap vector intoequations (47) and (48), yields

(w, ρtn+∆t(1

∆t2+ η

2∆t)xtn+∆t

n )Ωtn+∆t

= (∇g · δxtn+∆t,tn+2∆tn )Γc(tn+∆t) (52)

(w, ρtn+∆t(1

∆t2+ η

2∆t)xtn+∆t

τ )Ωtn+∆t

= (∇g · δxtn+∆t,tn+∆tτ )Γc(tn+∆t) (53)

6 The Solution of the Fric-tional Contact BoundaryBased on Point to Seg-ment Gap Vector Projec-tion Operator

Let point xtn+2∆t be the slave node on the contactboundary, namely, xtn+2∆t

s ; and let point xtn+2∆tm be

the projection point of the slave node on the the mas-ter segment of the contact boundary. Thus, basedon the finite element theory, the projection pointxtn+2∆t

m can be described by

xtn+2∆t =X

k

Nk(ξ)(xkm)tn+2∆t (54)

5American Institute of Aeronautics and Astronautics

The gap vector gtn+2∆ts for the PTS gap vector

model is expressed as

gtn+2∆te (xtn+2∆t) = xtn+2∆t

s −X

k

Nk(ξ)(xkm)tn+2∆t

(55)

which can be re-written as

gtn+2∆te (xtn+2∆t) = Gextn+2∆t (56)

where Ge is the projection operator for the con-tactable nodes to the gap vector, and xtn+2∆t ∈Γc(tn + ∆t) consists of the slave node xtn+2∆t

s andthe nodes of the master segment, (xk

m)tn+2∆tm . The

total projection operator [G] consists of the numberof nc element operators of slave-segment and is givenas

[G] =

nc

AeGe (57)

Therefore, the total PTS gap vector is obtained from

gtn+2∆t = [G]∆xtn+∆t+ gtn+∆t0 (58)

The variation of the gap vector gtn+2∆t and the gapvelocity gtn+∆t are obtained as

δgtn+2∆t = [G]δxtn+2∆t (59)

gtn+∆t = [G]vtn+∆t= [G] 1

2∆t∆xtn+∆t + evtn+∆t (60)

Substituting the variation of the gap vector, equa-tions (59) and (60), into the weak form of equations(47) and (48), and choosing w = ϕδxtn+∆t for theGalerkin approximation, yields

(ϕδxtn+∆t, ρtn+∆t(1

∆t2

+ η2∆t

)ϕ∆xtn+∆tn )Ωtn+∆t

= ([G]δxtn+∆t, tn+2∆tn )Γc(tn+∆t) (61)

(ϕδxtn+∆t, ρtn+∆t(1

∆t2

+ η2∆t

)ϕ∆xtn+∆tτ )Ωtn+∆t

= ([G], δxtn+∆t, tn+∆tτ )Γc(tn+∆t) (62)

where δxtn+∆t is the variation of ∆xtn+∆t atpoints xtn+∆t, and ∆xtn+∆t

n and ∆xtn+∆tτ is

for the points xtn+∆t ∈ Γc(tn + ∆t).

If xtn+∆t ∈ Γc(tn + ∆t), for all δxtn+∆t wehave,

[(ϕ, ρtn+∆t(1

∆t2+ η

2∆t)ϕ)Ωϕ ]∆xtn+∆t

n = [G]T tn+2∆t

n (63)

[(ϕ, ρtn+∆t(1

∆t2+ η

2∆t)ϕ)Ωϕ ]∆xtn+∆t

τ = [G]T tn+∆t

τ (64)

Without lose of generality, assume that nc is thenumber of the segments on the contactable bound-ary and one slave node corresponds to one mastersegment. Therefore, tn+2∆t

n and tn+∆tτ in the

above equations consist of nc nodal contact forces,

respectively.

