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Ductile damage evolution under triaxial state of stress: theory and experiments Nicola Bonora

a,*

, Domenico Gentile a, A. Pirondi b, Golam Newazc

a

Department DIMSAT, University of Cassino, Via G. Di Biasio 43, 03043 Cassino, Italy b Department DII, University of Parma, via dellÕUniversita` 12, 43100 Parma, Italy c Mechanical Engineering Department, Wayne State University, Detroit, 40202 MI, USA Received in ﬁnal revised form 17 June 2004 Available online 26 August 2004

Abstract In this paper, the continuum damage mechanics (CDM) model formulation proposed by [Eng. Fract. Mech. 58(1/2) (1997) 11] has been validated against ductile damage evolution experimentally measured in A533B low alloy steel under various stress triaxiality conditions. A procedure to identify the model parameters has been deﬁned ﬁrst. Then, the model has been used to simulate, via ﬁnite element analysis (FEA), tests on notched ﬂat rectangular bars with diﬀerent notch radii. The experiments and the FEA predictions are ﬁnally compared with each other. The results presented here conﬁrm the transferability of damage parameters deﬁnition and the potential of the proposed damage model in predicting ductile failure occurrence in structures and components under multi-axial state of stress loading conditions. 2004 Elsevier Ltd. All rights reserved. Keywords: Continuum damage mechanics; Stress triaxiality; Ductile failure; Damage measurement

*

Corresponding author. E-mail address: [email protected] (N. Bonora).

0749-6419/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2004.06.003

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1. Introduction Damage resulting from plastic deformation in ductile metals is mainly due to the formation of microvoids which initiate either as a result of fracturing or debonding from the ductile matrix of inclusions such as carbides and sulﬁdes. The growth of microvoids for an increasing strain progressively reduces the material capability to carry loads up to complete failure. A proper modeling of this micromechanism at the mesoscale is the basis for the prediction of ductile failure in real life components and structures (i.e., the macroscale). McClintock (1968) ﬁrstly recognized the role of microvoids in ductile failure process and tried to correlate the mean radius of the nucleated cavities to the overall plastic strain increment. Rice and Tracy (1969) analytically studied the evolution of spherical voids in an elastic-perfectly plastic matrix. In these pioneering studies, the interaction between microvoids, the coalescence process and hardening eﬀects were neglected and failure was postulated to occur when the cavity radius would reach a critical value speciﬁc for the material. Since then many diﬀerent models have been proposed in the literature. Today, all these formulations can be categorized in three main approaches: (I) abrupt failure criteria, (II) porous solid plasticity and (III) continuum damage mechanics (CDM). In the ﬁrst approach, failure is predicted to occur when one external variable, that is uncoupled from other internal variables, reaches its critical value as for the Rice and Tracy critical cavity growth criterion. In the second approach, the eﬀect of ductile damage, as proposed by Gurson (1977) and Rousselier (1987), is taken into account in the yield condition by a porosity term that progressively shrinks the yield surface. Later, Needleman and Tvergaard (1984) and Koplik and Needleman (1988) extended the initial formulation proposed by Gurson in order to include the acceleration in the failure process induced by void coalescence (GTN model). More recently, a number of ﬁnite element unit cell based micromechanical studies have been performed in order to correlate voids evolution and interaction with the resulting macroscale material yield function. Tvergaard and Niordoson (2004) investigated the role of smaller size voids in a ductile damage material using on-local plasticity model as proposed by Acharya and Bassani (2000). Schacht et al. (2003), used the 3D voided unit cell based approach to investigate the role and the eﬀects associated with the crystallographic orientation of the matrix material, ﬁnding a substantial dependency of the growth and coalescence phase with the anisotropy of the material surrounding the voids. Bonfoh et al. (2004) used the GTN damage deﬁnition to model damage evolution initiated by secondary included particles debonding in a polycrystalline material. The GTN model, although extensively used to study ductile failure and crack propagation (Xia and Shih, 1995; Mahnken, 2002; Besson et al., 2003), is also known to suﬀer from a number of limitations: – the identiﬁcation of a large number of material constants (up to nine) is required for the material under investigation (Brocks and Bernauer, 1995). Consequently, it is diﬃcult to evaluate possible mutual inﬂuence of the parameters;

