A hardening-based damage model for fast-evolving microstructures: Application to Ni-based single crystal superalloys

International Journal of Plasticity xxx (xxxx) xxx–xxx

Contents lists available at ScienceDirect

International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

A hardening-based damage model for fast-evolving microstructures: Application to Ni-based single crystal superalloys Jean-Briac le Graverenda,b,∗ a b

Texas A&M University, Department of Aerospace Engineering, TAMU 3141, College Station, TX, 77843, USA Texas A&M University, Department of Materials Science Engineering, TAMU 3141, College Station, TX, 77843, USA

ARTICLE INFO

ABSTRACT

Keywords: Nickel base single crystal superalloy Continuum damage mechanics Viscoplasticity Non-isothermal loading Multiaxial loading Anisotropic damage

A phenomenological hardening-based damage density function built on a Rabotnov-Kachanov's formulation and coupled to a microstructure-sensitive viscoplastic crystal plasticity model is developed for ductile materials experiencing fast-evolving microstructures. The model seats on the fact that classical continuum damage models, that only consider discontinuities of matter; viz. voids and cracks; as damage, are not able to predict the non-isothermal creep and isothermal dwell/fatigue lifetimes of materials exposed to temperature/stress conditions that trigger microstructure evolutions faster than the nucleation and growth of voids. The hereby-proposed damage model depends on microstructure-sensitive hardening variables that were previously tailored to model changes in the mechanical behavior of a Ni-based single crystal superalloy during non-isothermal loading. Furthermore, the model also proposes to predict lifetime scattering during creep by considering the initial volume fraction of pores obtained by X-ray tomography. Isothermal and non-isothermal uniaxial creep and dwell/fatigue experiments on a Nibased single crystal superalloy are used to calibrate and validate the model. Finite element simulations are also carried out to confirm the lifetime predictive capabilities of the model on an in-plane multiaxial creep test and on multiaxial torsion and tension/torsion dwell/fatigue tests. The numerical results show that the damage model well predicts the lifetime of isothermal and non-isothermal creep as well as dwell/fatigue experiments in uniaxial and multiaxial states of stresses.

1. Introduction Damage prediction has always been a subject of great importance for the accurate and realistic modeling of inelastic deformation and failure behavior of engineering materials. It is why many damage models have been already proposed to analyze the fracture behavior of metals (Basaran and Lin, 2008; Beese et al., 2010; Needleman and Tvergaard, 1984; Stoughton and Yoon, 2011; Tvergaard, 1982; Wierzbicki et al., 2005; Xue, 2009). Two main approaches can be distinguished. The first one is a micromechanicsbased damage model that was originally proposed by Gurson (1975) and further modified in Brünig et al. (2014), Gologanu et al. (1993), Needleman and Tvergaard (1984), Neil et al. (2010), Sinha and Ghosh (2006), Tvergaard (1981), Zhang et al. (2000), for instance. In this approach, damage evolution is described by void nucleation, growth and coalescence. The other approach to damage analysis is Continuum Damage Mechanics (CDM). CDM was initially developed in Kachanov (1958), Rice and Tracey (1969) and improved in Basirat et al. (2012), Besson (2010), Brünig and Gerke (2011), Chaboche et al. (2006), Lemaître and Krajcinovic (1987),

∗

Texas A&M University, Department of Aerospace Engineering, TAMU 3141, College Station, TX, 77843, USA. E-mail address: [email protected].

https://doi.org/10.1016/j.ijplas.2019.03.012 Received 16 October 2018; Received in revised form 26 March 2019; Accepted 27 March 2019 0749-6419/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Jean-Briac le Graverend, International Journal of Plasticity, https://doi.org/10.1016/j.ijplas.2019.03.012

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Malcher and Mamiya (2014), Naderi et al. (2013). This method recognizes only voids, cavities, and micro-cracks as damage and models their effects by decreasing the cross-section area. Therefore, both approaches only consider discontinuities of matter as the only reason for the changes in the load carrying capacity. This has recently confirmed its limitations when microstructure is likely to quickly evolve. Thus, with the envisioned extended lifespans in many applications and in particular the ones dealing with thermomechanical loading, such as in aerospace and in energy production, there is a need to still improve the predictive capabilities. As suggested by Dyson et al. (Ashby and Dyson, 1984; McLean and Dyson, 2000b), experimentally obtained by Reed et al. (2007) who observed no difference in terms of creep lifetime between isostatically-hot-pressed (HIP) and non-HIPed specimens, and recently pointed out by the author when doing ex-situ X-ray tomography on a Ni-based single crystal superalloy subjected to high temperature/low stress creep (le Graverend et al., 2017a), the increase in the plastic strain rate during the tertiary creep stage is not solely related to the increase in the void volume fraction. Other prime examples of the CDM limitations are: the improvement of the creep lifetime at 1050 °C when a unique thermal jump from 1050 °C to 1200 °C in 4s for 30s is introduced (le Graverend et al., 2014a), longer thermal jumps lead to longer lifetimes (Cormier et al., 2007, 2008), and cyclically cooling specimens leads to increase the plastic strain rate and to shorten the lifetime (Raffaitin et al., 2007; Viguier et al., 2011). These experimental results call for advanced models capable of predicting such unusual phenomena that are clearly due to fast modifications of microstructures and to large changes in the internal state of stresses. Despite an extensive modeling work on coupling damage and inelastic deformation at the macro- (Kyaw et al., 2016; Lemaitre, 1985a; Nguyen et al., 2015; Roy Chowdhury and Roy, 2019; Voyiadjis and Park, 1999; Zhang et al., 2018; Zhu and Cescotto, 1995) and micro-scales (Ekh et al., 2004; Feng et al., 2002; Potirniche et al., 2007; Zghal et al., 2016; Zhao et al., 2018), the depicted counter-intuitive experimental results can clearly not be predicted by the usual constitutive and continuum damage models that fail to consider transient behaviors and are based on “hotter is shorter”. However, a few have been done for non-isothermal conditions (Cormier and Cailletaud, 2010; Egner, 2012) and only recently Mattiello et al. (2018) have proposed a rate-sensitive threshold for the damage process to occur in a microstructure-sensitive framework, but with the equivalent stress-dependent isotropic damage law developed by Lemaitre (1985b). Many damage models have also been developed for superalloys (Cormier and Cailletaud, 2010; Cornet et al., 2011; Mattiello et al., 2018; Qi and Bertram, 1998, 1999; Tinga et al., 2009; Trinh and Hackl, 2014; Wang et al., 2019; Yue et al., 1995), but, similarly to other damage models, they either do not focus on non-isothermal damage, are applied-stress dependent, or are not fully coupled with the mechanical behavior. The objective of this work is, therefore, to formulate and validate a damage model within a CDM framework that acknowledges the role played by the microstructure in the damage evolutions to better predict mechanical response and lifetime scattering as well as thermo-mechanical loading conditions that trigger fast microstructure changes. The elements of novelty are (i) the development of a void-volume-fraction-dependent damage material parameter and (ii) the use of microstructure-sensitive internal state variables, also employed to depict the mechanical behavior, as driving forces for the damage kinetics. To achieve the paper objective, an effective stress concept will be used (the load carrying capacity is decreased by the degraded microstructure, i.e., the microstructure that does not have the same initial mechanical property anymore). This concept comes from Lemaitre who proposed the strain equivalence principle which allows a modification of constitutive equations of an undamaged material to describe a damaged material (Lemaitre, 1972, 1985a). This is an extension of the effective stress formulation of Kachanov (Kachanov, 1958, 1960) and Rabotnov (Rabotnov, 1968, 1969) who introduced a scalar damage parameter D (evolving from 0: undamaged to 1: fully damage) to represent the loss of a load-carrying cross-section due to the nucleation, growth, and coalescence of microcracks. According to this concept, the constitutive equations of any damaged material (microcracks and microvoids) can be obtained by replacing the Cauchy stress in the constitutive equations of the corresponding undamaged material with an adequately defined effective stress ~ related to the surface that effectively resists the load: ¯ ˜ = (I D) 1: (1) where is the applied stress tensor and D is the fourth-order damage tensor which has been proved to allow a consistent use of the equivalence principle for anisotropic damage (Keller and Hutter, 2011). By replacing the applied stress by the effective stress, the flow rule for steady creep stage can be extended to the tertiary creep stage and a strong coupling between plasticity and damage is obtained, as in Abu Al-Rub and Voyiadjis (2003), Hesebeck (2001), Mahnken (2002), Menzel et al. (2005), Tvergaard (1982), Voyiadjis and Deliktas (2000). The developed damage model is implemented in the author's viscoplastic microstructure-sensitive phenomenological model written in a crystal plasticity framework (le Graverend et al., 2014b). The damage model is validated with isothermal/non-isothermal uniaxial/multiaxial monotonic/cyclic experiments performed on a first-generation Ni-based single crystal superalloy that shows a strong anisotropic and non-linear mechanical behavior. In addition, to be consistent with the anisotropy of the damage process, the developed damage density function is implemented in an anisotropic damage formulation. The damage model is formulated in the context of internal state variable theory. The study is limited to damage mechanics, namely the onset of damage, which makes fracture mechanics out of the scope. 2. A coupled viscoplastic-damage formulation 2.1. Crystal plasticity model The microstructure-sensitive crystal plasticity framework previously developed and calibrated by the author (le Graverend et al., 2014b) only considers the octahedral slip systems. The constitutive model is based on a partitioning of the macroscopic strain into an 2

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

elastic and a viscoplastic part as well as on a finite strain theoretical setting, as in Asaro (1983), Hutchinson (1970), Kocks (1970), Méric et al. (1991). The relationships defining the elasticity are written at the macroscale, whereas the viscoplastic constitutive equations are written at the microscale, viz. at the slip system level. It is, therefore, a micro-macro approach that is, by definition, anisotropic due to the formulation at the slip system level. s is the viscoplastic shear strain rate on a given slip system s (see Eq. (2)) s in which the effective resolved shear stress eff is obtained by the tensorial product between the macroscopic effective stress tensor and the orientation tensor ms , calculated knowing the normal to the slip system plane ns and the slip direction in this plane ls (see Eq. ¯ ¯ ¯ (3)). The non-linear kinematic x s (Eqs. (4) and (5)) and isotropic r s (Eq. (6)) hardening variables on the system s are microstructure r sensitive through D and Orowan , respectively. They also possess dynamic recovery terms. s

s eff

= Cvisco sinh

s eff

xs K

1 s (n 2

= : mswith ms =

x s = a Cx

s

with

s

a

Nsyst

hsj j=1 r

s

j

+

T

n

s eff

(a

s eff

sign (

ls + ls

= [sign (

D = Dx e fs Kx and a =

r s = r0s + bQ

rs

x s)

x s)

