Force-depth relationships with irradiation effect during spherical nano-indentation: A theoretical analysis

Journal of Nuclear Materials 531 (2020) 152012

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Force-depth relationships with irradiation effect during spherical nano-indentation: A theoretical analysis Xiazi Xiao, Cewen Xiao, Xiaodong Xia* Department of Mechanics, School of Civil Engineering, Central South University, Changsha, 410075, PR China

h i g h l i g h t s
A model is proposed for the force-depth relationship of ion-irradiated materials.
Three distinguishing deformation stages for the force-depth curve are characterized.
Weakened pop-in events and irradiation hardening are addressed.
Theoretical results can match well with corresponding experimental data.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 31 October 2019 Received in revised form 28 December 2019 Accepted 17 January 2020 Available online xxx

A mechanistic model is developed for the force-depth relationship of ion-irradiated materials, which is conducted by spherical nano-indentation. With irradiation effect, the pop-in phenomenon almost disappears that is ascribed to the irradiation-induced defects serving as dislocation nucleation sites that facilitate the generation of new dislocations. After materials yielding, the evolution of statistically stored dislocations, geometrically necessary dislocations and irradiation-induced defects mutually contributes to the force-depth relationships with irradiation effect. Thereinto, the increase of loading force originates from the impediment of slipping dislocations by irradiation-induced defects. By comparing with the experimental data of Fee12Cr alloy, a reasonable agreement is achieved. © 2020 Elsevier B.V. All rights reserved.

Keywords: Force-depth relationship Pop-in Theoretical model Ion irradiation Spherical indentation

In recent years, nano-indentation has became the most widely applied technique analyzing the incipient plastic deformation of ion-irradiated materials [1e7]. By recording the applied load with the continuous stiffness measurement technique [8], it is now available and convenient to characterize the preliminary forcedepth relationship, based on which, one can further address the irradiation effect on the intrinsic mechanical properties, e.g. the depth-dependent hardness affected by the inhomogeneously distributed defects [9,10]. Therefore, a thorough comprehension of the force-depth relationships with irradiation effect plays a dominant role in understanding the fundamental deformation mechanisms of ion-irradiated materials. Generally speaking, there exist three deformation stages for the force-depth relationships of unirradiated materials, including the elastic stage, pop-in stage and elasto-plastic stage, as illustrated in

* Corresponding author. E-mail address: [email protected] (X. Xia). https://doi.org/10.1016/j.jnucmat.2020.152012 0022-3115/© 2020 Elsevier B.V. All rights reserved.

Fig. 1. In the initial stage, materials deform purely elastically that can be well characterized by the Hertzian contact theory [11]. With increasing indentation depth, deformation transforms from the initially elastic stage to the irreversible elasto-plastic stage. At the onset of the elasto-plastic transition, a sudden displacement excursion on the force-depth curves, i.e. the pop-in phenomenon, has been widely observed in nano-indentation experiments [12e17], which is believed to be determined by the homogeneous nucleation of dislocations when there exist limited dislocations in the highly stressed region or by the heterogeneous activation of pre-existing dislocations [18,19]. After ion irradiation, two distinguishing differences could be identified for the force-depth relationships, including the dramatically mitigated pop-in phenomenon and increase of the loading force during the elasto-plastic deformation stage [20e23]. For the former, the impact of high energy particles generates numbers of irradiation-induced defects like interstitials and vacancies, which serve as dislocation nucleation sources that lead to the increase of

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X. Xiao et al. / Journal of Nuclear Materials 531 (2020) 152012

that facilitate the nucleation of dislocations and inhibit the pop-in 1 events, therefore, it is generally observed that hirr p ≪hp [20e23]. With the increase of the loading force, either the nucleation of new dislocations or activation of slipping dislocations can be stimulated when the resolved shear stress reaches a critical value that the pop-in phenomenon occurs initially [27,28]. Following the standard Hertzian analysis [11], the critical force when a pop-in event occurs is expressed as

F¼

 3 2 max 3 p R S 6E2r

t



 h1p  h < h2p ;