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– the material constants are not physically based and cannot be directly measured for a material. Typically, an iterative calibration procedure, involving ﬁnite element simulations and experimental data, is necessary. For instance, Prahl et al. (2002), determined the damage parameters, in microalloyed, thermo-mechanically treated steel S460M, for both TGN and the Rousselier model using period homogenization approach. Similarly, Springmann and Kuna (2003) developed a non-linear optimization identiﬁcation procedure for the determination of the Rousselier damage model parameters. However, in none of these cases the portability of the set of the identiﬁed damage parameters has been proved to work for geometries diﬀerent than that used in the identiﬁcation stage. – The transferability to other geometries (i.e., diﬀerent state of stress) is not always satisfactory: material parameters identiﬁed with uniaxial tensile tests result in a very poor predicted response of notched or cracked components (Brocks and Bernauer, 1995). A posteriori adjustment of the model parameters is necessary to correctly simulate ductile failure under diﬀerent geometric constraints, Faleskog et al., 1998); – similarly to other formulations with damage softening in the yield function, also the Gurson model requires a length scale (Addessio and Johnson, 1993). In the case of rate independent ﬂow stress formulation, the length scale is given by the element mesh size. Consequently, stable predictions cannot be achieved and the damage model constants become a function of the mesh size. In the third approach, damage is assumed to be one of the internal constitutive variables that accounts for the eﬀects on the material constitutive response induced by the irreversible processes that occurs in the material microstructure. Starting from the early work of Kachanov (1958, 1986), the CDM framework for ductile damage was later developed by Lemaitre (1985) and Chaboche (1984). In the last two decades, a number of CDM based formulations have been proposed. In this context, the complete set of constitutive equations for a ductile damaged material is derived assuming TaylorÕs strain equivalence between micro and mesoscale and the existence of a damage dissipation potential, similar to the one adopted in the theory of plasticity (Lemaitre, 1986). More recently, starting from the consideration that the gradient eﬀect is important when the characteristic dimension of the plastic deformation or damage is of the same order of the material intrinsic length scale, a number of so-called non-local theories have been proposed (Fleck et al., 1994; Fleck and Hutchinson, 1997; Bammann et al., 1999). Brunig (2003) proposed an anisotropic CDM model using the porosity as a deﬁnition for the damage variable while Voyidajis et al. (2004) developed a coupled non-local viscoplasticity and non-local viscodamage model using the Kachanov damage deﬁnition. Relative to the initial framework proposed by Lemaitre, several damage models, based on the use of special expressions for the damage dissipation potential, have been derived by diﬀerent authors (Chandrakanth and Pandey, 1993; Tai and Yang, 1986; Tai, 1990). Unfortunately, these models also show major limitations:

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– the proposed choice for the damage dissipation potential is, in many cases, speciﬁc of the particular material. For instance, assuming a damage dissipation potential expression so that its partial derivative with respect to the associated damage internal variable Y is independent on the accumulated plastic strain, p, may be appropriate for copper but not for carbon steels or aluminum alloys. Consequently, moving to a diﬀerent material would require a new determination of the damage dissipation potential. – Damage evolution laws are often validated only with experimental data obtained under uniaxial stress. Therefore, the transferability of parameters to multiaxial stresses is not always proven. – Other relevant issues, such as possible stress triaxiality inﬂuence on damage parameters and kinetic law of damage evolution, are often neglected. – Similarly to the porosity models, also the standard CDM formulations, as a consequence of the substitution of the stress tensor with the eﬀective stress one, result in a softening term in the material plastic ﬂow stress deﬁnition that requires a length scale and is responsible for a mesh eﬀect in the numerical simulations. In real components, multiaxial state of stress loading condition, induced both by geometry changes and external loading, represents the rule rather than the exception. Consequently damage evolution and model performance should be directly validated in these loading conﬁgurations and not only limited to the uniaxial test case. Lammer and Tsamakis (2000) compared diﬀerent CDM models with reference to homogeneous and inhomogeneous deformations providing a formulation generalization to ﬁnite deformation. Recently, Bonora (1997) proposed a damage model formulation in the framework of CDM that has been shown to overcome most of the limitations previously discussed. This formulation is material independent, allowing the description of diﬀerent damage evolution laws with plastic strain without the need to change the choice of the damage dissipation potential. Furthermore, stress triaxiality eﬀects, such as the progressive reduction of material ductility with increasing stress triaxiality factor (i.e., TF = rm/req) are accurately predicted (Bonora, 1998). Later, Pirondi and Bonora (2003), starting form the consideration that it is not possible to separate in a real test damage softening from plasticity hardening eﬀect, pointed out how the ﬂow stress curve, as experimentally determined, does not requires the substitution of the eﬀective stress in its expression. This has the major consequence to avoid the presence of an explicit softening term and, consequently, to preclude mesh eﬀect. In this paper, ductile damage evolution under diﬀerent stress triaxiality regimes has been investigated through an extensive experimental program performed at room temperature on A533 grade B steel, which is commonly used in the nuclear industry. The damage parameters required in the CDM damage model formulation proposed by Bonora have been identiﬁed under the uniaxial tensile test and used to predict the damage evolution experimentally measured in the multiaxial state of stress regimes. The present paper has been organized as follows: in Section 2, for clarity purpose the model formulation is brieﬂy summarized; in Section 3, the experimental procedure used to measure damage evolution in A533B steel is given in

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details. In Section 4, damage evolutions under diﬀerent stress triaxiality regimes, predicted on the basis of the parameters identiﬁed in the uniaxial case, are presented, compared with the experimental data and discussed. In Section 5, the major outcomes of the work are summarized in the conclusions.