Let

[m∗] = (ϕ, ρtn+∆t(1

∆t2+

η

2∆t)ϕ)Ωϕ (65)

be the lumped mass matrix, thus we have

[m∗]∆xtn+∆tn = [G]T tn+2∆t

n (66)

[m∗]∆xtn+∆tτ = [G]T tn+∆t

τ (67)

Hence for the PTS gap vector model, the variationalinequality for the contact constraints in terms of theforward incremental displacement becomes

n − tn+2∆tn T

[G](∆xtn+∆

n + ∆xtn+∆tτ

+∆xtn+∆tf ) + gtn+∆t

0 i≥ 0 (68)

τ − tn+∆tτ T

1

2∆t[G]∆xtn+∆t

n

+∆xtn+∆tτ + ∆xtn+∆t

f + 2∆tevtn+∆ti≤ 0 (69)

ψn(tn+2∆tn ) ≥ 0 (70)

ψτ (tn+∆tτ ) ≥ 0 (71)

Alternatively, in terms of the contact forcestn+2∆t

n and tn+∆tτ , the variational inequality

become

− tn+∆tn T

[G][m∗]−1[G]T tn+2∆t

n

+tn+∆tτ + [G]∆xtn+∆t

f + gtn+∆t0

i≥ 0 (72)

τ − tn+∆tτ T

1

2∆t[G][m∗]−1[G]T tn+2∆t

n

+tn+∆tτ + [G] 1

2∆t∆xtn+∆t

f

+evtn+∆ti≤ 0 (73)

ψn(tn+2∆tn ) ≥ 0 (74)

ψτ (tn+∆tτ ) ≥ 0 (75)

Definition 1Let tn+2∆t

n i and tn+∆tτ i be the normal contact

force and the frictional force of the ith point-segmentpair of the PTS gap vector model, respectively. Theoperator [A] = [G][m∗]−1[G]T is termed as the pri-mary orthogonal operator for vector tn+2∆t

n andvector tn+∆t

τ if the following holds,

tn+2∆tn T

i [A]tn+∆tτ i = 0 (76)

Lemma 1For the projection operator [G] of the PTS gap vec-

tor model, the operator [A] = [G][m∗]−1[G]T havethe following properties,

1. Operator [A] is positive definite for vectortn+2∆t

n ,

tn+2∆tn T

i [A]tn+2∆tn i > 0

∀ tn+2∆tn i 6= 0

(77)

6American Institute of Aeronautics and Astronautics

2. Let tn+2∆tn i and tn+∆t

τ i be the normalcontact force and the frictional force of theith point-segment pair gap vector, respectively.The operator [A] = [G][m∗]−1[G]T is the pri-mary orthogonal operator for vector tn+2∆t

n and vector tn+∆t

τ ,tn+2∆t

n Ti [A]tn+∆t

τ i = 0 (78)

7 The Decoupled PTS GapVector Model

Introduce the assumption that the normal contactforces of the ith point-segment pair is independent ofthe frictional contact forces of the jth point-segmentpair, such that

tn+2∆tn i[A]tn+∆t

τ j = 0 (79)

With this assumption, the coupled PTS gap vectormodel can be degenerated to the following decoupledPTS gap vector model.

Problem 3Find tn+2∆t

n and tn+∆tτ by the following two

steps.

1. Find the normal contact forces tn+2∆tn , such

that

n − tn+2∆tn T

[A]tn+2∆t

n +egtn+∆t

n i≥ 0 (80)

ψn(tn+2∆tn ) ≥ 0 (81)

2. After the normal contact forces tn+2∆tn

are known, find the tangential friction forcestn+∆t

τ , such that

τ − tn+∆tτ T

1

2∆t[A]tn+∆t

τ +egtn+∆t

τ i≤ 0 (82)

ψτ (tn+∆tτ ) ≥ 0 (83)

where

egtn+2∆tn

= P n

[G]∆xtn+∆t

f + gtn+∆t0

(84)

egtn+∆t

τ = P τ

[G] 1

2∆t∆xtn+∆t

f + evtn+∆t

(85)

and P n is the projection operator to project the gapfunction onto the normal contact direction at theconfiguration at time tn + ∆t, P τ = I − P n is theprojection operator to project the gap velocity ontothe tangential contact direction at the configurationat time tn + ∆t.

Remark 7.1The decoupled PTS gap vector model has the follow-ing features:

1. The operator [A] is positive definite for bothstep 1 and step 2.

2. The constraint is convex in step 1 and concavein step 2, respectively.

3. The nonlinear iterations in step 1 and step 2 areunconditionally convergent regardless the valueof the frictional coefficient µc.

4. The solution of the decoupled PTS gap vectormodel satisfies the constraint approximately.

8 ARC Formulation for Fi-nite Deformation Fric-tional Contact Dynamicwith FIDCD Scheme andDecoupled PTS Gap Vec-tor Projection Model

With the solutions for the normal contact force andthe tangential friction force for the decoupled PTSgap vector model, the ARC formulation for the finitedeformation frictional contact dynamic with FIDCDscheme is readily obtained.