(2)

where h2p indicates the indentation depth when the elasto-plastic deformation occurs for unirradiated materials. S is the indentation Schmid factor [29,30] that is affected by the crystallographic orientation with respect to the indentation loading direction, and tmax indicates the maximum resolved shear stress that is determined by the dislocation nucleation and activation mechanisms, i.e. Fig. 1. Illustration of the force-depth relationships of metallic materials with and without irradiation effect. The pop-in event occurs obviously when the indentation depth ranges from h1p to h2p for unirradiated materials. With irradiation effect, the pop1 in phenomenon almost disappears, i.e. hirr p ≪hp .

the nucleation and activation probability of dislocations, and facilitates the initiation of plastic deformation [6,24,25]. Consequently, the pop-in event can hardly be observed in the nanoindentation test of ion-irradiated materials. In addition, the defect clusters are usually sessile that impede the slipping of dislocations within the plastic zone, therefore, during the elasto-plastic deformation stage, additional external force is required to enforce the slipping dislocations to overcome the impediment of these obstacles, i.e. irradiation hardening [9]. In this work, we intent to propose a mechanistic model for the force-depth relationship of ion-irradiated materials under spherical nano-indentation. The model mainly addresses the mechanisms concerning the irradiation effect on the pop-in events. In addition, the elasto-plastic deformation after the pop-in events is also characterized, which involves the influence of elastic deformation on the evolution of geometrically necessarily dislocations and hardening contribution induced by irradiation-induced defects. By comparing the theoretical results with the experimental data of ion-irradiated Fee12Cr alloys [21], the rationality and accuracy of the proposed model are verified. Under spherical nano-indentation, the force-depth relationship before the pop-in event is fully reversible, and can be characterized by the Hertzian contact theory [11], i.e.

tmax ¼ ttheo ,Dnuc þ tcrss ,Dact ;

(3)

where ttheo and tcrss denote, respectively, the theoretical strength for materials containing limited defects and critical resolved shear stress for the activation of existing dislocations [28], and in general ttheo [tcrss . Dnuc and Dact are the coefficients characterizing the difficulty of dislocation nucleation and activation, respectively. Without irradiation effect, both Dnuc and Dact are high as there exist limited dislocations in the stressed region, and the homogeneous dislocation nucleation mechanism plays a dominant role [31e33]. After ion irradiation, dislocation slipping is impeded by irradiationinduced defect clusters that results in the increase of tcrss . Whereas, considering the increasing number of dislocation nucleation sites induced by irradiation damage, Dnuc and Dact with irradiation effect could be much lower than that before ion irradiation. Therefore, it is generally observed that the critical loading force when the pop-in behavior initially occurs (or say tmax ) decreases with irradiation effect, and the pop-in phenomenon can hardly be observed for ionirradiated materials [20e22]. After the pop-in events, both elastic and plastic deformation contribute to the force-depth relationships, and the loading force is related to materials hardness H and contact area Ac , i.e.

F ¼ H,Ac ;

(4)

where H is influenced by statistically stored dislocations, geometrically necessary dislocations and irradiation-induced defects. In addition, for the spherical indenter, Ac yields as

Ac ¼ pa2 z2pRh; 4 pffiffiffiffiffiffiffiffi F ¼ Er Rh3 ; 3

(1)

where F is the loading force, and Er indicates the reduced indentation modulus. R represents the radius of the indenter tip, and h is the indentation depth. As irradiation damage has limited effect on the elastic deformation [26], Eq. (1) holds for both unirradiated and ion-irradiated materials when h < h1p and h < hirr p , respectively. Thereinto, h1p and hirr p , respectively, indicate the indentation depth when the pop-in event occurs without and with irradiation effect (as illustrated in Fig. 1). After ion irradiation, numbers of irradiation-induced defects serve as dislocation nucleation sites

(5)

pffiffiffiffiffiffiffiffiffi where a is the contact radius that can be approximated as az 2Rh when h≪R. In the following, the derivation of H with and without irradiation effect is deduced in details. For as-received materials, the hardness without irradiation effect, i.e. Huni , is mainly determined by the evolution of existing dislocations within the plastic zone. Following the Tabor relationship and Mises flow rule [34,35], Huni is given as

pffiffiffiffiffiffi Huni ¼ 2:8sflow ¼ 2:8M mba rG þ rS



 h  h2p ;

