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Analysis of surface effects on the deformation of a nanovoid in an elasto-plastic material
Applied Mathematical Modelling 39 (2015) 5091–5104
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Analysis of surface effects on the deformation of a nanovoid in an elasto-plastic material J.X. Liu ⇑ Faculty of Civil Engineering and Mechanics, Jiangsu University, 301 Xuefu Road, Zhenjiang, Jiangsu 212013, China
a r t i c l e
i n f o
Article history: Received 10 May 2014 Received in revised form 8 March 2015 Accepted 1 April 2015 Available online 23 April 2015 Keywords: Nanovoid Surface energy Plastic deformation Size effect Tension–compression asymmetry Pre-history yielding
a b s t r a c t Surface effects, usually embodied as surface tension or surface energy, become considerable in nanostructured materials. With attention to static finite deformations, we present a theoretical analysis about the surface energy effects on the evolution of a nanovoid in a plastic material. The following improvements have been incorporated based on the previous studies: (a) the initial configuration is assumed to be the equilibrium state, which is reached by applying the surface tension onto a fictitious stress-free configuration; (b) both cases of applying compressive and tensile hydrostatic stress is discussed. This illustrative solution provides some reasonable physical interpretations of the following ‘‘unconventional’’ phenomena: (a) when the void radius is smaller than some critical value, finite plastic flow happens in the process from the fictitious stress-free configuration to the initial configuration, which will dramatically influence the subsequent responses; (b) size effects exist, i.e., the growth of nanovoids is closely related to their sizes; (c) the tension–compression asymmetry becomes stronger with decreasing the void size; (d) void shrinkage instability can arise. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Deformations of voids on the nanometric scale are studied with the consideration of surface energy, whose importance has been widely recognized [1,2]. The main characteristic of this study is to consider the prehistory elasto-plastic deformation caused by the surface energy on the void surface, during the investigation of its response under external loadings.
1.1. A brief literature review on surface effects Surface effects in the mechanics of nanostructured elements, including nanoparticles, nanowires, nanobeams, nanofilms and materials containing nanoscale inhomogeneities such as cavities, have attracted considerable research attention, e.g. [1,3]. Here, we focus on the basic concepts of surface tension and composition as related to growth of nanovoids in plastic materials. The first two concepts are surface tension and surface energy. The idea of surface tension is proposed by Cabeo [4] and is stated more explicitly by Segner [5]. Shuttleworth [6] derives the relationship between surface tension and surface energy. We adopt his definitions in this research: ‘‘The surface energy is the work necessary to form unit area of surface by a process ⇑ Tel.: +86 18851280096. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apm.2015.04.007 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.
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of division; the surface tension is the tangential stress (force per unit length) in the surface layer; this stress must be balanced either by external forces or by volume stresses in the body’’. Zhou and Huang [7] and Tang et al. [8] argue that the surface tension depends on the competition between electron redistribution and the lower coordination on surfaces. For simplicity, the surface tension and surface energy are assumed to be constant in the present analysis, and commonly denoted by c, which has values 0–5 N m1 for most metals as shown by Vitos et al. [9]. Another important concept, the atomic structure of the surface, also deserves attention. From an atomistic viewpoint, it is actually a surface layer within a few atomic spacings of the outmost surface atom layer [6]. For example, molecular dynamics (MD) simulations were used by Diao et al. [10] to model the free relaxation process of a gold nanowire from a fictitious unrelaxed configuration to a physically realistic configuration. The outmost two layers of atoms decrease upon relaxation. The formation of such a layer is driven by unequilibrium interatom forces arising when a new surface is made, for instance, by cutting operation. Furthermore, plastic deformation in nanostructured elements under external loadings will generally change the density and number of the atoms at a surface [3]. Processes of surface formation and deformation are complicated and usually associated with atomic rearrangement. Nevertheless, this complexity adds no difficulty to the current study because surface tension (energy) is in principle a continuum-type concept. The value of surface energy is largely dependent on the atomistic spacing in the surface layer. For metals, finite deformations are mainly achieved by dislocations instead of changes in atomistic spacing. Thus, we assume the constant atomistic spacing and therefore the constant