2. Non-linear damage model The CDM framework for ductile damage was initially proposed by Lemaitre (1985). Here, damage is a thermodynamics state variable that takes into account the progressive loss of load carrying capability of the material as a result of irreversible modiﬁcations, such as, but not exclusively, the nucleation and growth of microvoids, initiated under material plastic deformation. From the physical standpoint, damage can be deﬁned as the reduction of the nominal section area of a given reference volume element (RVE) as a result of the nucleation and growth of microcavities, microcracks, etc.: ðnÞ

DðnÞ ¼ 1

Aeff

ðnÞ

A0

ð1Þ

; ðnÞ

ðnÞ

where, for a given normal n, A0 is the nominal section area of the RVE and Aeff is the eﬀective resisting one reduced by the presence of micro-ﬂaws and their mutual interactions. Rigorous formulation naturally leads to the deﬁnition of a damage tensor, as proposed either by Chaboche (1984) or Murakami (1987) who have shown how large plastic ﬂow is responsible for inducing anisotropy. In this case, damage parameters experimental identiﬁcation becomes a very diﬃcult and complicated task with very few examples in the published literature. However, assuming isotropic damage in many cases is not too far from reality, at least in the deformation range up to the maximum engineering stress, as a result of the random shapes and distribution of the included particles and precipitates that trigger damage initiation and growth. In this case, it is easy to deﬁne an eﬀective stress reﬀ: r : ð2Þ reff ¼ 1D In addition, for the strain equivalence hypothesis (i.e., the strain associated with a damage state, under a given applied stress, is equivalent to the strain associated with its undamaged state, under the corresponding eﬀective stress (Lemaitre, 1985), it follows that the damage variable D can be given as: D¼1

Eeff ; E0

ð3Þ

where E0 and Eeﬀ are the YoungÕs modulus of the undamaged and damaged material, respectively. The complete theoretical framework for a damage material can be deﬁned by means of the following additional assumptions:

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The existence of a damage dissipation potential fD, similarly to the one used in plasticity theory, is assumed; The absence of any coupling between the damage and plasticity dissipation potentials. The total dissipation potential is given by linear superposition of the dissipation potential associated with plastic deformation, fp (function of stress, isotropic and kinematic hardening, R and X, respectively), and the damage dissipation potential, fD (function of the internal variable associated to damage, Y, and of the accumulated eﬀective plastic strain, p): f ¼ fp ðr; R; X ; DÞ þ fD ðY ; p; DÞ:

ð4Þ

The damage variable, D, is coupled with plastic strain. The relation between damage and the irreversible strain in the mesoscale is included in the kinetic law of damage evolution, where the rate of the plastic multiplier, k_ is proportional to the rate of the eﬀective accumulated plastic strain, p_ The nucleation and growth of micro-cavities and microcracks is related to plastic strain; therefore, the damage dissipation potential has to depend on the accumulated plastic strain. Damage processes are highly localized in the material micro-scale. At the beginning, damage eﬀects remain conﬁned in the mesoscale until the complete failure of several elementary volume elements results in the formation of a macroscopic ductile crack. The same set of constitutive equations for the virgin material can be used to describe the damaged material replacing the nominal stress with the eﬀective one and assigning an evolution equation for D. It is worth noticing that the damage associated variable Y is related to the elastic strain only through the material stiﬀness matrix, that is Y is equal to one half of the variation of the elastic strain energy with damage at constant stress (Lemaitre, 1985). Therefore, if Y depends on elastic strains only, the presence of damage through the eﬀective stress in the yield condition is questionable. As a matter of fact, the plastic ﬂow response experimentally measured is the result of the concurring action of both hardening and damage eﬀects that cannot be separated in a test. Moreover, the macroscopic tensile stress–strain curves of most metals do not exhibit softening even when applying the Bridgman correction for the necking. This implies that the evolution law for the plastic deformation rate of a damaged material should follows the standard plasticity theory without coupling with damage. At the same time this consideration implies that the material ﬂow stress does not explicitly exhibits a softening term avoiding all the mesh size eﬀect previously mentioned. A detailed discussion on this issue can be found in Pirondi and Bonora (2003). In Bonora (1997) the following expression for the damage dissipation potential was proposed, " # 2 1 Y S0 ðDcr DÞða1Þ=a fD ¼ : 2 S0 1D pð2þnÞ=n

ð5Þ

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Here, Dcr is the critical value of the damage variable for which ductile failure occurs. S0 is a material constant and n is the Ramberg–Osgood 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 matrix–inclusion bond that triggers voids nucleation and growth. According to these assumptions the following complete set of constitutive equations can be written for an isotropic hardening material (no kinematic hardening) as follows: Total strain decomposition eTij ¼ eeij þ epij :

