Coupled plastic damage model for low and ultra-low cycle seismic fatigue

11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver and A. Huerta (Eds) COUPLED PLASTIC DAMAGE MODEL FOR LOW AND ULTRA-LOW CYCLE SEISMIC FATIGUE LUCIA G. BARBU*, SERGIO OLLER†, XAVIER MARTINEZ† AND ALEX H. BARBAT† *,†International Center for Numerical Methods in Engineering (CIMNE) Universidad Politécnica de Cataluña Campus Norte UPC, 08034 Barcelona, Spain e-mail: [email protected] Key Words: LCF, ULCF, Plastic damage model, Seismic fatigue Abstract. This paper presents the theoretical framework for a coupled plastic damage constitutive model valid for materials subjected to cyclic loads that lead to low and ultra-low cycle fatigue. Two numerical examples were presented in order to illustrate the behaviour of the model and its capabilities. 1. INTRODUCTION The fatigue phenomenon produces a loss of material strength as a function of the number of cycles, load amplitude, reversion index, etc. This loss of strength induces the material to inelastic behaviour, micro-cracking followed by crack coalescence, leading to the final collapse of structural parts. When dealing with low and ultra-low cycle fatigue (LCF and ULCF), characterized by levels of stress superior to the elastic limit, this collapse occurs for a number of cycles below 105 and it is due to both damage and plasticity effects. The most common procedures used to simulate LCF and ULCF are based on counting the number of cycles that can be applied to the material for a given plastic strain. Examples of those approaches are the Coffin-Manson rule, or the enhanced rule proposed by Xue [1]. However, one of the main drawbacks of these formulations is that they require regular cycles to predict the material failure and, often, this regularity does not exist. Furthermore, these approaches do not take into account a change in material stiffness that can influence the estimated fatigue life for low cycle fatigue. This work proposes the coupling of a plastic model with a damage model to simulate Low and Ultra Low Cycle Fatigue. The constitutive model is based on the work of Luccioni et al. [2] for monotonic loads. The hardening behaviour of the material is formulated in terms of the dissipated energy as presented by Martinez et al. in [3]. With this approach, the energy required in each hysteresis cycle by both the damage process and the plastic process is measured, as well as the available remaining energy of the material. Failure takes place when all the fracture energy of the material has been dissipated. The effect of the accumulation of number of cycles of loading can be taken into account as proposed by Oller et al. in [4]. This implies the incorporation of a new internal variable for the Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 2 model, the number of cycles, which affects the damage and plasticity yield surfaces. Furthermore, it can make use of the load-advancing strategy proposed by Barbu et al. in [5] ensuring a reasonable computational time for materials that fail in a range of 104-105 cycles. 2. PLASTIC DAMAGE MODEL FORMULATION LCF and ULCF mechanical processes cannot be modelled using traditional fracture mechanics and fatigue models because they are often accompanied by large inelastic strains (damage and/or plasticity) which may invalidate stress intensity-based K  or typeJ approaches [6]. Furthermore, the induced loading histories are extremely random in the case of ULCF, difficult to adapt to conventional cycle counting techniques such as rain flow analysis [7, 8] or strain life approaches. Finally, K  or typeJ   methods require an initial sharp crack or flaw, which is absent in many structural details. These limitations, together with the large strain advance finite-element formulation methods, require understanding the underlying ULCF and LCF processes and the development of improved models to predict them. The study of LCF should start from the basic fact that it is not a phenomenon associated to the classic concept of plasticity. Typically, a progressive loss of strength occurs depending on the number of stress cycles that induces local plasticity and/or damage effects. In addition, all these phenomena are usually coupled with thermal effects. These phenomena will be simulated with a constitutive model that couples plasticity, driven by a yield function, with a mechanical damage affected by the number of cycles. The damage model takes into account the pore formation and the subsequent coalescence. The Bauschinger effect [9, 10] is taken into account by means of the kinematic hardening incorporated in the plasticity formulation. The strength reduction produced at a very low number of the cycles is a consequence of the softening caused by kinematic effects on the plastic yield function during cyclic plasticity (Bauschinger effect [9,10] ). Following these ideas, a phenomenological model is formulated to couple plastic and damage behaviours. This model activates itself with the accumulation of the number of cycles leading to the loss of strength at each point for a wide range of fatigue processes. The decrease of the strength shows a crack in which the total fracture energy available at each