ICF10B

ICF10 Honolulu (USA) 2001 Vol. B

ORAL/POSTER REFERENCE: ICF100602OR DOUBLE COHESIVE ZONE MODEL AND PREDICTION FOR MICRO-SCRATCH TEST ALONG SOLID SURFACE M. Zhao1, Y. Wei1,2 and J. W. Hutchinson2 1LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China 2Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA ABSTRACT A new double cohesive zone model describing the failure of ductile film in the micro-scratching test is presented in this paper. The failure behavior and adhesion work in the micro-scratch test is simulated and predicted based on the model. In the present analyses, the thin film is treated as the elastic-plastic material, and substrate is elastic material, and three-dimensional elastic-plastic finite element method is adopted. In order to simplify the analyses, the total problem will be divided into two sub-problems. One problem is that elastic-plastic large bend deformation of scratched film is considered, and an analytical solution can be obtained. Another problem is that the thin film is delaminated plastically along the interface with the elastic substrate, for which a special three-dimensional finite element method is used. The parameter relations of the horizontal driving force for the scratch test with the separation strength of thin film/substrate interface and the material shear strength, as well as the material parameters are developed. As an example of the application, the prediction result is applied to a scratch test for the Pt/NiO material system given in the literature, and both results are fairly agreement with each other. KEYWORDS Micro-scratch test, driving force, double cohesive zone model. INTRODUCTION The micro-scratch test is an important approach for determining the interfacial strength, toughness and adhesion properties for the thin film or coating layer on the substrate interface [1]. Its principle can be described as follows: On the material or specimen surface along the vertical direction an indentation force is exerted and indenter tip penetrates inside the material, then the indenter is moved in the horizontal and vertical directions simultaneously according to a fixed proportion. When the indenter tip moves near the film/substrate interface, a region of the thin film or coating layer near the indenter tip will be delaminated along the interface. Through measuring driving forces, the scratch depth, and the failure geometry, one will obtain the material or interface adhesion properties. According to usual experimental observations, there are two main kinds of failure characters in the scratch tests [1-5] depending on the material property of thin film or coating, whether ductile or brittle. One kind of failure character can be described as that for the ductile film case, a delaminated film strap is formed before the end of the scratch test and the delaminated film will be curved into a circular shape. The geometry of the delaminated area is fan shaped. Another failure character is that when film is brittle. A damage zone is formed near the indenter tip, inside which the film will be pressed to break into many small pieces and also delaminated from the substrate. In the present research, our attention will be focused on the metal film/ceramic substrate case. The ductile failure character will be simulated and analyzed in detail. On the research of the material surface properties and adhesion work and strength of thin film or coating

layer along the substrate interface, many experimental researches based on the scratch methods have been carried out in past decade [1-5]. However, theoretical analyses connected with the scratch experiments are very few [1]. This is because that any theoretical study must deal with on the complicated failure geometry of the scratch test. It is obvious that a three-dimensional elastic-plastic deformation problem must be solved, and a robust theoretical model for describing the scratch failure behavior is needed. Must theoretical researches have been based on the simple geometry of the scratch failure strap and the simple mechanics equilibrium to simulate the scratch failure behavior [1-5]. However, it is difficult to use a simple model to describe the strong influence of plastic deformation on the micro-scratch behavior. It is well known that plastic deformation has a strong shielding effect on the interface cracking [6-8]. So that in an elastic-plastic failure process more energy is dissipated than that in a pure elastic