COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 40 (1983) 327-360NORTH-HOLLAND

A NUMERICAL ANALYSIS OF A CLASS OFPROBLEMS IN ELASTODYNAMICS WITH FRICTION*

l.A.C. MARTINS and l.T. aDENThe University of Texas at Austin, Austin, TX 78712, U.S.A.

Received 25 May 1983

1. Introduction

This study is concerned with the numerical analysis of a class of problems in elastodynamicsin which friction effects are taken into account. Finite element methods are used herein toanalyze the dynamic behavior of linearly elastic bodies which have a part of their boundariessubjected to friction and to prescribed normal tractions. Error estimates are derived, analgorithm for the solution of two-dimensional problems is presented, and several numericalexperiments are described.

The numerical scheme developed is based on the variational formulation originally in-vestigated by Duvaut and Lions [121. These authors established a variational principle, in theform of a variational inequality, for this class of problems and they were also able to proveexistence and uniqueness of the solutions.

This class of problems is a subclass of general types of problems involving the response ofan elastic body that has come in contact with other elastic bodies or rigid foundations alongrough dry surfaces. This general class of problems remains one of the most difficult in solidmechanics. The issue of existence and uniqueness of solutions, even in the absence of friction,is still open.

These problems are inherently nonlinear since a priori it is not known which part of theboundary of a body is, at each instant, in contact with another body or a foundation. Inaddition, it is also not known in advance which parts of the actual contact surfaces are slidingor are adherent. As a consequence, unilateral contact conditions and friction conditions thatinvolve unknown stresses, displacements, and velocities must be imposed on the (candidate)contact surfaces.

Additional difficulties arise from the hyperbolic nature of the equation of linear momentumbalance to be solved. In fact, the impact between portions of the boundaries of elastic bodies isexpected to produce propagating stress and velocity discontinuities. Other discontinuities arealso expected to occur if the sliding and adhesion on the contact surfaces is governed by thediscontinuous Coulomb's friction law.

Owing to these difficulties, the few analytical solutions found in the literature. applicable

* Research sponsored by the Air Force Office of Scientific Research (ASFC) under contract F-49620-83-C-0058.One author (l.A.C.M.) also expresses gratitude for support from the Funda~iio Calouste Gulbenkian. Lisboa, andthe Instituto Superior Tecnico of the Universidade Tecnica de Lisboa for a portion of his work on this project.

0045-7825/83/$3.00 © 1983. Elsevier Science Publishers B.V. (North-Holland)

328 J.A.c. Martins. J.T. Oden, Class of problems in elastodynamics

only for specific situations and simple geometries, are obtained under simplifying assumptions,commonly the absence of friction or the complete adhesion on the contact surface. Surveys ofthe literature on these types of solutions as well as of those obtained with various numericalmethods may be found in the papers by Kalker [16, 17].

Several authors have attempted the numerical solution of dynamic contact problems usingfinite element methods. Among them it is important to mention the work by Hughes et al. [14]who presented a finite element formulation for linearly elastic contact-impact problems validfor perfect frictionless or perfect adhesion conditions on the contact surface. The componentsof the traction vector at the contact nodes are taken as additional unknowns of the problem.The Newmark method or the central-difference technique are used to integrate the resultingequations of motion. Careful procedures, compatible with wave propagation theory, are usedto enforce linear momentum balance when the impact or the release of the contact nodesoccurs. Several successful applications to frictionless problems are described and a goodagreement with analytic results is shown in bar impact problems. An extension of this work tolarge deformations is presented by Hughes, Taylor and Kanoknukulchai [15].

Applications of the finite element method to solve a formally derived variational inequalitywhich governs the dynamic behavior of linearly elastic bodies in unilateral contact withfrictionless supports were presented by Lee and Kamemura [19]. Panagiotopoulos [25] andTalaslidis and Panagiotopoulos [30] also derived, formally, variational inequalities for thedynamic analysis of solids in unilateral contact with foundations. These foundations areassumed to satisfy a general holonomic monotone constitutive law which is designed to modelrigid, elastic or plastic situations and also friction conditions. In the applications, the problemssolved involve linearly elastic bodies in unilateral contact with elastic foundations withoutfriction or with prescribed friction stresses.

Although not directly concerned with the unilateral contact problems in elasticity, severalworks have been published on related areas which are worth mentioning here: the book ofGlowinski, Lions and Tremolieres [13] in which the authors discuss the numerical solution ofvariational inequalities, in particular those corresponding to evolution problems; the papers ofSchatzman [28, 29], Amerio [1, 2], Amerio and Prouse [3] and Citrini [10, 11] in whichexistence and uniqueness of solution for several problems with vibrating strings with impact onobstacles is proved; the papers of Cabannes [5, 6] also on vibrating strings but now subjectedto friction; and the works of Lotstedt [20-22] in which the dynamics of systems of rigid bodieswith unilateral constraints with or without friction is numerically analyzed.

In the present investigation, a formal mathematic statement of the class of problems studiedherein is done in the beginning of the Section 2. In the same section, the correspondingvariational formulation and the major results on existence and uniqueness of solutions areestablished. There it is shown that the frictional effects are included in the variationalformulation through a non-differentiable functional which represents the virtual power of thefriction forces. As a consequence of this non-differentiability, which is associated with thenon-conservative friction forces, the variational principle has the form of an inequality. Thestudy then turns to the construction of a regularized problem in which an approximation of thefriction functional by a convex differentiable functional is introduced. This regularizationprocess involves a family of perturbed regularized variational principles which have the formof an equation. Choosing a particular form for the regularized friction functional, it is provedat the end of Section 2 that the sequence of solutions of the regularized problems converges,

. \

...

..

J.A.c. Martins, 1.T. Odell, Class of problems in elastodynamics 329

as the perturbation parameter e tends to zero, to the solution of the original problem in thesense of the deformation and kinetic energy norms.

Section 3 is concerned with the discretization of the regularized problem. The finite elementmethod is used for the spatial discretization and an a priori error estimate for the convergenceof the semi-discrete approximation of the regularized problem is derived.

In Section 4, time discretizations using Newmark's and central-difference techniques aredescribed and an algorithm for the solution of two-dimensional problems is presented. Thensome numerical applications are described. These include an analysis of an elastic slabsubjected to impact or harmonic forces and to prescribed normal tractions over a portion of itsboundary whereon frictional forces can be developed. The resulting displacements, velocities.and friction stresses obtained with different time integration schemes are presented anddiscussed.

2. A class of contact problems in elastodynamics

2.1. Formal statement of the problem

The class of contact problems considered in this study is characterized by the followingsystem of equations and inequalities:

(1) Linear momentum equation:

a'ij(tl),i + fi = pili in n x (0, T) .

(2) Boundary conditions:

(a) prescribed displacements

Ui = Vi on r0 x (0, T) ;

(b) prescribed tractions

(c) prescribed normal tractions on the contact boundary

(d) friction conditions on the contact boundary

IuTI < vlFnl ~ lh= 0,

IuTI = VIFnl ~ 3 A >0: lh = -AUT.

(3) Initial conditions:

(a) initial displacements

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

'30 J.A.c. Marlins, 1.T. Gden, Class of problems ill elastodynumics

tI = tlo in fl. for 1 = 0 ;

(b) initial velocitics

ti = til in fl. for ( = 0 .

(2.6)

(2.7) . ,

In this system, the following conventions and notations werc used:fl.: an open bounded Lipschitz domain in R N (with N typically 2 or 3) representing thc

.nterior of a linearly elastic body. Points (particles) in n with Cartesian coordinates Xj.

I:::;; i :::;;N, relativc to a fixcd coordinate frame are denoted x (= (x" Xz, ... , XN)) and thcvolumc measure by dx (=dxl dxz' .. dXN )).

(0, T): the open time intcrval over which the motion of the body is described, (0, T) c.lR.tI = (lill liz, ••. , liN): the displacement vector; tI = tI(x, 1) is a function defined on n x [0, T].(Tjj: Cartesian components of the stress tensor. The notation (Tjj(tI) in (2.1) signifies that thc

CTij are to be given in terms of u or gradients of u by the constitutive equations for thematerial, as is made clear below.