surface energy in the finite deformation processes in metals. The last concept is the specific surface area (SSA), namely the ratio of the surface area to the volume of the bulk [11]. The high SSA value of nanostructured materials is generally considered to be the cause for their mechanical behaviors, which can be very different from the behaviors of the macro-bulk. This conclusion is true for nanofilms, nanowires, nanobeams and nanoparticles, but is not true for nanovoids because the SSA value is close enough to zero when a nanovoid is embedded in infinite bulk materials. The unique characteristic of nanovoids is that the bulk is outside of the void surface, unlike in nanoparticles in which the bulk is surrounded by its surface. Instead of revisiting detailed studies on effects of the SSA in the fields of materials science and mechanics [3], here we reflect its association to Torricelli’s trumpet, an ‘‘unthinkable’’ idea first observed by Evangelista Torricelli, a student of Galileo [12]. This trumpet has a finite volume, V ¼ p, but an infinite surface area, S ¼ 2p ln xj1 1 ¼ 1. This can be only mathematically true. Actually, this infinite surface cannot be manufactured in the real physical world. If we consider that everything is composed of atoms, the maximum surface area will reach 10
¼ 20pln 10 if the atom radius is equal to 1010 m. By the way, there is a continuous increase in SSA along S ¼ 2p ln xj10 1 the length of the trumpet and it can be a potential idea for making a functionally graded material. The purpose of mentioning the Torricelli’s trumpet is to show the atomic composition nature of materials, because it becomes crucial for accurate descriptions of material composition especially at the nanometer level. The above reflection on Torricelli’s trumpet indicates that mathematics and physics do not always coincide with each other, and should be a note on the Gibbs idealization [13] describing the surface layer as a ‘‘mathematical surface’’ with a zero thickness, as well as the feasibility of continuum theories, which is questioned by some researchers such as Huang et al. [14]. Although the present study is conducted within the continuum framework, the validity of continuum concepts deserve a great awareness. Nanovoids with the surface effects have been studied intensively in the elastic regime. Yang [15] discusses the effective bulk and shear moduli of heterogeneous materials containing spherical nanovoids at dilute concentrations. Duan et al. [16] discover that nanoporous materials can be made stiffer than non-porous counterparts by surface modification. He and Li [17] examine the effect of surface stress on stress concentration near a spherical void in an elastic medium in the framework of continuum surface elasticity. Unfortunately, these theoretical works are confined to systems of simple geometry, therefore numerical simulation techniques have also been widely applied to more complex geometries. For example, Wei et al. [18] study the size-dependent mechanical properties of nanostructures by developing a kind of surface element to take into account the surface elastic effect. Chen et al. [11] propose a continuum model by introducing the surface energy to the total potential energy, then implement the model using the finite element method. To date, plastic deformation with the inclusion of nanovoids has been studied much less compared with the elastic response, mainly because both surface atom rearrangements and atom exchange can occur during finite plastic deformations. Some efforts in this area have been made within the framework of continuum mechanics. Within the small deformation regime, Zhang et al. [19] study the plastic deformation of nanoporous materials and nanocomposites with consideration of both the surface residual stress and the surface elasticity. Huo et al. [20] assume a constant surface energy and study its effects on the growth of a spherical shell or a thick cylindrical column, concluding that for typical metals, the surface effect is negligible for voids larger than 100 nm, but it becomes significant when the void size is on the order of 10 nm. Considering the size effect due to the plastic strain gradient which becomes important from hundreds of microns to hundreds of nanometers, Gao et al. [21] and Huang et al. [22] propose the mechanism-based strain gradient plasticity and then use this theory to solve the void growth problem. Nevertheless, the strain gradient effect is not possible to increase unlimitedly with decreasing the characteristic size of plastic deformation. For example, Bazˇant [23] addresses that the questionable small-scale asymptotic behavior can exist. That is why the concept of maximum allowable dislocation density has been later mentioned [24,25]. Based on this fact, we are not going to consider the gradient effect because we argue that the surface effect is more dominant than the gradient effect when the void size is smaller than 20 nm. We admit this ignorance makes the result somehow less accurate, but we would say that the bottom line is that the tendencies found in this study is qualitatively