(6)

where M is the Taylor factor. m and b represent the shear modulus

X. Xiao et al. / Journal of Nuclear Materials 531 (2020) 152012

and magnitude of the Burgers vector, respectively. rG and rS are, respectively, the average density of geometrically necessary dislocations and statistically stored dislocations, and the strength coefficient is a. In order to accommodate the plastic strain gradient beneath the indenter tip, geometrically necessary dislocations are generated from the contact surface, and the elastic deformation plays a dominant role in affecting rG at the onset of the elasto-plastic deformation [36]. Moreover, it should be noted that irradiationinduced defects may not necessarily affect the characteristic length scale of most nuclear structural materials [37], therefore, the consideration of irradiation effect on the derivation of rG is not addressed in this work. Through the integration of the total length of dislocation loops within the plastic zone [36,38,39], the length of geometrically necessary dislocations can be calculated as

l¼

ða 0

4pr 2 h  h2e 4pa3 dr ¼ 1 bR h  h 3bR e

1

h2e h

!



 h  h2p ;

(7)

where h2e indicates the elastic indentation depth during the elastom1 plastic loading stage. As noted in Ref. [36] that h2e ¼ h1e ðh=h1e Þ 1 with he the maximum elastic indentation depth during the Hertzian loading stage, and m the coefficient characterizing the actual geometrical morphology of the indenter tip [40] and elastic deformation ability of the materials. When dividing l by the volume of the plastic zone, rG as a function of h yields as

l

2 1 h2 rG ¼ ¼ 3 1 e V bf R h

!



 h  h2p ;

where V ¼ 2pðfaÞ =3 and f is a proportional coefficient that is around 2 for materials with modulus to hardness ratio of  100 [21]. Eq. (8) indicates that rG increases with the decrease of R, which is the well known indentation size effect for spherical indentation [39,41]. Moreover, rG /0 when h/h1e indicating that the elastic deformation dominates the evolution of rG at the onset of elasto-plastic deformation. With the increase of h, rG keeps increasing until it reaches a stable value that plastic deformation becomes the dominant deformation mechanism. Besides the hardening contribution induced by geometrically necessary dislocations, the evolution of statistically stored dislocations also plays an important role in affecting Huni . Following the work of [41], rS can be derived from the uniaxial stress-strain relationship by applying the Tabor law [42], i.e.



 2

s F εp rS ¼ ref M mba



h  h2p



;

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u" pffiffiffiffiffiffi!#2  u R h2e 2h t 1 Funi ¼ 2pRh,H0 þ F pffiffiffi R h 5 R

(9)

where sref and Fðεp Þ indicate, respectively, the reference stress and non-dimensional function that are obtained from the uniaxial stress-strain curve. Hereinto, pffiffiffiffiffiffiffiffiεffi p is the indentation plastic strain defined as εp ¼ a=ð5RÞ ¼ 2Rh=ð5RÞ [43], and sref ,Fðεp ¼ 0Þ gives the yield stress for the uniaxial stress-strain relationship due to the initial statistically stored dislocations [41]. By submitting Eqs. (5)(9) into Eq. (4) and after some derivations, the final expression of Funi is expressed as



 h  h2p ; (10)



where H0 ¼ 2:8sref and R ¼ 2bðM maÞ2 =ðs2ref f 3 Þ. The first and second item under the square root of Eq. (10) indicate, respectively, the contribution of statistically stored dislocations and geometrically necessary dislocations to the loading force without irradiation effect. For the latter, elastic deformation plays a dominate role in determining the hardening contribution induced by geometrically necessary dislocations at the initial elasto-plastic deformation, i.e. h/h2e . When h[h2e , it can be reduced to the classical indentation size effect model for spherical nano-indentation, i.e. materials hardening scales inversely with R [39,41]. After ion irradiation, the irradiation-induced defect clusters are usually sessile that impede the slipping of dislocations within the plastic zone, therefore, the hardness with irradiation effect, i.e. Hirr , should be recast as

Hirr ¼ 2:8M mb

qffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ðrG þ rS Þ þ b2 Ndef ddef



 h > hirr p ;

(11)

where hirr p denotes the indentation depth when the elasto-plastic deformation occurs for ion-irradiated materials. Ndef and ddef are, respectively, the average number density and size of defects with the strength coefficient of b. Considering the inhomogeneously distributed defects induced by ion irradiation, N def is calculated by dividing the total number of defects within the plastic zone by its volume [36], i.e.