/;: components of the body force, assumed to be given functions defined in fl. x [0, T].p: mass density of the material of which the body is composed. It is assumed to be a given

function of x satisfying thc following conditions:

P E L"'(fl.), p (x) ~ po > 0, Vx E fl. .

r = to U I\U Fe: smooth boundary of the body.r D (rF): portions of the boundary on which thc displaccmcnts (tractions) are prcscribed.re: the contact surface where the normal tractions arc prescribed and the friction condition

are to be satisfied. It is assumed that Fe n to = 0.U/: componcnts of the prescribed displacement assumed to be givcn functions defined on

rn x [0, T].1;: components of the surface traction assumed to be given functions defined on rF x [0, T].(Tjjllj = (Tollj + (TTl: Ilj are the components of the unit vector outward and normal to the boundary

f; (Tn is the normal stress on the boundary, whose value at a displacement u is (To(u) = (Tjj(u) I1jllj

and (TTi are the components of the stress vector tangent to r, whose value at a displacement u, isthus

Fn: prescribed normal tractions on the contact boundary, assumed to be given functions ' .jefined on re x rO, T].

tiT: the tangential velocity on the boundary; tiT = Ii - linll; lin = ti • 11 is the normal velocity')n the boundary.

v: friction cocfficient of the contact boundary; assumed to be a given function of x'iatisfying the following conditions:

p(x) ~ Po> 0, Vx E r..

J.A.C. Martins, J.T. Oden. Class of problems in elaslodynamics 331

uo(u(): initial displacements (velocities) of the body, assumed to be given functions definedin n.

Throughout this study, standard indicial notation and the summation convention areemployed. Superposed dots C) indicate differentiation with respect to time and commas ( ).kdenote partial differentiation with respect to Xk. Thus,

au;lij =Tt, au;

Uj.k = aXk ' etc.

We confine our attention to problems for which the material body n is linearly elastic andthe stress is determined by the gradients of displacement by the generalized Hooke's law.

ajj(u)=E;jklUk.l. (2.8)

Here E ;jkl are the elasticities of the material and are assumed to satisfy the followingconditions:

E;jkl = Ejjkl = Ejj1k = Elkjj, Ejjkl E L"'(n) ,

I~Vk~r;'N liE ijk 1 II", ~ M, (2.9)

E;jklAjjAkl ~ nzA;jAjj for every symmetric array Ajj (= Ajj). M, III > 0 .

It is clear that in (2.3)-(2.5) we mean

Some comments on the friction conditions (2.5) are in order. Physically, these conditionsdictate that a particle of the body on the contact surface will have no movement if themagnitude of the tangential stress is smaller than a critical value equal to the product of thefriction coefficient and the magnitude of the normal stress. However, if the magnitude of thetangential stress reaches this critical value, the particle will move and the tangential stress willhave the same direction of the tangential velocity but an opposite sense. Roughly speaking.this implies the following: if a particle on the contact boundary has no tangential movement(constant displacement and zero velocity and acceleration) the friction stress acting on it willbe the stress required to obtain the static equilibrium of the particle on the tangential plane; ifthe particle moves on the tangential plane, the frictional stress will have a known magnitudeand its direction and orientation are obtainable from the kinematic variables which aregoverned by the dynamic equations of motion of the particle.

We also note that, for simplicity, we shall not distinguish between the so-called static andkinematic friction coefficients and these are, therefore, assumed herein to have the samevalue.

332 J.A.c. Martins, J. T. Oden, Class of problems in elastodynamics

2.2. Variational formulation of the problem

A variational formulation of the problem introduced in the previous section has beenpresented by Duvaut and Lions [12] and will be reviewed briefly in this section together withresults on existence and uniqueness of solutions.

To simplify notation, we shall, without loss in generality, assume momentarily that p == 1and that Vi = 0 so that Uj = 0 on r0 for all IE (0, T).

In addition to the assumptions and the notations previously introduced, the followingdefinitions are now needed:

V: the space of admissible displacements (velocities)

a(u, v): virtual work (power) produced by the action of stresses (J'jj(u) on the strains (strainrates) e··(v) = -2

1 (v .. + v·):/J I.J J./

(2.10)

where u is a displacement and v is a displacement (velocity).j(v): virtual power done by the frictional forces on the velocity v:

where g = vlFnl.(r/J, v): virtual power done by the external forces on the velocity v:

(ljJ,v)= ( f·vdx+ ( t'y(v)ds+ ( FnYn(v)ds.In JrF Jrc

(2.11 )

(2.12)

In these definitions, HI(D) is the usual Sobolev space of classes of functions with L 2-partialderivatives of order 1; y is the trace operator mapping (H1(fl)t onto (H1/2(fl)t which maybe decomposed into normal components Yn(V) and tangential components Yrtv):

(2.13)

Here, for smooth functions v and a well-defined unit normal to r,

In (2.12), ( ... ) denotes duality pairing on V' x V where V'is the topological dual of the spaceV which is equipped with the norm 1I·lb that will be defined below. To give meaning to theintegrals in (2.12), it is sufficient to assume that for each t E [0, T], fi E L2(D), Ii E L2(rF), andFn E e(rc). We shall elaborate on the assumed regularity of these data later in this section.

J.A.c. Martins, J. T. Oden, Class of problems in elastodynamics 331

IlO(UI): initial displacements (velocities) of the body, assumed to be given [unctions definedin n.

Throughout this study, standard indicial notation and the summation convention areemployed. Superposed dots C) indicate differentiation with respect to time and commas ( ),kdenote partial differentiation with respect to Xk. Thus,

au;ll; = at' au; etc.= ,U;.k iJXk

We confine our attention to problems for which the material body n is linearly elastic andthe stress is determined by the gradients of displacement by the generalized Hooke's law,

(2.8)

Here E;jkl are the elasticities of the material and are assumed to satisfy the followingconditions:

1"~k~':N IIE,jklll.., ~ M, (2.9)

ElikIA;jAkl;:;a: mA;jAij for every symmetric array Aij (=Aj,), M, m >0.

It is clear that in (2.3)-(2.5) we mean

an( ll) = E ijkl U k.1tlillj ,

aT; = aT;(ll) = aIJ(u) Ilj - a'o(ll) tI; = E;lkIUk.11l1 - EmlkIUUnmlljl1;.

Some comments on the friction conditions (2.5) are in order. Physically, these conditionsdictate that a particle of the body on the contact surface will have no movement if themagnitude of the tangential stress is smaller than a critical value equal to the product of thefriction coefficient and the magnitude of the normal stress. However, if the magnitude of thetangential stress reaches this critical value, the particle will move and the tangential stress willhave the same direction of the tangential velocity but an opposite sense. Roughly speaking.this implies the following: if a particle on the contact boundary has no tangential movement(constant displacement and zero velocity and acceleration) the friction stress acting on it willbe the stress required to obtain the static equilibrium of the particle on the tangential plane; ifthe particle moves on the tangential plane, the frictional stress will have a known magnitudeand its direction and orientation are obtainable from the kinematic variables which aregoverned by the dynamic equations of motiol1 of the particle.

We also note that, for simplicity, we shall not distinguish between the so-called static andkinematic friction coefficients and these are, therefore, assumed herein to have the samevalue.

332 J.A.C. Martins, J.T. Oden, Class of problems in elastodynamics

2.2. Variational formulation of the problem

A variational formulation of the problem introduced in the previous section has beenpresented by Duvaut and Lions [12] and will be reviewed briefly in this section together withresults on existence and uniqueness of solutions.

To simplify notation, we shall, without loss in generality, assume momentarily that p = 1and that Vi = 0 so that Ui = 0 on To for all IE (0, T).

In addition to the assumptions and the notations previously introduced, the followingdefinitions are now needed:

V: the space of admissible displacements (velocities)

a(u, v): virtual work (power) produced by the action of stresses (Tjj(ll) on the strains (strainrates) e·(v) = 2!(V .. + v .. ):u ~/ ~l

where u is a displacement and v is a displacement (velocity).j(v): virtual power done by the frictional forces on the velocity v:

j(v) = f gIYT(V)ldsJrcwhere g = vlFnl.