failure process. Therefore, it is important to develop a reasonable mechanics model for scratch test simulation. The failure characteristics of the scratch test for ductile thin film materials[2-4] are somewhat similar with the thin film peeling problems. Therefore, in micro-scratch test research, the analytical method for the thin film peeling problem [8] is relevant. It is important to obtain a reasonable relation between the critical driving force and the parameters of the materials and scratch strap geometry. In the present research, based on the three-dimensional character of failure strap, a new mechanics model describing the interface separation and the thin film shear failure, i.e., a double cohesive zone model will be presented. Using the new model, a relation between the scratch horizontal driving force and the parameters of the materials will be set up and used to predict the scratch work. Finally, the simulation results will be applied to an experimental result for Pt/NiO from [4]. FUNDAMENTAL DESCRIPTION AND SIMPLIFICATION From failure characteristics for ductile film scratching, the scratch test process can be described by figure 1. This process can consist of two stages. One stage is a normal scratch before thin film delamination occurs along interface. With the indenter moving forward and downward with scratch depth increase, especially when indenter tip is near the interface, a region of thin film or coating layer near the indenter tip will be delaminated from interface. Thereby, the scratch process is transferred to another stage. The failure character changes from the indenter tunnel growth to the delaminated film strap formation and growth (or post-scratch process). For simplifying the analysis, the problem is divided into two sub-problems. One problem is "plate bend" under elastic-plastic large deformation for the delaminated thin film part BCD, see figure 1. This sub-problem has been solved successfully in [8]. Another problem is a three-dimensional delaminating problem for a part of thin film BA and jointed substrate. In the present research, our attention will focus on the latter problem. The solution of the former problem [8] will be taken as the boundary condition and exerted on the section B directly. For simplification, we present and adopt a new double cohesive zone model to simulate the film failure process and the scratch work for the post-scratch process. The new model is shown in figure 2. In this model, there are three cohesive zones, one is the separation-dominated cohesive zone and other two are shear-dominated cohesive zones. DOUBLE COHESIVE ZONE MODELS AND MECHANICS DESCRIPTIONS In figure 2, the thin film delaminates from interface of thin film/substrate (plane x2=0), and this failure process can be simulated by a separation-dominated cohesive zone surface. Simultaneously, the curved film layer is cut off from two sides of the delaminated region (planes x3=-W and 0). The cutting process for each plane can be described by the shear-dominated cohesive zone deformation. In figure 2, and cn δ ct δ are critic relative displacements for the separation and shear cohesive zone surfaces, respectively. The separation cohesive zone model under plane strain case has been widely adopted and completely formulated in [6-8]. In the following, we shall discuss and give a brief description and generalization for the two kinds of cohesive zone models for the three-dimensional case.

Figure 1: Scratch test geometry and ductile failure Figure 2: Double cohesive zone model Let ( ）be relative displacements at each cohesive surface along direction ( ), respectively, and define a normalized displacement quantity 1 2 3 , , δ δ δ 1 2 3 , , x x x 2 3 2 2 2 1 1 λ =δ δ +δ +δ − c , （1） The critical condition for the cohesive zone is, 1 λ = . For the separation-dominated cohesive zone case, , while for the shear-dominated cohesive zone case, c cn δ =δ ct cδ =δ . The traction relations, ( ) σ λ and ( ), on the cohesive zone surfaces are sketched in figure 2. τ λ The traction component expressions can be formulated in detail as follows. Define a potential function δδ δ =δ σλ′λ′ ∫λ d cn ( ) ( , , ) 0 1 2 3 Π （2） then, one will derive out the traction expressions easily ( , , ) ( ) ) , , , , ) ( 1 2 3 1 2 3 1 2 3 δ δ δ λδ ( σ λ = ∂δ ∂ ∂δ ∂ ∂δ ∂ = cn T T T Π Π Π （3） Similarly, for the shear-dominated cohesive surface, one reads ( , , ) ( ) , , ) 1 2 3 1 2 3 δ δ δ λδ ( τ λ = ct T T T （4） Adhesion work