ð Rp h

(8)

3

3

Ndef ¼

0

#

p R2p  w2 Ndef ðw; hÞdw ;

V

(12)

where Rp ¼ f ,a is the radius of the plastic zone that is assumed to be a hemisphere. Ndef ðw; hÞ indicates the distribution of defect density (along the w-axis direction) during the elasto-plastic deformation stage, which is dominated by the initial distribution of irradiation damage and defect annihilation induced by the dislocation-defect interaction, and can be specified as

 ffi pffiffiffi pffiffiffiffiffi ffi h  hirr p

p 2h p R

Ndef ðw; hÞ ¼ N ydef ðwÞ,e5



 h > hirr p ;

(13) y

where h indicates the defect annihilation coefficient, and Ndef ðwÞ is the initial distribution of defect density before the dislocationdefect interaction is activated [9], i.e.

n 8 w > < N0 def Ld Nydef ðwÞ ¼ > : 0

ðw  Ld Þ

;

(14)

ðw > Ld Þ

where N0def is the maximum defect density at the boundary of the irradiated region with the depth of Ld . n characterizes the distribution profile of defects within the irradiated region. As indicated by Eq. (13), the initial distribution of defect density is generally inhomogeneous that is determined by both the irradiation conditions and materials properties [9]. Once h > hirr p , the defect annihilation mechanism becomes activated that results in the decrease of defect density within the plastic zone [21]. For a detailed derivation

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X. Xiao et al. / Journal of Nuclear Materials 531 (2020) 152012

of Eq. (13), please refer to the Appendix. By submitting Eqs. (13) and (14) into Eq. (12), one can finally deduce Ndef when the irradiated materials yield q with the plastic ffiffiffiffiffiffiffiffiffiffiffiffiffi zone locating within the irradiated region (or say 2Rhirr p < Ld = f ), i.e.

N def ðhÞ ¼



where A ¼ 3N0def ddef bR b2 f 3 =ð2a2 Þ is affected by irradiation conditions and materials properties. By comparing the third term with the first two terms under the square root of Eqs. (16) and (18), the increase of h may not necessarily lead to an obvious irradiation hardening phenomenon, especially when the plastic zone extends

0

pffiffiffiffiffiffiffiffiffi !n 8 pffi pffiffiffi pffiffiffiffiffi  irr 3N0def 2Rhf >  p2ffih h  hp > > ,e 5 R > > L ðn þ 1Þðn þ 3Þ > d > > < > > > > > > > > :

3N 0def 2

"

B irr Bh < h  @ p

3 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi !3 pffi pffiffiffi pffiffiffiffiffi  irr 2h 1 1 2Rhf 2Rhf 7 7,e5pffiR h  hp  5 n þ 1 Ld nþ3 Ld

and by further combining qffiffiffiffiffiffiffiffiffiffiffiffiffi Eqs. (4), (5), (8), (9), (11) and (15), one can have Firr when 2Rhirr p < Ld =f

1 L2d 2Rf

0

C C

2A

1

;

(15)

2 C B Bh > Ld C @ 2Rf 2 A

into the unirradiated substrate that both defect annihilation and unirradiated substrate mitigate the irradiation effect, and the difference between the force-depth relationships with and without

