28 To further validate the tensorial aspect of the **elastic** − **plastic** model, without coupling **with** **damage**, simulations are performed on a rectangular plate **with** a hole, which involve heterogeneous stress / strain states within the plate. The geometry, dimensions and loading conditions for this test are all specified in Figure 5. The plate is subjected to a prescribed displacement of 2 mm along the length direction. The mixed hardening model available in ABAQUS, which is based on the Voce isotropic hardening combined **with** the Armstrong − Frederick kinematic hardening and the von Mises yield surface, is adopted in these simulations. The associated elasticity and hardening parameters are taken from Table 3. Due to the symmetry, only one quarter of the plate is modeled. Figure 6 shows the simulation results, in terms of load − displacement responses, as obtained **with** the UMAT subroutine and the built-in ABAQUS model. These numerical results reveal the excellent agreement between the two models. Moreover, Figure 7 displays the distribution of the equivalent **plastic** strain within the plate, as predicted by the developed UMAT and the built-in ABAQUS model. As clearly shown by this figure, the distribution of the equivalent **plastic** strain is well reproduced by the implemented model when compared **with** the reference ABAQUS results.

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2.1. Anisotropic **elastic**-**plastic** model coupled **with** **damage**
Continuum **damage** mechanics was first introduced by Kachanov (1958) and slightly modified later by Rabotnov (1969). This concept was then further developed (Lemaitre, 1992; Chaboche, 1999), based on a continuous variable, d , related to the density of defects or mi- cro-cracks in the material in order to describe its deterioration. Leckie and Hayhurst (1974), Hult (1974), and Lemaitre and Chaboche (1975) adopted this approach to solve creep and creep-fatigue interaction problems. Later, it was applied to ductile **plastic** fracture (Lemaitre, 1985) and also to other various applications (Lemaitre, 1984). Note that most of the available presentations on the concept of continuum **damage** mechanics were concerned **with** metals as can be found in many treaties and research papers (Altenbach and Skrzypek, 1999; Doghri, 2000; Brünig, 2002; Abu Al-Rub and Voyiadjis, 2003; Brünig, 2003; Brünig et al., 2008). In the context of **damage** mechanics in metal matrix composites, more recent publications can also be found, which are more specifically devoted to composite materials (see, for instance, Voyiadjis and Kattan, 1999).

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In his initial analysis of localized necking, Hill (1952) used a simple constitutive model (rigid-**plastic**, J 2 flow theory) **with** a plane stress condition, showing that localized necking of a thin sheet occurs along the line of minimum straining. As an improvement, Benallal (1998) proposed a three-dimensional analysis of localized necking by studying the properties of the **plastic** hardening modulus at the point of strain localization **with** J 2 **elastic**-**plastic** flow theory. Dudzinski and Molinari (1991) proposed a linear perturbation analysis as an alternative to the bifurcation theory to predict **plastic** instabilities, while Barbier et al. (1998) showed a relation between this approach and the bifurcation one. Hora et al. (1996) proposed an improved ver- sion of the Swift criterion and applied it to the prediction of FLDs, while Brunet and Morestin (2001) used this criterion **with** an advanced **elastic**-**plastic** **damage** constitutive model and fi- nite element analysis. Recently, Kuroda and Tvergaard (2004) studied the development of shear bands by using non-associative plasticity within the framework of a finite element analysis, while Kristensson (2006) used a micromechanical model **with** void effects (size and distribution in the representative volume element) to study the formability of metal sheets **with** voids.

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The pressure dependence of the acoustic velocity in these cemented samples is mostly associated **with** the adhesive elasto-**plastic** nature of the cemented contact between the grains. In glass bead packs, external load- ing would develop a stiffer grain-grain contact, whereas in PMMA bead packs it may increase the size of the ce- mented contact. By unloading cemented granular sam- ples prepared at high prestress, acoustic velocities de- crease. The correlation technique of the codalike scat- tered waves shows that this irreversible behavior is likely associated **with** the heterogeneously fractured structure caused by tensile stress, due to the strongly inhomoge- neous residual stress. On the contrary, the cemented ma- terial prepared at low prestress develops a more progres- sive **damage** process under unloading, likely due to the cracks at the cement-grain interfaces distributed more evenly inside the cemented contact network. More in- vestigation is necessary for understanding the respective contributions from the interfacial adhesion/fracture and the **plastic** behavior or breakage of the cementing mate- rials.