per unit area along the cohesive surface can be written as ) ˆ (1 2 1 2 1 0 = σδ +λ −λ cn Γ （5） for the separation zone, and ) ˆ (1 2 1 2 1 0 = τδ +λ −λ τ ct Γ （6） for the shear cohesive zone. Earlier work has shown that the shape parameters (λ and ) of cohesive zone model have the secondary influence on the analytical results. In the present analysis, we take . Moreover, for reducing the number of governing parameters, we take 1 2λ ( , ) (0.15, 0.5) 1 2 λ λ = c ct cn δ =δ =δ , then from (5) and (6), one have . = τ σ τ ˆ / ˆ / 0 0 Γ Γ ENERGY BANLANCE AND ELASTIC-PLASTIC MECHANICS METHOD The double cohesive zone model has been sketched by figure 2. The variation equation for the total system can be written as

∫ ∫ ∫ ε −ε ε = δε δε σ = δε V p kl kl e ijkl ij V e kl e ijkl ij V ij ij dV D D dV dV ) ( . . . = | . | . } . { 1 3 1 ∆ + δ + δ δ ∑ ∫ ∫ = − + u tdS u tdS WQ k S i i S i i k k | . | | | . | | 1 3 1 ∆ + δ =− δ − ∑ ∫ = + − u u T dS WQ k S i i i k = （7） | . | | | . | | 1 3 1 ∆ + δ − δ δ ∑ ∫ = T dS WQ k S i i k Where （k are cohesive surfaces, ( are the displacement and traction components on the cohesive zone surfaces, kS 1,3) = , ) i i u t ( , ) i iT δ are the relative displacement and traction on the cohesive surfaces, see formulas（1）to（4）. 1∆ is the displacement of the point acted by horizontal driving force . Q Based on (7), one can develop the finite element method for scratch test problem. The incremental constitutive relation of plasticity usually is expressed as kl kl ij e kl ij jl ik ij H E E ε       σ′ σ′ +ν σ + δ δ − − ν ν δ δ + +ν = & & 2 [1 (2/3)(1 ) / ] (3/2) 1 2 1 Ω σ （8） ij σ′ is deviator stress, /2 3 ij ij e σ′ σ′ σ = is effective stress; for plastic loading 1=Ω , otherwise 0 =Ω . H is plastic modulus. In uniaxial tension the film material has ε for σ ； ，for （9） / ,E =σ Y <σ N Y Y E 1/ ( / )( / ) ε = σ σ σ Y σ≥σ so that （10） 1 1/ 1 1} / ) {(1/ )( − − − σ σ = N e Y H E N Strap advance is assumed to occur in steady-state such that the stress and strain increment components can be expressed as / ) , / ( ( , ) 1 1 x x V ij ij ij ij σ ε = ∂σ ∂ ∂ε ∂ & & （11） where V is velocity of crack tip during film delamination in direction. The formula（8）is independent of V. Plastic strain components can be expressed by stress and total strain as 1x ε （12） kl e ijkl ij p ij D = ε − σ −1 A numerical method [9] which employs iteration to satisfy condition (11) is used to directly obtain the steady-state solution. Similarly, in the present analyses, adopting the fundamental relations of tensors and matrixes, (7) can be changed into the finite element relations. The steps of solving the problem can be described as follows: (1) Adopting a plastic strain distribution (in first step, take ), find displacement and strain. (2) Find stress distributions in plastic zone and unloading zone using (8), (11) and yielding condition ( 0 ε = p ij Y eσ = Y is current flow stress). (3) Find plastic strain by (12). Repeat procedures until a convergent solution is obtained. Consider that the substrate material is elastic and Young's modulus and Poisson ratio are and sE sν respectively. For further simplification, neglect the effect of mismatch of film and substrate materials, so that we take in the present analysis. During the steady-state advance of delaminated film strap, total work per unit length is ; dissipated work per unit length along the separation cohesive surface is ; and along two shear cohesive surfaces it is . Let plastic dissipation work be ( , ) ( , ) ν = ν E Es s QW W0Γ τ 0 2 Γt P WΓ . According to energy balance under steady-state advance, we have P Wt Γ Γ Γ + = + τ 0 2 0 Q （13） For elastic case, 0= PΓ . In principle, the interface separation work (interface fracture toughness) 0Γ and the material shear work (or material shear strength) could be determined by experimental measurement. The normalized horizontal driving force Q during the steady-state advance of failure strap can be τ 0Γ