(2)

ns) D

(3)

S] s

(4)

1) (5)

2 Gb with 3 w

s

= (1

b s)

s

rOrowan

(6)

and are time where Cvisco , K, n, Cx, Dx, b, and Q are temperature-dependent material parameters, Kx is a material parameter, constants accounting for the evolution of the kinematic hardening with the temperature history, fs is the volume fraction of the small γ′ precipitates, namely the tertiary γ′ precipitates, formed when a temperature change is introduced above 975 °C (Cormier and Cailletaud, 2010), fl is the volume fraction of the large γ′ precipitates, namely the secondary γ′ precipitates, formed after the heat treatment, [h] is the interaction matrix whose form has been described in Franciosi (1985) and is assumed to have all its components equal to 1, r0s is the initial radius of the yield curve, i.e., the initial critical resolved shear stress on the slip system s, G is the shear modulus (GPa), b is the magnitude of the Burgers' vector, w is the width of the γ channels (nm) whose the rate-sensitive constitutive equations are presented in le Graverend et al. (2014b), and j is the isotropic state variable on the slip system j that models the evolution of the dislocation density on each slip systems and is, therefore, related to dislocation hardening. 2.2. Anisotropic formulation The damage process in metals is generally anisotropic, even if the material is initially isotropic (Chow and Wang, 1987; Hayhurst, 1972; Lemaitre et al., 2000). Indeed, it is, for instance, well-known that the creep process of metals is accompanied by the nucleation, growth, and coalescence of microcracks that have a directional characteristic in the three-dimensional states of deformation (Ai et al., 1992; Kassner and Hayes, 2003; Nix, 1983). In addition, it was already shown during low cycle fatigue experiments on FCC austenitic stainless steel that microcracks initiate and orient along slip planes (Heino and Karlsson, 2001). This was used to legitimize the formulation of a damage model at the crystal scale (Ekh et al., 2004). Thus, an isotropic damage model is strongly limited and has intrinsic limitations for multi-dimensional loading conditions. Even if plenty of anisotropic formulations have been proposed, viz. second-rank tensor (Abu Al-Rub and Voyiadjis, 2003; Murakami and Ohno, 1981; Wulfinghoff et al., 2017) and fourth-rank tensor (Chaboche, 1982, 1984; Olsen-Kettle, 2018; Rajhi et al., 2014), it has been chosen to use the fourth-rank tensor developed by Chaboche (Chaboche, 1982, 1984). It is due to a more complex use and identification of second-rank tensors for the description of the behavior since it requires measurements of defects at the microstructural level. On the contrary, the identification of the corresponding evolution equations for fourth-rank tensors is easier because of the use of the effective stress concept. The developed damage density function D, which depicts the magnitude of damage, has been implemented in the Chaboche's fourth-rank anisotropic damage tensor D via a fourth-rank tensor whose formalism and physical meaning will not be explained here, but can be obtained in Chaboche (1982), (1984).

D=

(7)

D

= ( ) = (R R ) 0 (R R)T is a fourth-order tensor defining, in the actual effective stress state, the distribution of where damage, that is the preferential directions of cracking with R = R( ) the rotation matrix from the coordinate system defined by the directions of the principal stresses to the reference system. In the general case, Ω depends on stress because on the non-coincidence between the chosen axis system and the principal stress one: = ( ) for non-proportional loading (time-dependent anisotropy), = 0 otherwise. 0 is a linear combination of the isotropic case and of a particular case of anisotropy, e.g., the one corresponding to planar microcracks, all parallel, developing perpendicular to the direction of the maximum principal stress: 0

= (1

)

(8)

+ I

where the anisotropy tensor Γ is a fourth-order tensor obtained from elastic analysis results and which depends on the Poisson's ratio 3

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

ν and a material parameter ξ (Delameter and Herrmann, 1974). η is a material parameter which corresponds to an anisotropy factor: isotropic damage if η = 1 and pure anisotropic damage if η = 0 and I is the fourth-order identity matrix. This anisotropic formulation assumes that the principal stresses are responsible for the damage growth, and the anisotropy of the damage evolution depends on the principal directions of the stress and damage tensor. We consider that the scalar damage is an invariant of D and its law is given by identification with the simple tensile case, attempting to describe the = 0 since the considered multiaxial loadings are all proportional. isochronous surfaces in the stress space. In the following, Based on the work done by Lesne & Savalle (Lesne and Savalle, 1987) on the Hayhurst's equivalent stress (Hayhurst, 1972) ( ( ) = J0 ( ) + J1 ( ) + 1

J2 ( ) with α and β material parameters) on Ni-based superalloys, η is equal to 0.3. The ma-

terial parameter used in has been taken equal to 0.1 because of the results in torsion (le Graverend et al., 2018) that highlighted a slower kinetics of damage than in tension. 2.3. Damage density function The damage density function defined by D is coupled to the viscoplastic behavior via the kinetics of hardening variables, viz. x i and r i , and has the same formalism than a Rabotnov-Kachanov's law (Eq. (9)). The formulation in Eq. (9) is not expressed at the slip system level, contrary to what was done by Cormier & Cailletaud (Cormier and Cailletaud, 2010) who only used a kinematic variable as a driving force for the damage evolution but not for the yield surface evolution. This is equivalent to say that each activated slip systems contributes to the damage evolution of all the slip systems. It is already known that slip systems interact with each other when it comes to mechanical behavior (Cuitino and Ortiz, 1993; Friedel, 1964; Hirth, 1961; Kocks and Mecking, 2003; Shenoy et al., 2000; Tabourot et al., 1997). So, it is hereby considered that it is also the case with regards to damage evolution. 12

D= i=1

ri xi + A B

r (Fv0)

(1

D)

k

(9)

where Fv0 is the initial volume fraction of voids in the material which will play a major role in predicting the scattering in the experimental results (see section 3.1), A, B, and k are temperature-dependent material parameters. Based on Chaboche's work that used an order-2 polynomial function of σ for the parameter k to predict the non-linear accumulation of damage with the stress level (Chaboche, 1978), the author expressed the parameter r as an Fv0-dependent material parameter that takes the form given in Eq. (8). The Fv0 values used in the present paper are the initial volume fractions of pores obtained in le Graverend et al. (2017a). It allows, at minima, to predict the lifetime scattering of creep tests: the larger Fv0 is, the lower the value of r is, and the shorter the lifetime is. This relation can provide the initial volume fraction of pores that leads to a failure at the end of the loading. For instance, 6% of pores, namely r = 0.3, leads to a direct failure of a specimen at 1050 °C/160 MPa.

r = r0 e

Fv0

and

(10)

Fv 0, r > 0

where r0 and are material parameters. r0 is temperature dependent while is not, for now, because of the lack of experimental data at temperatures above 1050 °C. The additive form in the first bracket in Eq. (9) is because the formulation is based on a kinetics of damage driven by components of the internal stress. Indeed, the form of the macroscopic mechanical response of a uniaxial strain-controlled test for a constitutive model with an isotropic and a kinematic hardening is given by: = v + R0 + R + ROrowan + X . The viscous stress does not affect the internal

kinetics of damage based on the observations made by the author who did strain-controlled varying tensile tests (le Graverend et al., 2016). Indeed, the magnitude of the drop in the viscous stress when switching the strain rate from 10−3 s−1 to 10−5 s−1 between a specimen initially subjected to a thermal jump from 1050 °C up to 1200 °C in 4s for 30s under no stress and an as-received specimen is the same. This result clearly demonstrates that the viscous flow is not affected by microstructural changes, which is indubitably the case for the hardenings which showed saturated behaviors (perfectly plastic behavior) contrary to the as-received specimen. The initial yield stress (R0) and the Orowan stress (ROrowan ) are not considered either. The Orowan stress was excluded because of recent experimental observations that showed that a thermal jump may improve the lifetime and that a longer thermal jump leads to longer lifetime (Cormier et al., 2007; le Graverend et al., 2014a). If a thermal jump is introduced, the γ channel width will first increase then decrease when the temperature will go back to its nominal value because of the fine γ’ precipitation. Hence, the Orowan stress will first decrease (w increases) then increase (w decreases), which would respectively decrease and increase the damage rate if the Orowan stress would have been included. These changes in the kinetics of damage would not be consistent with the experimental results previously mentioned. The author wants to emphasize that microstructure evolutions, such as rafting and dissolution/precipitation, modify the rate of damage because of the microstructure-sensitive formulation of the crystal plasticity model used. Indeed, the kinematic hardening is directly dependent on the volume fraction of the tertiary or small precipitates, viz. fs , via the material parameter D∗. It is, however, not the case of r . Nonetheless, the mechanical response depends on the volume fractions of a bimodal γ′ phase distribution, viz. fs and fl , and the state of coarsening w due to the directional and isotropic coarsening of the γ′ phase (see Eqs. (5) and (6)). This modifies s that is used to compute the two internal state variables s and s . Therefore, the damage density function D does not only account for the decrease of the cross-section area due to the growth of voids, as done by the commonly-used Rabotnov-Kachanov's model, but also 4

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

the softening and hardening in the mechanical behavior due to microstructural changes, e.g., dynamic recoveries included in the formulations of j and i which are subsequently modifying the damage kinetics. As a reminder, only the kinetics of the hardening variables, that are known to bring to light microstructure evolutions (Aifantis, 1984; Fribourg et al., 2011; Moosbrugger and McDowell, 1990), are of interest: the isotropic hardening is related to the density of dislocations or flow arrests and the kinematic hardening is related to the state of internal micro-stress concentration. Thus, alloys designed with large hardening to stabilize the deformation and to extend the homogeneous deformation via slowing down the localization during plastic deformation will not damage faster than other alloys having the same hardening kinetics. The use of the hardening variables instead of the applied stress is also motivated by the need to have a damage variable able to predict the mechanical cycling response. In fact, if there is no opening in the hysteresis loop nor ratcheting effect, which is responsible for reducing fatigue life during LCF (Kang et al., 2009; le Graverend, 2013; Mataveli Suave et al., 2016; Shenoy et al., 2005) and TMF (Kupkovits and Neu, 2010; Shi et al., 1998; Xiao et al., 2006) by accelerating damage accumulation, the hardening variables will hardly evolve which will lead to a very slow damage kinetics, i.e. extended lifetime. In addition, contrary to the isotropic hardening, the kinematic hardening can be negative in compression. Therefore, the formulation in Eq. (9) will lead to a slower damage kinetics (