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reasonable. Recently, the same issue is revisited in the appendix of the work by Reina et al. [26], providing a relation between surface energy, external hydrostatic pressure and initial yield strength within the elastic range. 1.2. Challenges Our attempt is to address the following challenges based on the existing investigation of nanovoids. Challenge No. 1: what can happen during the free relaxation process driven by surface tension? Even before any external loading is applied, nanostructured materials have already deformed to balance the surface tension. Therefore, Huang and Wang [27] and Zhang et al. [19] emphasize that it is necessary to construct a nonstress configuration that they call ‘‘a fictitious stress-free configuration’’ based on which stress tensors are calculated within the infinitesimal deformation range. Plastic deformation possibly arises during free relaxation. Some proofs are listed as follows. Firstly, Mackenzie and Shuttleworth [28] study the calculation of the decrease in the radius of a spherical pore, surrounded by a shell of incompressible but shearable material, when a surface-tension induced pressure is applied inside the pore. They solve the viscous dissipation problem by analogy with the theory of plasticity. Diao et al. [10] employ atomistic simulation and explore that a h 1 0 0 i gold nanowire yields under surface stresses when the cross-sectional area is less than 2.45 nm 2.45 nm. Zhang et al. [19] also predict the possible surface-tension-induced yielding using their proposed nonlinear micromechanics theory. Although the pre-existing deformation field is generally considerable, it is extremely complicated to determine it exactly, because it is closely related to manufacturing techniques associated with chemical and/or thermal processes. Therefore, we will adopt the approximating strategy proposed by Huang and Wang [27], who implement the free relaxation process by fixing the material properties and applying surface forces onto the ‘‘fictitious stress-free configuration’’. Challenge No. 2: Is the isotropic continuum theory too simplistic to describe the surface effects?. Surface tension/energy is a continuum-level concept, which can become meaningless with extremely small sizes. Huang et al. [14] argue that discrete dislocation models should be employed rather than the continuum plasticity theory especially at the nanometer level. Needleman and Van der Giessen [29] and Balint et al. [30] address the same concern. Molecular dynamics simulations are an alternative for obtaining accurate results. Another fact is that surface energy depends on the lattice orientation of the surface [9]. Nevertheless, Zhang et al. [19] demonstrate that the isotropic continuum theory still gives plausible predictions if the void radius is on the order of several nanometers and above. We emphasize that our current analysis cannot be extended to too small a size without jeopardizing the rationality of continuum theory. 5 nm is the minimum void radius in Huo et al. [20], and that in Zhang et al. [19] is 2 nm. In the present study, results will be provided when the void radius is no less than 2 nm. 1.3. Present purpose We investigate surface effects on the plastic deformation of a spherical void in a plastic medium. As an effort to make a further step forward based on previous investigations, the following factors are considered simultaneously. 1. Pre-history yielding. 2. Finite deformation. 3. Nanovoid shrinkage and expansion. When compared with the work of Huo et al. [20], we further consider the first one of above factors and the shrinkage response. In consequence, we point out that the void expanding process under external loading is intrinsically an unloading-loading process, as well as tension–compression asymmetry and the shrinkage instability. While the research in Zhang et al. [19] is within the small deformation framework, we show that plastic deformations due to surface effects can be considerably large at least under the current material parameter settings. The effect of surface elasticity is neglected in this study. The same simplification is adopted by Huo et al. [20]. Zhang et al. [19] include such an effect into their micromechanical model, but conclude that the surface elasticity effect is weak. 1.4. The paper structure The remainder of this paper is organized as follows. In Section 2, the theoretical formulation is presented within the continuum plasticity framework for both the free relaxation process and the subsequent process under external hydrostatic pressure. These solutions are analyzed using a series of illustrative diagrams in Section 3. The paper concludes in Section 4. 2. Finite plastic deformation of a nanovoid in a plastic material Consider an isotropic, homogeneous and incompressible elasto-plastic shell that sustains uniform hydrostatic tensile traction, rm . As shown in Fig. 2.1, the initial inner radius is R0 , while the outer radius is R1 . This problem has a physical
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Fig. 2.1. A spherical void in a finite sphere with R0 and R1 the inner and outer radii in the fictitious stress-free configuration R, and R0 and R1 are the radii in the initial configuration R, while r 0 and r 1 are the radii in the external load-driven deformed configuration r.