expressed by the related independent parameters as follows       ν β σ τ σ σ σ + = σ τ = + , , , , , ˆ , ˆ , ) ˆ ˆ )( 2 1 ( 0 0 0 N W t R t E f W t Q Y Y Y P Γ Γ Γ (14) In（14）a reference length parameter has been introduced, whose definition is , characterizing the plastic zone size in small scale yielding. ) ] /[3 (1 2 2 0 0 Y R E π −ν σ = Γ SOLUTIONS AND ANALYSES Three-dimensional elastic-plastic finite element calculation for the scratch test problem is carried out based on the concepts implemented in previous sections. The results and analyses are shown in the following. Figure 3 shows the relation of the normalized horizontal driving force (or applied work per unit area) to the maximum strength of separation and shear for two kinds of cohesive zones. The driving force changes with shear strength and separation strength are very sensitive, especially for large values of separation strength or Figure 3: Scratch work vs. material parameters Figure 4: Scratch work vs. material parameters for low hardening film for high hardening film shear strength. For the lower separation strength case, the horizontal driving force changes with the material shear strength slowly as shear strength increases, then sharply increases. For the higher separation strength, even a lower material shear strength will make the horizontal driving force increase very quickly. In the figure 3, the results of two different ratio of thin film thickness with the delaminated film width are compared. Clearly, the horizontal driving force changes sensitively with the ratios. The results correspond to the lower strain hardening exponent material, i.e., N=0.1. The separation cohesive energy along interface is taken as the normalized quantity. When the shear strength of material is zero and the elastic case is considered, the value of the normalized horizontal driving force (applied work per unit area) equals to unity. When the Figure 5: Scratch work vs. film thickness and material shear strength is not zero, the normalized indenter angles horizontal driving force will change linearly with the

shear strength from (14) for elastic case. Obviously, from figure 3, the energy contributed from plastic deformation is very high. Figure 4 shows the results for the higher hardening thin film material case, N=0.3. The variations of the horizontal driving force with the material shear strength and the interface separation strength are similar to those of the lower hardening material. Clearly, the work dissipated in the plastic deformation for higher hardening material is lower than that for lower hardening material. Figure 5 shows the relation of the driving force (applied work) changing with normalized thin film thickness. The curves are for different direction angles of indenter surface, , , , and 165 . In the figure, the experimental results for the Pt/NiO film/substrate system and for the two kinds of techniques annealed at 500 o 90 β= o 120 o 135 o 150 o oC and 800oC from [4] are also shown. From figure 5, the horizontal driving force increases as thin film thickness increases and as the indenter angle increases. The indenter shape with the largest β corresponds to the high driving force. The simulation results are roughly consistent with the experiment results. CONCLUDING REMARKS By the detailed analyses in the present research, some important conclusions are obtained as follows: (1) Thin film plastic deformation has the important influence on the advance of delaminated film strap in the scratch test. (2) The interface separation strength and material shear strength have important influence on the failure of thin film/substrate system. (3) The horizontal driving force depends on the thin film or coating layer thickness. With the thin film thickness increase, the horizontal driving force increases and asymptotes to a stable value, which corresponds to the small scale yielding case. When either the interface separation strength or the material shear strength is large, a strong shielding effect from plastic deformation can be produced when the failure strengths are increased. In other words, with any cohesive strength increase, it is difficult or even impossible to make a film failure strap advance due to the strong plastic shielding. Such a prediction from using the conventional elastic-plastic theory seems somewhat contradictory. Actually, for the strong separation strength of interface or for the high shear strength, or for both, a strong plastic strain gradient effect could dominate the crack tip fields [10]. A reasonable simulation for this behavior might be obtained by using the strain gradient plasticity theory. A success application of the strain gradient plasticity to the crack growth problem has been shown in [11]. Acknowledgements The work is supported by National Science Foundations of China through Grants 19891180 and 19925211; and jointly supported by Fundamental Research Project from Chinese Academy of Sciences through Grant KJ951-1-201 and "Bai Ren" Project. The work is also supported by NSF Grants CMS-96-34632 in USA. References 1. Blees, M.H., Winkelman, G.B., Balkenende, A.R et al.