<f/I, v): virtual power done by the external forces on the velocity v:

(f/I,v)=l f·vdx+ r t'y(v)ds+ f FnYn(v)ds.n J~ J~

(2.10)

(2.11 )

(2.12)

In these definitions, HI(D) is the usual Sobolev space of classes of functions with L2-partialderivatives of order 1; Y is the trace operator mapping (H1(f1))N onto (HII2(f1))N which maybe decomposed into normal components Yn(v) and tangential components y-r(v):

(2.13)

Here, for smooth functions v and a well-defined unit normal to T,

In (2.12). ( . , . ) denotes duality pairing on V' x V where Viis the topological dual of the spaceV which is equipped with the norm 11'1\, that will be defined below. To give meaning to theintegrals in (2.12), it is sufficient to assume that for each IE [0, T], f E L2(fl), Ii E L2(TF), andFn E L2(rc). We shall elaboratc on thc assumed rcgularity of these data later in this section.

J.A.c. Martins, J.T. Odell, Class of problems in elastodynamics 335

The conditions listed in Theorem 2.1 deserve some elaboration. It is not difficult to establishthat conditions (2.18) hold whenever

fi, j;, /; E L2(L2(a)) ,

ti, ii, i; E L2(L2(rF)),

Pm Fn, Fn E L2(L2(rc)).

(2.24)

However as g = vlPnl is assumed to be independent of time, and since v is also independent oftime, we must have Fn = Fn = O. Physically, conditions (2.21) and (2.22) mean that the initialdisplacements of the body must be such that no tangential stresses on the contact boundaryare required to 'equilibrate' that initial state and, accordingly, the initial tangential velocities onthe contact surface must be zero.

2.3. A perturbed regularized variational problem

The technique used in this study to approximate problem (2.17) was suggested by the proofof Theorem 2.1. Although the details need not be presented here, it is convenient tosummarize the major steps of the proof of the existence of solution in order to introduce aperturbed regularized variational problem and to introduce some intermediate results of thatproof that wi\1 be used subsequently.

The first step in the proof of Theorem 2.1 consists of considering a convex regularization ofthe friction functional j, i.e., a convex and Gateaux differentiable functional j" : V ~ R given by

(2.25)

where E is a real positive parameter and j" approaches j as E ~ O.With this regularized functional j", a solution for the following approximate regularized

version of problem (2.17) is sought:

Find the function t ~ u,,(~) of [0, T} ~ Vsuch that Vt E (0, T] and Vv E V,

(2.26)'with the initial conditions

Here (Dj,,(u,,(t)), v) is the Gateaux derivative of j" at u,,(t) in the direction v. It is importantto note that, due to the Gateaux differentiability of the functional j" the variational statementof this problem is now an equation and not an inequality as (2.17) was.

One proof of the existence of a solution u. to (2.26) involves the construction of a sequence

336 J.A.c. Martins, J.T. Oden, Class of problems in elastodynamics

of finite-dimensional approximations of (2.26) in finite-dimensional subspaces Vm of V andthen proving that unique solutions of these finite-dimensional problems exist and are boundedby constants independent of m. From this it follows that some subsequence of these solutionscan be found that converges in a weak sense to a solution of (2.26). An application of thistechnique to the corresponding frictionless problem in elastodynamics is given in [12, pp.127-130]. We must then establish that such solutions are uniformly bounded in e in ap-propriate norms, and this, in particular, involves showing that there exist constants Lt, L2,

L3 > 0 (independent of e and t) such that for every t E [0, T],

(2.27)

(2.28)

This means that UI! and Ill! exist in bounded sets of L "'(V) and iis exists in a bounded set ofL"(L2(fl)N). Consequently there exists a subsequence ue,. of Ue such that as e -+0:

U e,. -+ u, ti e,. -+ U weakly star in L"(V) ,

iie,.-+ ii weakly star in L"«L2(fl»N).

The final step of the proof consists of showing that the function U obtained in this mannersatisfies the variational inequality (2.17).

2.4. A regularization of the friction functional

The regularization is of the frictional functional i used in this work is the same used byCampos, Oden and Kikuchi [8] in the corresponding elastostatics problem with Coulomb'sfriction and prescribed normal tractions on the contact boundary. Their study employed thefollowing choice of the function r/1e in (2.25):

{I vTI- !e , for IvTI > E ,

r/1e (IVTI) = 1 22e IVTI, for IVTI ~ E,

for given e > 0, where, for simplicity, VT denotes 'Yr(v). It can be easily seen that with thischoice of r/1e, is is indeed convex and Gateaux differentiable on all V for all E > O.

The Gateaux derivative of is at U in the direction v is given by

(2.29)if IUrI>E,

The following lemma essentially proved in [26, pp. 39-501, establishes some properties of isthat are useful in subsequent developments (see also [7, pp. 18, 26]).

J.A.c. Martins, J.T. Oden, Class of problems in elastodynamics 337

(./2

"""""/

Ixrl ,,/"",,

//

""//,

"/,/

Fig. I. Graphs of the functions 1/1. and c/J •.

LEMMA 2.2. Let j, je: V -+ R be given by (2.11) and (2.25), respectively, with t/J~ 01.10 '}'T: V ~L I(rd given by (2.28) and gEL OC(Td. g ~ 0 a.e. Then for all u, v E V

(i)

(ii)

(iii)

(iv)

(2.30)

(2.31 )

(2.32)

(2.33)

These results furnish us with sufficient tools to establish the following theorem on theconvergence of the solutions u£ of the regularized problem (2.26) to the solution of (2.17) as E

tends to zero.

THEOREM 2.3. Let u be the solurion of problem (2.17) and u£ be the solurion of (2.26) for fixedE > 0 with t/Jc in j, given by (2.28). Then, there exists a constant c > 0 independent of E and t suchthat, for every t E In, T],

Ilti,(I) - li(t)II~ + a(u£(t) - u(t), u,(I) - u(t)) ~ CE . (2.34 )

338 J.A.c. Martins, J. T. Oden, Class of problems in elastodynamics

\1oreover, if meas(r D) > 0, there exist constants a, C > 0 independent of £ and t such that, for~very t E [0, T],

II,i,,{t)- u(t)II~+ allu.(t)- u(t)II~~ C£. (2.35)

PROOF. Setting v = u,(t)- u(t) in (2.26) and v = ue(1) in (2.17) and subtracting yields the'ollowing inequality.

(u,(t)- u(t), u,(t)- ti(t)) + a(u.(t)- u('), li,(t)- u(t))~

~j(li,(t)) - j(u(t)) - (Dj.(u, (t)), u,(t)- li(t)

Nhere we have used the fact that ii(t) and ii.(t) are in (L2(fl)t by virtue of Theorem 2.1.;ince js is convex and Gateaux differentiable

(Dj, (u, (t)), u(t) - u, (t) ~ j, (u(t)) - j, (Ii, (t))md we have

~~ [IIu,(1)- li(t)1I5+ a(u,{t)- u(t). u,(t) - u(t))] ~

~[j(Ii, (t)) - j£ (Ii. (t)) 1 - [j (ti (t) - j,(Ii (t))] .

Now, using the estimate (2.31) of Lemma 2.2 to obtain an upper bound for the secondnember of this inequality and integrating in time from 0 to t, the following inequality is)btained:

Illi,(t)-li(t)II~+ a(ue(t)- u(t), u,(t) - u(t))~

~ lIu,(O) - u(O)II~+ a(u,(O) - u(O), u,(O) - u(O)) + 4(meas rdllgliL-(rd£ .

The assertion (2.34) follows immediately from this result and the initial u,(O) = u(O) = Un

md u,(O) = u(O) = UI and the fact that t is bounded by T.Finally, (2.35) follows from (2.34) by applying the V-ellipticity property (2.16) of the bilinear

'orm a ( . , . ).

REMARK 2.4. The estimate (2.35) in the previous theorem implies that, for meas(r d> 0,

U, -+ U strongly in L""( V) ,Ii, -+ U strongly in L""«L 2(fl)t)·

3. Approximation and numerical analysis

1.1. Finite element approximation

We shall now consider semi-discrete finite element approximations of (2.26). The formal)rocedure is standard: we partition ti into a mesh of finite elements over which piecewise

J.A.c. Martins, J. T. Oden, Class of problems in elastodynamics 339

polynomial approximations of the displacement field u. at each time t are introduced. Forconforming elements, this process can lead to the construction of a family {Vh} of finite-dimensional subspaces of the space V of admissible displacements. Here h denotes the meshparameter, typically the largest element diameter in a given mesh, and members of the family{Vh} are generated by quasi-uniform refinements of the mesh.