background for studying the void growth [20]. The surface tension on the outer shell surface is zero because it is not a true material surface but an imaginary one in the original problem, as pointed out by Huo et al. [20]. If the surface effect is considered, any physical experiment is conducted on specimens that are initially stressed, say, due to the self-driven reconstruction in order to form the surfaces. Therefore, as shown in Fig. 2.1, there arise three configurations: the fictitious stress-free configuration, R, the observed initial configuration, R, and the final configuration, r, under external loadings and surface effects. Correspondingly, this study divides the whole deformation process into two stages: stage 1 and stage 2 (refer to Fig. 2.1). In this section, we consider the cases in which the void expands during stage 2, say, r > R. Cases of void shrinkage (r < R) are studied in Section 3. 2.1. Stage 1: from a fictitious stress-free state, R, to the initial configuration, R During this stage, the loading is related to the surface energy. That is, the traction on the ‘‘bulk surface’’ attached to the void surface is given by
Ps ¼
2c : R0
ð2:1Þ
Even though later the void surface-related virtual work is calculated directly via c instead of P s , Eq. (2.1) provides a more descriptive picture of how the void is driven to shrink. For simplicity, we assume the Poisson’s ratio is equal to 0.5. Such an incompressibility condition can be expressed in the form
R3 R30 ¼ R3 R30 ;
ð2:2Þ
where the spherical surface with radius R in the fictitious configuration can be mapped to the surface with radius R in the initial configuration, as shown in Fig. 2.1. Eq. (2.2) immediately gives the radial and tangential logarithmic strain components, eR and eT , as follows
eR ¼ 2eT ¼ 2 ln
R ; R
ð2:3Þ
where subscripts ‘‘R’’ and ‘‘T’’ mean variables during stage 1, while subscripts ‘‘r’’ and ‘‘t’’ represent variables during stage 2 as will be shown in Section 2.2. In this study, we have adopted the logarithmic strain and the corresponding work-conjugate as the strain and stress measurements, therefore, the present formulation is robust for finite deformations. To set up the equilibrium equation according to the virtual work principle, assume an infinitesimal variation:
dR0 < 0:
ð2:4Þ
The positive work related to the surface shrinkage is assumed to be completely transferred into the deformation energy of the material, leading to
Ps 4pR20 dR0 ¼
Z
R1
ðrR deR þ 2rT deT Þ4pR2 dR ¼
R0
where Ps is given by Eq. (2.1).
Z
R1
R0
ðrR rT ÞdeR 4pR2 dR;
ð2:5Þ
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Eqs. (2.2) and (2.3) provide the above related variational quantities in the form
deR ¼ 2deT ¼ 2
R20 R3
ð2:6Þ
dR0 :
The Von Mises effective strain and stress are, respectively, written as
re ¼ jrR rT j ¼ rR rT ; ee ¼ jeR j ¼ eR :
ð2:7Þ
Finally, Eq. 2.5 becomes
2c ¼ R0 ry where
Z
eR1
eR0
eR0 ¼ 2 ln
e
f Ið RÞ 32eR
e
1
R0 R0
,
8 eR < ey f I ðeR Þ ¼ N : eR ey
deR ;
eR1 ¼ 2 ln
ð2:8Þ R1 R1
when
eR 6 ey ;
when
eR > ey ;
, and the flow rule f I ðeR Þ is written in the form
ð2:9Þ
where N is the hardening exponent. Because now the Von Mises effective strain is equal to eR , as shown in Eq. (2.7), the flow rule in Eq. (2.9) is exactly the elastic-power-law-hardening uniaxial stress–strain relation, which has been used widely, e.g. [31]. Now, Eq. (2.8) can be used to determine the void radius, R0 , in the fictitious stress-free configuration, R. Additionally, the elastic part among the total effective strain, eR , can be written as
eelast ¼ ey f I ðeR Þ; R
ð2:10Þ
which will be used in Section 2.2. 2.2. Stage 2: from the initial configuration, R, to the final deformed configuration, r The external loading, rm , is applied on the surface with radius R1 , say the outer surface of the spherical shell. Again, note that in this section only void expansion is considered, while void shrinkage under external loadings will be discussed in Section 3. For void expansion, we assume a variation
dr 0 > 0:
ð2:11Þ
Then, again according to the impressibility requirement, we have
r21 dr 1 ¼ r 20 dr0 :
ð2:12Þ