(2000). Thin Solid Films, 359,1. 2. Pistor, C.and Friedrich, K. (1997). J. Appl. Polymer Science, 66, 1985. 3. Maekawa, H., Ikeda, T. and Horibe, H., et al.(1994). Quart. J. Japan Welding Society, 12, 262. 4. Venkataraman, S., Kohlstedt, D.L. and Gerberich. W.W. (1996). J. Mater. Res., 11, 3133. 5. Thouless, M.D. (1998). Eng. Frac. Mech., 61, 75. 6. Tvergaard, V. and Hutchinson, J.W. (1993). J. Mech. Phys. Solids, 41, 1119. 7. Wei, Y. and Hutchinson, J.W. (1997). J. Mech. Phys. Solids, 45, 1137. 8. Wei, Y. and Hutchinson, J.W. (1998). Int. J. Fracture, 93, 315. 9. Dean, R.H. and Hutchinson, J.W. (1980). In: Fracture Mechanics, ASTM STP700, p.383. 10. Fleck, N.A. and Hutchinson, J.W. (1997). Advances in Applied Mechanics, 33, 295. 11. Wei, Y. and Hutchinson, J.W. (1997). J. Mech. Phys. Solids, 45, 1253.

DUCTILE DAMAGE ACCUMULATION UNDER CYCLIC DEFORMATION AND MULTIAXIAL STATE OF STRESS CONDITIONS N. Bonora1 and A. Pirondi2 1DiMSAT – Dept. of Mechanics, Structures and Environment, University of Cassino 03043 Cassino, Italy 2Industrial Engineering Department, University of Parma 43100 Parma, Italy ABSTRACT The possibility to develop reliable predictive tools for the design of components undergoing plastic deformation is connected to the capability to incorporate damage mechanics into the constitutive model of the material. Even though many damage models for ductile failure are available in the literature, none of them, as far as the authors are aware of, is extended to reversal plastic flow occurring under compressive stress states. In this paper the damage model proposed by Bonora (1997) has been reformulated in order to account for compressive loading introducing a new internal variable associated to damage. The model has been implemented in a commercial finite element code and used to predict single element performance under cycling loading and damage accumulation in a round notch tensile bar. Some preliminary experimental results are also presented. KEYWORDS Ductile damage, cyclic loading, fatigue, CDM INTRODUCTION In the last decades it has been shown that local approaches have a great potential in predicting the occurrence of failure in specimens, components and structures. Today it is well assessed that ductile failure occurs as a result of microvoids nucleation and growth at inclusions. The local approach is based on the assumption that, if the microscopic mechanism of failure is known, the modification of the material constitutive response can be predicted from micromechanical considerations. Consequently, direct transferability from material to structure, without any geometry effect, would be one of the key features. Many theoretical models have been proposed in the literature that can be grouped in two main sets: continuum damage mechanics (CDM) based models and porosity models. Porosity models, derived from the Gurson type model, are based on the modification of the yield function as a result of the increasing porosity with plastic strain. Here, porosity plays the role of a softening variable that progressively implodes the yield surface in order to account for damage effects. CDM models are developed in the framework of continuum mechanics. Here, damage effects are accounted by a thermodynamic variable, D, that reduces material stiffness. Thus, the complete set of constitutive equations for the material undergoing damage is derived. Both approaches suffer major limitations. Porosity models usually require a large number of material parameters, none of which has a physical

meaning, that have to be identified using coupled numerical simulation and experiments. On the other side, CDM model performance depends on the assumed form for the dissipation potential from which damage evolution law can be derived by normality rule. All the models proposed in the literature show material dependency, lack of performance under multi-axial state of stress conditions and temperature and strain rate effect is usually neglected. In 1997 Bonora [1] proposed a new non-linear CDM model for ductile failure that overcome material dependency and stress triaxiality effects. The model resulted successful in predicting notched and cracked components response using only information, such as damage parameters, identified in simple uniaxial state of stress condition, [2]. Later, Bonora