Algorithm 1After the starting step, vtn , vtn , xtn , xtn+∆t, and

tn are known, and vtn+∆t, vtn+∆t, ∆xtn+∆t, andtn+∆t are obtained by the following procedures.

1. Predictor:

xtn+∆t = xtn + ∆xtn+∆tevtn+∆t =1

2vtn +

1

4∆tvtn

evtn+∆t= − 1

∆tvtn − 1

2vtn

2. Perform the contact search and form the gapvector gtn+∆t

0 .

3. Stress update:

tntn+∆t = (J tn+∆t

tn)−1Ftn+∆t

tntn(Ftn+∆t

tn)T

(Etmid)tn+∆ttn

=1

2

n(F

tmidtn

)T Ftmidtn

−hF

tmidtn

(Ftmidtn

)Ti−1

trialtn+∆t = tn

tn+∆t + J tn+∆ttmid

(Ftmidtn

)−1C : (Etmid)tn+∆t

tn

(F

tmidtn

)−T

tn+∆t = trialtn+∆t − λp∇σF(trial

tn+∆t)

F(trialtn+∆t) ≤ 0, λp ≥ 0

λpF(trialtn+∆t) = 0

7American Institute of Aeronautics and Astronautics

4. Solve the dynamic equation without frictionalcontact for ∆xtn+∆t

f ,

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆xtn+∆t

f )Ωtn+∆t

= (w, btn+∆t)Ωtn+∆t + (w,f tn+∆t)Γf (tn+∆t)

− (w, ρtn+∆tevtn+∆t+ ρtn+∆tηevtn+∆t)Ωtn+∆t

− (D(w),tn+∆t)Ωtn+∆t

5. Compute the normal contact force vectortn+2∆t

n ,

tn+2∆tn = max0, ψstick

n n

ψstickn = −m∗

eff

2n ·h∇g(∆xtn+∆t

f )

+ gtn+∆t0

i6. Compute the tangential friction force vectortn+∆t

τ ,

tn+∆tτ = minµc ‖ tn+2∆t

n ‖, ‖ stickτ ‖

stickτ

‖ stickτ ‖

stickτ = −m∗

eff∆t∇τg(1

2∆t∆xtn+∆t

f

+evtn+∆t)

7. Solve the complete dynamic equation for theforward incremental displacement ∆xtn+∆t,

(w, ρtn+∆t(1

∆t2+

η

2∆t)∆ttn+∆t)Ωtn+∆t

+ (w, ρtn+∆tevtn+∆t+ ρtn+∆tηevtn+∆t)Ωtn+∆t

+ (D(w),tn+∆t)Ωtn+∆t = (w, btn+∆t)Ωtn+∆t

+ (w,f tn+∆t)Γ(tn+∆t)

+ (δg,tn+2∆tn + tn+∆t

τ )Γc(tn+∆t)

8. Update the velocity vector vtn+∆t and the ac-celeration vector vtn+∆t,

vtn+∆t = evtn+∆t +1

2∆t∆xtn+∆t

vtn+∆t = evtn+∆t+

1

∆t2∆xtn+∆t

9. Update the reference configuration, the stresstensor,

xtn ← xtn+∆t, vtn ← vtn+∆t

vtn ← vtn+∆t, tn ← tn+∆t

tn ← tn + ∆t

Remark 8.11. The PTP gap vector model can be employed by

the same computational procedures with minorchanges on the formation of the gap vector.

2. With the PTP gap vector model the contactconstraints are satisfied exactly.

3. With the decoupled PTS gap vector model thecontact constraints are satisfied approximately.

9 Numerical Examples

9.1 Two Identical Bars Impact

The numerical study of two identical bars impactingwith each other with an initial velocity of v0 = 202.2is performed. The problem configuration is shown inFigure 1. The dimension of each bar is 1 × 1 × 10and meshed by 4× 4× 20 elements. The initial gapdistance between the two bars is 2g = 0.02. Thematerial properties are given by: Young’s modu-lus E = 3.0E + 7, Poisson ration ν = 0.0, densityρ = 7.337E − 4. The time step is taken as ∆t =0.7274E − 6, and the result is out put every 10 timesteps. The impact time and the releasing time ob-tained by the degenerated 1-D exact solution is givenby timpact = g/v0 and trelease = g/v0 + 2L