For clarity, again, subscripts ‘‘R’’ and ‘‘T’’ are for stage 1, while ‘‘r’’ and ‘‘t’’ are for stage 2. For example, eR is the radiusdirection strain during stage 1, while er is that during stage 2. It is worth noting that the variational quantities der and det will not change no matter which configuration is chosen as the reference configuration, R or R, because of the following fact
r r R r ¼ d 2 ln ; der ¼ d 2 ln ¼ d 2 ln 2 ln R R R R
ð2:13Þ
where d 2 ln Rr refers to configuration R, while d 2 ln Rr refers to R. Similarly, for the tangential component of the logarithmic strain, et , there is
r r det ¼ d ln ¼ d ln : R R
ð2:14Þ
The final expression takes the form
der ¼ 2det ¼ 2
r20 dr 0 < 0: r3
ð2:15Þ
According to the virtual work principle, there is
4prm r 21 dr 1 ¼
Z
r1
r0
ðrr rt Þder 4pr 2 dr þ 4pcdr 20 ;
ð2:16Þ
where ðrr rt Þder in the integrand of the integral on the right side denotes the variation in the strain energy density. There is a significant difference between the current study and the work by Huo et al. [20]. Huo et al. [20] say that ðrr rt Þ is negative during stage 2. Here, ðrr rt Þ is positive at the early period of stage 2, then becomes negative with
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the further expansion of the void. This is easy to understand: During stage 1, ðrR rT Þ is positive as given in Eq. (2.7). Because the stress state should evolve continuously, there should be a transition period, say from positive ðrR rT Þ to negative ðrr rt Þ. In Section 3.2, this is explained thoroughly as an intrinsic unloading–reloading process. ðrr rt Þ and the flow rule are related to each other in the form
rr rt ¼ sign eelast þ er f II ðeR ; er Þ; R ry where
ð2:17Þ
eelast is given by Eq. (2.10), and the flow rule f II ðeR ; er Þ is written as R 8 f I ðeR Þ þ eeyr > > > < er f II ðeR ; er Þ ¼ ey f I ðeR Þ > > > : eR er 2f ðe Þ N I R ey
when
when
er 2 eelast ;0 ; R when er 2 eelast max eelast ; ey ; eelast ; R R R
ð2:18Þ
er 2 1; eelast max eelast ; ey : R R
The Von Mises effective strain has to be calculated according to its original definition
(R
ee ¼
R
stage 1
jdeR j
stage 1
jdeR j þ
R R
stage 2
jder j
stage 2
jder j 2eelast R
er 2 eelast ;0 ; R when er 2 1; eelast : R
when
ð2:19Þ
With the help of Eqs. (2.3) and (2.15), we rewrite Eq. (2.19) as
(
2 ln Rr
ee ¼
er 2 eelast ;0 ; R when er 2 1; eelast : R when
2 ln rR 2eelast R R2
ð2:20Þ
Finally, Eq. (2.16) becomes
rm 2c ¼ ry r 0 ry where
er0 ¼ 2 ln
Z
er1
sign
er0
r0 R0
,
eelast þ er R
er1 ¼ 2 ln
f II ðeR ; er Þ 3
e2er 1
der ;
ð2:21Þ
r1 . R1
In order to compare Eq. (2.21) with previous studies, we provide the formula as follows: (1) Eq. (2.21) reduces to the solution from [32]:
rm ¼ ry
Z
er1
er0
f II ð0; er Þ 3
e2er 1
der
ð2:22Þ
by letting c ¼ 0 and eR ¼ 0. Here c ¼ 0 means that the surface energy is eliminated, while configuration transfer is not considered. (2) Eq. (2.21) reduces to the solution from [20]:
rm 2c ¼ þ ry r 0 ry by letting
Z
er1
er0
f II ð0; er Þ 3
e2er 1
eR ¼ 0 suggests that the R ! R
der
ð2:23Þ
eR ¼ 0.
3. Discussion and implications In the following analyses, the material parameters are set with the ranges shown in Table 1, where the magnitude of the surface energy is within the range provided by Vitos et al. [9] for metals. 3.1. Pre-history plastic deformation During stage 1, with the help of Eq. (2.8), the surface tension can produce considerable plastic flow measured by the Von Mises effective strain, ee ¼ eR . For instance, before any external loading is applied, under the parameter setting in Fig. 3.1, : : near the void surface, the Von Mises effective plastic strain is ee ¼15ey when the surface energy is c ¼ 1:0Nm1 ; ee ¼82ey when : the surface energy is c ¼ 1:5 N m1 ; ee ¼586ey when the surface energy is c ¼ 2:0 N m1 .
Table 1 Ranges of material properties. Initial void radius R0 (nm) Shell outer radius R1 (R0 ) Initial yield stress ry (MPa)
5–106 3–1 25–200
Hardening exponent N
0–0.5
Surface energy c (N m1) Initial yield strain ey
0.5–5.0 0.001–0.02
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Fig. 3.1. Distribution of the dimensionless Von Mises effective strain, eeye , in relation to the radius-direction position, R, under different surface energies, c.