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able to simulate a **plastic** material **with** damaged **elastic** materials.
Our functionals in the regime corresponding to α = 0 and 0 < β < ∞ are essentially those intro- duced by Ambrosio and Tortorelli in [ 4 ] (see also [ 3 ]) to approximate the Mumford-Shah functional for image segmentation. Here we extend the study of the limit behavior of these functionals to the other regimes α > 0, or β = 0, or β = ∞; we provide an approximation of the functional ( 1.6 ), different from those studied in [ 2 ] and [ 10 ].

In this type of problem, it is therefore very important to capture the variations of the **elastic** (unloading) stiﬀness of the material upon mechanical loading. Continuum **damage** is the right theoretical framework for that. But continuum **damage** models cannot, alone, be implemented. The material undergoes also some irreversible deformations during loading. As sketched in Fig. 2 , even if **damage** or plasticity models taken separately are capable of capturing the same material response upon monotonic loading, both theories do not capture the evolution of unloading stiﬀness accurately. At a given point on the stress–strain response, neglecting **plastic** strains according to a pure **damage** approach would result in an artiﬁcial increase of **damage** (secant unloading slope). Neglecting **damage** eﬀects in a pure **plastic** model would result in a perme- ability that does not evolve, or is not described objectively when it is a function of the applied strains or stresses [4] . By objective, we mean that for the same value of strain (or stress) invariant combinations, diﬀerent values of permeability are to be expected. Hence, coupled **damage**-plasticity models are a requisite in hydro-mechanical problems dealing **with** concrete structures.

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Mohamed Selmani ∗ and Tayeb Messaoudi
Abstract. We consider a mathematical model for the process of a fric- tionless contact between an **elastic**-viscoplastic body and a reactive foun- dation. The material is **elastic**-viscoplastic **with** internal state variable which may describe the **damage** of the system caused by **plastic** defor- mations. We establish a variational formulation for the model and prove the existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear equations **with** monotone operators, on parabolic type inequalities and ﬁxed point.

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In light of these encouraging results, in this paper, a quantitative calorimetric analysis of the fretting **damage** is proposed. The heat sources are derived from thermal images using a local expression of the heat diffusion equation. A specific 2D image processing technique is then introduced to separately estimate the dissipated energy associated **with** irreversible microstructural transformations and the thermoelastic coupling induced by the reversible thermal sensitivity of the material. The maximal local evolution of the intrinsic dissipation as function of the maximal shear stress is then coupled **with** the onset of micro-**plastic** phenomena responsible of the fretting **damage**. Finally, these results were used to build the **elastic** shakedown boundary.

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A robust and efficient computer implementation has been performed in a finite element code, in order to apply the developed models to forming process simulation. Direct and sequential rheological tests have been simulated in order to validate the computer implementation, **with** very good results. The application of the **elastic**-**plastic** model to a springback analysis highlighted the impact of advanced hardening

Department of Mechanical and Aerospace Engineering, University of Florida, REEF, 1350 N. Poquito Rd, Shalimar, FL 32579, USA.
ABSTRACT
Modeling of ductile **damage** is generally done using analytical potentials, which are expressed in the stress space. In this paper, for the first time it is shown that strain rate potentials which are exact conjugate of the stress-based potentials can be instead used to model the dilatational response of porous polycrystals. A new integration algorithm is also developed. It is to be noted that a strain-rate based formulation is most appropriate when the **plastic** flow of the matrix is described by a criterion that involves dependence on all stress invariants. In such cases, although a strain-rate potential is known, the stress-based potential cannot be obtained explicitly. While the proposed framework based on strain-rate potentials is general, for comparison purposes in this work we present an illustration of the approach for the case of a porous solid **with** von Mises matrix containing randomly distributed spherical cavities. Comparison between simulations using the strain-rate based approach and the classical stress- based Gurson’s criterion in uniaxial tension is presented. These results show that the model based on a strain-rate potential predicts the dilatational response **with** the same level of accuracy.