ð6Þ

Elastic strain 1 þ m rij m rkk dij : eeij ¼ E 1D E 1D

ð7Þ

Plastic strain rate ofp 3 s_ ij ¼ k_ : e_ pij ¼ k_ 2 req orij

ð8Þ

Plastic multiplier ofp _ _ ¼ k ¼ p: r_ ¼ k_ oR Kinetic law of damage evolution

ð9Þ

1=a

ofD ðDcr D0 Þ D_ ¼ k_ ¼a oY lnðef =eth Þ

f

rm req

p_ ðDcr DÞða1Þ=a : p

ð10Þ

In Eq. (9), the eﬀect of stress triaxiality is accounted by the function f(rm/req) given by, 2 rm 2 rm f ; ð11Þ ¼ ð1 þ mÞ þ 3 ð1 2mÞ 3 req req that derives from the assumption that ductile damage mechanism is governed by the elastic strain energy (Lemaitre, 1985; Chaboche, 1984). The ratio of the hydrostatic pressure and von Mises equivalent stress is often indicated in the literature as triaxiality factor, TF, and used to characterize notched geometries. The kinetic law of damage evolution requires the actual value of the function f(rm/req), for varying strain. However, it can be directly integrated in the simple cases of uniaxial loading, for which f(rm/req) = 1.0, or in the case of proportional loading where f(rm/req) is constant through the entire deformation process. The solution for the uniaxial loading, a lnðe=eth Þ D ¼ D0 þ ðDcr D0 Þ 1 1 ð12Þ lnðef =eth Þ

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is the simplest and allows a ﬁrst general veriﬁcation of the model. The proportional loading condition, where the stress triaxiality ﬁeld remains constant, independent of the applied load level, is peculiar of some geometries. In axisymmetric round notch tensile bar (RNB(T)) specimens, for example, the stress triaxiality distribution along the minimum section, which shows a minimum (TF = 0.333) on the free surface and a maximum at the inner center, does not vary much with the increasing plastic strain level across the notch region, while the value of the maximum TF is determined by the notch radius (i.e., the geometry) and the material plastic ﬂow hardening exponent (Bonora et al., 1996). This statement holds for rate independent constitutive models, which may incorporate damage, without softening in the plastic ﬂow curve as in the present case. On the contrary if the damage induces softening in the material plastic ﬂow curve, as for the rate independent Gurson and Rousselier models, the stress triaxiality along the minimum section is no longer constant and increases, with the increasing of the plastic strain, with a rate that is function of the derivative of the ﬂow stress model with respect to strain. Under the assumption of proportional loading, the damage evolution is given according to the following expression

a lnðp=pth Þ rm D ¼ D0 þ ðDcr D0 Þ 1 1 f ; lnðef =eth Þ req

ð13Þ

where the threshold strain under multiaxial stress, pth, is taken equal to that in the uniaxial case, eth (see Bonora, 1997 for more details). Thus, a proportional loading conﬁguration such as a RNB(T) specimen is the simplest case where to test the capability of the model to predict damage evolution under diﬀerent stress triaxiality and therefore to verify the damage parameters transferability between diﬀerent geometries. As formulated, the model requires the knowledge of ﬁve material parameters which can be reduced to four assuming the initial damage equal to zero (D0 = 0) for the virgin material. The damage parameters are: eth, that is the damage threshold strain at which damage processes are initiated. Thomson and Hancock, 1984) experimentally observed that, in metals, this value is scarcely sensitive to stress triaxiality. On the other hand, strain threshold measurements performed under a uniaxial state of stress can show a consistent scatter band due to point-to-point microstructure variations, such as diﬀerent inclusions shape, dimension and orientation. This scatter can be strongly reduced performing measurements with a superimposed stress triaxiality that hides local microstructural eﬀects. ef is the theoretical strain to failure that a ductile material would exhibit under uniaxial stress. This value does not coincide with the failure strain of a standard tensile test, since the stress triaxiality induced by the necking, which generally develops, aﬀects the material ductility. Dcr is the critical damage value at which complete failure occurs. Theoretically this parameter should be equal to one for all materials, but since the damage variable accounts also for void and crack interaction, in some cases it can be less. Lemaitre (1992), estimated Dcr as 1 rR/ru where, rR and ru are the engineering rupture and maximum uniaxial stresses, respectively, stating that typical Dcr values for steels

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are in the range 0.2–0.5. However, there are experimental evidences, as reported in this work based on direct damage measures with stiﬀness loss, that demonstrate how damage values of the order of 0.4–0.5 can be observed before the initiation of the onset necking and, in term of strain, well ahead the occurrence of complete failure. In addition, the damage variable can be related to the development of porosity in the reference volume element as: D ¼ P 2=3 N 1=3 ;

ð14Þ

where P is the dimensionless porosity and N is the void number density per unit volume. If Eq. (14) is used with the experimental measures performed by Thissel et al. (2003) on half hard 10100 copper in spall test, results in Dcr @ 0.8 that is consistent with the values indirectly determined by Bonora et al. (2003) on OFHC copper. Finally, a is the damage exponent that determines the shape of the damage accumulation law with plastic strain. Usually, low values of a give a low damage rate, that increase steeply when the failure strain is approached. This behavior corresponds to the nucleation and early growth of a few voids that coalesce rapidly when the strain is close to the failure value. On the other hand, high values of the exponent corresponds to a large number of voids nucleated from inclusions, that do not grow much until complete failure occurs. These parameters can be identiﬁed from direct damage measurements, given by the elastic modulus reduction as stated in Eq. (3), as a function of the accumulated inelastic strain. Due to the possible scatter associated with the damage threshold strain, several procedures which make use of failure data obtained on diﬀerent specimen geometries have been proposed and can be found elsewhere (Bonora et al., 2004).