In order to study certain qualitative features of the approximation, we assume that thesubspaces Vh are endowed with standard asymptotic interpolation properties as h --+ 0 (see,e.g. [9,23]). In particular, if the shape functions forming the basis of Vh contain completepolynomials of degree ~K and if a vector-valued function u is given in (Hm (n))N n V, m > 0,then there exists a constant C, independent of u and h, and an element Vh E Vh such that

Ilu - vhlls:;:;; ChJlllullm, µ = min(K + 1- s, m - s) (3.1)

where II· lis denotes the norm on the Sobolev space (HS (J2)t of order s. In addition, it isassumed that n and the spaces Vh are such that the traces of functions on the boundary Tsatisfy the estimate

(3.2)

where 11·llp,Fe denotes the norm on the pth order (fractional) Sobolev space (HP(Tc))N and p isa real number possibly negative (see [4]).

A typical member of V" is a vector v" with components of the form

Vhi(X, t) = L vf(t) cP,(X)1

x E ii, t E [0, T] where cPt denote basis functions spanning Vh and vf(t) are the values of Vh atnodal point 1. Thus, time derivatives of Vh are

Vhi = L vf(t) cPl(X) .1

Vh; = L vf(t) cPl(X),1

etc.With these approximations the semi-discrete finite element approximation of the

regularized variational problem (2.28) is characterized as follows:

Find the function t-+ u~(t) of [0, T]-+ Vh

such that 'V t E (0, T] and 'V vh E Vh ,

(ii~(t), vh)+ a(u~(t), vh) + (Dj,,(li~(t)), vh) = (ljI(t), vh)

and u~ satisfies the initial conditions(3.3)

(u~(O). vh) = (uo, vh

),

Obviously, u~(t), u~(t), ii~(t) (E Vh): tih --+ IRN are the approximate regularized displacements,velocities and accelerations at time t.

340 J.A.c. Martins, J.T. Ode". Class of problems in elastodynamics

(3.4)

Let now Nh denote the number of nodes of the finite elements mesh. Then the variationalproblem (3.3) is clearly equivalent to the following dynamical problem in RNXN~:

Find the function t ~ r(t) of [0, T) -+ R NxN~

such that V t E (0, T]:

Mr(t) + J(;(t») + Kr(t) = R(t)

and r(t) satisfies the initial conditions

r(O) = Po, r(O) = PI.

Here we have introduced the following matrix notations:ret), ;(t), r(t): the column vectors of nodal displacements, velocities and accelerations at

time t.M: the standard consistent mass matrix.K: the standard linear elastic stiffness matrix.R(t): the vector of consistent nodal applied forces (body forces, prescribed tractions on rF

and prescribed normal tractions on rc) at time t.J(;(t)): the vector of consistent nodal friction forces at time t.po, PI: the initial nodal displacements and velocities, respectively.

Clearly, r(t) depends upon the parameters E and h. The system of equations (3.4) is anonlinear system of second-order ordinary differential equations, the nonlinearity deriving, ofcourse, of the velocity dependent vector of friction forces J, the components of which aregiven by

K = 1,2, ... , N:' , i = 1,2, ... , N (3.5)

whereN~: the number of nodes on the contact boundary rc;CP~I: the ith component of the restriction to rc of the (global) basis function of Vh

associated with the Kth node of rc;cP,,: the nonlinear function defined in (2.29).Due to the form of cP", the vector J may be written as the sum

J(;(t» = C(;(t)) r(t)+ FU(t))

where the components of C and F are given by:

Ph( ) i -I.h u~n(t) d ( ")J/ t = - g'PJ; I .h ()I s no summatIon on t ,rc(» U"r t

J. L = 1,2, .... N;" i, k = 1,2, ... , N

(3.6)

(3.7)

(3.8)

J.A.c. Martins, J. T. Odell. Class of problems in elastodynamics 341

where Fd:!;;;) and Fd» are the portions of Fe where Ili~T(t)I:!;;; E and IU~T(t)1 > E. respectively.The vector F is the vector of the friction forces associated with the points on Fe where thetangential velocity has modulus greater than E ('slipping' occurs) and its dependence on thetangential velocity is only with respect to the direction and orientation, at each point of rc, ofli~(t)liu~(t)l· The matrix C is a damping matrix associated with the points on Fe where thetangential velocity has its modulus lower or equal to E (the points that are ·stuck').

Hereafter, we confine our attention to plane problems (N = 2), and we consider two typesof finite element approximations: four-node, bilinear (Ot) elements and nine-node, biquadratic(02) elements. In implementing thesc elements in a computer code designed for solving (3.4),the surface integrals appearing in (3.5), (3.7), and (3.8) are evaluated using numericalintegration. the trapezoid rule being employed for Qt-elements and Simpson's rule forOrelements. It is important to note that, since the integration points for these rules are thenodes, no coupling between nodes exists in computing the damping matrix C; in particular,entries in C are of the form

J, L = 1,2, ... , N~, i, k = 1, 2

where WJ is a quadrature weight associated with the contact node 1. Furthermore, if the linere is parallel to one of thc coordinate axes, no off-diagonal contributions to the dampingmatrix will occur.

Again in the two-dimensional case, it is clear that the direction of the tangcntial velocity ateach point of re is always tangent to re. Consequently, if the integration rules referred toabove are used, the components of F at each contact node (with the modulus of the tangentialvelocity at that node greater than E) will depend only on the orientation of that velocity.

In summary, the particular formulation adopted here has the convenient features: (i) thecomputation of the values of the damping coefficients of C and the friction components of Fneed only to be performed once; (ii) the 'slipping' or 'sticking' of a node is easily checked bycomparing the absolute value of its tangential velocity with E; (iii) if Ivorl :!;;;E, the correspondingcoefficient(s) on C must be introduced; if IVTI > E, the corresponding component(s) of F withappropriate choice of sign (opposing the motion) must be introduced.

3.2. Semi-discrete error estimate

This section is devoted to the derivation of an a priori error estimate for the semi-discrete finiteelement approximation u: of u". An auxiliary result needed for this purpose is recorded in thefollowing dynamical stability lemma.

LEMMA 3.1. Let conditions (2.18}-(2.22) of Theorem 2.1 hold, let u" be the solution of (2.26).and let tl~ be the solution of (3.3) for fixed E > 0 and h >O. If there exist constants Kh K2,

K3 >0, independent of E and h, such that

(3.9)

\42 J.A.c. Martins, J. T. Oden, Class of problems in elastodynamics

hen there exist constants MJ, M2, M3 > 0, independent of e, hand t, such that 'V t E [0, T]

(3.10)

~ROOF. The first step of this proof consists of showing that there exists a constant P > 0,ndependent of e, hand 1 such that

(3.11)

lIld the second step consists of proving that there exists a constant Q > 0, also independent of" h and I, such that

(3.12)

Ne shall only give the proof of (3.12) since (3.11) is established using similar argument.We begin by differentiating (3.3) with respect to time and setting v'' = ii:(1). This differen-

iation is meaningful since the terms a (ll~ (I), v") and <l/1(I), v") are clearly differentiable andhe term <DjE(u~(t)), v") is Lipschitz continuous. Consequently, for almost every 1E (0, T) thelerivative of the remaining term in the equation, (ii~(I), v"), is well defined. Then noting thathe monoticity of the function 4>£ implies that (see [12, p. 159])

)Ie are able to deduce that

~:1 [llii~(t)II~+ a(,i~(I), li~(I))] ~ (.jJ(t), 'l~(1) .

\l'ext, integrating in time from 0 to t, and integrating the second member by parts, we arrive athe inequality

+ 2(.jJ(t). ti~(I) - 2(rb(0), ti~(O) - 2 fa' <-ii(T), U~(T) dT.

By hypothesis, rb, -ii E L2( V') (see Theorem 2.1) and, by construction, U~(I) E Vh C V. Con-;equently,

1<l/1(t),,i~(I)1 ~ Ilrbll·II'i~(I)lh '

l<rb(O), ,i:(O)1 ~ IIrb(O)II·llu:(O)II. ,

/(-ii(T), u~(T»1 ~ 11-ii(T)II·II,iZ(T)II.