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1. Introduction
Sheet metal forming is one of the most used processes in manufacturing industries. This process involves **plastic** deformation of metallic sheets and is designed to obtain complex parts **with** fast cadence. Nevertheless, it happens that localized necking occurs in the drawn part before the end of forming operations. The onset of this localized necking represents the ultimate deformation that the drawn part can undergo, since this phenomenon is often precursor to material failure. Hence, efficient and reliable prediction of the occurrence of localized necking is required to help in the calibration of the process controlling parameters. The most common representation for the necking limit strains relies on the concept of forming limit diagram (FLD), which was initially proposed by Keeler and Backofen (1963). The prediction of such diagrams requires the combination of a **plastic** instability criterion and a constitutive model that describes the mechanical behavior of the studied sheet. Our attention in this paper is focused on materials exhibiting **plastic** anisotropy. Such anisotropic behavior is due to rolling operations, which are performed before the forming process. It is expected that **plastic** anisotropy plays a crucial role in the prediction of localized necking in sheet metals. Hence, accurate predictions of strain localization are needed, especially for anisotropic materials and for complex loading paths. The onset of localized necking may occur as a bifurcation from a homogeneous deformation state or it may be triggered by some assumed initial imperfection. Accordingly, two main classes of strain localization criteria can be found in the literature:

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I =1
( I )J( I )B T ( I )·C·B( I ) (13)
where J ( I ) is the Jacobian of the transformation between the unit reference configuration and the current configuration of an arbitrary hexahedron. It is important to underline that, although undertaken within a displacement-based approach, the investigation of hourglass modes conducted in this section and the associated stabilization procedure given in Section 2.4 are quite general as long as at least two integration points are used. This is further detailed in Appendix B where it is shown that the proposed stabilization still applies after the projection technique is implemented within the assumed strain framework, for which the underlying variational principle is presented in Section 2.5. Note also that two integration points are sufficient for both providing a rank sufficient element and dealing **with** **elastic** problems, as will be shown through the numerical examples given in Section 3. It has also been revealed, from illustrative test problems, that a minimum of five integration points should be used when dealing **with** **elastic**–**plastic** applications. Table I gives the coordinates and the associated weights of the Gauss points, which represent the roots of the Gauss–Legendre polynomial, in the case of five integration points along the thickness direction.

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2 Author name / Structural Integrity Procedia 00 (2016) 000–000
1. Introduction
Sheet metal forming is one of the most used processes in manufacturing industries. This process involves **plastic** deformation of metallic sheets and is designed to obtain complex parts **with** fast cadence. Nevertheless, it happens that localized necking occurs in the drawn part before the end of forming operations. The onset of this localized necking represents the ultimate deformation that the drawn part can undergo, since this phenomenon is often precursor to material failure. Hence, efficient and reliable prediction of the occurrence of localized necking is required to help in the calibration of the process controlling parameters. The most common representation for the necking limit strains relies on the concept of forming limit diagram (FLD), which was initially proposed by Keeler and Backofen (1963). The prediction of such diagrams requires the combination of a **plastic** instability criterion and a constitutive model that describes the mechanical behavior of the studied sheet. Our attention in this paper is focused on materials exhibiting **plastic** anisotropy. Such anisotropic behavior is due to rolling operations, which are performed before the forming process. It is expected that **plastic** anisotropy plays a crucial role in the prediction of localized necking in sheet metals. Hence, accurate predictions of strain localization are needed, especially for anisotropic materials and for complex loading paths. The onset of localized necking may occur as a bifurcation from a homogeneous deformation state or it may be triggered by some assumed initial imperfection. Accordingly, two main classes of strain localization criteria can be found in the literature:

