NUMERICAL SIMULATION OF THE QUASISTATIC CONTACT PROBLEM WITH DAMAGE

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1 NUMERICAL SIMULATION OF THE QUASISTATIC CONTACT PROBLEM WITH DAMAGE NICOLAE POP 1, CONSTANTIN GHIŢĂ Abstract. This paper eals with the numerical analysis of a quasistatic contact problem with friction between an elastic boy an a rigi obstacle. The contact is frictional an is moele with a version of normal compliance conition an the Coulomb s law of ry friction, incluing the evelopment of material amage, cause by opening an growth of micro-craks an micro-cavities, which results from internal compression or tension. The material amage is taken into account in the constitutive law. The amage fiel varies between one an zero at each point in boy. The variational form of this problem is a couple system which consists of a first-kin variational inequality for the isplacement fiel an a linear parabolic variational equation for the evolution of amage fiel. The iscrete scheme of the couple system is introuce base on the finite element metho to approximate the spatial variable an an Euler scheme to iscretize the time erivative, see [3, 4, 7]. The algorithm use is an iterative-alternative. The novelty of this work consists in the propose numerical algorithm, an its implementations for a numerical solution of the quasistatic elastic contact problem with friction an amage. Key wors: quasistatic frictional contact, amage, liniarelastic material, weak solutions, finite elements, numerical simulations. 1. INTRODUCTION This paper stuies the numerical analysis of a couple problem which consists from an elasto-quasistatic contact problem with friction, an an inclusion parabolic equation for the evolution of the amage fiel that results from internal compression or tension. The mathematical backgroun of the theory of contact with friction is still incomplete. The quasistatic moel of contact with friction, without inertial effects has been propose by Klarbring, Mikelic an Shillor [1]. This moel consists of the incremental formulation obtaine from the approximation with finite ifference of the quasistatic variational inequation. The proof of existence an uniqueness of solution for quasistatic frictional problem is base on the assumption that the isplacements satisfy some constraint qualifications an the friction coefficient is sufficiently small. The amage material is cause by the opening an growth of micro-cracks an micro-cavities with lea to ecrease in the loa carrying capacity of the boy. 1 Department of Mathematics an Computer Science, North University of Baia Mare, Romania Department of Mathematics, Valahia University of Târgovişte, Romania Rev. Roum. Sci. Techn. Méc. Appl., Tome 56, Nº 3, P , Bucarest, 11

2 6 Nicolae Pop, Constantin Ghiţă The moeling of material amage uses the concept of amage fiel, see Fréman [6]. In this moel, the amage fiel, ζ, may have values between one an zero at every point of the boy. For ζ = 1 the material is amage free, for ζ = the material is completely amage, an for < ζ < 1 the material is partially amage. The evolution of the amage fiel is escribe by an inclusion parabolic equation with a amage source function. This source function results from a mecanical compression or tension provie from elastic contact problem. The novelty of this paper consists in the propose algorithm an its implementation for a quasistatic elastic contact problem with friction an amage.. CLASICAL AND VARIATIONAL FORMULATION We will consier an elastic boy which occupies the omain Ω, = 1,,3, with the bounary Γ = Ω which we consier Lipschitz continuous, an let the time interval of interest be [, T], T >. The bounary Γ= Ω is assume to be Lipschitz continuous, an it is ivie into three isjunct measurable parts Γ D, Γ N, Γ C, where meas( Γ D ). The volume force f acts on Ω T =Ω (, T ). For x Γ, we enote by ν ( x) an τ ( x) the unit outwar an tangential vector to Γ. The boy is clampe on Γ D an, in consequence the fiel of isplacement will vanish there, the volume forces ensity f acts in Ω T, the surface tractions with intensity f N will act on Γ N an on Γ C the boy is suppose to be in unilateral or bilateral contact with friction on a founation, see Campo et al. [3]. We will enote by u the isplacement fiel, σ the stress tensor an εu ( ) the linearize tensor of eformation. We enote by ζ the amage fiel, efine on Ω T which measures the intensity of the micro-cracks in the boy. The boy is assume to be elastic with the following constitutive law: ( ε( u) ) σ= ζ A, where A is stipulate linear function. The amage fiel, which measures the ecrease in the strength of the material, will be enote ζ = ζ ( x, t) which is the Eeff ratio ζ = ζ ( xt, ) =, where E eff is the effective elasticity moulus, at the time t E an E is the elasticity moulus of the amage free boy. From this efinition results that the amage fiel is restraine to the values ζ 1. The evolution of microscopic cracks an cavities responsible for the amage is escribe by a parabolic (inclusion) ifferential equation, accoring to Fréman an Nejar [7].