J.A.c. Marlins, J.T. Odell, Class of problems ill elastodYllamics 343

where 11·11·denotes the norm on the dual space V'. Applying now Young's inequality to eachof these products

V 8.>0. I(tb(t). u~(t)1 ~ 411

IItb(t)lf. + 8111u~(t)ll~ '

and employing inequalities (2.15) we get, for arbitrary positive numbers 81• 82, 83, A:

Ilii~(t)lI~+ allu~(t)II~ ~ A IIli~(t)ll~ + llii:(O)II~+ Mllu~ (O)II~+ 21. IItb(t)II: + 28111u~(t)II~

+ 2~2I1tb(O)II:+ 282I1u~(0)11~+ L [2~311.;i(T)II~+ 2831Iu~(T)llndt.

(3.13)

But since

li~(t)= U~(O)+ (' ii~(T)dT,Jo

it can be proved that there exists a constant d > 0 such that

If now, with no loss of generality, 81 is chosen such that a - 281 = 1, 83 is chosen such that283 = Ad, hypotheses (3.10) are used, and if we note that tb, .;i E L2(V') implies that tb ECO([O, T]; V') (see [12, p. 130]), then the following result is obtained:

Ilii~(t)II~+ Ilu~(t)ll~ ~ a + b foT (IItb(T)II~+ 11.;i(T)II:)dT+ C 50' (lIii~(T)II~+ Ilu~(T)llndT.

(3.14)

Here, a, b, c are positive constants independent of E. hand t and the constant a depends onlyon the initial state (K2, K3• Iltb(O)II·).

Finally, applying Gronwall's lemma to (3.14), we obtain

Ilii~(t)II~+ Illi~(t)lI~ ~ Aec1

344 J.A.c. Martins, J.T. Odell, Class of problems in elastodynamics

where the constant A denotes the first two terms on the right-hand side member of (3.14).Since t ~ T the second and third assertions of (3.10) follow immediately.

Under mild additional assumptions, we are now able to establish the following estimate forthe rate of convergence of u~ to u., as h tends to zero.

THEOREM 3.2. Let the conditions of Theorem 2.1 and Lemma 3.1 hold together with theinterpolation properties (3.1) and (3.2). and suppose that the solution u" of (2.26) is of sufficientregularity that

u." II" E LOO(V) n L2«H2(!1)t '

iiI! E LOO«L2(!1))N).

Moreover, let the solution u~ of problem (3.3) for fixed E > 0 be such that

(3.15)

(3.16)

Then there exist constants a, C > 0, independent of hand t, such that for all t E [0, T],

(3.17)

PROOF Setting v = u" - u~ in the first member of (2.26) and adding and subtracting (ii~,u" - u~) and a(u:, U"- u~) to the resulting equation, we obtain

(" "h' 'h)+ ( h' 'h)_U" - U", U" - u" a u" - U", U" - U" -

(.,.' . h) ("h . h ') (h' h ') (D' (') .h ')= .,., UI! - 11« + U", II I! - II" + au", U I! - U" + JE U" , U" - U" . (3.18)

Letting now vh = i5h- u: in (3.3), adding the resulting equation to (3.18) and, for simplicity of

notation, renaming vh for i5\ the following equation is valid V vh E Vh:

(" "h' 'h)+ ( h' 'h)U" - u", U" - u" a u" - u '" U" - U" =

( "h h ') + (h h ') (.,. h ')= u ", v - u" au", v - u" - .,., v - U"

(D· ('h) h 'h) (D' (' 'h ')+ JE U" , v - u" + JI! II", U E - U"

where the dependence on t is understood.Rewriting the first member as a derivative with respect to time, adding and subtracting

(ii,,(t), vh- ti,,(t)) and a(u,,(t), vh

- u,,(t)) and noting that equation (2.26) holds for every v E V,namely for v = vh - UE(/), the following equation holds V vh E Vh

4 Jt lIIu,,(/) - u:(t)W + a(u,,(t) - u:(t), u,,(t) - U:(/))] =

= (Il:(t) - ii,,(t), vh- li,,(t)) + a(u:(t) - uE(/), vh

- liE(t))

+ (Dj,,(u:(t)), vh - li:(t» + (Dj" (uE(t)), u:(t) - u,,(1)- (Dj" (U,,(/)), vh - U,,(/) .(3.19)

J.A.c. Martins, J. T. Oden, Class of problems in elastodynamics

The three last terms in this equation can be estimated as follows

(Dj,,(u~(t)), vh- u~(t) + (OJ,,(u,,(t)), u~(t) - u,,(t) - (Dj" (ll" (t)), vh

- ll,,(t):s:;

:s:;j,,(vh) - j,,(u:(t)) + je(ll~(t)) - j,,(ll,,(t)) + I(Dj" (lle(t)), vh

- Ile(t)1

,.::::1111 II h 'h II (JU"T(t)'(V~-U"T(t))ld--= g I/Ue V T - U"T -l/Ue + Jre g IU "T(t)1 S

:s:;21\g1II12.rel\V~ - Ii "T( t)1\-\fUe

:s:;DIllgllll2.rellu"T(t)113I2.reh2

:s:;D21Igllln.rellll£(t)lb,nh2 .

345

In arriving at this result, we have used:(i) the convexity and Gateaux differentiability of j£.

(ii) the estimate (2.33) in Lemma 2.2,(iii) the definition (2.29) of (D j" (u), v), in particular the fact that O:s:;cP£ (IUTI):s:;1,(iv) the Schwartz inequality,(v) the estimate (3.2),

(vi) the Trace Theorem.Applying now the Schwartz inequality to the first term of the right-hand side of (3.19) andusing (3.1) and the third estimates of (2.27) and of (3.10) the following inequalities aresuccessively obtained:

l(ii~(t) - ii,,(t), v: - u,,(t))I:s:; lIii:(t) - i;,,(t)llo IIvh- u,,(t)llo

:s:;(M3 + L3)\Iu" (1)lb h2•

Finally, using the continuity of the bilinear form a(·,·), the estimate (3.1) and the Young'sinequality, we obtain

a(U:(I) - U£(I), vh- u,,(t)):S:; Mllu~(t) - u,,(t)lh Ilv~- u£(t)111

:s:;MI3I\II~(t) - u,,(I)\Ii + ~ \Ivh- u£(I)\Ii

:s:;MI3I\Il:(t) - 11£ (t)lli + ~ Ilu£(t)ll~ h2

where 13 is an arbitrary positive constant.Using these estimates and integrating the inequality (3.19) from 0 to t yields

II'l,,(t) - li~(t)II~+ a (ll,,(t) - ll~(t), ll,,(t) - ll~(t)) :s:;

:s:;\Iu,,(O)- Il~(O)\I~+ a(u,,(O)- 1l~(O), 11,,(0)- u~(O))

+ 2h2[(M3 + L3) + D2\1g\lln,rel L'llu,,(T)\I2dT

M (' ('+ h2 213 Jo \Iu,,(T)II~dT+ 2M13 Jo \IU~(T) - u,,(T)llidT.

l6 J.A.c. Martins. J. T Ode", Class of problems in elastodYllamics

ince, by hypothesis, ur E U«H2(f2)t), we must have

L Ilu£(T)lbdT ~ D311IirIlL2((H2(f))N),

L Illi£(T)II~dT ~ Ilurlli2((H2(!l»N).

ince 'V A > 0, 3 a > 0 such that

10reover, since

he following result holds, for arbitrary positive [3 and A,

Ilu£(t) - li~(t)116 + O'l\u£(1) - u~(t)ll~ =:;;

~llu£(O)- u~(0)115+ a(u£(O)- u~(O), u£(O)- u~(O)) + Allu£(O)- u~(O)II~+ D41Iucllh(H2UI»N)h

2 + Dsllli£IIL2((1/2(f))H)h2

+ f [Adlllir(T) - ti~'(T)lI~+ 2M[3l\ur(T) - ll~(T)lm(h.

Finally, choosing A and [3 such that Ad = 1 and 2M[3 = a, using the continuity of a(· .. ) andhypothesis (3.16), and applying Gronwall's inequality, the desired estimate is obtained.

REMARK 3.3. It is easily seen that the constant a in Theorem 3.2 is independent not only ofh and I but also E. However, the constant C, under the conditions of Theorem 3.2 may bedependent on E. Its dependence may be due to a possible dependence on E of the constants Coand C, governing the convergence of the initial displacements and velocities u~(O) and U~(O)and a possible dependence on E of Iltlrlk2«(H2(f)))H). To be sure of the convergence of the solutionof the finite element regularized problem (3.3) to the non-regularized problem (2.17) in theenergy norms appearing in the error estimates (2.35) and (3.17), the constant C in (3.17)should be independent of E. In such a situation the combined error estimate for every timet E [0, T], would be of the form (if meas r0> 0)

with A, B > 0 independent of E, hand t.