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1. Introduction
Sheet metal forming is one of the most used processes in manufacturing industries. This process involves **plastic** deformation of metallic sheets and is designed to obtain complex parts **with** fast cadence. Nevertheless, it happens that localized necking occurs in the drawn part before the end of forming operations. The onset of this localized necking represents the ultimate deformation that the drawn part can undergo, since this phenomenon is often precursor to material failure. Hence, efficient and reliable prediction of the occurrence of localized necking is required to help in the calibration of the process controlling parameters. The most common representation for the necking limit strains relies on the concept of forming limit diagram (FLD), which was initially proposed by Keeler and Backofen (1963). The prediction of such diagrams requires the combination of a **plastic** instability criterion and a constitutive model that describes the mechanical behavior of the studied sheet. Our attention in this paper is focused on materials exhibiting **plastic** anisotropy. Such anisotropic behavior is due to rolling operations, which are performed before the forming process. It is expected that **plastic** anisotropy plays a crucial role in the prediction of localized necking in sheet metals. Hence, accurate predictions of strain localization are needed, especially for anisotropic materials and for complex loading paths. The onset of localized necking may occur as a bifurcation from a homogeneous deformation state or it may be triggered by some assumed initial imperfection. Accordingly, two main classes of strain localization criteria can be found in the literature:

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and in Figures 4 and 5 for the second model **with** change in material parameters (Eqs. (30–31)). The comparison of the volumetric **damage** evolution in Figs. 2 and 4 exhibits that ignoring the induced change in material parameters leads to a limit in the irreversible change in volume (see the two final constant steps in Fig. 2), whereas if this change is taken into account the volumetric **damage** evolves continuously during triaxial cyclic loading. This mechanical response can be also observed **with** the stress-strain responses in Figs. 3 and 5: for the former model a unique curve is reached for about λ = 1.5 and unloading parts of the response become closer and closer as depicted in Fig. 3. For the latter model, Fig. 5 shows that the material stiffness evolves in a regular manner and the result- ing cyclic response is similar to the one encoun- tered **with** **damage**-like constitutive equations, see for example Chagnon et al. (2004).

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M. The Mechanics and Thermodynamics of Continua. Effects of hydrogen on the properties of iron and steel. Hydrogen trapping models in steel. Modelling hydrogen-induced [r]

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Université Pierre et Marie Curie-Paris 6, Laboratoire Jaques Louis Lions, 4 place jussieu 75005 Paris.
Abstract
Earlier works in engineering, partly experimental, partly computational have revealed that asymptotically, when the excitation is a white noise, **plastic** deformation and total deformation for an elasto-perfectly-**plastic** oscillator have a variance which increases linearly **with** time **with** the same coefficient. In this work, we prove this result and we characterize the corresponding drift coefficient. Our study relies on a stochastic variational inequality governing the evolution between the velocity of the oscillator and the non-linear restoring force. We then define long cycles behavior of the Markov process solution of the stochastic variational inequality which is the key concept to obtain the result. An important question in engineering is to compute this coefficient. Also, we provide numerical simulations which show successful agreement **with** our theoretical prediction and previous empirical studies made by engineers.

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To cite this version :
Lotfi MANSOURI, Hocine CHALAL, Farid ABED-MERAIM, Tudor BALAN - **Plastic** Instability Based on Bifurcation Analysis: Effect of Hardening and Gurson **Damage** Parameters on Strain Localization - Key Engineering Materials - Vol. 504-506, p.35-40 - 2012

5.4 **Elastic**-**Plastic** crack growth prediction
This experiment has two very interesting outcomes: First the crack path which is rather complex, and second the fact that the crack stops during a significant time and the starts again to propagate. We first compare the evolutions of the experimental crack tip length to the computed ones obtained for different yield equivalent stresses. The crack length as a function of time is displayed of figure 9 and compared to the computed ones. One can see that the crack first propagates at constant speed then stops between 0.25ms to 0.32ms and then propagates again at roughly the same speed. **With** **elastic** or reasonable yield equivalent stress of 80 MPa, the introduction of an **elastic**-**plastic** behavior doesn’t really change the global computed response of the specimen (figure 9). The crack path are very similar to the experimental one (figure 11). It can hence be concludes that crack propagates **with** small scale yielding.

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Fig. 4. Sequential FLDs for an IF-Ti steel (at left) and for a Dual Phase steel (at right)
In Fig. 4, FLDs are plotted **with** two different pre-strains for the two steels. Only a qualitative comparison can be made since no experimental or reference results are available, as ArcelorMittal’s FLD model is not designed to predict formability when two-stage or more complex loading paths are considered. Nevertheless, Haddad [17] has shown that during a second loading path, the FLD is shifted in the direction of the first path. For example, in the case of uniaxial tension pre- strain the curve is expected to be translated along the preloading direction. This is clearly found **with** the present model.

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