12 x i i=1 A

ri

+

B

will be smaller in compression than in tension because of the negative value of the kinematic hardening) which has

been experimentally observed in Baldan (1991), Epishin et al. (2014), Kunz et al. (2012). Closed voids are not decreasing the loadcarrying capacity. Thus, the current model intrinsically considers the microdefects closure effect which corresponds to a partial closure of micro-cracks that increases the effective load-carrying capacity. One can mention that if no load is applied or if the load is removed, D 0 which might seem wrong since D = 0 with the Rabotnov-Kachanov formulation and because it is commonly believed that loading is the only way to damage materials. In addition to what Dyson et al. clearly stated (Ashby and Dyson, 1984; McLean and Dyson, 2000b), i.e., damage can be strain-, thermally-, or environmentally-induced, the author wants hereby to introduce other counter examples that prove that not applying a load is not necessary synonym for no damage accumulation. When the author performed thermal jumps without an applied stress, the lifetime was decreased (le Graverend et al., 2016), which cannot be predicted if D = 0 when applied = 0MPa. Indeed, thermal exposure or aging modifies microstructural states which may lead to decreased lifetimes (Acharya and Fuchs, 2004; Jiang et al., 2018; Liu et al., 2010). For instance, if a load is removed above 0.1 ± 0.03% of deformation at 950 °C in the CMSX-4®, rafting will continue (Matan et al., 1999b) whereas it was shown by many researchers that rafts perpendicular to the loading direction have a deleterious effect on tension creep (Nathal and Mackay, 1987; Shui et al., 2007), on monotonic mechanical behavior (Gaubert, 2009; MacKay and Ebert, 1985; Pessah-Simonetti et al., 1992), and on strain-controlled cyclic response (Ott et al., 1999). Furthermore, when a material is aging, extra phases can precipitates, like TCP phases (Rae et al., 2000) or carbides (Liu et al., 2003). Those phases degrade the mechanical behavior (Pessah-Simonetti et al., 1992) and lead to shorter lifetimes. Therefore, considering that the kinetics of damage should be equal to 0 when the load is removed is clearly incorrect, especially because oxidation also continues to proceed, i.e., injects vacancies in the materials that form sub-scale pores and, therefore, lead to a decrease in the load-carrying capacity. Thus, the rationale behind the presented formulation is based on the fact that CDM is only valid when microstructures hardly evolve, which is not the case at higher temperatures, i.e., when voids are the only source of damage. It is the reason why a microstructure-sensitive formulation is so important for fast-evolving microstructures. The positivity of the dissipation potential is ensured by Y the dual variable associated with the damage D and defined as: e

Y=

D

=

1 C: ( 2

e

k T I): (

e

k T I)

(11)

Y = 2 Cijkl ije kle for and T constant.where ρ is the density of the material, e is the elastic free energy, C is the stiffness matrix, the elastic strain tensor, and kΔT is the thermal dilatation. Thus, the potential of dissipation is given by: 1

= :

p

YD

X:

Rp

micro

1 q grad T T¯

e

is

(12)

where p is the plastic strain tensor, X is the kinematic hardening tensor, is the kinematic associated tensor, R is the isotropic hardening variable, p is the accumulated plastic strain, and micro is the potential of dissipation associated with the microstructural degradation which is changing the Orowan stress (see Eq. (6)). Under the hypothesis that the intrinsic mechanical and thermal dissipations are not coupled and the second principle of thermodynamics is verified, it comes:

:

p

YD

X:

Rp

micro

(13)

0

As shown in Lemaitre and Chaboche (1990), this inequation is enforced if:

:

p

X:

Rp

micro

0 and

YD

(14)

0

This article is not aimed at demonstrating the positivity of the intrinsic mechanical potential of dissipation, but to show that the intrinsic damage potential of dissipation is always positive. This dissipation is positive only if D 0 (Desmorat, 2006) since Y is a positive-defined quadratic function (see Eq. (11)). D is a positive-defined function which verifies the irreversible thermodynamic approach. 5

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 1. Microstructure of the MC2 single crystal superalloy after the standard heat treatments.

3. Application to Ni-based single crystal superalloys Nickel-based single-crystal superalloys are widely used in the manufacturing of turbine blades and are, therefore, subjected to non-isothermal high-temperature creep and dwell-fatigue loading for a wide range of temperature/stress regimes during in-service operations or severe certification procedures (Caron and Khan, 1983; Cormier et al., 2015; Reed, 2006). Such extreme thermomechanical environments lead to phase transformations as well as stress and microstructure gradients which dramatically alter the mechanical properties (Antolovich et al., 1979; Epishin et al., 2008; Nathal and Mackay, 1987; Ott et al., 1999; Shui et al., 2007). The high creep resistance of Ni-based single crystal superalloys is induced by the precipitation of a high volume fraction (close to 70%) of the long-range ordered L12 γ’ phase which appears as cubes coherently embedded in a face-centered cubic (FCC) solid solution γ matrix (Caron, 2005) (see Fig. 1) and evolves into platelets by directional coarsening during high temperature mechanical testing (T > 850 °C) (Pollock and Argon, 1994). This volume fraction is modified during thermal changes above ∼850 °C, which has a tremendous effect on creep life (Murakumo et al., 2004), and depends on the temperature, the hold time and the applied load (Giraud et al., 2013). The mechanical behavior of the Ni-based single crystal superalloy at 1050 and 1200 °C was previously calibrated in le Graverend et al. (2014b). Only the parameters associated with the developed damage density function are obtained in section 3.1. Furthermore, a Rabotnov-Kachanov damage model (Eq. (15)) has also been calibrated to compare with the developed damage model and to discuss the advantages of it.

D=

( ) r (1 S

D)

k

(15)

) J2 ( ) is the Hayhurst equivalent stress (Hayhurst, 1972) with J0 ( ) the principal stress, where ( ) = J0 ( ) + J1 ( ) + (1 J1 ( ) = 3 H , J2 ( ) the von Mises equivalent stress, α and β are two material parameters whose values have been provided by Lesne & Savalle (Lesne and Savalle, 1987) in the case of Ni-based superalloys, namely α = 0 and β = 0.3. r, k and S are material parameters. The code has been implemented in the commercial software Z-set®. The Z-sim option has been used in sections 3.1 to 3.4. Z-sim is a constitutive equation driver, allowing the user to load any representative volume element (RVE) and thus perform fast simulations on material elements without using any FEA. 3.1. Calibration of the model at 1050 and 1200 °C The models have been calibrated at the two levels of temperature that will be used for introducing a thermal jump (see Table 1): 1050 and 1200 °C. The two models, namely the Rabotnov-Kachanov and the one developed, have been calibrated to provide the same results at 1050 °C/140 MPa (Fig. 2a) and 1200 °C/100 MPa (Fig. 3a) with Fv0 = 0.02%. The initial volume fractions used in this paper were obtained by ex-situ tomography in le Graverend et al. (2017a). In Fig. 2b, three experiments (le Graverend et al., 2012) have Table 1 Values of parameters used in the damage law for 1050 and 1200 °C. Temperature (°C)

New Model Rabotnov-Katchanov

1050 1200 1050 1200

Parameters k

r

A

B

Fv0

7.0 2.0 10.0 3.0

5.564 11.494 7.4 6.9

770 204 1,475 376

230 40

0.2 0.2

6

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 2. Simulation of tension creep tests with two damage models, namely the “New Damage Model” and the “Rabotnov-Kachanov”, for different stress levels: (a) 140 MPa, (b) 160 MPa, (c) 180 MPa, (d) 200 MPa, (e) 230 MPa at 1050 °C. Tests performed at 140, 180, 200 and 230 MPa are extracted from Cormier (2006), le Graverend et al. (2014b).

been used to show the discrepancy in terms of plastic strain rate during the secondary creep stage and of lifespan. Experiment 1 is the one with the longest lifetime while Experiment 3 is the one with the shortest. Experiment 2 is the one which showed the highest plastic strain rate during the secondary creep stage. From Figs. 2 and 3, the new damage model appears to provide better results in all temperature/stress conditions except at 1050 °C/200 MPa. The evolution of the damage density function D has been plotted for the simulations performed at 1050 °C (see Fig. 4). It can be noticed that a non-linear accumulation of damage is highlighted: the higher the stress is, the more abrupt the damage evolution is (Chaboche, 1978). The same phenomenon can be modelled by a Rabotnov-Kachanov damage variable only if the parameter k (see Eq. (9)) is function of the applied stress, as done by Chaboche in Chaboche (1974) who used an order-two polynomial function for modeling the non-linear accumulation. Therefore, contrary to what was done before, the non-linear accumulation is intrinsically taken into account and does not require extra calibrations. Going back on D 0 when the load is removed with the new damage model (as discussed in section 2.3), the following condition has been simulated in Fig. 5: creep at 1050 °C/140 MPa for 150 h, no load for 500 h at 1050 °C, and creep again at 1050 °C/140 MPa until rupture. Fig. 5a clearly shows that the new damage model predict a decrease in the lifetime compare to the no-aging simulation, namely 249 h with aging when the load is applied compare to 558 h with no aging. On the contrary, the Rabotnov-Kachanov model does not predict any reduction in the lifetime compare to no-aging: 558 h with aging when the load is applied. Therefore, the Rabotnov-Kachanov model is not able to account for lifetime changes when the material is just exposed to high-temperature when already deformed. Looking at Fig. 5b, the kinetics of damage of the Rabotnov-Kachanov curve is the same as the one without aging when the load is re-applied which is not the case for the new-damage-model curve. In fact, the kinetics of this last is faster than the other two leading to a shorter lifetime. The material parameter Fv0 has also been modified in the range that was found in le Graverend et al. (2017a) by doing ex situ tomography, i.e., [0.015; 0.034]%. Fig. 6 shows the effect of varying the parameter Fv0 when keeping all the other parameters the same. If Fv0 is increased to 0.034 the lifetime is highly decreased and becomes 93.8 h instead of 185.8 when Fv0 is equal to 0.02. 7

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 3. Simulation of tension creep tests with two damage models, namely the “New Damage Model” and the “Rabotnov-Kachanov”, for different stress levels: (a) 100 MPa, (b) 80 MPa, (c) and 67 MPa at 1200 °C. The experiments are extracted from Cormier (2006).