3. Experimental and numerical methodologies 3.1. Material The material used in this investigation is an A533 class 1, grade B low alloy steel commonly used in the nuclear industry. The brittle/ductile transition temperature (NDT) is 23 C that means fully ductile behavior at room temperature. A detailed characterization of this material can be found in Nauss et al. (1988). The chemical composition and the typical mechanical properties at room temperature are summarized in Tables 1 and 2, respectively. Firstly, the elastic–plastic material behavior has been determined with a standard tensile test procedure that makes use of a 12 mm

Table 1 Typical chemical composition for the A533B Cl. 1 steel C

Mn

P

S

Si

Mo

Ni

0.19

1.28

0.012

0.013

0.21

0.55

0.64

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Table 2 Summary of typical A533B mechanical properties Minimum yield strength (MPa)

Failure stress (MPa)

Elongation over 50 mm (min) (%)

345

550–690

18

wide, 5.0 mm thick ﬂat rectangular specimen. Engineering strain has been measured with a clip-gauge with 25.4 mm reference length (see Fig. 1). The plastic work hardening behavior has been ﬁtted with a Ramberg–Osgood type law: m ep r ; ð15Þ ¼c r0 e0 where r0 is the yield strength, E the Young modulus, e0 = r0/E and m the stress hardening exponent. Since the tests are performed at RT and the strain rates experienced

40 mm

R= 12 mm

20 mm 50 mm

HG(T)

8 mm

R= 12 mm

R= 2 mm

8 mm R= 4 mm

FRNB(T) R= 10 mm

Clip gauge knives locations (L0=25.4 mm)

Fig. 1. Specimen geometries used for the evaluation of tensile properties and damage evolution.

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by the material in the present investigation are low, no rate eﬀect on the plastic ﬂow curve has been taken into account. In Figs. 2(a) and (b), the experimental true stress vs true plastic strain data points are compared with the multi-slope ﬁt proposed by Nauss et al. (1988) and the Ramberg-Osgood power law ﬁt used in the present ﬁnite element simulations. The Young modulus of the materials is E = 200 GPa and the yield stress at RT is r0 = 400 MPa, approximately. With this values c = 7.316 and m = 4.739 were found, statistical parameters for the linear ﬁt are given in Fig. 2(b). The agreement between the measured data and the reference values reported in the literature conﬁrms the quality of the material used in the present investigation.

Fig. 2. (a) A533B work hardening parameters determination at 25 C. (b) Comparison of the measured work hardening with reference data for A533B steel at 25 C.

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3.2. Specimen geometries and testing procedure The possibility to continuously monitor ductile damage evolution, even though conceptually simple, is not an easy task. The evaluation of damage under uniaxial state of stress has been performed by testing of the HG(T) geometry given in Fig. 1. In order to evaluate the eﬃciency of the CDM model in describing damage evolution under triaxial state of stress, ﬂat rectangular notched bar specimens (FRNB) have been also tested. Three diﬀerent values of the notch radius, to which correspond a diﬀerent nominal stress triaxiality value, were examined. The corresponding geometries have been named FRNB(T)2, FRNB(T)4 and FRNB(T)10 in order to identify specimens with a 2, 4 and 10 mm notch radius, respectively. In Fig. 1, the specimen geometries together with the dimensions are given. The same experimental testing procedure has been adopted for both HG(T) and FRNB(T) specimen. It is subdivided in the following steps: (a) 3.0 · 6.2 mm2 strain gauge is attached to the specimen in correspondence of the minimum cross-section using a two component epoxy glue; the strain gauge resistance is 350 X and the nominal deformation limit ±20.0%; (b) the specimen is mounted on a MTS 100 KN servo-hydraulic testing machine and a ﬁrst loading ramp, imposing a remote displacement, is performed in order to adjust the specimen into the loading ﬁxtures; (c) the test is performed under displacement control with a series of partial unloadings to monitor the elastic slope changes with the increasing strain. In addition, a clip-gauge, with a reference length of 25.4 mm, is positioned across the minimum section in order to have a further check of the strain measured with the strain gauge. As far as concerns the HG(T) specimen, even though the stress distribution along the specimen longitudinal axis is expected to vary as a result of hourglass shape, the stress cross the minimum section is practically uniform due to the smooth geometry variation. (d) The process is carried on until the deformation limit for the strain gauge is reached; (e) the specimen is then unloaded and removed from the machine and the strain gauge is replaced with a new one; (f) a cross-check loading-unloading ramp within the elastic regime, in order to verify the correspondence of the slope with that of the last unloading measured with the previous strain gauge, is performed; (g) the points b to e are repeated until the occurrence of necking. In the post necking regime, unloading ramps are no longer performed, since it became almost impossible to balance a new positioned strain gauge, thus the specimen strained up to failure. Additional longitudinal strain measurement has been performed on FRNB(T) specimens by placing a clip gauge with a reference length L0 = 25.4 mm across the minimum section, Fig. 1, on the frontal surface. The use of the clip gauge allows one to verify the localization of damage eﬀects: at the beginning, no damage is meas-