3 3 An elastic frictional quasistatic contact problem with amage 63 Because of some technical reasons we will choose a positive small constant ζ in orer to restrain the amage function to the values ζ ζ 1. If we have ζ approaching zero the boy will have ense micro cracks an moeling this elastic boy is senseless. That is, we will use the truncating operator 1 if 1 ζ η ( ζ) = ζ if ξ ζ 1 ξ if ζ ξ specifying that there will be no ifference between ζ an η. Let us enote by S the space of the secon orer symmetric tensors on, by " " the inner prouct on or S an the Eucliean norms on these spaces. We also efine the following variational spaces: 1 { } H = L ( Ω ), Y = L ( Ω ), K = ξ H ( Ω); ξ 1 ae.. Ω { ν 1 H ( ) ; ν on D, νv v ν on C} { τ ( ij ) i }, j= 1 ( ) ; ij ji,, 1,. V = Ω = Γ = = Γ Q= = τ L Ω τ =τ i j = Moreover, for a Banach space X, let (,) X enote its inner prouct an associate norm. its X Let u an ζ be the initial values of the isplacement an amage fiels, respectively, an assume that the inertia effects are negligible an so the process is quasistatic. The classical form of the couple system of the quasistatc contact problem with friction an of the evolution of micro-cavities an of cracks responsible for the amage is: Problem P. Fin a isplacement fiel u : Ω [, T ], a stress fiel σ : Ω [, T ] S, an a amage fiel ζ : Ω [, T] [,1] such that, Divσ+ f = in ΩT, (1) ( ε u ) σ= ζ A ( ) in ΩT, () ζ κ ζ + I ( ζ) [,1] Φ( εu ( ), ζ ) in ΩT, (3) ζ = on Γ (, T ), (4) ν u = on Γ (, T ), (5) D

4 64 Nicolae Pop, Constantin Ghiţă 4 The initial conitions are given by The contact conition are given by: where if an if u ν > g it results σν = f on Γ (, ). (6) N N T u( x,) = u, ζ( x,) = ζ in Ω. (7) m ν ν ν + ν σ = c ( u g) on Γ (, T), (8) u ν C = g σ =, (9) τ mτ στ cτ ( un g) + u τ = m τ σ τ cτ( uν g) + ( ) λ> s.t. u τ = λσ τ, (1) where cν = cν( s), cτ = cτ( s), m ν an m τ are parameters which characterize the contact interface. These parameters can be euce experimentally. The parameter g, g is the initial gap between Γ C an the founation measure along the outwar normal irection to. We enote the positive part of the argument by () + which is recovere if m Γ C. The friction law from Problem P, is a generalization of the Coulomb s law, ν = mτ. In such case, µ = cτ cν is the usual coefficient of friction, positive an enough small. This law escribes a epenence of the friction coefficient on normal contact pressure. Furthermore in orer to have well efine integrals on the bounary, Γ C, it is necessary for the following relations to hol cn(), s ct() s L ( Γ C), 1 mν, mτ < if = an 1 mν, mτ 3 if = 3. Here, eq. (8) represent the normal compliance law, where u ν = u τ enotes the normal isplacement, στ = ( τ σν) τ an u τ = ( τ u ) τ are the tangential components of the stress an velocity files. The constant κ > is the amage iffusion constant, Φ is the amage source function, I[,1] enotes the subifferential of the inicator function I [,1] which enforces the constraint ζ 1. Assuming that it oes not exist amage flux on bounary Γ, that is, ζ ν = on Γ where η is the exterior normal unit vector on the bounary Γ, see [13,14]. In the stuy of the mechanical problem (1)-(7), we assume essentially that the elastic operator A an the amage source function Φ are Lipschitz continuous operators an that A is strictly monotone. Let the boy forces an surface tractions have the regularity