Fig. 4. Evolution of the damage variable D as a function of the time fraction t/tR at 1050 °C. tR is the time at rupture.

Fig. 5. (a) Simulation of a creep test at 1050 °C/140 MPa for which the load was removed after 150 h for 500 h and put back until rupture. The new damage model and the Rabotnov-Kachanov model were used to simulate the condition and the creep strain evolution for a regular creep test at 1050 °C/140 MPa simulated with the new damage model has been added as a matter of comparison. (b) Evolution of the damage density function D for the tests simulated in (a) as a function of the time fraction t/tR. tR is the time at rupture. 8

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 6. Effect of the parameter Fv0 on the predicted lifetime at 1050 °C/160 MPa.

Similarly, the lifetime is increased to 215.9 h when Fv0 is decreased to 0.017. Therefore, the parameter Fv0, which has been defined as the initial volume fraction of voids, provides a way to probe discrepancies between specimens in terms of lifetime and plastic strain rate. 3.2. Isothermal stress-strain curves In the damage density functions already developed by other authors, the damage rate is proportional to the accumulated plastic strain rate p and enhanced by the stress triaxiality (Besson, 2010; Krairi and Doghri, 2014; Lemaitre, 1985b; Srivastava and Needleman, 2015). In many of them, the higher the stress level the lower the ductility which forced authors to come up with ratedependent parameters (Johnson and Cook, 1985; Naumenko et al., 2010) or rate-dependent damage threshold (Mattiello et al., 2018; Simo and Ju, 1987; Wolff et al., 2015). Stress-strain curves are simulated at 1050 °C and three different strain rates, namely from 10−4 to 10−6 s−1, for 001-oriented specimens (Fig. 7). Fig. 7 highlights a rate sensitivity of the predicted ductilities: 34.1%, 11.0%, and 5.4% from the fastest to the slowest rate. Looking at the curves at 10−5 s−1, one can notice that the shapes are not the same, but the softening rates are. It was, therefore, not necessary to introduce a rate-dependent parameter, or threshold, to get a rate sensitivity of the ductility, since it is intrinsically taken into account by the damage model. The author wants also to point out that there is no softening predicted at 10−3 s−1. The model was proposed to account for the evolution of the microstructure, i.e., when the microstructure evolves as fast or faster than the voids, which is clearly not the case at 10−3 s−1. Indeed, the test duration is not long enough to allow for microstructure evolutions. 3.3. Non-isothermal creep loading Two stress levels are investigated, namely 140 (Cormier, 2006; Cormier and Cailletaud, 2010; Cormier et al., 2008) and 160 MPa (le Graverend, 2013; le Graverend et al., 2014b). The experiments presented thereafter consist of introducing a unique thermal jump from 1050 to 1200 °C in 5s and maintaining it for either 30 (Fig. 8a and Fig. 9) or 90s (Fig. 8b and c) somewhere in the creep life of the specimen. The specimen is then cooled down to its nominal temperature, viz. 1050 °C. Fig. 8 shows that introducing a thermal jump of 90s after 206 h at 1050 °C/140 MPa leads to an extended lifetime compared to a 30s thermal jump (see Fig. 8a and b), viz. 560 h instead of 275 h. Cormier et al. (2007) explained that this phenomenon is due to a better dynamic recovery of the dislocation

Fig. 7. Stress-strain curves at 1050 °C and different strain rates for 001-oriented specimens. The experimental curve at 10−5 s−1 is from Cormier (2006). 9

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 8. Creep tests at 1050 °C/140 MPa for which a thermal jump of (a) 30s and (b) 90s has been introduced after 206 h of testing. In (c), the 90s thermal jump has been introduced after 120 h. (d) is the evolution of the damage density function for the simulated curves in (a) and (b).

Fig. 9. Creep tests at 1050 °C/160 MPa for which a thermal jump of 30s has been introduced after (a) 0 h, (b) 2 h, (c) 18.6 h, and (d) 129 h of testing.

networks in the case of longer thermal jump which subsequently leads to more stable networks that prevent the γ’ particles from cutting. This clearly states that the level of the applied stress is not enough for describing the evolution of damage during nonisothermal loading during which static and dynamic recovery processes leading to highly non-linear evolutions of damage operate. The same results is qualitatively obtained with the new damage model, namely 320 h instead of 275 h. This is highlighted in Fig. 8d showing a change in the damage kinetics after the thermal jump between the two durations. It is due to the use of the kinematic hardening kinetics (X : n /Cx ) in the damage density function D whose formulation has been specifically set up in le Graverend et al. (2014b) for fast and very-high (close to the solvus of the material) thermal jump by making the kinetics of the kinematic hardening D 10

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 10. Creep life predictions compared to experimental ones for the non-isothermal conditions presented in Figs. 8 and 9.

dependent on the volume fraction of the small precipitate fs . It can be noticed that the Rabotnov-Kachanov model is always predicting a failure during the thermal jump, contrary to the new damage model. The author wants here to point out three things with the use of Fig. 8c: the residual lifetime is smaller than when a 90s thermal jump is introduced after 206 h which was hypothesized to be due to the value of the constrained lattice misfit when the thermal jump is introduced (le Graverend et al., 2014a), the 90s thermal jump leads to a very large strain jump compared to Fig. 8b which is not described by the model, and the plastic strain rate is increased subsequently to a thermal jump for a stress of 140 MPa (see Fig. 8) which is not the case at 160 MPa (see Fig. 9). This last goes against what is known about when a particle is sheared or by-passed (Mohles et al., 1999) and is not correctly understood for now. As just expressed, the simulations at 160 MPa are providing not as good results as for 140 MPa (see Fig. 9). It is due to the fact that the plastic strain rate is not increased after a thermal jump contrary to 140 MPa. Furthermore, some experiments have longer lifetime (up to a factor 1.6) compared to the longest isothermal experiment performed at 1050 °C/160 MPa, viz. 218 h (le Graverend et al., 2014a). Even if the current model strives to consider the microstructural evolutions in order to have better predictive capabilities, a lack in the description of the mechanical behavior leads to poorer lifetime predictions. Therefore, the formulation of the isotropic and kinematic hardenings needs to be improved, as already initiated in le Graverend et al. (2014b), for better considering the static and dynamic recovery processes happening at very-high temperatures and which may lead to improve the current lifetime predictions. However, the non-isothermal life predictions have been compared to the experimental ones (see Fig. 10). The total non-isothermal creep life, i.e. the sum of the creep time before and after the thermal jump, is well predicted by the model since almost all the points are comprised between the 1.5-factor dotted lines. The two points that are not well predicted are those for which a thermal jump improved the lifetime. This qualitative result cannot be easily predicted since it is going against the common belief that hotter is shorter. Finally, the new damage model is tested with regards to its capability to predict changes in the damage kinetics when the applied stress is modified during a creep test for which a thermal jump is introduced at t = 0 h (see Fig. 11). It can be observed that the experimental plastic strain rate is increased when the applied stress is switched from 140 to 160 MPa after 60 h which is well described by the new model, even if this one overestimates the final lifetime. 3.4. Isothermal dwell/fatigue interactions It is particularly important to improve the lifetime prediction of components subjected to dwell/fatigue loadings since creep and fatigue strongly interact and lead to shorter lifetime than pure creep, as shown in le Graverend et al. (2016), Perruchaut (1997) for

Fig. 11. Complex test consisting of a 30s thermal jump at t = 0 h followed by a creep test at 1050 °C/140 MPa switched to 160 MPa after 60 h. 11

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 12. Mechanical cycle used for doing the dwell/fatigue test in tension showed in Fig. 13.

single crystals and in Billot et al. (2010), Flageolet et al. (2005) for polycrystals. Fig. 12 shows the type of mechanical cycle that has been used for performing the dwell/fatigue experiments carried out in Fig. 13 at 1050 °C. Indeed, the mechanical cycle shown is the one in tension: the one in compression has a hold time in compression and a reversed loading in tension. Fig. 13 exhibits that the new damage model respectively predicts a failure after 16.6 h and 29.3 h in tension and compression, whereas the experimental specimens failed after 15.7 h and 22.8 h. The Rabotnov-Kachanov model does not predict a failure even after 50 h in tension because it does not accumulate damage in compression, contrary to the present damage model. One can argue that the simulated shapes of the curves are not consistent with the experimental ones. The damage model cannot be taken responsible for it, the flow rule can. In fact, even if the flow rule used in the model is an hyperbolic sine (Chaboche and Gallerneau, 2001; Miller, 1976), which improves the range of plastic strain rate predictability, it is clearly not enough when comparing the experimental and simulated opening of the hysteresis loops (le Graverend, 2013; le Graverend et al., 2016). Despite a notperfect prediction of the mechanical behavior in dwell/fatigue, the difference in terms of lifetime between tension and compression as well as the final predicted lifetimes are in good agreement. The evolution of the damage during the simulated dwell/fatigue test in tension (see Fig. 14) has also been plotted. The damage continuously evolves which is consistent with the physics of damage, viz. void nucleation and growth as well as microstructure evolution that are continuously evolving even when a mechanical cycling is applied. Furthermore, the effect of dwell time on the result (see Fig. 15) has been tested: the longer the hold time is, the smaller the number of cycles to failure. This qualitative result has been also experimentally obtained in Goswami (1999), Yandt et al. (2012), Zrnı́k et al. (2001). The new damage model is, therefore, capable of qualitatively and quantitatively predicting the lifetime of isothermal creep and dwell/fatigue experiments. The author wants here to point out that the previous result is, for now, restricted to dwell/fatigue loading since the strain mechanisms are different in fatigue and the damage mechanisms are highly influenced by oxidation, which is out of the scope (Pineau and Antolovich, 2009). However, the formulation of the new damage model, which is able to predict the dwell/ fatigue interaction with one damage variable instead of two (one for the creep damage and the other for the fatigue damage), lays the ground for the use of damage formulations that are not broken down into several damage variables dedicated to specific loads (Chaboche and Gallerneau, 2001; Miller et al., 1992) and degrading processes (McLean and Dyson, 2000a), even if these types of model largely contributed to the field.