and Milella [3] extended the damage model in order to incorporate temperature and strain rate effects. Up to now, very little attention have been given to the mechanics of ductile deformation and damage under compressive state of stresses. This issue becomes very important in order to understand and predict component life under low cycle fatigue regime or under intense dynamic loading in which damage accumulation is related to the bouncing motion of strain waves into the body. Bonora and Newaz [4] demonstrated the possibility to predict low cycle fatigue life at ductile crack growth initiation discussing possible integration scheme for the non-linear damage law. At the moment, as far as the authors are aware of, no attempt to extend CDM model formulation to cyclic loading under variable stress triaxiality loading conditions has been made. In this paper, for the first time, the non-linear damage model proposed by Bonora has been extended to negative stress triaxiality loading condition, based on simple physical considerations, introducing a new internal variable associated to damage D. The model, implemented in form of user subroutine in the finite element code MSC/MARC, has been tested on single FEM element under simple loading conditions such a as normal and shear stress. Successively, it has been applied to round notched bar specimens loaded in tension. At the present time, an extensive experimental program is under investigation. Here, the promising preliminary results are presented and discussed. NON-LINEAR CDM MODEL FOR DUCTILE FAILURE Lemaitre [5] firstly defined the CDM framework for plasticity damage. Damage accounts for material progressive loss of load carrying capability due to irreversible microstructural modifications, such as microvoids formation and growth, microcracking, etc. From a physical point of view, damage can be expressed as D A A n eff n n ( ) ( ) ( ) = -1 0 (1) where, for a given normal n, An 0 ( ) is the nominal section area of the RVE and Aeff n( ) is the effective resisting one reduced by the presence of micro-flaws and their mutual interactions. Even though this definition implies a damage tensor formulation, the assumption of isotropic damage leads to a more effective description where the scalar D can be simply experimentally identified. Additionally, this assumption is not too far from reality as a result of the random shapes and distribution of the included particles and precipitates that trigger plasticity damage initiation and growth. The strain equivalence hypothesis gives the operative definition of damage as: D E E eff = -1 0 (2) where E0 and Eeff are the Young’s modulus of the undamaged and damaged material, respectively. The complete set of constitutive equation for the damage material can be derived assuming that: - a damage dissipation potential fD, similarly to the one used in plasticity theory, exists; - no coupling between damage and plasticity dissipation potentials exists; - damage variable, D, is coupled with plastic strain; -the same set of constitutive equations for the virgin material can be used to describe the damaged

material replacing only the stresses with the effective ones and assigning a state equation for D. Bonora [1] proposed the following expression for the damage dissipation potential, ( ) n n cr D p D D D S S Y f + - - × ú ú û ù ê ê ë é - ÷÷ × ø ö çç è æ = - 2 1 0 2 0 1 2 1 a a (3) where, Dcr is the critical value of the damage variable for which ductile failure occurs, S0 is a material constant and n is the material hardening exponent. a is the damage exponent that determines the shape of the damage evolution curve and is related to the nature of the bound between brittle inclusions and the ductile matrix. Thus, the constitutive equation set for isotropic hardening material is given by: strain decomposition p ij e ij T ij e e e & & & = + (4) elastic strain rate & & & e n s n s d ij e ij kk ij E D E D = + - - - 1 1 1 (5) plastic strain rate eq ij ij p p ij D s f s l ¶s ¶ e l 1 21 3 - = = & & & & (6) plastic multiplier (1 ) p D R f r p =- = = - & & & & l ¶ ¶ l (7) kinetic law of damage evolution Y f D D ¶ ¶ l& & =- = ( ) ( ) p p D D D D cr eq H f th cr f & × × - ÷ ÷ ø ö ç ç è æ × - × - a a a s s e e a 1 1 0 / ) ln( (8) Detailed description on the derivation of these Equations can be found elsewhere, [1]. In Equation (9) stress triaxiality effects are accounted by the function f(sH/seq) defined as, ( ) ( ) 2 3 1 2 1 3 2 ÷ ÷ ø ö ç ç è æ × = + + × - ÷ ÷ ø ö ç ç è æ eq H eq H f s s n n s s (9) that is derived assuming that ductile damage mechanism is governed by the total elastic strain energy, Lemaitre [5]. Here, sH = skk/3 is the hydrostatic part of the stress tensor and n is the Poisson’s ratio. The model requires five material parameters in order to be applied. The strain threshold (in uniaxial monotonic loading) eth, at which damage processes are activated; the theoretical failure strain ef, at which ductile failure under completely uniaxial state of stress conditions occurs; the initial amount of damage present in the material, D0; the critical damage, Dcr, at which failure occurs and the damage exponent, a, that control the shape of damage evolution with plastic strain. Experimental procedure for damage parameters identification can be found elsewhere, [6]. EXTENSION TO REVERSAL PLASTIC FLOW Ductile damage formulations available in the literature always address tensile loading configuration, since it is well known that positive stress triaxiality enlarges nucleating voids in the material microstructure. The possible effect on damage variable due to compressive loading is usually neglected in the theoretical formulations. The major consequence of this limitation is that the damage variable, D, has to be associated to the total effective accumulated plastic strain, usually indicated with p, that plays the role of the associated internal variable. In the literature, little attention is given to the effects on the

material constants, or constitutive response, due to plastic deformation under pure compressive loading. This knowledge is critical in order to develop a predicting model capable to account for plastic strain reversal as for low cycle fatigue. Few attempts based on the cyclic accumulation of damage, or its associated variable, always resulted in predicted very short lives as a consequence of the fact that p usually accumulates quickly. Porosity models, such as the Gurson model, are incapable to predict material failure since porosity effects are fully recovered during compressive loading resulting in a unrealistic healing-material behavior, [7]. These premises clearly indicate that additional hypotheses must be formulated in order to describe properly material behavior under compressive stress states. If ductile damage can be imputed to the formation and growth of microcavities that have the effect to reduce the net resisting area, and consequently material stiffness, thus the following scenarios can be speculated. Scenario a). The material is initially stress-free and it is assumed that no strain history has modified its status from the one of “virgin material”. Let us assume to start to load a material reference volume element, RVE, under pure compressive uniaxial state of stress avoiding any buckling phenomena. In this configuration, microvoids cannot nucleate since the ductile matrix is compressed around the brittle inclusions. If the local stress in the particle overcomes the particle strength, the particle itself can eventually break. This kind of damage should not affect material stiffness since no reduction of the net resisting area is occurred. The only effect that we would expect is probably an anticipated microvoids nucleation, due to an early void opening since the particle is broken, when the stress state is reversed in tension, (i.e. a lower strain threshold value). Even though an irreversible process such as particle breaking will eventually occur under compressive loads, the stiffness should remain unaffected indicating no damage in compression. Scenario b). Let say that the virgin RVE is initially loaded in tension until some amount of damage. Then, the load is reversed in compression developing additional plastic strain. In this case, during the unloading from positive stress-state to zero, microvoids can close controlled by the large ductile matrix volume movement, (here, potential buckling of microcavities is neglected). Voids implode back to the particle from which they have nucleated. Void closure can eventually close to the zero displacement condition. During this phase the net resisting area is restored and the stiffness should be the same as for the virgin material. Continuing in the compressive ramp the stiffness, once again, should remain unaffected. Further compressive loads, will eventually breaks some particles, but no effects are expected on E. A new reload in tension would see both the opening