5 5 An elastic frictional quasistatic contact problem with amage 65 f C ([, T]; H) an N ([, T];[ L ( ΓN)] ) f C efine the element F ( t) V given by ( F( t), ν) V = ( f( t), ν) H + ( fn( t), ν ) [ L ( Γ N )], an let µ : ΓC [, + ) be such that µ L ( Γ C ), the friction coefficient. Let a: H1( Ω ) H1( Ω ) be the bilinear form , Ω a( ξ, ξ ) =κ ξ ξ x, ξ, ξ H ( Ω) an we enote by jν : V an jτ : V the functionals m jν( u, v) = cν( uν g) + ν vν S, ν V ; Γ C m jτ( u, v) = cτ ( un g) + τ vτ S, ν V. Γ C By choosing test functions from V an K, applying the Green s formula an using the conitions an the notation above, we obtain the following formulation. Problem VP. Fin a isplacement fiel u : ΩT V, a stress fiel σ : ΩT Q an a amage fiel ξ : ΩT K, such that u() = u, ξ () =ξ an for a.e. t [, T], σ= ζ A ( ε( u) ), σ(), t ε( w u ()) t + j u(), t w u + j u(), t w u F(), t w u () t, w V, ( ) ( ) ( ) ( ) Q ( ζ t ζ t ) a( ζ t ζ t ) ( εut ζ t ζ t ) ( ), ξ ( ) + ( ), ξ ( ) Φ( ( ( )), ( )), ξ ( ), ξ K. Y ν τ The existence of a unique solution to Problem VP an its regularity are summarize in the following theorem. Theorem 1. If the initial conitions are accomplishe such that u V an ξ K, then Problem VP has a unique solution for u C 1 ([, T]; V) ( Ω ) 1(, ; ), ; 1( ). ξ H T Y L T H Y V an The proof of Theorem 1 is obtaine following [9] an it is base on monotone operator theory an classical results on parabolic equations. 3. NUMERICAL APPROXIMATIONS In this section we use a finite element algorithm for solving Problem VP an obtain an error estimate on the approximate solutions. For convenience, we rewrite the variational problem VP in terms of the velocity fiel v( t) = u ( t) given by

6 66 Nicolae Pop, Constantin Ghiţă 6 t u() t = ν( s)s+ u. For this, we will approximate this variational problem in two steps. First, we consier two finite-imensional spaces V h V an Bh H 1 ( Ω ) approximating the spaces V an H 1 ( Ω ), respectively, an let Kh = K B h. Here, h > enotes the iscretization parameter. The unknowns are the velocity fiel an the amage fiel. Seconly, we will iscretize the time erivatives, by consiering an uniform partition of the time interval [, T ], enote by = t < t1 < < tn = T an let k be the time step size, k = T N. For a continuous function g( t ), let gn = g( tn) an, for a sequence { w n} n N =, we let δ wn = ( wn wn 1) k be its corresponing ivie ifferences. The fully iscrete approximation of Problem VP, base on the forwar Euler scheme, is as follows: N h Problem VP. Fin ν = { ν n } n= V an ξ = { ξn } n= K, such that ξ h =ξ an for all ξh K h, w h V h an n= 1,,, N, N h ( 1 ( ( 1)), ( h ) ) ( 1, h ) ( 1, h n A εun εw νn j Q ν un w νn jτ un w ν n ) ( F, h n w νn ), V ( δξ, h ) (, h ) ( ( ( 1), 1), h n ξ ξ n + a ξn ξ ξn Φ εun ξn ξ ξn), ξ + Y N h where the iscrete isplacement fiels u = { u n } n= V are efine by u n = un 1 + k ν n for n= 1,, N, an u h = u an ξ h are appropriate approximations of the initial conitions. Similarly, the iscrete amage fiels ξ = { ξ} N h n n= K are efine by ξ n =ξ n 1 + k ξ n for n= 1,, N. Using stanar arguments for variational inequalities [5,8], we euce the existence an uniqueness of the solution to Problem VP. Y 4. NUMERICAL SOLUTION OF PROBLEM VP FOR TWO DIMENSIONAL CASE Let n {1,, N} an assume that u n 1 an ξ n 1 are known, an for the next step is use a penalty-uality algorithm introuce in [], in orer to obtain the iscrete amage fiel, the iscrete velocity fiel, strain fiel an stress fiel, by using numerical algorithms available in the literature [1,8,9,1]. In the two-imensional case, the elastic stress tensor σ= ζ A ( ε( u) ), uner plane stress hypothesis has the following form: ( A τ) Er ( 11 ) E αβ = τ +τ δ, 1,, S 1 r αβ + τ 1 r αβ α β τ, +