4. Algorithms and numerical experiments

4.1. Time discretization

The full discretization of the regularized finite-element problem (3.4) is obtained by

I •

J.A.C. Martills. J. T. Odell. Class of problems ill elastodynamics 347

introducing a partition P of the time domain [0, T} into M intervals of length At such that0= to < tl < ... < tM = T, with tK+1 - tK = At. In addition, the displacement and its timederivatives at thc time points of P (dcnoted by rK, rK and rK, K = 0, I, ... , M) are relatedthrough difference formulas designed to approximate as At ~ O. the actual relations betweenr(t), r(t) and r(t) for t E [0, T]. In this work the well-known Newmark's and central-differenceschemes will be used for this purpose.

Using the Newmark's formulas, the displacements and accelerations at time tK may beobtained as functions of the displacements, velocities, and accelerations at time t K-I and ofthe velocities at time tK by the following relations:

rK = rK-I + At(l-li) r _ + 1At2(1- Yi) r" + ~ .'V K 12K-I rKI "Y "Y

(4.1 )

where f3 and "Yare two parameters that govern the stability and accuracy of the method.For f3 = t 'Y = ~ the average acceleration method is recovered and for f3 = 0, "Y = ~ the

central-difference method is recovered.Introducing the formulas (4.1) into the dynamic equilibrium equations (3.4) at time tK the

following equation is obtained, where the only independent unknown function is the velocityat time tK:

Here,

R~=RK-M[(l-l)rK-I- ~trK-I]"Y 'Y .

- K [ r K -I + At ( 1- ~) r K -I + 1At2 ( 1 - ¥) r K -I] .

(4.2)

(4.3)

The fully discretized version of problem (3.4) consists of seeking the function tK ~ rK ofP ~ R NxN~ such that 'V 1 ~ K ~ M equation (4.2) holds, with rK-I and rK-I in Rk obtained bythe recurrence formulas (4.1), with the initial conditions:

ro = po, (4.4)

and the initial acceleration being computed from the following equation

Mro+ Kro = Ro. (4.5)

REMARK 4.1. Consistent with the hypothesis (2.21) of Theorem 2.1. no initial friction forcesare considered in the initial dynamic equilibrium equations (4.5).

Equation (4.2) is, of course, nonlinear since a priori the actual contribution of the frictionstresses to the damping matrix CK and to the vector FK is not known and will depend on thesolution rK.

348 J.A.c. Martins, J.T. Odell, Class of problems ill elastodYllamics

If the central-difference method is used, equations (4.2) and (4.3) simplify to:

(4.6)

(4.7)

As noted earlier, the absence of contributions from different nodes in the definition of thedamping matrix CK, leads to a diagonal matrix. To produce a fully explicit scheme, we mustuse a diagonal mass matrix in the central-difference scheme. For this purpose, we employ theprocedure of Rock and Hinton [27] and use a diagonal mass matrix with entries for eachelement of

J = 1,2, ... , N\:'), i = 1,2, .... N (no summation on i or J)wherc

N~~): thc number of nodes of the element (e);n\~): the area of the element (e);¢~,(~):the ith component of the restriction to 12\:') of the basis function of Vh associated with

the Jth node of the clement (e) ..This procedure concentrates the mass of the element on the nodes in such a way that the

proportions between nodal masses equal the proportions existing between the correspondingdiagonal components of the consistent mass matrix and the total mass of the element ispreserved.

4.2. Algorithms

A remaining difficulty which must be resolved is the handling of the nonlinear system (4.2).To resolve this problem, at each time step an iterative procedure is needed in order to obtainthe solution rK. The iterativc scheme used in this study is a straightforward method ofsuccessive approximations analogous to those used by Kikuchi and Oden [18] and Pires [26] instatic contact problems.

Let the superscript (i) denote the iteration counter at time tK. To start the iteration process,an initial guess r~) for the velocity vector is required. That starting value is taken to he the(converged) velocity at time (K-l:

Accordingly, the starting values for the damping matrix and the vector of friction forces are:

J.AC. Martins, J.T. Oden, Class of problems in elastodynamics 349

At the ith iteration of the process 0 = 1, 2, ... ), the ith approximation for the velocity attime tK("'~) is obtained by solving the following system of linear equations

[~ K + --L M + C~-I)l ,..~)= R~ + F~-I).'Y 'Y t:. t

With this velocity vector a new distribution of tangential velocities on rc is computed and,accordingly, new vector of friction forces F~) and new damping matrix C~ are obtained usingequations (3.7) and (3.8). The iterative process terminates at some iteration (i) if, for thatiteration,

C~ = C~-I), (4.8)

In that situation, equation (4.2) is exactly satisfied by ,..~ and consequently

(4.9)

The displacements rK and accelerations rK are then computed using equations (4.1) and thestresses in the body are calculated using standard finite element procedures. The frictionstresses on the contact surface rc are calculated, according to (2.29), using the followingexpressions:

./,tl e1'K . fl' II I- g -, . I. " 1 tl eTK > E ,tleTK

(4.10)

It is important to note that conditions (4.8) that govern the termination of the iterativeprocess at some iteration (i) are satisfied if and only if all the contact nodes that are 'adherent'at iteration 0 - 1) remain 'adherent' after iteration 0) and all the contact nodes that are'sliding' in some sense at iteration 0 - 1) remain 'sliding' in the same sense after the ithiteration.

REMARK 4.2. The iterative scheme described above has been slightly modified in order toobtain convergence of the solution when the unloading of the contact nodes occurs. Thechange introduced is the following: if at some iteration 0 - 1) some contact node is sliding inone sense and at the next iteration 0) the same node is sliding in the opposite sense, then.instead of introducing in F~ the corresponding force component(s). what tentatively isassumed is that at iteration (i) the node became 'adherent' and the corresponding dampingcoefficient(s) are introduced in C~). Using this procedure, the tentative assumption turns out tobe, most of the times, correct and the node remains 'adherent' in the next iterations and,eventually, a solution of equation (4.2) is obtained. The iterative scheme described above withthe modification presented in this remark has always converged to a solution of equation (4.2).Without this modification the iterative scheme may not converge in an 'unloading' situation.

\50 1.A.c. Martins. 1.T. Odell, Class of problems ill elastody"amics

[t is then observed that the node oscillates, in successive iterations, between two 'sliding' statesn opposite senses, both not satisfying equation (4.2) and the true 'adherent' solution is neverlchieved.

REMARK 4.3. The numerical experiences carried out also suggest an additional restriction"elated to unloading at nodes on the contact surface: time steps lit immediately after the:angential velocity of each contact node becomes smaller than e need, in general, to be smaller·:han the time steps used in the remaining of the analysis. In fact, the results obtained suggestthat in these situations the tangential movement of the node has the same characteristics of theoverdamped movement of a particle acted by a linearly elastic spring and a linear damper.However, for small e (large damping), if the time step is not sufficiently small, the quicklyoverdamped behavior of the node cannot be captured by the numerical time integrationscheme used and some slightly damped oscillations of the node occur. The node velocityoscillates about the values that are obtained when a smaller time step is used.

These oscillations have almost no influence on the tangential displacement of the node (thatremains approximately constant) since the velocity is smaller than e and the amplitude of its,)scillation is also smaller than e. However, these oscillations are clearly observed in theLangential acceleration of the node and on the tangential stress (note that for I,i £TKI ~ e thevelocity oscillations induce stress oscillations amplified by the factor g/e, according to equation~.1O).

As a final note on the computational algorithm, we mention that the significant difference in:he implementation of the central-difference method from that described above is that the.;olution is advanced in time by simple explicit function evaluations such that it is not necessaryto employ band-solver routine as it is when the Newmark's method is used. Of course, in such~xplicit schemes, numerical stability is a concern and limitations on the time step size must be~nforced.

4.3. Numerical experiments

To test the methods described earlier the dynamic behavior of a slab of dimensions 8 x 1Jnits, of unit thickness, made of an elastic material with mass density p = I units of mass per.mit volume, a Young's modulus E = 1000 units of force per unit area and a Poisson's ratiou = 0.3 was analyzed. The slab is assumed to be in a state of plane strain.

The slab is simply supported on one part of its boundary (/'D) and subjected to a uniformlyjistributed and constant in time normal pressure Fn = 300 stress units acting through a;rictional surface (I'd, with a friction coefficient JI = 0.3. The slab is also subjected to a timejependent uniformly distributed force:

tx = to· O(t) (4.11)

)n one of its ends (rF). In (4.11) t" was taken to be equal to 150 stress units and O(t), a:unction of time, was taken to be either the Heaviside step function H(t) or the periodic,'unction sin 81ft.

The initial conditions used were the following: the initial velocities in all the slab equal to

J.A.c. Martins. J.T. Odell, Class of problems ill elastodynamics 351

71

~8 °t

( 0)

,.I

8~

( b)

E • 1000 stress units

µ. • 0.3

P • I units of moss per unit volume

V • 0.3

Fn' 300 stress units

t. ·150 l 8(t) stress Units

Fig. 2. Undeformed (a) and initial (b) configurations of an clastic slab with prescribed normal stress on a surface(fc) on which Coulomb's law of friction holds.

(t) • H(t)( • O.OIJOI

C~ntra 1 dl HerenceHax. ilt • 0.001

1.000

0.950

0.900

0.850

III~ 0.800zUJ~UJ O. 750ua:...J

~ 0.700-00.650

O. 600

0.550

0.500I I

o.OlHl a.Dlo 0. IlHl 0.150 0.'" G.UO

TIME

81

8.100 O. sso 0.4.00 0...$0 a.loa

Fig. 3. Horizontal displacements using central-difference method.

352 J.A.c. Martins. J. T. Oden, Class of problems in elastodynamics

zero and the initial displacements equal to the static equilibrium displacements of the slabacted by the normal pressure Fo alone (no friction on Fe and no applied tractions on rF).

The results that will be presented in the following were obtained with a mesh of 16nine-node isoparametric quadratic elements as illustrated in Fig. 2.

In a first series of experiments, the case O(t) = H(t) was analyzed using the central-difference method, with a maximum time step ~t = 10-3 units of time. The regularizationparameter e was taken, successively, to be equal to 1, 10-2 and 10-4.

The results obtained with e = 10-4 are shown in Figs. 3, 4 and 5. The computed distributionof frictional stresses is shown in Fig. 5. In this figure, the distribution of frictional stresses

81

71

P(tl • lI(t)c • 0.0001

Central difference11ax 6t • 0.001

100

88

76

en 64wenen 52wa:l-en 40z0;: 28ua: 16lL.

4

-8

-20

a.GOD 0.050 0.100 O.ISO 0.* o.no O.tOO o.no 0.4000."500.100

TIME

Fig. 4. Friction stresses using central-difference method.

€ = 0.0001'0 = 150

SIalic I '. = 210

Dynamic It. = '0' H( I)

Cenlral difference

Mox AI' 0.001

87

I

\ ;. ,V:\

I1\I x

6"'0.042:' 1=0.011

65432

0Q

en~wen~~l-en

~~....u~2

0

0~ 0

Fig. 5. Distribution of friction stresses on rc.

J.A.c. Martins. 1.T. Odet!, Class of problems it! e/astodynalllics 353

obtained in a static analysis of the same slab with an applied force tx = 300 stress units (see [8])is also plotted for comparison. The comparison of the results obtained with the different valuesof e indicated above is done in Figs. 6, 7 and 8.

From these results, it is clear that, for e sufficiently small (e = 10-2 or 10-4) the adhesion of a

point on Fe when the modulus of its tangential stress is smaller than g (=90 stress units). isaccurately simulated with the regularization procedure used.

1.000

0.950

- O. 900CD

w 0.B5000z.... O.BOOcren 0.750....zW:I: 0.700wucr...J 0.650Q...

en...0 0.600

0.550

0.500I..-

e • 0.0001

~

~(t) • Ii( t)Central differenceIlax hI • 0.001

I. D$O D.IOO o. no 6. too a~1'0 0,100 0.110 O. tOO O. tlO 0.100

T/HE

Fig. 6. Effect of the regularization parameter on the tangential displacements.

0.00

-0.50

-1.00

-1.50

-2.00

;;;-2.50

~ -3.00 ~ I I \Ie: 0.01< • 0.0001

0-

"":::-3.50;:

~-4.00l~ott) • Ii( tlCentral difference" .. f>l • 0.001

-4.50

-5.00

O.Doa 0.010 0.120 0.. no O.2tO •• toO a. "0 a. no a. tla

TIHE

Fig. 7. Effect of the regularization parameter on the tangential velocities.

354

lOa

88

a; 76uJD0 64z....a: 52U'luJU'l 40U'luJa:.... 28U'l

z0 16-....w- 4a:lI..

-8

-20

J.A.c. Martins, J. T Oden, Class of problems in elaslodynamics

e(t) • H( t)Central differenceI,.x ilt • 0.001

D. ODD 0. aSCI 1.100 L liD 0.100 "'10 G,tao LUO 0. lao O.tla 0.100

TlHE

Fig. 8. Effect of the regularization parameter on the friction stresses.

It is also important to observe the rapid variation of the friction stress that occurs when themodulus of the tangential velocity of a point on rc becomes smaller than e. That variation isassociated with a rapid decreasc of the modulus of the tangential acceleration of the point.This rapid decrcasc of tangential acceleration approximates the discontinuity that necessarilyoccurs in the tangcntial acccleration, solution of the non regularized problem, at a point on rcat the time it stops sliding and starts to be adherent.

Finally, a plot of the evolution of the total mechanical encrgy is presented in Fig. 9. This

-167.0

-168. I

-169. 2 ~ \ e(t) • lI(t)C • 0.0001

Centr.1 dfrference)0- ltax ilt • 0.001l:la: -190.3uJzuJ

.J -191. 4a:u- -192.5za::I:u -193.6wr.J -194.7a:....0... -195.6

-196.9

-196.0

0.000 G.cso 0.. 100 O. 110 D. lOG 0.250 O. !OO 0.510 O. foOO 0.450 O. SOD

TIME

Fig. 9. Dissipation of mechanical energy.

J.A.c. Martins, J.T. Oden, Class of problems in elastodynamics 355

energy, at time tK, is computed as

-11K +"'M' RI1TK - 2rK rK 2r" r" - KrK.

A second series of experiments was done again with O(t) = H(t) and e = 10-4 but now usingthe Newmark's method with a maximum time step tlt = 5 X 10-3

.

The displacements and friction stresses obtained with (3 = 0.25 and 'Y = 0.5 (no artificialnumerical damping) and with a reduction of time step to 10-7 in the 10 time steps immediatelyafter each node becomes 'adherent', are plotted in Figs. 10 and 11.

8.1

r(t) • IHt!£ y 0.0001

HewN rk •S I'll'thada • O.2S Y' 0.5H.. M • O.OOS

Q.l$O D. loa a. SID G. COO

1.000

0.950

0.900

0.850

VI~ 0.800zw::t:w 0.750u~..J0.. 0.700VI-0

0.650

0.600

0.550

O. 500I I I I I

0.000 o,oso 0, lOG •• 110 •• 100

TIHEFig. 10. Horizontal displacements using Newmark's method.

100

88

76

VI 64wVIVI 52wa:~VI 40z0- 28~u-a: 16l1...

4

-8

-20

.. 000 .. oso L 11110 D. 110 Do 200 eo nCi ... toO Go1$0 ... 400

TIME

Fig. II. Friction stresses using Newmark's method.

356 l.A.C. Marrins, l. T. Oden. Class of problems in e1astodynamics

In Figs. 12 and 13 the results obtained in this manner are comparcd with the results that arcobtained without any reduction of time step, i.e., if a constant time step M = 5 X 10-3 is usedtogether with the same values of {3 and 'Y (0.25 and 0.5, respectively). As already pointed outin Remark 4.3, the tangential stresses of the 'adherent' nodes present spurious oscillations thatare attributed to the loss of accuracy on the numerical integration of the overdamped behaviorof these nodcs. The influence of these oscillations on the displacements is very small as it canbe seen in Fig. 13.