Fig. 13. Simulation of a tension dwell/fatigue test (blue diamond) and a compression dwell/fatigue test (red triangle) at 1050 °C and with the type of mechanical cycle shown in Fig. 12. The “New Damage Model” has been used for simulating the two tests (red and blue lines) while the “RabotnovKachanov” model has only been used for simulating the dwell/fatigue test in tension. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 12

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 14. Evolution of the damage density function D as a function of the time fraction t/tR for the dwell/fatigue test in tension at 1050 °C. tR is the time at rupture. A magnification around t/tR = 0.6 has been included as an inset to evaluate the evolution of the damage density over 3 cycles.

Fig. 15. Evolution of the number of cycles to failure depending on the dwell time used in Fig. 12.

3.5. FE simulations 3.5.1. In-plane multiaxial loading: a notched specimen A creep test at 1050 °C/F = 506 N/σnominal = 138 MPa performed on a notched specimen developing an in-plane multiaxial state of stresses has been simulated using the mesh and boundary conditions shown in Fig. 16. The mesh is 16,641 C3D4-type elements corresponding to 3,893 nodes and 11,679 degrees of freedom. The simulation provides a lifetime of 16.7 h which have to be compared to the experimental one of 17.8 h and to the Rabotnov-Kachanov model that gave a lifetime of 0.22 h. Just after the loading ramp, the stress goes as high as 324 MPa and then relaxes to reach 240 MPa at the notches at failure for the new damage model. The large predicted lifetime difference between the two models is because the RK model is stress driven and, therefore, much more sensitive to high stress levels. On the contrary, plastic deformation is a requirement with the new damage model in order to have a non-zero kinetics leading to a limited accumulation of damage. Even if the lifetime predicted by the simulation is in good agreement with the experiment, they underestimate the lifetime more than the results may show. Indeed, a large lattice rotation experimentally occurs in the multiaxial area (le Graverend et al., 2014b; le Graverend et al., 2017b) leading to a softening of the mechanical behavior and, therefore, a shorter lifetime. This phenomenon also happens during uniaxial isothermal loading (Ardakani et al., 1998; Ghighi et al., 2012; Ghosh et al., 1990; Matan et al., 1999a), but is not yet considered in the actual version of the model. Therefore, the model should be overestimating the lifetime, which it does not, in order to take into account the softening that occurs in the multiaxial area. Fig. 17d shows the evolution of the displacement at a node located on the top cross section where the pressure is applied. It is observed that the plastic strain rate during the secondary creep stage is not well predicted by the simulation whereas it was the case for the uniaxial loading conditions (see Figs. 2–6, 7–9, 11, 13). Therefore, it is supposed that the yield criterion has to be changed from von Mises, namely an isotropic yield criterion, to Hill (Hill, 1948, 1950), viz. an anisotropic yield criterion, in order to better consider shear stresses. In addition, it appears that the predicted ductility does not match the experimental one. It is probably due to the experimental activation of new slip systems, such as cubic, may be activated to accommodate the large plastic deformation. 3.5.2. Multiaxial loading: tubes Isothermal torsion and tension/torsion have been performed on tubes in le Graverend et al. (2018). The tube dimensions and the boundary conditions used for the simulations are presented in Fig. 18. The mechanical cycle in Fig. 19a and the longitudinal thermal gradient in Fig. 19b measured by K-type thermocouples welded in the strain gage area on a test tube have been used to do the FE simulations. Therefore, the model has also been calibrated at 800 °C. Torsion and tension/torsion have been respectively performed 13

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 16. Dimensions of the bi-notched sample use for the experiment as well as the boundary conditions and mesh used for the FEM simulation.

with a von Mises stress of 160 and 113 MPa. The mesh used for the simulations is 20,130 C3D8-type elements having 26,455 nodes (total number of degrees of freedom (dof): 79,365). This mesh has been employed based on a mesh sensitivity analysis presented in Fig. 20. It can be seen that the number of cycles to failure reaches an asymptote after a dof around 70,000. As already discussed in Bonnand (2006), Nouailhas and Cailletaud (1995), Nouailhas et al. (1993), a torsion test analysis requires FE simulations because of the complex stress redistributions due to the anisotropy of the behavior related to the crystalline anisotropy of the material. Fig. 21 shows the distribution of the accumulated plastic strain (see Fig. 21a) and the damage level (see Fig. 21b) at fracture. It can be observed that there are hard and soft areas (see Fig. 19a) which is consistent with what has been already observed in Méric and Cailletaud (1991), Nouailhas and Cailletaud (1995), Nouailhas et al. (1993). The simulation in torsion provided a lifetime of 1,106 cycles which have to be compared to the experimental one of 1,284 cycles. Therefore, the modeling underestimates the lifetime, but the predicted lifetime is very close to the experimental one. An isotropic formulation would have underestimated even more the lifetime to failure since the anisotropic the anisotropic formulation leads to an effective torsional stress of D ) which clearly leads to a smaller effective damage variable and consequently to a longer /(1 D × [(1 ) + ]) , i.e., of /(1 lifetime in torsion compare to the isotropic formulation. A FE simulation has been performed for tension/torsion at 113 MPa with the anisotropic damage that gave a lifetime of 855 cycles which have to be compared to the experimental one of 706 cycles (see Fig. 22). Even if the simulation overestimates the number of cycles to failure, the lifetime is shorter than the pure torsional case having a higher von Mises stress applied. Therefore, as already mentioned in Section 3.4, the current damage formulation which utilizes the kinetics of the hardening variables well takes into account the strong interactions between loading types. 14

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 17. (a) damage, (b) accumulated plastic strain, and (c) von Mises stress distribution of an isothermal test at 1050 °C/σnominal = 138 MPa on a notched specimen using the new damage model in its anisotropic formulation. (d) comparison between the experimental data (le Graverend et al., 2017b) and the simulation in terms of displacement.

One should keep in mind that the amount of porosity can vary from one specimen to another and Fv0 = 0.02 was used for each simulation. To improve the lifetime predictions or to be sure of the parameters used in the anisotropic damage formulation, it would have been required to do X-ray tomography characterizations on the tubes prior to testing in order to use the real Fv0 for each of the specimens. However, one can notice that the Fv0 range used in section 3.1 allows to have lifetime longer in torsion by having a Fv0 smaller than 0.02 and shorter in tension-torsion by having a Fv0 larger than 0.02. Furthermore, the differences between experiments and simulations may be due to a thickness effect that has been already observed in tension (Cassenti and Staroselsky, 2009; Doner and Heckler, 1985; Srivastava et al., 2012) and not yet studied in torsion. In fact, tube walls are 1 mm thick which has to be compared with the 1.5 mm and 1.2 mm respectively used for the creep and the dwell/fatigue experiments. From the results above, it is clear that multi-mechanical interactions have to be better understood.

15

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 18. Boundary conditions used for the torsion and tension/torsion FEM simulations on tubes. The mesh is also shown.

Fig. 19. (a) Mechanical cycle with 160 MPa-hold stresses and (b) thermal gradient which have been used to do the isothermal torsion and tension/ torsion FE simulations.

16

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 20. Evolution of the number of cycles to failure for the torsion test as a function of the number of degrees of freedom used for the mesh.

Fig. 21. (a) Accumulated plastic strain and (b) damage distribution at rupture (Nf-predicted = 1,106) for a tube subjected to torsion at 1050 °C/ σMises = 160 MPa with the mechanical cycle and the thermal gradient respectively presented in Fig. 19a and b.

4. Conclusion A hardening-based damage model has been developed to encompass not only the effect of voids but also the effect of microstructure degradations on the kinetics of damage in a sole variable that can be used for monotonic and cyclic loading. The damage model uses microstructure-sensitive hardening variables to better predict the damage evolution occurring during high-temperature isothermal and non-isothermal creep and dwell/fatigue experiments, i.e., during conditions for which the microstructure is likely to quickly evolve. It is, indeed, essential to consider microstructural degradation when evolving faster than the nucleation and growth of voids. The damage density function has been implemented in a microstructure-sensitive crystal plasticity framework, embedded in an anisotropic formulation, and applied on a Ni-based single crystal superalloy. It has been tested on uniaxial as well as FE multiaxial loading and showed its capability to predict the lifetime in conditions for which the Continuum Damage Mechanics approach based on discontinuity of matter, through the use of the Rabotnov-Kachanov model, fails. The developed model also intrinsically takes into account the non-linear accumulation of damage and the scattering in the experimental results in terms of lifetime and plastic strain rate during the secondary creep state by considering the effect of the initial volume fraction of voids.

17

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Fig. 22. (a) Accumulated plastic strain and (b) damage distribution at rupture (Nf-predicted = 855) for a tube subjected to tension/torsion at 1050 °C/ σMises = 113 MPa with the mechanical cycle and the thermal gradient respectively presented in Fig. 19a and b.

Acknowledgments The author is particularly grateful to Prof. Jonathan Cormier (ISAE-ENSMA) for his interest in this work and for stimulating discussions. The simulations were performed using the computing resources from Laboratory for Molecular Simulation (LMS) and High Performance Research Computing (HPRC) at Texas A&M University. The author is grateful to Mr. James Fillerup and the financial support from AFOSR through Award No.: FA9550-17-1-0233 to carry out this study. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijplas.2019.03.012. References Abu Al-Rub, R.K., Voyiadjis, G.Z., 2003. On the coupling of anisotropic damage and plasticity models for ductile materials. Int. J. Solids Struct. 40, 2611–2643. Acharya, M.V., Fuchs, G.E., 2004. The effect of long-term thermal exposures on the microstructure and properties of CMSX-10 single crystal Ni-base superalloys. Mater. Sci. Eng., A 381, 143–153. Ai, S.H., Lupinc, V., Maldini, M., 1992. Creep fracture mechanisms in single crystal superalloys. Scripta Metall. Mater. 26, 579–584. Aifantis, E.C., 1984. On the microstructural origin of certain inelastic models. J. Eng. Mater. Technol. 106, 326–330. Antolovich, S.D., Domas, P., Strudel, J.-L., 1979. Low cycle fatigue of René 80 as affected by prior exposure. MTA 10, 1859–1868. Ardakani, M.G., Ghosh, R.N., Brien, V., Shollock, B.A., McLean, M., 1998. Implications of dislocation micromechanisms for changes in orientation and shape of single crystal superalloys. Scripta Mater. 39, 465–472. Asaro, R.J., 1983. Micromechanics of crystals and polycrystals. Adv. Appl. Mech. 23, 1–115. Ashby, M.F., Dyson, B.F., 1984. Creep Damage Mechanics and Micromechanisms. Fracture 84 Pergamon, pp. 3–30. Baldan, A., 1991. Rejuvenation procedures to recover creep properties of nickel-base superalloys by heat treatment and hot isostatic pressing techniques. J. Mater. Sci. 26, 3409–3421. Basaran, C., Lin, M., 2008. Damage mechanics of electromigration induced failure. Mech. Mater. 40, 66–79. Basirat, M., Shrestha, T., Potirniche, G.P., Charit, I., Rink, K., 2012. A study of the creep behavior of modified 9Cr–1Mo steel using continuum-damage modeling. Int. J. Plast. 37, 95–107. Beese, A.M., Luo, M., Li, Y., Bai, Y., Wierzbicki, T., 2010. Partially coupled anisotropic fracture model for aluminum sheets. Eng. Fract. Mech. 77, 1128–1152. Besson, J., 2010. Continuum models of ductile fracture: a review. Int. J. Damage Mech. 19, 3–52. Billot, T., Villechaise, P., Jouiad, M., Mendez, J., 2010. Creep–fatigue behavior at high temperature of a UDIMET 720 nickel-base superalloy. Int. J. Fatigue 32, 824–829. Bonnand, V., 2006. Etude de l'endommagement d'un superalliage monocristallin en fatigue thermo-mecanique multiaxiale (Damage evolution of a single crystal superalloy during multiaxial thermo-mechanical loading). Ecole Nationale Supérieure des Mines de Paris, France. Brünig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. Int. J. Plast. 27, 1598–1617.