8 > > > > > > > > > <

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u" pffiffiffiffiffiffi!#2 pffiffiffiffiffiffiffiffiffi !n  pffi pffiffiffi pffiffiffiffiffi  u irr 2h R h2e A 2h 2Rhf pffi Firr ¼ 2pRh,H0 t F p ffiffiffi 1 þ þ ,e5 R h  hp > ðn þ 1Þðn þ 3Þ L R h > 5 R d > > > > > > > : vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! u" pffiffiffiffiffiffi!#2 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi !3 #  pffi pffiffiffi pffiffiffiffiffi  u irr 2h R h2e A 1 1 2h 2Rhf 2Rhf pffi t F p ffiffiffi 1 þ þ  ,e5 R h  hp 2 n þ 1 Ld nþ3 Ld R h 5 R

Otherwise, when the plastic zone extends beyond the irradiated qffiffiffiffiffiffiffiffiffiffiffiffiffi region, i.e. 2Rhirr p  Ld =f , the expression of N def is deduced as

! N def h

¼

3N0def 2

and Firr yields as

"

hirr p

h>

L2d


L2d 2Rf 2

! 2pRh,H0

! :

2Rf 2

(16)

irradiation effect gradually vanishes in the bulk region. In this work, a mechanistic model is proposed for the forcedepth relationship with irradiation effect. As a comparison with

3 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi !3 pffi pffiffiffi pffiffiffiffiffi  irr 2h 2Rhf 2Rhf 7 1 1 7 , e5pffiR h  hp  5 Ld n þ 1 Ld nþ3

0

1

B C Bh > hirr C; p A @

(17)

the previous work [28,36], the effect of irradiation damage on the pop-in mechanism and hardening mechanism is addressed that it

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " ! u" pffiffiffiffiffiffi!#2 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi !3 #  pffi pffiffiffi pffiffiffiffiffi  u irr 2h 2h 2Rhf 2Rhf R h2 A 1 1 pffi 1 e þ Firr ¼ 2pRh,H0 t F pffiffiffi þ  ,e5 R h  hp Ld 2 n þ 1 Ld nþ3 R h 5 R



 h > hirr p ;

(18)

X. Xiao et al. / Journal of Nuclear Materials 531 (2020) 152012

becomes available to theoretically characterize the force-depth relationships of ion-irradiated materials under spherical nanoindentation. In order to verify the rationality and accuracy of the proposed theoretical model, several sets of experimental data have been collected in the literature, including Ni-irradiated Molybdenum [6], Au-irradiated sapphire [20], Ar-irradiated ferritic/martensitic T-91 steel [2], He-irradiated tungsten [22] and Fe-irradiated Fee12Cr alloys [21]. Thereinto, only the work of [21] simultaneously offers the unirradiated data with indentation size effect and irradiated data characterizing the weakened pop-in phenomenon and irradiation hardening behavior, therefore, the experimental data of ion-irradiated Fee12Cr alloys [21] is considered in this work. As informed by the experimental work [21], polycrystalline Fee12Cr alloy is 99.999% pure with little carbon impurities, and the average grain size is 147 mm. 6 displacements per atom (dpa) irradiation with 2 MeV Feþ ions is applied, and the irradiation temperature is 593 K. The pop-in behavior of unirradiated Fee12Cr alloy occurs when h1p < h < h2p with h1p ¼ 58 nm and h2p ¼ 151 nm, as indicated in Fig. 2 (a). The radius of the spherical indenter is 10 mm. When h < h1p , Er ¼ 178.7 GPa is obtained following the Hertzian contact

Fig. 2. Force-depth relationships of (a) unirradiated and (b) ion-irradiated Fee12Cr alloy are compared between the experimental data [21] and theoretical results (given by Eqs. (10) and (18)). h1p ¼ 58 nm and h2p ¼ 151 nm when the pop-in event occurs. hirr p ¼ 6:67 nm in inset.