7 7 An elastic frictional quasistatic contact problem with amage 67 where E an r represent the Young s moulus an the Poisson s ratio of the material, respectively, an δ αβ enotes the Kronecker symbol. The amage source function use here has the form where λ, λ u an 1 ξ Φ( εu ( ), ξ ) =λ 1 λ u ( ( )) +λw η ( ξ) εu, (11) λ w are process parameters, an if the value of (()) εu > q then (()) εu = q. A truncation values of q = 1, ξ =.1 are consiere as lower limit for the amage. The incremental form is obtaine by the variational formulation of the Problem VP through the approximation of the temporal erivatives of the isplacements with finite ifferences. The quasistatic problem will be solve step by step, such that at each step we shall calculate small strains, small isplacements an amage fiel, an we will a to the previously calculate result, for small changes of the applie forces. Obviously, both the contact area an the contact state are changing (open contact, fixe contact an sliing contact). In the paper we use the penalize augumente Lagrangian metho which combines penalty-uality metho in orer to impose on the contact bounaries intense tensions which serve as penalties, for the case when the contact conition are violate, with Lagrange s multipliers metho, together with Uzawa s algorithms type is one of the most use methos for solving the iscretize contact problems with friction, using the finite element metho [11, 15,, 1]. Also, Newton- Raphson s metho is ieal for solving the iterative an incremental contact linearize problem with friction, presente in the form of the penalize Lagrangian [16], an for solving the mechanical problem of quasistatic amage evolution in an elastic boy. EXAMPLE 1. THE CONTACT PROBLEM BETWEEN A PLANE PLATE AND A RIGID FOUNDATION This example, presente in Fig.1, was consiere by Raous [17] an has the avantage that is simple an contains all the contact states (open, stick an sliing). Five variants are presente with ifferent loas an ifferent friction coefficients. Also, an increasing traction is acting on a rectangular plate plane an we stuy the stress istribution an amage, at time t = 1, in the center of the boy an near the contact bounary. The following ata were use in this example: T = 1s, F = an mm, E = 1 an mm, r =.3, λ =.1, λ u = 1, λ w =, ξ =.1, κ =.1, ξ = 1. Next, we use a triangulation with finite triangular elements, with linear interpolation functions, an the contact surface was moele with finite contact

8 68 Nicolae Pop, Constantin Ghiţă 8 elements with 3 noes [11, 18], an with liniar triangular finite element for the amage fiel. Is observe the linear convergence of the algorithm with respect to h+ k. Fig. 1 The geometry ( h = 4 mm) an the loaing. Table 1 Contact states for ifferent loaing cases µ F [an/mm ] f [an/mm ] Open zone AB [mm] Sliing zone BC [mm] Stick zone CD [mm] The area near the contact bounary was finest iscretizate for a better approximation an a more precise elimitation of the various contact states. These are in goo agreement with the one from [17]. EXAMPLE. THE CONTACT PROBLEM BETWEEN TWO CYLINDERS AND A RIGID FOUNDATION This example represents an axial, symmetric contact problem, which consists of two cyliners which come in contact at the bases. The partial geometry an the loaing are shown in Fig.. The following ata were use in this example: T = 1s, F = 5 an mm, E = 1 an mm, r =.3, λ =.5, λ = 1, λ =, ξ =.1, κ=.1, ξ = 1. u w