18

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Brünig, M., Gerke, S., Hagenbrock, V., 2014. Stress-state-dependence of damage strain rate tensors caused by growth and coalescence of micro-defects. Int. J. Plast. 63, 49–63. Caron, P., 2005. A propos des superalliages à base de nickel. Université Paris XI Orsay. Caron, P., Khan, T., 1983. Improvement of creep strength in a nickel-base single-crystal superalloy by heat treatment. Mater. Sci. Eng. 61, 173–184. Cassenti, B.N., Staroselsky, A., 2009. The effect of thickness on the creep response of thin-wall single crystal components. Mater. Sci. Eng., A 508, 183–189. Chaboche, J.-L., 1974. Une loi différentielle d'endommagement de fatigue avec cumulation non linéaire. Revue Francaise de Mecanique. pp. 50–51. Chaboche, J.-L., 1982. Le Concept de Contrainte Effective Appliqué à l’Élasticité et à la Viscoplasticité en Présence d’un Endommagement Anisotrope. In: Boehler, J.-P. (Ed.), Mechanical Behavior of Anisotropic Solids/Comportment Méchanique des Solides Anisotropes. Springer Netherlands, pp. 737–760. Chaboche, J.-L., 1984. Anisotropic creep damage in the framework of continuum damage mechanics. Nucl. Eng. Des. 79, 309–319. Chaboche, J.-L., Boudifa, M., Saanouni, K., 2006. A CDM approach of ductile damage with plastic compressibility. Int. J. Fract. 137, 51–75. Chaboche, J.L., 1978. Description thermodynamique et phenomenologique de la viscosite cyclique avec endommagement. Chaboche, J.L., Gallerneau, F., 2001. An overview of the damage approach of durability modelling at elevated temperature. Fatigue Fract. Eng. Mater. Struct. 24, 405–418. Chow, C.L., Wang, J., 1987. An anisotropic theory of elasticity for continuum damage mechanics. Int. J. Fract. 33, 3–16. Cormier, J., 2006. Behavior of the Single Crystal Superalloy MC2 during High and Very-High Non-isothermal Creep Loading. PhD Thesis. ENSMA - Université de Poitiers, France. Cormier, J., Cailletaud, G., 2010. Constitutive modeling of the creep behavior of single crystal superalloys under non-isothermal conditions inducing phase transformations. Mater. Sci. Eng., A 527, 6300–6312. Cormier, J., Mauget, F., le Graverend, J.B., Moriconi, C., Mendez, J., 2015. Issues related to the constitutive modeling of Ni-based single crystal superalloys under aeroengine certification conditions. AerospaceLab J. 9. Cormier, J., Milhet, X., Mendez, J., 2007. Non-isothermal creep at very high temperature of the nickel-based single crystal superalloy MC2. Acta Mater. 55, 6250–6259. Cormier, J., Milhet, X., Mendez, J., 2008. Anisothermal creep behavior at very high temperature of a Ni-based superalloy single crystal. Mater. Sci. Eng., A 483–484, 594–597. Cornet, C., Zhao, L.G., Tong, J., 2011. A study of cyclic behaviour of a nickel-based superalloy at elevated temperature using a viscoplastic-damage model. Int. J. Fatigue 33, 241–249. Cuitino, A.M., Ortiz, M., 1993. Computational modelling of single crystals. Model. Simulat. Mater. Sci. Eng. 1, 225. Delameter, W.R., Herrmann, G., 1974. Weakening of elastic solids by doubly-periodic arrays of cracks. In: Zeman, J.L., Ziegler, F. (Eds.), Topics in Applied Continuum Mechanics: Symposium Vienna, March 1–2, 1974. Springer, Vienna, Vienna, pp. 156–173. Desmorat, R., 2006. Positivité de la dissipation intrinsèque d'une classe de modèles d'endommagement anisotropes non standards. Compt. Rendus Mec. 334, 587–592. Doner, M., Heckler, J.A., 1985. Effects of Section Thickness and Orientation on Creep-Rupture Properties of Two Advanced Single Crystal Alloys. SAE International. Egner, H., 2012. On the full coupling between thermo-plasticity and thermo-damage in thermodynamic modeling of dissipative materials. Int. J. Solids Struct. 49, 279–288. Ekh, M., Lillbacka, R., Runesson, K., 2004. A model framework for anisotropic damage coupled to crystal (visco)plasticity. Int. J. Plast. 20, 2143–2159. Epishin, A., Link, T., Nazmy, M., Staubli, M., Klingelhoffer, H., 2008. Microstructural degradation of CMSX-4: kinetics and effect on mechanical properties. In: Reed, R.C., Green, K.A., Caron, P., Gabb, T.P., Fahrmann, M.G., Huron, E., Woodard, S.A. (Eds.), International Symposium on Superalloys. TMS, pp. 1–7. Epishin, A.I., Link, T., Fedelich, B., Svetlov, I.L., Golubovskiy, E.R., 2014. Hot isostatic pressing of single-crystal nickel-base superalloys: mechanism of pore closure and effect on Mechanical properties. In: MATEC Web of Conferences 14 08003. Feng, L., Zhang, K.-S., Zhang, G., Yu, H.-D., 2002. Anisotropic damage model under continuum slip crystal plasticity theory for single crystals. Int. J. Solids Struct. 39, 5279–5293. Flageolet, B., Jouiad, M., Villechaise, P., Mendez, J., 2005. On the role of γ particles within γ' precipitates on damage accumulation in the P/M nickel-base superalloy N18. Mater. Sci. Eng., A 399, 199–205. Franciosi, P., 1985. The concepts of latent hardening and strain hardening in metallic single crystals. Acta Metall. 33, 1601–1612. Fribourg, G., Bréchet, Y., Deschamps, A., Simar, A., 2011. Microstructure-based modelling of isotropic and kinematic strain hardening in a precipitation-hardened aluminium alloy. Acta Mater. 59, 3621–3635. Friedel, J., 1964. Dislocations. Pergamon. Gaubert, A., 2009. Modélisation des effets de l'evolution microstructurale sur le comportement mécanique du superalliage monocristallin AM1, PhD Thesis. Ecole Nationale Supèrieure des Mines de Paris, (France). Ghighi, J., Cormier, J., Ostoja-Kuczynski, E., Mendez, J., Cailletaud, G., Azzouz, F., 2012. A microstructure sensitive approach for the prediction of the creep behavior and life under complex loading paths. Technishe Mechanik 32, 205–220. Ghosh, R.N., Curtis, R.V., McLean, M., 1990. Creep deformation of single crystal superalloys - modelling the crystallographic anisotropy. Acta Metall. Mater. 38, 1977–1992. Giraud, R., Hervier, Z., Cormier, J., Saint-Martin, G., Hamon, F., Milhet, X., Mendez, J., 2013. Strain effect on the γ' dissolution at high temperatures of a nickel-based single crystal superalloy. Metall. Mater. Trans. 44, 131–146. Gologanu, M., Leblond, J.-B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754. Goswami, T., 1999. Low cycle fatigue—dwell effects and damage mechanisms. Int. J. Fatigue 21, 55–76. Gurson, A.L., 1975. Plastic Flow and Fracture Behavior of Ductile Materials Incorporating Void Nucleation, Growth, and Interaction. Brown University. Hayhurst, D.R., 1972. Creep rupture under multi-axial states of stress. J. Mech. Phys. Solids 20, 381–390. Heino, S., Karlsson, B., 2001. Cyclic deformation and fatigue behaviour of 7Mo–0.5N superaustenitic stainless steel—slip characteristics and development of dislocation structures. Acta Mater. 49, 353–363. Hesebeck, O., 2001. On an isotropic damage mechanics model for ductile materials. Int. J. Damage Mech. 10, 325–346. Hill, R., 1948. A Theory of the yielding and plastic flow of anisotropic metals. In: Proceeding of the Royal Society of London. The Royal Society, pp. 281–287. Hill, R., 1950. The Mathematical Theory of Plasticity. Clarendon Press. Hirth, J.P., 1961. On dislocation interactions in the fcc lattice. J. Appl. Phys. 32, 700–706. Hutchinson, J.W., 1970. Elastic-plastic behaviour of polycrystalline metals and composites. Proc. R. Soc. Lond. A. Math. Phys. Sci. 319, 247–272. Jiang, X.W., Wang, D., Wang, D., Liu, X.G., Zheng, W., Wang, Y., Xie, G., Lou, L.H., November 2018. Microstructural degradation and the effects on creep properties of a hot corrosion-resistant single-crystal Ni-based superalloy during long-term thermal exposure. Metall. Mater. Trans. 49 (11), 5309–5322. Johnson, G.R., Cook, W.H., 1985. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech. 21, 31–48. Kachanov, L.M., 1958. Time of the rupture process under creep conditions. Izvestiya Akademii Nauk SSSR. Otdelenie Tekhnicheskikh Nauk 8, 26–31. Kachanov, L.M., 1960. Creep Theory. Fizmatgiz. Kang, G., Liu, Y., Ding, J., Gao, Q., 2009. Uniaxial ratcheting and fatigue failure of tempered 42CrMo steel: damage evolution and damage-coupled visco-plastic constitutive model. Int. J. Plast. 25, 838–860. Kassner, M.E., Hayes, T.A., 2003. Creep cavitation in metals. Int. J. Plast. 19, 1715–1748. Keller, A., Hutter, K., 2011. On the thermodynamic consistency of the equivalence principle in continuum damage mechanics. J. Mech. Phys. Solids 59, 1115–1120. Kocks, U.F., 1970. The relation between polycrystal deformation and single-crystal deformation. Metall. Mater. Trans. 1, 1121–1143. Kocks, U.F., Mecking, H., 2003. Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48, 171–273. Krairi, A., Doghri, I., 2014. A thermodynamically-based constitutive model for thermoplastic polymers coupling viscoelasticity, viscoplasticity and ductile damage. Int.