point is dissipated. Thus, the structural fracture is expressed as a succession of points that have suffered loss of strength during fatigue. The coupling between plasticity and damage, together with the addition of the sensibility to the number of cycles in both formulations, provides a general fatigue model capable of working in a wide field, ranging from ultra-low cycle fatigue to high cycle fatigue. The inclusion of the number of cycles into the plastic and damage formulation, through its internal variables, allows the incorporation of the accumulative damage in the fatigue process, requiring no special or additional constitutive rule [11]. Thus, the constitutive model proposed herein is able to account for non-linear damage accumulation problems that occur when a structural part is subjected to cycles of different load amplitudes and duration (low and high number of cycles). Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 3 2.1 Quasi-static elasto-plastic damage model. Mechanical formulation. The theories of plasticity and/or damage can simulate the material behaviour beyond the elastic range, taking into account the change in the strength of the material through the movement of the yield and/or damage surface (isotropic and kinematic) due to the inelastic behaviour (plasticity and damage) of each point of the solid. However they are not sensitive to cyclic load effects. In this work the standard inelastic theories are modified to introduce the fatigue effect coupled with non-fatigue material behaviour. It is assumed that each point of the solid follows a damage-elasto-plastic constitutive law (stiffness hardening/softening) [2, 12 and 13] with the stress evolution depending on the free strain variable and plastic and damage internal variables. The formulation proposed herein studies the phenomenon of stiffness degradation and irreversible strain accumulation through the combined effect of damage and plasticity. Since this work is oriented towards mechanical problems with small elastic strains and large inelastic strains, the free energy additively hypothesis is accepted ep   [14, 15]. The elastic e  and plastic p  parts of the free energy are written in the reference configuration for elastic Green strainsܧ௜௝ ௘ൌܧ ௜௝ െܧ ௜௝ ௣; the last variable operates as a free field variable [2, 13, 15, and 16]. The free energy is thus written as   )(C 2 1 )1()(),( ppe kl o ijkl e ij o ppe ij eEE m ddE   (1) Considering the second thermodynamic law (Clausius-Duhem inequality – [14, 17, 18], the mechanical dissipation can be obtained as [15] 0        d d m ES po p ij ij      (2) The fulfilment of this dissipation condition (Equation 2) demands that the expression of the stress should be defined as (Coleman method; see [18])   kl o ijkl ij o ij Ed E mS C)1(      (3a) Also, from the last expressions, the secant constitutive tensor can be obtained as:  o ijkl e kl e ij e o e kl ij s ijkl d EE m E S dC)1(C 2       (3b) where o m is the material density, p ijij e ij EEE ,, are the elastic, total and plastic strain tensors, 1 ini dd is the internal damage variable enclosed between its initial value ini d and its Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 4 maximum value 1, p  is a plastic internal variable, 0 Cijkl and CS ijkl are the original and secant constitutive tensors and ij S is the stress tensor for a single material point. 2.2 Yield and potential plastic functions The yield function P Faccounts for the residual strength of the material, which depends on the current stress state and the plastic internal variables and, in the formulation proposed herein, it is sensitive to the fatigue phenomenon. This P F function has the following form, taking into account isotropic and kinematic plastic hardening (Bauschinger effect [9, 10]): 0),,()(),(  NSKSfSF p ij P ijij Pp ij P  (4) where )( ijij PSf   is the uniaxial equivalent stress function depending of the current value of the stresses ij S, ij  is the kinematic plastic hardening internal variable,   NSK p ij P,,  is the plastic strength threshold and p  is the plastic isotropic hardening internal variable [2, 12, 13]. N is the number of cycles of the stress in the point of the solid and p  is a symbolic notation for all the plastic variables involved in the process. The evolution law for the plastic strain is ij P P ij S G E     , being  the consistency plastic factor and P G the plastic potential. Kinematic hardening accounts for a translation of the yield function and allows the representation of the Bauschinger effect in the case of cyclic loading. This translation is driven by the kinematic hardening internal variable ij  which, in a general case, varies proportionally to the plastic strain of the material point. One of the laws that define the evolution of this parameter is P ijkij Ec    ,with kk hc 3 2  for Von Mises (5) where k h is a material property to be determined by particular tests for the Prager and Melan kinematic hardening [15]. The evolution of isotropic hardening is controlled by the evolution of the plastic hardening function P K , which is often defined by an internal variable p  . The rate equation for these two functions may be defined, respectively, by pp p kk Eh S G hH hHK                  kkk :   (6) where k denotes scalar and k stands for a tensor function. Depending on the functions defined to characterize these two parameters, different solid performances are obtained. Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 5 2.3 Threshold damage function Onset of damage depends on the current stress state, the internal damage variable and, with the current formulation, it also depends on the number of cycles. The threshold damage function is defined as (see [18, 19]) .)(),( 0),,()(),( cteSgdSG NdSKSfdSF ij D ij D ij D ij D ij D   (7) where () Dij f S is the equivalent stress function in the undamaged space, ),,( NdSK ij D is the damage strength threshold, and 0 t dddt   the damage internal variable. The evolution of the damage strength threshold is analogous to that of the plastic strength threshold, depending on the internal degradation variable d  dh hK dd dd k D       k   (8) In equation 8, d k h is a scalar function with scalar arguments and d hk is a scalar function with tensorial arguments as shown by [2, 12, and 13]. 2.4 Coupled plastic-damaged response and tangent constitutive law From the simultaneous consistency conditions for the plastic ( 0 P F  ) and damage ( 0 D F ) problems, the evolution of the plastic strain and damage variables can be obtained. The secant constitutive law and the stress rate are )(C)1( p kl kl o ijklij EEdS   p kl p ijkl kl e ijkl p kl kl o ijklij EEEEd t S CC)(C)1(     kl o rsij rs D D D s ijkl e ijkl S S g g G d                      C )1( 1 CC 0 (9) Considering the stress rate as kl ep ijkl ij ES  C, the analytical expression of the tangent constitutive tensor is rs P s mnrs mn P tu P tur p r P tu P tu P k e mnkl mn P rs P s ijrs e ijkl ep ijkl S G S F S G h F S GF c S F S G                     C)(: CC CC  (10) Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 6 2.5 Algorithm for the numerical implementation of the plastic-damaged model For this model, plasticity and damage equation must be integrated simultaneous. This is done with the following Euler-backward algorithm. Between two-equilibrium configurations t and tt the formulation is updated as follows:  t ij P tt p t p S G EE ijij                    t p ij tt p t pSH ),(      t ij P k tt ij t ij S G c                t ttt ddd     (11) The stress state is updated according to the secant constitutive law shown in equation 9 and its reduction to the damage and plastic yield surfaces is done simultaneously leading to the following non-linear system of equations: 0),,()(),(),(  NSKSfSFdH p ij P ijij Pp ij P tt P  0),,()(),(),(  NdSKSfdSFdH ij D ij D ij D tt D  (12) This system of equations can be solved by Newton-Raphson procedure [20] 1 1 1 1),( ),(                                                                                        k tt Dtt P k t D t Dt P t P k t t k t t dH dH d HH d HH dd      (13) Despite having the analytical expression of the tangent constitutive tensor, equation (10), the calculation of this tensor is extremely costly and, depending on the yield and damage functions used, its approximation does not provide correct results. To overcome this drawback this tensor will be calculated numerically by a perturbation method. This is obtained as ab ij ep ijab E S C     (14) with ab E   an infinitesimal perturbation applied to the mechanical strain tensor, and ij S   the stress variation produced by the strain perturbation. With this procedure, it is necessary to apply twice a×b perturbations to obtain the complete tangent tensor. However, despite the computational cost, it provides an accurate approximation that improves the global convergence of the problem [21]. 3. PLASTIC DAMAGE MODEL ORIENTED TO FATIGUE ANALYSIS The effects of a cyclic load on the constitutive behaviour of a material range from the Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 7 accumulation of plastic strain in the case of ULCF to the reduction of material stiffness when dealing with high cycle fatigue. LCF induces changes in the material that are a combination of the aforementioned phenomena. In the following the methodology and motivation for taking into account the effects generated by the cyclical load will be presented. 3.1 Ultra-low cycle fatigue The model is able to account for ULCF effects by incorporating a new law, especially developed for steel materials, that has been designed to reproduce their hardening and softening performance under monotonic and cyclic loading conditions (Figure 1). This law depends on the fracture energy of the material.  