of the previously grown voids plus the opening of the new ones nucleated at the broken particles. However, at this stage it can be assumed that compressive damage does not modify damage developed under positive stress states. It follows that ductile damage can accumulate under positive stress state only, while total plastic strain will accumulate under both. Consequently, the associate damaged variable has to be a redefined as an “active accumulated plastic strain” p+, i.e. the plastic strain that accumulates if and only if, the actual stress triaxiality is greater than zero. Similarly, the damage effect on material stiffness will also be activated if and only if the current stress triaxiality is positive. According to this, the damage model proposed by Bonora can be modified in terms of active damage D+ and active plastic strain p+ as follows: Y f D D ¶ ¶ l& & =- + = ( ) ( ) + + - + × × - ÷ ÷ ø ö ç ç è æ × - × p p D D D D cr eq H f th cr f & a a a s s e e a 1 1 0 / ) ln( (10) ( ) m eq H f D p s s l / ) (1 + + - = & & (11) ( ) m eq E E D H f s s/ ) (1 ~ + = - (12) where

( ) î í ì ³ < = 0 / 1 0 / 0 / m eq m eq m eq H f s s s s s s (13) FINITE ELEMENT ANALYSES Damage model performance under cycling loading has been firstly checked on a single axisymmetric 4 node element. The element size is 1.0 x 1.0 mm. The material is a 22NiMoCr37 steel of German production for which damage parameters were previously determined by Bonora et al. (1998). A cyclic imposed displacement with zero mean value dm=0 and amplitude da=0.2 mm has been applied until element failure. Response under isotropic and kinematic hardening is given in figure 1 together with the displacement evolution with time. Here, time is a fictitious variable since viscoplastic and time dependent behaviors have been neglected. Figure 1 – Single element response under cyclic loading: a) axial stress vs axial strain; b) equivalent plastic strain vs time; c) stress triaxiality vs time; d) active damage vs time; e) active strain vs time; f) imposed displacement vs time In figure 1 it is shown how the effective plastic strain, together with active damage, accumulates only when stress triaxiality is positive. In this case, where the imposed nominal strain amplitude is 20%, failure is expected to occur after 5 cycles. Subsequently, the model has been used to investigate damage evolution in round notched tensile bar (R = 2mm ) under reversal plastic flow loading conditions, figure 2a. At the present time a single test has been performed on SA 537 steel. An initial monotonic ramp up to 0.2 mm has been applied followed by sinusoidal cycling with an amplitude of 0.125 mm. The cycling frequency was 0.0125 Hz. Local deformation field across the notch has been monitored using an extensometer with a gage length of 10 mm. In figure 2b, the comparison between the finite element results and the experimental data is given in term of applied load versus displacement at the gauge. It is important to note that the FEM model incorporating damage is capable to reproduce the load cycle in the near notch region pretty well if kinematic hardening is used. Isotropic hardening results in a very narrow

hysteresis loop. The major difference is given by the macroscopic ratcheting. A posteriori it has been found that this phenomenon can occur in SA537 steel under very low strain rate since at this deformation rate viscoplastic behavior becomes manifested. Experimental test has been stopped after 20 cycles. According to finite element prediction only a limited amount of damage should be generated in the specimen minimum section without failure in this cycling. In figure 3 the deformed mesh showing damage contours after 7 cycles is given as a sample. Figure 2a – RNB(T) specimen geometry and dimensions 2b- cyclic response: comparison between FEM results and experimental data Figure 3 – Damage contours along the minimum section after 7 cycles. REFERENCES 1. Bonora, N., (1997), Eng. Frac. Mech.,, 58, pp. 11-28 2. Bonora, N., (1998), Int. J. Frac., 88, pp.359-371 3. Bonora, N., and Milella, P.P. (2001), J. of Impact Engng., to appear. 4. Bonora N., and Newaz, G., (1998), Int. J. Solids and Struc., 35 , pp. 1881-1894 5. Lemaitre, J., (1986), Engng. Fract. Mech., 25, No. 5/6, pp.523-537 6. Bonora, N, (1999), J. Strain Analysis, 34, 6, pp. 463-478 7. Brocks, W., Bonora, N., O’Dowd, N. and Steglich, D., (2000), personal discussion at GKSS

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