5

theory [11]. In order to characterize the hardening contribution induced by statistically stored dislocations, one can have Fðεp Þ ¼ 0:492 þ ðεp Þ0:563 by fitting the uniaxial stress-strain curve of unirradiated Fee12Cr alloy, as illustrated in Fig. 3. Thereinto, the solid symbols indicate the predicted results by the elasticviscoplastic self-consistent method [44,45]. Moreover, the Fðεp Þ  εp relationship follows well the Taylor-type parabolic hardening law, which fundamentally indicates the deformation mechanism is dominated by the evolution of statistically stored dislocations. After irradiation with 2 MeV Feþ ions, the irradiation depth Ld reaches about 0.7 mm, and the indentation strain εirr p equals 0.0073 when plastic deformation occurs for ion-irradiated irr Fee12Cr alloy. ffiffiffi 2 Furthermore, it can be calculated that hp ¼ pffiffiffi irr p ð5 R εp = 2 Þ ¼ 6.67 nm. The rest parameters characterizing the force-depth relationships with and without irradiation are listed in Table 1. We firstly address the force-depth relationship without irradiation effect, which is compared between the experimental data and theoretical results predicted by Eqs. (1), (4), (5) and (10). Corresponding model parameters are listed above. As presented in Fig. 2 (a): (1) the theoretical results (solid lines) match well with the unirradiated experimental data (empty symbols) in both the elastic deformation stage (i.e. h < h1p ) and elasto-plastic deformation stage (i.e. h > h2p ). (2) A pop-in event (h1p < h < h2p ) is obviously noticed for unirradiated Fee12Cr alloy when R ¼ 10 mm. Moreover, the predicted pop-in load, by using Eq. (10) with the calibrated parameters, equals 12.25 mN which is close to the experimental data 11.96 mN when h ¼ h2p . Generally speaking, there exist limited dislocation nucleation sites within the stressed region for unirradiated materials as illustrated in Fig. 4(a), therefore, increasing loading force (close to the theoretical strength) is required for the nucleation of dislocations either homogeneously beneath the indenter tip or heterogeneously from the contact surface [42]. (3) When the pop-in event occurs, the loading force almost keeps a constant. It is mainly ascribed to the remaining of indentation hardness and contact radius with the increase of indentation depth [46]. Actually, during this transition process, the surface profile of the indented materials varies with the occurrence of slip bursts, which ensures that the contact radius does not change significantly

Fig. 3. The stress-strain relationship of unirradiated Fee12Cr alloy, which is predicted by the elastic-viscoplastic self-consistent method (solid symbols) [44,45]. The red line indicates the fitted result with the relationship y ¼ a þ bxc . (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

6

X. Xiao et al. / Journal of Nuclear Materials 531 (2020) 152012

Table 1 Model parameters characterizing the force-depth relationships of unirradiated and irradiated Fee12Cr alloy [36]. Unirradiated

Irradiated



h1e (nm)

H0 (GPa)

R (mm)

m

0.5

1.59

2.8

1.63

A

n

f

h

6.2

0.01

1.8

50

Fig. 4. Schematic of the spherical nano-indentation of (a) unirradiated and (b) ionirradiated materials. ⊥ indicates the dislocation nucleation sites. For unirradiated materials, limited dislocation sites exist within the highly stressed region, which results in the high probability for the occurrence of a pop-in event. With irradiation effect, numbers of dislocation nucleation sites exist in the irradiated region, which facilitates the generation of new dislocations and mitigates the pop-in phenomenon.

[46]. (4) When h > h2p , the nucleation and activation mechanisms of dislocations are both activated, thus the increase of rS and rG mutually contribute to materials hardening. One should note that the dislocation construct illustrated in Fig. 4 is presented to help comprehend the occurrence of the pop-in events, and useful for the model evaluation. After the pop-in event, the actual indentation plastic deformation takes place by the shear-induced slipping dislocations. With irradiation effect, the elasto-plastic deformation of ionirradiated Fee12Cr alloy initiates at a small indentation depth, i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffi hirr 2Rhirr p ¼ 6.67 nm. Moreover, it is informed that p z Ld = f with R ¼ 10 mm, Ld ¼ 0.7 mm and f ¼ 1.8, therefore, the combination of Eqs. (4), (5) and (18) can be applied to characterize the force-depth relationship of ion-irradiated Fee12Cr alloy when h > hirr p . As illustrated in Fig. 2 (b), a good agreement is achieved by comparing the theoretical results (dashed lines) with the experimental data (empty symbols) with irradiation effect. By further comparing Fig. 2 (a) and (b), one can see that the force with irradiation effect is