IUU

88

76~~ 64

t 52V>u;V>

~ 40c.:t-V>

~ 28t-~ 16<Y."-

4

o (tJ : H( t J£ • 0.0001

r~eW1la rk' 50 mt'!thod~ : 0.25 Y' 0.5II.. , 6t : 0.005

-8

-20

0.000 O.oso 0.100 0.150 0.700 0.250 O. SOD O. !SO 0, coo o. no o. sao

TIME

EfTect of the reduction of time step on the friction stresses.

111111111111

~'" '" "''''''''

_ with lime step reduction.fUf constant lime step •

O.!50 0.400 o. no 0.5.00

e (t) • II( t)£ • 0.0001

N... OIdrk·s .... thoda • 0.25 Y • 0.5M.. lit • 0.005

O. '00

I II" 111111; 1111111111111111 II .. "" II II

.....0.150 0.1000.1000.e50

Fig. 12.

1.000

0.950

0.900

a; 0.850....00:z: O.BOO,-e>:V> O. 750!::~L.;

0.700u<::..Jc..V> 0.650C

0.600

0.550

0.500I

•. oaa

TIHE

Fig. 13. Effect of the reduction of time step on the tangential displacements.

J.A.c. Martins. J.T. Oden, Class of problems ill elasrodynamics 357

Obviously these spurious oscillations may be damped out if the parameters f3 and 'Yarechosen so as to introduce an artificial numerical damping in the time integration procedure.The well-known consequence of such a procedure is the artificial damping of the higher-ordermodes on the solution. These effects are illustrated in Figs. 14 and 15 where the friction stress-es obtained with f3 = 0.25, 'Y= 0.5, the maximum time step At = 5 X 10-3 and the reduced timestep equal to 10-7

, are compared with those obtained with f3 = 0.3025, 'Y= 0.6 and a constanttime step At = 5 X 10-3

.

100

88

76

64

52

a; 40...-1

I e( t) , H( t)C>

t • 0.0001'" 28 rlewmdrk's methodz:I- H.x At ' 0.005..:::: 16'"'"...'"t; 4z:~t; -8

'"...R • 0.25 Y' 0.5 with time steo reductto

++i++ 6,0.3025 y • 0.6 constanl tt.,.. steo

-20

Q. 000 o. asa 0.100 O.I~O o. lOa O.1S0 o. )00 o. no Q. tOO

11M(

Fig. 14. Eflect of the numerical damping on the friction stresses at node 81.

6' 0.25 Y' 0.5 with ttme"Step reduction......... 9 • 0.3025 y .. O.fi constant time step

Q. ]50 0.400

e(t) • H(t)c • 0.0001

rlewmark's methods11a, lit • 0.005

O.OSO 0.100 a.ISo 0.200 0.150 0.300

100

88

C- 76wCl<:) 64z0-cr: S2(f'l

w40(f'l

(f'l.....0=

280-(f'l

z16<:)

;::u- 4 f.0=...

-8

-20I

o. 000

T I HE

Fig. 15. Effect of the numerical damping on the friction stresses at node 71.

358 J.A.c. Martins, J.T. Oden. Class of problems in elastodynamics

Finally, the case of a periodic loading 8(t) = sin 8TIt is analyzed using the central-differencenet hod, with a maximum time step At = 10-3 units of time and the regularization parameter/;= 10-4. The plots of the evolution of the horizontal displacements at the nodes 81 and 85 and)f the friction stresses at the nodes 71 and 81 on rc are presented in Figs. 16 and 17. Thejistribution of friction stresses on rc at several instants is presented in Fig. 18.

1. so

1. 40

1. 30

1. 20

V)

l- 1.10zw:0:W 1. 00ua:.JQ.. 0.90V)

~0.60

0.70

0.60

0.50 r-,--,0.000 a. usa 0.100

o(t) • sin &"tE • 0.0001

Central differenceflax At • 0.001

-r--O. 110 0.100 0. no D. JOO 0. sn o. '00 o.no D. SOD

TIHE

Fig. 16. Horizonlal displacements due to periodic loading.

100

60

60

V) 40wV)V) 20wa:l-V) 0z0

20 ,8(t) • sin 8.t- - E : 0.0001I-U Central difference- o Max lit • 0.001a: -4.....

-60

-60

-100

0.000 a.GSO 0.100 •• 1$0 0.2000.2'50 O.toO 0."0 0.'00 0•• 50 0.500

TIHE

Fig. 17. Friction stresses due to periodic loading.

..

1.A.c. Martins, 1.T. aden, Class of problems in elastodynamics 359

8t I I' sin 81T t(.. 0.0001Centro I dlflarenceMOl. 61' 0.001

oQ

oCD

oII)

enOwNenIJ)

werOI-CIl

zoQNI-U

~~

oCD

oQ

•___0....

time0.0400.0820.1240.1640.1950.235

..~.

2 3

\

"\~

\

0--0. •I \

' ... ,... ..+ • \ \r I ·b

/ I \

f,' \I 6 I

I .

f I 'II,' .,+ I .

I! \

t\\

References

Fig. 18. Distribution of friction stresses on Fe due to periodic loading.

(1) L. Amerio, Su un problema di vincoli uniiaterali per I'equazione non omogenea della corda vibrante. Inst.App!. Calc. Mauro Picone Pub!. Ser. D 109 (1976) 3-11.

[2] L. Amerio, On the motion of a string through a moving ring with a continuously variable diameter, AttiAccad. Naz. Lincei Rend. 62 (1977) 134-142.

[3] L. Amerio and G. Prouse, Study of the motion of a string vibrating against an obstacle, Rend. Mat. 2 (1975)563-585.

[4J I. Babuska and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method, in:A.K. Aziz, ed .. The Mathematical Foundations of the Finite Element Method with Applications to PartialDifferential Equations (Academic Press. New York, 1972) 1-359.

[5] H. Cabannes. Mouvement d'une corde vibrante soumise a un frottement solide, C.R. Acad. Sci. Paris 287A(1978) 671-673.

[6] H. Cabannes, Propagations des discontinuites dans les cordes vibrantes soumises a un frottement solide. C.R.Acad. Sci. Paris 289B (1979) 127-130.

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[8] L.T. Campos, J.T. Oden and N. Kikuchi, A numerical analysis of a class of contact problems with friction.Comput. Meths. Appl. Mech. Engrg. 34 (1982) 821-845.

[9] P.G. Ciarlet, The Finite Element Method for EIIiptic Problems (North·Holland. Amsterdam. t978).[10] C. Citrini. SulI'urto parzialmente elastico 0 aneIastico di una corda vibrante contro un ostacolo. Atti Accad.

Naz. Lincei Rend. 59(6) (1975) 667-676.[11] C. Citrini, The energy theorem in the impact of a string vibrating against a point shaped obstacle, Atti Accad.

Naz. Lincei Rend. 62 (1977) 143-149.[12] G. Duvaut and J.L. Lions. Inequalities in Mechanics and Physics (Springer-Verlag. Berlin, 1976).

J.A.c. Martills, J.T. Odell, Class of problems in elastodY1lamics

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14] T.1.R. Hughes, R.L. Taylor, J.L. Sackman, A. Cumicr and W. Kanoknukulchai, A finite clement method for aclass of contact-impact problems, Comput. Meths. Appl. Mech. Engrg. 8 (1976) 249-276.

15J T.J.R. Hughes, R.L. Taylor and W. Kanoknukulchai, A finite element method for large displacement contactand impact problems, in: Bathe et aI., cds., Formulations and Computational Algorithms in Finite EIcmentAnalysis (MIT, Cambridge, MA, 1977) 478-495.

16J J.J. Kalkcr, 11le mechanics of the contact between deformable bodies in: de Pater and Kalker, eds., Proc.Symp. IUTAM (Delft Univ. Press, Delft 1975).

17] J.1. KaIkcr, A survcy of thc mechanics of contact betwecn solid bodies, Z. Angcw, Math. Mech. 57 (1977)T3-T17.

18J N. Kikuchi and J.T. Oden, Contact problems in elasticity, TICOM Rept. 79-8, University of Texas at Austin,1979.

19) J.K. Lee and K. Kamcmura, Analysis of elastodynamics with unilateral supports, Proc. Third Engrg. Mech.Division Specialty Conf. (ASCE, New York, 1979) 777-780.

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22] P. LOtstedt, A numerical method for the simulation of mechanical systems with unilatcral constraints, Dcpt.Num. Anal. Compo SC., Royal Inst. Tech., Stockholm, TRITA-NA-7920, 1979.

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J •