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ured by the clip gauge while the strain gauge is recording the progressive material stiﬀness reduction; at larger deformation, when the extension of the damaged area becomes even larger of the notched region, the clip gauge measurement measures the damage eﬀects averaged on the clip reference length. This latter measure can be a useful datum for further ﬁnite element results assessment. At least three specimens, for every geometries, have been tested in order to have indication on the damage measurement reproducibility. 3.3. Finite element analysis Flat rectangular specimen geometries do not realize either fully plane stress or plane strain condition. An accurate study of the evolution of stress triaxiality with plastic strain across the minimum section requires 3D ﬁnite element simulation. Here, a ﬁnite element study on HG(T) and FRNB(T) has been performed in order to determine the evolution of stress triaxiality under increasing plastic strain. Only one fourth of the entire geometry has been modeled, because of the symmetry, using eight-noded, isoparametric, hexahedral element. Simulations have been performed using ﬁnite strain, large displacement and rotation formulation with Lagrangian update. In order to have a detailed map of stress and damage, the minimum section was meshed with evenly shaped bricks with a side length of 200 lm.

4. Results and discussion 4.1. Stress triaxiality distribution in HG(T) and FRNB(T) specimens The ﬁrst ﬁnding is that stress triaxiality in HG(T) specimen is not uniform across the minimum section. Alves et al. (2000) pointed out the diﬃculty in evaluating the eﬀective Young modulus using hourglass shaped specimen geometry. Here, a inner core of slightly higher stress triaxiality is located at the center of the minimum section, due to the Poisson eﬀect, similarly to the RNB(T). At the center, the stress triaxiality value increases with the plastic deformation level involving larger portions of the section only after the starting of the necking process. Anyhow, the dominate portion of the minimum section remains at stress triaxiality levels close to 0.333, that is the uniaxial reference value, for the entire strain range monitored for damage measurements. In Figs. 3(a)–(d), the stress triaxiality contours are given at diﬀerent strain levels across the minimum section. Here, only a 1/4 of the total minimum section is depicted. The corresponding points on the strain gauge curve of each picture are also given for clarity purpose. In Fig. 4, the evolution of the stress triaxiality with plastic strain is given for three locations: the center core of the minimum section (point O), and two points on the free surface, point A, located at the center of the minimum section and point B at half thickness. According to this, the assumption of almost uniaxial state of stress condition (i.e., TF = 0.333) for the HG(T) specimen can still be made for damage measurement up to necking initiation. Further strain level increase will require stress triaxiality corrections on damage measurements.

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Fig. 3. Stress triaxiality distribution on the minimum section of HG(T) specimen at diﬀerent strain levels showing the domination of the uniaxial reference value (0.333) up to the occurrence of necking. Contour plots are relative to 1/4 of the entire section.

Similarly to HG(T) specimen geometry, stress triaxiality distribution across the minimum section of FRNB(T) specimen is also not uniform. Here, as a consequence of the presence of the notch, the average stress triaxiality level is higher than the uniaxial case (i.e., 0.333). In Fig. 4 the evolution of stress triaxiality as a function of plastic strain at the center of the minimum section is given for all three notched specimen geometries. This plot shows that, in all cases, stress triaxiality, after a ﬁrst increases, sharply drops in correspondence with the yielding of the entire minimum section at which local stress redistribution as well as a notch stress concentration effect reduction occurs. After this point, stress triaxiality starts to increase almost linearly as a function of the increasing plastic strain, with a rate that is peculiar to the notch radius (see Fig. 5). 4.2. Identiﬁcation of damage parameters Due to the severe necking occurring in the minimum section and the consequent impossibility to apply a new strain gauge, damage measurements could be performed

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Fig. 4. Stress triaxiality evolution as a function of local plastic strain at three locations along the minimum section of a HG(T) specimen. Incipient necking in this specimen conﬁguration occurred at 40% of strain, approximately. Note that up to 40% of strain the stress triaxiality at location A and B remains practically constant ad equal to the uniaxial value, TF = 0.333. In the sketch on the right, the locations of the measure point are given as well as the area occupied by the strain gauge in the real case. The contours show the TF distribution at 40% strain.

up to a 40% strain, approximately. The value of Dcr was taken equal to 1. In Fig. 6, the load versus strain diagram measured on HG(T) specimen is given for reference purpose. For this sample, ﬁve strain gauges were used to explore the deformation range up to the sudden failure of the last strain gauge, the multiple unloading used

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Fig. 5. Triaxiality at the specimenÕs center for three diﬀerent FRNB(T) geometry.

Fig. 6. Load versus strain diagram for HG(T) specimen: strain is measured at the minimum section, nominal section area 40 mm2.

for damage measurement are clearly visible. Here, only the measures obtained with the ﬁrst four gauges are reported. In Fig. 7, the close up of an elastic unloading ramp is given showing a limited hysteretic loop. According to the prescriptions suggested in Lemaitre and Chaboche (1990), the Young modulus has been measured during the unloading ramp. However, the uncertainty in the estimation of the Young modulus during either the unloading or the elastic reloading is less than 2%. In absence of experimental data relative to proportional loading geometry conﬁguration, a trial

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Fig. 7. Detail of elastic unloading for damage measurement: material stiﬀness, measured (dashed lines indicate linear ﬁtting) during unloading or re-loading, shows limited scatter.