Figure 1: Evolution of the equivalent stress The equivalent stress state shown in Figure 1 has been defined to match the uniaxial stress evolution shown by most metallic materials. This curve is divided in two different regions. The first region is defined by fitting a curve to a given set of equivalent stress-equivalent strain points. This curve is a polynomial of any given order and is fitted by using the least squares method. The data given to define this region is expected to provide an increasing function, in order to obtain a good performance of the formulation for cyclic analysis. The second region is defined with an exponential function to simulate softening. The function starts with a null slope that becomes negative as the equivalent plastic strains increases. The exact geometry of this last region depends on the fracture energy of the material. The exact formulation of the constitutive law can be found in Martinez et al. [3]. Characteristic of this type of fatigue is the Bauschinger effect that is taken into account in the constitutive model by combining isotropic hardening with kinematic hardening. The energy dissipated in each hysteresis loop is monitored and failure under cyclical loads is reached when the total available fracture energy of the material is spent. The plastic damage formulation presented in this paper can be used to improve result accuracy when simulating the softening behaviour under ULCF loads. 3.2 Low cycle fatigue The behaviour of a material subjected to cyclical loads that induce low cycle fatigue (LCF) exhibits both accumulation of plastic strain and a reduction of stiffness (Figure 2). While Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 8 ULCF can be described exclusively by plastic models and HCF by damage models, LCF should be modelled with coupled plastic damage models. It is often difficult to predict at which moment in the material life stiffness reduction begins, since the boundaries between these types of fatigue are rather arbitrary. The model this paper proposes aims at making a contribution in correctly assessing the fatigue life for materials subjected to ULCF and LCF and is particularly effective for the transition zone between these two phenomena. Figure 2: Experimental stress-strain curve for X52 steel [22] In the context of the hardening law proposed for ULCF, the plastic damage model presented in this work activates itself in the softening region. This is justified by the physical implications behind the damage phenomenon, as damage induces porosity that leads to stress relaxation. This implies that region 1 in Figure 1 is governed by plasticity ensuring that only the cyclical loads that last a long enough number of cycles get to experience damage effects. This is important as the formulation is meant to guarantee that, for a material life clearly in the ULCF range (dozens of cycles or less), the constitutive equations governing should be those of plasticity. By regulating the extension of region 1 with respect to region 2 discrimination is made between materials that exhibit more sensitivity to ULCF with respect to LCF or the opposite. 4. PERFORMANCE OF THE FORMULATION In the following we included the results obtained for several simulations conducted to illustrate the performance of the formulation presented. These simulations prove the ability of the formulation to characterize mechanical softening behaviour, under monotonic loading conditions. The main aim of all these simulations is to show the response obtained with the proposed constitutive model. Therefore, for the sake of simplicity, and to reduce the computational cost of the simulations, all of them have been conducted on a single hexahedral finite element. The element is fixed in one of its faces and the load is applied to the opposite face as an imposed displacement. -6e+008 -4e+008 -2e+008 0 2e+008 4e+008 6e+008 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.04 5 Stress(Pa) Strain (-) Experimental curve Lucia G. Barbu, Sergio Oller, Xavier Martinez and Alex H. Barbat 9 4.1 Simulation of the mechanical performance under monotonic loads – softening behaviour The first simulation presents the capabilities of the formulation when dealing with materials that exhibit softening behaviour. The material characteristics are given in Table 1. Table 1: Material characteristics Young Modulus 2.01·105 MPa Poisson Modulus 0.30 Elastic Stress ( eq Y  ) 838.9 MPa Fracture Energy 0.1 MN·m/m2 Figure 3 shows the evolution of the stress for the plastic damage model as compared with the same simulation conducted with a classic plasticity model and with a classical scalar damage model such as that proposed by Kachanov [23]. Three different simulations were made with the coupled model. The first one assumes equal distribution of energy between the two interconnected phenomena: 50% for damage and 50% for plasticity. A different one was made with 10% for damage and 90% for plasticity and the last one allotted 90% for damage and 10% for plasticity. Unloading was made at approximately the middle of the loading history in order to observe better the material behaviour. For the numerical simulations using the two uncoupled models, the entire available fracture energy was used (100%). Figure 3: Stress evolution for the plastic damage model as compared to the classical plasticity and damage models It can be seen in Figure 3 that, unless the material is unloaded, the behaviour is the same in all the simulations ran. When unloading, the loss of stiffness and the accumulated plastic strain are in accordance with the percentage of energy given to each one of the models. For 0 1e+008 2e+008 3e+008 4e+008 5e+008 6e+008 7e+008 8e+008 9e+008 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Stress (Pa) Strain (-) Kachanov damage (Uncoupled) Uncoupled plasticity Coupled 50%d - 50%p Coupled 10%d - 90%p Coupled 90%d - 10%p