higher than that without irradiation effect due to the irradiation hardening mechanism induced by irradiation-induced defects. Whereas, this difference gets weakened with the increase of h owing to the annihilation of defects within the irradiated layer and softening effect induced by the unirradiated substrate [21]. Another obvious feature as presented in Fig. 2 (b) is that the pop-in event can hardly be observed after ion irradiation. Similar experimental phenomena have also been reported in Ni-irradiated Molybdenum [6], Au-irradiated sapphire [20], Ar-irradiated ferritic/ martensitic T-91 steel [2] and He-irradiated tungsten [22], etc. Previous studies have indicated that pre-existing point defects or defect clusters can act as heterogeneous nucleation sites for dislocations [24,25]. After ion irradiation, the original defect-free sample contains numbers of irradiation-induced point defects (like interstitials and vacancies) and defect clusters in the irradiated region, as illustrated in Fig. 4 (b). Therefore, even though the resolved shear stress is much lower than the theoretical strength at a small indentation depth, the nucleation of dislocations in ion-irradiated materials is still possible as there exist plenty of dislocation nucleation sites within the highly stressed region [6]. After the onset of dislocation nucleation, dislocation avalanches are activated that facilitate the smooth transition from the elastic deformation to the elasto-plastic deformation, and the evolution of dislocations and irradiation-induced defects both contribute to the force-depth relationships with irradiation effect. To summarize, a mechanistic model is proposed to characterize the force-depth relationship with and without irradiation effect. The effect of irradiation-induced defects on the weakened pop-in mechanism and increase of loading force, i.e. irradiation hardening, is systematically addressed within the theoretical model. For unirradiated materials, the force-depth relationship is divided into three deformation stages, thereinto, the elastic response can be addressed by the Hertzian contact theory. When it transforms from the elastic deformation to elasto-plastic deformation, a pop-in event occurs that is mainly dominated by the dislocation nucleation and activation mechanisms. During the elasto-plastic deformation stage, the evolution of statistically stored dislocations and geometrically necessary dislocations both contribute to the forcedepth relationship. For the latter, the elastic deformation plays an important role in determining the density of geometrically necessary dislocations at the onset of the elasto-plastic deformation. After ion irradiation, the irradiation-induced defects can not only act as dislocation nucleation sites that facilitate the generation of new dislocations and weaken the pop-in events, but also serve as obstacles that result in irradiation hardening. The accuracy and rationality of the developed model have been verified by comparing the theoretical results with the experimental data of ion-irradiated Fee12Cr alloy, and a good agreement is achieved. This work was supported by the National Nature Science foundation of China (NSFC) under Contract No. 11802344 and 11902365, and Natural Science Foundation of Hunan Province, China (Grant No. 2019JJ50809). X.Z. thanks the initial funding supported by Central South University. Appendix Once the dislocation-defect interaction mechanism is activated, irradiation-induced defects could be annihilated through the interaction with glissile dislocations. Thus, the distribution of defect density during the elasto-plastic deformation stage [36] follows as irr y Ndef ðwÞ ¼ N def ðwÞ,ehðεp εp Þ ;

(A1)

X. Xiao et al. / Journal of Nuclear Materials 531 (2020) 152012 y

where N def ðwÞ is the initial distribution of defect density induced by ion irradiation before defect annihilation. εp and εirr p are the indentation strain and indentation yield strain with irradiation effect, respectively, which can be expressed as

1 εp ¼ 5

rffiffiffiffiffiffi 2h R

and

εirr p

1 ¼ 5

sffiffiffiffiffiffiffiffiffiffi 2hirr p

(A2)

R

where hirr p is the indentation depth when materials yielding occurs with irradiation effect. By submitting Eq. (A1) into Eq. (A1), Ndef ðw; hÞ as a function of h yields as

 ffi pffiffiffi pffiffiffiffiffi irr

p

y

Ndef ðw; hÞ ¼ N def ðwÞ,e

5p2ffiRh

h

hp

:

(A3)

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