pρ/E,

respectively, where L is the length of the bar andL = 10. The displacement and velocity of one of thebar’s at the center of the impact surface is shownin Figure 2 and comparing with the degenerated 1-D exact solution. The displacement result demon-strates that the numerical result of the proposed for-mulation satisfies the normal impenetrability contactconstraint exactly. The velocity result matches thedegenerated 1-D exact solution exactly during theimpact. The oscillation of the velocity after the re-leasing is due to the stress wave induced by the im-pact while the degenerated 1-D exact solution doesnot account for the effect of the stress wave insidethe body.

9.2 Hypoelasto-Plastic Bar Im-pact on Rigid Base

The problem selected is the dynamic impact of a cir-cular bar with length L = 0.8 and circular sectionwith radius r = 0.2. The initial velocity is v0 = 1000.The constitutive model of material is characterizedby the Von Mises yield criterion. The material prop-erties are: Young’s modulus E = 200, Poisson ra-tio ν = 0.28, yield stress σy = 0.2, and hardeningh′ = 100. The density in the initial configurationis taken as ρ = 7.8× 103. Figure 3 shows the initialconfiguration and meshes with 1140 nodal points and810 finite elements . The Figure 4 shows the config-uration at the time t = 264µs. The time step isadaptively calculated by the program as the criti-cal time step of the FIDCD scheme. The exampledemonstrates that the contact boundary conditionsare clearly satisfied when employing the proposedformulation.

8American Institute of Aeronautics and Astronautics

10 Concluding Remarks

An effective explicit formulation for frictional con-tact/impact problems is proposed in the exposition.The formulation employed the FIDCD scheme forthe equation of motion, and also the ARC-end stressupdate formulation for the constitutive equation andincorporating a newly developed decoupled PTS gapvector projection model for the contact boundaryconstraints. The resulting frictional contact con-straint equations obtained by the proposed formu-lation are easy to solve. The normal impenetrabilitycontact constraint and the tangential Coulomb fric-tion constraint is readily satisfied by the proposedformulations. Two numerical examples are shown todemonstrate the robustness and the effectiveness ofthe formulation.

ACKNOWLEDGMENTS

The authors are very pleased to acknowledge sup-port in part by Battelle/U. S. Army Research Of-fice (ARO) Research Triangle Park, North Carolina,under grant number DAAH04-96-C-0086, and bythe Army High Performance Computing ResearchCenter (AHPCRC) under the auspices of the De-partment of the Army, Army Research Laboratory(ARL) under contract number DAAD19-01-2-0014.Dr. Raju Namburu is the technical monitor. Thecontent does not necessarily reflect the position orthe policy of the government, and no official endorse-ment should be inferred. Support in part by Dr.Andrew Mark and Dr. Raju Namburu of the IMTand CSM Computational Technical Activities andthe ARL/MSRC facilities is also gratefully acknowl-edged. Special thanks are due to the CIS Directorateat the U.S. Army Research Laboratory (ARL), Ab-erdeen Proving Ground, Maryland. Other relatedsupport in form of computer grants from the Min-nesota Supercomputer Institute (MSI), Minneapolis,Minnesota is also gratefully acknowledged.

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[4] J. Chung and G. Hulbert. A Time Integra-tion Method for Structural Dynamics With Im-

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[17] J. O. Hallquist. An Implicit, Finite Deforma-tion, Finite Element Code for Analying Staticand Dynamic Response of 2-D Solids. TechnicalReport UCID-19677, Lawrence Livermore Lab,1983.

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Figure 1: Problem configuration

Time(sec)

Dis

pla

cem

ent

0 5E-05 0.0001 0.00015 0.0002-0.02

-0.01

0

0.01

NumericalExact

(a) Displacement

Time(sec)

Vel

ocity

0 5E-05 0.0001 0.00015 0.0002

-200

-100

0

100

200

300

NumericalExact

(b) Velocity

Figure 2: Comparison results of the numerical simulation and the degenerated 1-D exact solution

10American Institute of Aeronautics and Astronautics

Figure 3: Initial configuration and meshes of a bar impacting rigid base

Figure 4: The configuration and meshes of a bar impacting rigid base at time t = 264µs

11American Institute of Aeronautics and Astronautics