19

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

J. Plast. 60, 163–181. Kunz, L., Lukáš, P., Konečná, R., Fintová, S., 2012. Casting defects and high temperature fatigue life of IN 713LC superalloy. Int. J. Fatigue 41, 47–51. Kupkovits, R.A., Neu, R.W., 2010. Thermomechanical fatigue of a directionally-solidified Ni-base superalloy: smooth and cylindrically-notched specimens. Int. J. Fatigue 32, 1330–1342. Kyaw, S.T., Rouse, J.P., Lu, J., Sun, W., 2016. Determination of material parameters for a unified viscoplasticity-damage model for a P91 power plant steel. Int. J. Mech. Sci. 115–116, 168–179. le Graverend, J.-B., 2013. Experimental Analysis and Modeling of the Effects of High Temperature Incursions on the Mechanical Behavior of a Single Crystal Superalloy for Turbine Blades. PhD Thesis. ISAE-ENSMA/ONERA, France, pp. 299. le Graverend, J.-B., Adrien, J., Cormier, J., 2017a. Ex-situ X-ray tomography characterization of porosity during high-temperature creep in a Ni-based single-crystal superalloy: toward understanding what damage is. Mater. Sci. Eng., A 695, 367–378. le Graverend, J.-B., Bonnand, V., Cormier, J., Pacou, D., 2018. Multiaxial thermo-mechanical loading at high temperature on a ni-based single crystal superalloy. Mater. Sci. Eng. A. le Graverend, J.-B., Cormier, J., Gallerneau, F., Kruch, S., Mendez, J., 2014a. Highly non-linear creep life induced by a short close γ′-solvus overheating and a prior microstructure degradation on a nickel-based single crystal superalloy. Mater. Des. 56, 990–997. le Graverend, J.-B., Cormier, J., Gallerneau, F., Kruch, S., Mendez, J., 2016. Strengthening behavior in non-isothermal monotonic and cyclic loading in a Ni-based single crystal superalloy. Int. J. Fatigue 91, 257–263. le Graverend, J.-B., Cormier, J., Gallerneau, F., Villechaise, P., Kruch, S., Mendez, J., 2014b. A microstructure-sensitive constitutive modeling of the inelastic behavior of single crystal nickel-based superalloys at very high temperature. Int. J. Plast. 59, 55–83. le Graverend, J.-B., Cormier, J., Kruch, S., Gallerneau, F., Mendez, J., 2012. Microstructural parameters controlling high-temperature creep life of the nickel-base single-crystal superalloy MC2. Metall. Mater. Trans. 43, 3988–3997. le Graverend, J.B., Pettinari-Sturmel, F., Cormier, J., Hantcherli, M., Villechaise, P., Douin, J., 2017b. Mechanical twinning in Ni-based single crystal superalloys during multiaxial creep at 1050°C. Acta Mater. Lemaitre, J., 1972. Evaluation of dissipation and damage in metals submitted to dynamic loading. Mech. Behav. Mater. 540–549. Lemaitre, J., 1985a. A continuous damage mechanics model for ductile fracture. J. Eng. Mater. Technol. 107, 83–89. Lemaitre, J., 1985b. Coupled elasto-plasticity and damage constitutive equations. Comput. Methods Appl. Mech. Eng. 51, 31–49. Lemaitre, J., Chaboche, J.-L., 1990. Mechanics of Solid Materials. Cambridge University Press, Cambridge. Lemaitre, J., Desmorat, R., Sauzay, M., 2000. Anisotropic damage law of evolution. Eur. J. Mech. A Solid. 19, 187–208. Lemaître, J., Krajcinovic, D., 1987. Continuum Damage Mechanics: Theory and Applications. Springer-Verlag. Lesne, P.M., Savalle, S., 1987. A differential damage rule with microinitiation and micropropagation. La Recherche Aerospatiale 33–47. Liu, J.L., Jin, T., Yu, J.J., Sun, X.F., Guan, H.R., Hu, Z.Q., 2010. Effect of thermal exposure on stress rupture properties of a Re bearing Ni base single crystal superalloy. Mater. Sci. Eng., A 527, 890–897. Liu, L.R., Jin, T., Zhao, N.R., Wang, Z.H., Sun, X.F., Guan, H.R., Hu, Z.Q., 2003. Microstructural evolution of a single crystal nickel-base superalloy during thermal exposure. Mater. Lett. 57, 4540–4546. MacKay, R.A., Ebert, L., 1985. The development of γ-γ' lamellar structures in a nickel-base superalloy during elevated temperature mechanical testing. Metall. Mater. Trans. 16, 1969–1982. Mahnken, R., 2002. Theoretical, numerical and identification aspects of a new model class for ductile damage. Int. J. Plast. 18, 801–831. Malcher, L., Mamiya, E.N., 2014. An improved damage evolution law based on continuum damage mechanics and its dependence on both stress triaxiality and the third invariant. Int. J. Plast. 56, 232–261. Matan, N., Cox, D.C., Carter, P., Rist, M.A., Rae, C.M.F., Reed, R.C., 1999a. Creep of CMSX-4 superalloy single crystals: effects of misorientation and temperature. Acta Mater. 47, 1549–1563. Matan, N., Cox, D.C., Rae, C.M.F., Reed, R.C., 1999b. On the kinetics of rafting in CMSX-4 superalloy single crystals. Acta Mater. 47, 2031–2045. Mataveli Suave, L., Cormier, J., Bertheau, D., Villechaise, P., Soula, A., Hervier, Z., Hamon, F., 2016. High temperature low cycle fatigue properties of alloy 625. Mater. Sci. Eng., A 650, 161–170. Mattiello, A., Desmorat, R., Cormier, J., 2018. Rate dependent ductility and damage threshold:application to Nickel-based single crystal CMSX-4. Int. J. Plast. McLean, M., Dyson, B.F., 2000a. Modeling the effects of damage and microstructural evolution on the creep behavior of engineering alloys. J. Eng. Mater. Technol. 122, 273–278. McLean, M., Dyson, B.F., 2000b. Modeling the effects of damage and microstructural evolution on the creep behavior of engineering alloys. J. Eng. Mater. Technol. 122, 273–278. Menzel, A., Ekh, M., Runesson, K., Steinmann, P., 2005. A framework for multiplicative elastoplasticity with kinematic hardening coupled to anisotropic damage. Int. J. Plast. 21, 397–434. Méric, L., Cailletaud, G., 1991. Single crystal modeling for structural calculations: Part 2—finite element implementation. J. Eng. Mater. Technol. 113, 171–182. Méric, L., Poubanne, P., Cailletaud, G., 1991. Single crystal modeling for structural calculations: Part 1 - model presentation. J. Eng. Mater. Technol. 113, 162–170. Miller, A., 1976. An inelastic constitutive model for monotonic, cyclic, and creep deformation: Part I—equations development and analytical procedures. J. Eng. Mater. Technol. 98, 97–105. Miller, M.P., McDowell, D.L., Oehmke, R.L.T., 1992. A creep-fatigue-oxidation microcrack propagation model for thermomechanical fatigue. J. Eng. Mater. Technol. 114, 282–288. Mohles, V., Ronnpagel, D., Nembach, E., 1999. Simulation of dislocation glide in precipitation hardened materials. Comput. Mater. Sci. 16, 144–150. Moosbrugger, J.C., McDowell, D.L., 1990. A rate-dependent bounding surface model with a generalized image point for cyclic nonproportional viscoplasticity. J. Mech. Phys. Solids 38, 627–656. Murakami, S., Ohno, N., 1981. A continuum theory of creep and creep damage. In: Ponter, A.R.S., Hayhurst, D.R. (Eds.), Creep in Structures. Springer Berlin Heidelberg, pp. 422–444. Murakumo, T., Kobayashi, T., Koizumi, Y., Harada, H., 2004. Creep behaviour of Ni-base single-crystal superalloys with various γ' volume fraction. Acta Mater. 52, 3737–3744. Naderi, M., Hoseini, S.H., Khonsari, M.M., 2013. Probabilistic simulation of fatigue damage and life scatter of metallic components. Int. J. Plast. 43, 101–115. Nathal, M.V., Mackay, R.A., 1987. The stability of lamellar γ-γ′ structures. Mater. Sci. Eng. 85, 127–138. Naumenko, K., Altenbach, H., Kutschke, A., 2010. A combined model for hardening, softening, and damage processes in advanced heat resistant steels at elevated temperature. Int. J. Damage Mech. 20, 578–597. Needleman, A., Tvergaard, V., 1984. An analysis of ductile rupture in notched bars. J. Mech. Phys. Solids 32, 461–490. Neil, C.J., Wollmershauser, J.A., Clausen, B., Tomé, C.N., Agnew, S.R., 2010. Modeling lattice strain evolution at finite strains and experimental verification for copper and stainless steel using in situ neutron diffraction. Int. J. Plast. 26, 1772–1791. Nguyen, G.D., Korsunsky, A.M., Belnoue, J.P.H., 2015. A nonlocal coupled damage-plasticity model for the analysis of ductile failure. Int. J. Plast. 64, 56–75. Nix, W.D., 1983. Introduction to the viewpoint set on creep cavitation. Scripta Metall. 17, 1–4. Nouailhas, D., Cailletaud, G., 1995. Tension-torsion behavior of single-crystal superalloys: experiment and finite element analysis. Int. J. Plast. 11, 451–470. Nouailhas, D., Pacou, P., Cailletaud, G., Hanriot, F., Rémy, L., 1993. Experimental study of the anisotropic behavior of the CMSX2 single-crystal superalloy under tension-torsion loadings. Adv. Multiaxial Fatigue 114, 244–258. Olsen-Kettle, L., 2018. Bridging the macro to mesoscale: evaluating the fourth-order anisotropic damage tensor parameters from ultrasonic measurements of an isotropic solid under triaxial stress loading. Int. J. Damage Mech 1056789518757293. Ott, M., Tetzlaff, U., Mughrabi, H., 1999. Influence of directional coarsening on the isothermal high-temperature fatigue behaviour of the monocrystalline nickel-base superalloys CMSX-6 and CMSX-4. In: Mughrabi, H., G., G., Mecking, H., Riedel, H., Tobolski, J. (Eds.), Microstructure and Mechanical Properties of Metallic High