Fig. 8. Damage parameters identiﬁcation: best ﬁt has been determined iterating on the choice of ef, here equal to 1.75.

failure strain ef was assumed until a proper ﬁtting of the damage measurement was obtained in the ln((Dcr D)/Dcr) vs. ln[ln(ef/e)] diagram, as given in Fig. 8. The resulting damage measurements are given in Table 3 while the identiﬁed damage parameters are given in Table 4.

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Table 3 Young modulus reduction, plastic strain and damage measured under uniaxial state of stress E (GPa)

ep (mm/mm)

D

170.8 165.6 161.0 161.7 155.6 151.3 147.0 141.0 136.5 129.9 118.6 114.8

0.05460 0.07218 0.09490 0.09490 0.11651 0.12890 0.14369 0.16850 0.19870 0.2340 0.31880 0.36000

0.15025 0.17612 0.19900 0.19552 0.22587 0.24726 0.26866 0.29851 0.32090 0.35373 0.40995 0.42886

Table 4 Identiﬁed damage parameters at RT for A533B steel eth

ef

Dcr

a

0.0195

1.75

1.0

0.535

4.3. Comparison of experimental and predicted damage evolution under multiaxial stress In order to account for the stress triaxiality eﬀect on the damage measurement and to deﬁne a reference stress triaxiality value characterizing each notch radius in FRNB(T) the following analysis is proposed. From ﬁnite element results, neglecting the values in the strain range region where stress redistribution occurs, stress triaxiality as a function of plastic strain can be written in the following form: rm ¼ ðA þ B pÞ; ð16Þ req where A and B are coeﬃcients speciﬁc for the geometry under investigation. Thus Eq. (11) becomes rm 2 f ¼ 23ð1 þ mÞ þ 3 ð1 2mÞ ðA þ B pÞ ¼ K 1 þ K 2 p þ K 3 p2 req and the damage rate equation becomes: 1=a ðDcr D0 Þ K1 ða1Þ=a D_ ¼ a þ K 2 þ K 3 p ðDcr DÞ p_ lnðef =eth Þ p

ð17Þ

ð18Þ

that can be still analytically integrated along the load history. According to this total accumulated damage can be calculated as the sum of three terms:

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Table 5 Summary of the constants for the calculation of stress triaxiality evolution with plastic strain in FRNB(T) specimen

K1 K2 K3 TF*

Notch radius R = 2 mm

Notch radius R = 4 mm

Notch radius R = 10 mm

1.40 0.84 0.73 0.667

1.17 0.57 0.60 0.501

1.11 0.41 0.18 0.450

Here, TF is the reference stress triaxiality under the assumption of proportional loading condition (i.e., f(rH/req) = K1).

D ¼ D1 þ D2 þ D3 ;

ð19Þ

where

a lnðp=pth Þ D1 ¼ D0 þ ðDcr D0 Þ 1 1 K 1 lnðef =eth Þ

ð20Þ

that represents the proportional loading term at TF = A that is used here to characterize notched specimen geometry a ðp pth Þ D2 ¼ D0 þ ðDcr D0 Þ 1 1 K 2 ; ð21Þ lnðef =eth Þ

a ðp2 p2th Þ D3 ¼ D0 þ ðDcr D0 Þ 1 1 K 3 lnðef =eth Þ

ð22Þ

are the linear and quadratic contributions on damage accumulation due to stress triaxiality variation, respectively.

Fig. 9. Damage evolution in FRNB(T) and HG(T): comparison with present model prediction.

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The Ki constants are given as follows: K 1 ¼ 23ð1 þ mÞ þ 3 ð1 2mÞA2 ; K 2 ¼ 6 ð1 2mÞA B; K 3 ¼ 3 ð1 2mÞB

ð23Þ

2

Fig. 10. Eﬀect of second order changes of TF with plastic strain on damage evolution law. Solid line: eﬀective TF variation, dashed line: approximated constant TF assumption.

Fig. 11. FRB(T)2 veriﬁcation of local geometry response: applied load versus axial strain at the strain gauge location (i.e., outer surface, center of the minimum section).

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The value of these constants for the three notched ﬂat bar geometries used in the experimental program have been determined through FEA and summarized in Table 5. Clearly, the larger the notch radius the lower the reference stress triaxiality. The reference stress triaxiality for notched ﬂat rectangular specimen can be given by the constant A that can be used for a ﬁrst rough damage evolution estimation using Eq. (16).

Fig. 12. FRB(T)2, veriﬁcation of global geometry response: applied load versus axial strain at the extensometer location (i.e., across the minimum section, reference length L0 = 25.4 mm).

Fig. 13. Specimen HG(T)-hg6, veriﬁcation of global geometry response: applied load versus axial strain at the extensometer location (i.e., across the minimum section, reference length L0 = 25.4 mm).