9 9 An elastic frictional quasistatic contact problem with amage 69 The most amage area there is near the contact bounary with founation. Fig. The geometry, the mesh an loaing. Table Calculate versus experimental normal isplacements µ F [an/mm ] calculate u N [mm] experimental u N [mm] CONCLUSIONS The elastic frictional quasistatic contact problem with amage is moele using a couple system which consists from a elasto-quasistatic equation with bounary coitions an contact conitions with friction, an an inclusion parabolic equation for the evolution of the amage fiel. In this paper, a quasistatic elastic contact problem was stuie. The contact was moelle using the Coulomb s friction law. Accoring to [15,16,], the effect of the amage was inclue into the moel. The variational formulation le to a couple system of two nonlinear parabolic variational inequalities. Then, a fully iscrete scheme, namely Problem VP, was introuce using the finite element metho an the Euler scheme for approximating the spatial variable an the time

10 7 Nicolae Pop, Constantin Ghiţă 1 erivative, respectively, error estimates were provie accoring to [3, 18, 19], from which, uner suitable regularity assumptions, the linear convergence of the scheme was erive. The main contribution of this paper concerne the numerical resolution of Problem VP, where a penalty of the frictional term was use. Then, a penalty-uality algorithm, introuce in [], was employe for solving the penalize problem. Finally, two numerical examples were performe to show the accuracy of the algorithm. From the numerical examples we conclue that the solution of the mechanical problem of quasistatic amage evolution is consierably more regular with only a minor increase in the assumptions on the regularity of the ata. It is obvious that the algorithm can be also ivergent, when the conitions are not satisfie. In the case when the stick contact surface, of two boies in contact, is much larger than the sliing contact zone, then the algorithm converges in a relatively small number of iterations. Critical situations may occur if we change the sliing contact with a rigi one or vice versa, this being the most important change in the behavior of the solution. Those ifficulties can be overcame if we reuce the incremental step (quasistatic problem) until two successive terms are sufficiently close. Receive on September, 11 REFERENCES 1. BARBOTEU, M., HAN, W., SOFONEA, M., Numerical analysis of a bilateral frictional contact problem for linearly elastic materials, IMA J. Numer. Anal.,, 3, pp ,.. BERMÚDEZ, A., MORENO, C., Duality methos for solving variational inequalities, Comput. Math. Appl., 7, pp , CAMPO, M., FERNANDEZ, F.R., VIANO, J.M., Numerical analysis an simulations of a quasistatic frictional contact problem with amage in viscoelasticity, Journal of Computational an Applie Mathematics, 19, pp. 3-39, CHAU, O., FERNÁNDEZ, J.R., HAN, W., SOFONEA, M., A frictionless contact problem for elastic-viscoplatic materials with normal compliance an amage, Comput. Methos Appl. Mech. Engrg., 191, pp ,. 5. DUVAUT, G., LIONS, J.L., Inequalities in Mechanics an Physics, Springer, Berlin, FRÉMOND, M., Non-smooth Thermomechanics, Springer, Berlin,. 7. FRÉMOND, M., NEDJAR, B., Damage, graient of amage an principle of virtual work, Internat. J. Solis an Structures, 33, 8, pp , GLOWINSKI, R., Numerical Methos for Nonlinear Variational Problems, Springer, New York, HAN, W., SHILLOR, M., SOFONEA, M., Variational an numerical analysis of a quasistatic viscoelastic problem with normal compliance, friction an amage, J. Comput. Appl. Math., 137, pp , HAN, W., SOFONEA, M., Quasistatic Contact problem in Viscoelasticity an Viscoplasticity, American Mathematical Society-Intl. Press, Provience, RI,.

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