20

International Journal of Plasticity xxx (xxxx) xxx–xxx

J.-B. le Graverend

Temperature Materials. Perruchaut, P., 1997. Etude des interactions fatigue-fluage-oxydation a 950°C dans l'endommagement du superalliage monocristallin AM1. Universite de Poitiers. Pessah-Simonetti, M., Caron, P., Khan, T., 1992. Effect of mu phase on the mechanical properties of a nickel-base single crystal superalloy. In: Antolovitch, S.D., Strusrud, R.W., Mackay, R.A. (Eds.), International Symposium on Superalloys. TMS, pp. 567–576. Pineau, A., Antolovich, S.D., 2009. High temperature fatigue of nickel-base superalloys – a review with special emphasis on deformation modes and oxidation. Eng. Fail. Anal. 16, 2668–2697. Pollock, T.M., Argon, A.S., 1994. Directional coarsening in nickel-base single crystals with high volume fractions of coherent precipitates. Acta Metall. Mater. 42, 1859–1874. Potirniche, G.P., Horstemeyer, M.F., Ling, X.W., 2007. An internal state variable damage model in crystal plasticity. Mech. Mater. 39, 941–952. Qi, W., Bertram, A., 1998. Damage modeling of the single crystal superalloy SRR99 under monotonous creep. Comput. Mater. Sci. 13, 132–141. Qi, W., Bertram, A., 1999. Anisotropic continuum damage modeling for single crystals at high temperatures. Int. J. Plast. 15, 1197–1215. Rabotnov, Y.N., 1968. Kinetics of creep and creep rupture. In: Parkus, H., Sedov, L.I. (Eds.), Irreversible Aspects of Continuum Mechanics and Transfer of Physical Characteristics in Moving Fluids. Springer Vienna, pp. 326–334. Rabotnov, Y.N., 1969. Creep rupture. In: Hetényi, M., Vincenti, W.G. (Eds.), Applied Mechanics. Springer Berlin Heidelberg, pp. 342–349. Rae, C.M.F., Karunaratne, M.S.A., Small, C.J., Broomfield, R.W., Jones, C.N., Reed, R.C., 2000. TCP phases in an experimental rhenium-containing single crystal superalloy. In: Pollock, T.M., Kissinger, R.D., Bowmann, R.R. (Eds.), International Symposium on Superalloys. TMS, pp. 767–776. Raffaitin, A., Monceau, D., Crabos, F., Andrieu, E., 2007. The effect of thermal cycling on the high-temperature creep behavior of a single crystal nickel-based superalloy. Scripta Mater. 56, 277–280. Rajhi, W., Saanouni, K., Sidhom, H., 2014. Anisotropic ductile damage fully coupled with anisotropic plastic flow: modeling, experimental validation, and application to metal forming simulation. Int. J. Damage Mech. 23, 1211–1256. Reed, R.C., 2006. The Superalloys: Fundamentals and Applications. Cambridge Univ. Press. Reed, R.C., Cox, D.C., Rae, C.M.F., 2007. Damage accumulation during creep deformation of a single crystal superalloy at 1150 °C. Mater. Sci. Eng. 448, 88–96. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress fields∗. J. Mech. Phys. Solids 17, 201–217. Roy Chowdhury, S., Roy, D., February 2019. A non-equilibrium thermodynamic model for viscoplasticity and damage: two temperatures and a generalized fluctuation relation. Int. J. Plast. 113, 158–184. Shenoy, M.M., Kumar, R.S., McDowell, D.L., 2005. Modeling effects of nonmetallic inclusions on LCF in DS nickel-base superalloys. Int. J. Fatigue 27, 113–127. Shenoy, V.B., Kukta, R.V., Phillips, R., 2000. Mesoscopic analysis of structure and strength of dislocation junctions in fcc metals. Phys. Rev. Lett. 84, 1491–1494. Shi, H.-J., Korn, C., Pluvinage, G., 1998. High temperature isothermal and thermomechanical fatigue on a molybdenum-based alloy. Mater. Sci. Eng., A 247, 180–186. Shui, L., Jin, T., Tian, S., Hu, Z., 2007. Influence of precipitate morphology on tensile creep of a single crystal nickel-base superalloy. Mater. Sci. Eng., A 454–455, 461–466. Simo, J.C., Ju, J.W., 1987. Strain- and stress-based continuum damage models—II. Computational aspects. Int. J. Solids Struct. 23, 841–869. Sinha, S., Ghosh, S., 2006. Modeling cyclic ratcheting based fatigue life of HSLA steels using crystal plasticity FEM simulations and experiments. Int. J. Fatigue 28, 1690–1704. Srivastava, A., Gopagoni, S., Needleman, A., Seetharaman, V., Staroselsky, A., Banerjee, R., 2012. Effect of specimen thickness on the creep response of a Ni-based single-crystal superalloy. Acta Mater. 60, 5697–5711. Srivastava, A., Needleman, A., 2015. Effect of crystal orientation on porosity evolution in a creeping single crystal. Mech. Mater. 90, 10–29. Stoughton, T.B., Yoon, J.W., 2011. A new approach for failure criterion for sheet metals. Int. J. Plast. 27, 440–459. Tabourot, L., Fivel, M., Rauch, E., 1997. Generalised constitutive laws for f.c.c. single crystals. Mater. Sci. Eng., A 234, 639–642. Tinga, T., Brekelmans, W.A.M., Geers, M.G.D., 2009. Time-incremental creep-fatigue damage rule for single crystal Ni-base superalloys. Mater. Sci. Eng., A 508, 200–208. Trinh, B.T., Hackl, K., 2014. A model for high temperature creep of single crystal superalloys based on nonlocal damage and viscoplastic material behavior. Continuum Mech. Therm. 26, 551–562. Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17, 389–407. Tvergaard, V., 1982. Material failure by void coalescence in localized shear bands. Int. J. Solids Struct. 18, 659–672. Viguier, B., Touratier, F., Andrieu, E., 2011. High-temperature creep of single-crystal nickel-based superalloy: microstructural changes and effects of thermal cycling. Phil. Mag. 91, 4427–4446. Voyiadjis, G.Z., Deliktas, B., 2000. A coupled anisotropic damage model for the inelastic response of composite materials. Comput. Methods Appl. Mech. Eng. 183, 159–199. Voyiadjis, G.Z., Park, T., 1999. The kinematics of damage for finite-strain elasto-plastic solids. Int. J. Eng. Sci. 37, 803–830. Wang, R.-Z., Zhu, S.-P., Wang, J., Zhang, X.-C., Tu, S.-T., Zhang, C.-C., 2019. High temperature fatigue and creep-fatigue behaviors in a Ni-based superalloy: damage mechanisms and life assessment. Int. J. Fatigue 118, 8–21. Wierzbicki, T., Bao, Y., Lee, Y.-W., Bai, Y., 2005. Calibration and evaluation of seven fracture models. Int. J. Mech. Sci. 47, 719–743. Wolff, C., Richart, N., Molinari, J.-F., 2015. A non-local continuum damage approach to model dynamic crack branching. Int. J. Numer. Methods Eng. 101, 933–949. Wulfinghoff, S., Fassin, M., Reese, S., 2017. A damage growth criterion for anisotropic damage models motivated from micromechanics. Int. J. Solids Struct. 121, 21–32. Xiao, L., Chen, D.L., Chaturvedi, M.C., 2006. Effect of boron and carbon on thermomechanical fatigue of IN 718 superalloy: Part I. Deformation behavior. Mater. Sci. Eng., A 437, 157–171. Xue, L., 2009. Stress based fracture envelope for damage plastic solids. Eng. Fract. Mech. 76, 419–438. Yandt, S., Wu, X.J., Tsuno, N., Sato, A., 2012. Cyclic dwell fatigue behaviour of single crystal Ni-base superalloys with/without rhenium. In: Huron, R.S., Reed, R.C., Hardy, M.C., Mills, M.J., Montero, R.E., Portella, P.D., Telesman, J. (Eds.), International Symposium on Superalloys. Seven Spings, PA, USA, pp. 501–508. Yue, Z.F., Lu, Z.Z., Zheng, C.Q., 1995. The creep-damage constitutive and life predictive model for nickel-base single-crystal superalloys. Metall. Mater. Trans. 26, 1815–1821. Zghal, J., Gmati, H., Mareau, C., Morel, F., 2016. A crystal plasticity based approach for the modelling of high cycle fatigue damage in metallic materials. Int. J. Damage Mech. 25, 611–628. Zhang, K., Badreddine, H., Saanouni, K., 2018. Thermomechanical modeling of distortional hardening fully coupled with ductile damage under non-proportional loading paths. Int. J. Solids Struct. 144–145, 123–136. Zhang, Z.L., Thaulow, C., Ødegård, J., 2000. A complete Gurson model approach for ductile fracture. Eng. Fract. Mech. 67, 155–168. Zhao, L.-Y., Zhu, Q.-Z., Shao, J.-F., 2018. A micro-mechanics based plastic damage model for quasi-brittle materials under a large range of compressive stress. Int. J. Plast. 100, 156–176. Zhu, Y.Y., Cescotto, S., 1995. A fully coupled elasto-visco-plastic damage theory for anisotropic materials. Int. J. Solids Struct. 32, 1607–1641. Zrnı́k, J., Semeňák, J., Vrchovinský, V., Wangyao, P., 2001. Influence of hold period on creep–fatigue deformation behaviour of nickel base superalloy. Mater. Sci. Eng., A 319–321, 637–642.

21