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Therefore, damage evolution prediction in notched geometry has been obtained analytically from Eq. (21) and compared with experimental data obtained on both HG(T) and FRNB(T) specimens, Fig. 9. Here, the model prediction is in very good agreement with experimental data. In Fig. 10, the damage evolution given by Eq. (20) and the one obtained under the assumption of proportional loading (i.e., neglecting contribution represented by Eqs. (21) and (22)) are compared. It is evident how the eﬀect on the damage accumulation law, due to the second order terms in the TF variation with increasing strain, is limited with a relative error that, in the case of R = 2, is less that 5%. Consequently, the FRNB(T) can be considered an appropriate geometry for the study of damage evolution under multiaxial state of stress and quasi-constant stress triaxiality. In order to have additional veriﬁcation about the damage model predicting capability performance, 3D ﬁnite element analysis of FRNB(T) geometry incorporating

Fig. 14. Damage contour map showing ductile failure evolution, from (a) to (d), across the minimum section. The depicted mesh is a detail of the 1/8 of the entire geometry, failed element across the minimum section are colored in white.

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damage has been performed. The damage model was implemented in a commercial ﬁnite element code (MSC/MARC2003) in form of a user subroutine. Only 1/8 of the entire geometry has been simulated for symmetry reason and a uniform traction test has been simulated imposing a axial displacement to the remote section. The results reported here are relative to R = 2 mm specimen geometry which shows a major stress triaxiality eﬀect. Firstly, both local and global geometry predicted response, in the form of applied load versus axial strain measured at the strain gauge and clip gauge locations, have been compared with available data. The comparison is given in Figs. 11 and 12, which shows a very good agreement in both cases. Eight node brick element are known to give a stiﬀer response with respect to reality, here a scatter in the load estimation of ±7.5% was found. The loss of stiﬀness due to damage at macro scale has been also veriﬁed simulating partial unloading as in the experimental case, in this case very close values have been found with an average relative error between

Fig. 15. (a) SEM image of the ductile fracture surface of FRNB (R = 2 mm) specimen. The white square box indicate probable initiation site. Here, (b), voids are larger showing coalescence.

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the measured and the calculated load/strain slope of the order of ±18%. Similar good agreement is also found in the simulation of the HG(T) specimen geometry. In Fig. 13, the comparison of the predicted global response, in terms of clip gauge strain vs applied load, together with the experimental measures is given as an example. Successively, the damage evolution across the minimum section has been analyzed in order to determine the ﬁrst ductile failure location. In Figs. 14(a)–(d), the contour plots of damage distribution are given. These results shows that ductile failure does not initiate at the center of the minimum section as it would be expected in analogy with RNTB geometry. Here, the competition of stress triaxiality and plastic strain accumulation, controlled by the notch eﬀect, determines failure initiation near the notch. From there, ductile fracture spreads across the minimum section. The failure of the elements along the minimum section occurs in few load increments conﬁrming that once a ductile crack is initiated by the failure of few RVEs catastrophic failure immediately follows. The calculated ﬁnal fracture area has been found to be 19.5 mm2 approximately, which is in a good agreement with the experimentally measured values that range between 16.1 and 17.7 mm2. In order to verify this result, the failed surface of the FRTB (R = 2 mm) specimen has been analyzed with scanning electron microscopy (SEM) which revealed that, at the location where FEM predicted failure to occurs ﬁrst, microvoids have higher density and larger dimensions together with the fact that most of them showed extensive coalescence, Figs. 15(a) and (b). On the contrary, in the center of the minimum section, microvoids have smaller size and very few of them show coalescence. Finally, in Fig. 16 the calculated failure surface is given in a 3D view. Here, the major

Fig. 16. FEM predicted ductile fracture surface. Failed elements have been removed in order to show ‘‘cup’’ surface.

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features observed in the real failed specimens, such as cup-cone fracture, hourglass shaped, etc. are also found.

5. Conclusions In this paper the eﬀect associated with stress triaxiality on ductile damage evolution in metals has been investigated from both experimental and theoretical points of view. Ductile damage has been modeled using the non-linear CDM formulation proposed by in Bonora, 1997). Stress triaxiality plays a major role on the damage evolution law, which is demonstrated by the progressive reduction of material ductility under increasing triaxial states of stress. Here, these eﬀects have been investigated testing notched ﬂat rectangular bar specimen with diﬀerent notch radii (FRNB(T)). The stress triaxiality evolution with plastic strain has been determined via FEA. The CDM model predictions are in a very good agreement with the experimental damage measurements conﬁrming the potential of this model to fulﬁll to the requirement of transferability between diﬀerent loading conditions. The FEA incorporating the proposed damage model show the potential to accurately predict both global and local geometry response together with all the major features of the overall ductile failure process. Finally, it resulted that the FRNB(T) geometry is appropriate also for the study of damage evolution under multiaxial stress since the eﬀect due to the increase of the stress triaxiality in the minimum section con damage measurements can be neglected, at least in a ﬁrst approximated estimation. According to the ﬁnite element analysis results presented here, calibration curves can be derived for these geometries in order to account for this eﬀect.

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