In Eq. (2.8), by letting eR0 ¼ ey , a critical void radius Rcrit 0 beyond which no plastic deformation will be caused by the surface tension can be expressed in the form
Rcrit 0 ¼
ry
R ecrit R1 ey
2c f I ð eR Þ 3e e 2 R 1
de
ð3:1Þ
;
where crit R1
e
2 x ¼ ln 3ey 3 x1þe2
! ð3:2Þ
R3
with x ¼ R13 . 0
crit crit R1 1 From Eq. (3.1), Rcrit 0 / c, R0 / ry . Fig. 3.2 shows the change of R0 along with the initial yield strain, ey . For every fixed R0 , crit Rcrit 0 decreases slightly with increasing ey , suggesting that the dependence of R0 on ey is almost negligible. On the other hand, R1 R1 R1 R1 when ey is fixed, Rcrit 0 decreases quickly with increasing R0 when R0 < 5, but it tends to become independent of R0 when R0 > 10.
3.2. The intrinsic unloading–reloading process There is a sharp transition in the stress state between stage 1 and stage 2, as briefly mentioned in SubSection 2.2. At the end of stage 1, there is ðrR rT Þ > 0; but in the late period of stage 2, ðrr rt Þ < 0. Therefore, there must be a continuous process implementing the change from ðrR rT Þ > 0 to ðrr rt Þ < 0, driven by the increasing rm . It is more convenient to explore this process in the space constructed by the Von Mises effective strain, ee , and stress, re , as shown in Fig. 3.3, which gives the re ee diagram for the material point at R ¼ 1:5R0 . Stage 1 corresponds to the O ! A ! B process. Point B is the end of stage 1, denoting the equilibrium state under the action of c. Then, the external hydrostatic stress, rm , is applied increasingly from zero, leading to a decrease in rr and an increase in rt during the B ! C ! B process. Briefly, it can be summarized as
0 6 re
8 > < rr rt ¼ 0 > : rt rr
during B ! C; at point C; during C ! B:
ð3:3Þ
Obviously, the above B ! C ! B process is just an unloading–reloading process in the re ee space. Eqs. (2.21) and (2.18) are valid for the process B ! C ! B ! D.
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Fig. 3.2. The critical void radius below which plastic flow arises in stage 1, Rcrit in relation to the initial yielding strain, ey , under changing geometrical ratio 0 R1 . R0
Fig. 3.3. Curve of the dimensionless effective stress in relation to the strain, for the material point at R ¼ 1:5 R0 in the initial configuration R.
With consideration of stage 1 and therefore the above unloading–reloading process, the loading–deformation curve is shown in Fig. 3.4 compared with the result by Huo et al. [20], say, Eq. (2.23). The following differences are seen: (1) when : the deformation indicator is Rr00 ¼ 1, the present theory, namely Eq. (2.21), gives rm ¼ 0, but (2.23) predicts rm ¼5:33ry . Eq. (2.23) fails to satisfy the condition, say rm ¼ 0 when Rr00 ¼ 1, because it is established based on an unequilibrated initial configuration. This is caused by neglecting stage 1. (2) Eq. (2.21) predicts a peak value of rm , which is much higher than that from Eq. (2.23). Pre-history plastic deformation leads to considerable hardening, which is an important contributing factor.
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3.3. Influences of the void size R0 and the geometrical ratio R1 =R0 We can see a considerable size effect, i.e. the dependence on the void radius, R0 , when the geometrical ratio, RR10 , is fixed, as shown in Figs. 3.5 and 3.6. In Fig. 3.5, the curve with R0 ¼ 106 nm represents the result with a negligible surface effect. Within the range R0 2 ð1; 100Þ nm, the loading–deformation response has a strong dependence on R0 . Namely, the maximum external loading, max ðrm Þ, decreases quickly with increasing R0 . Fig. 3.6 should be understood by referring to Eqs. (2.19) and (2.20). In other words, the differences among curves in Fig. 3.6 are related to eR , i.e., the pre-history deformation. The curve corresponding to R0 ¼ 5 nm is much higher than others because its pre-history plastic flow is several hundred times of ey and the strongest. By combining Figs. 3.5 and 3.6, for the cases of R0 ¼ 10 nm and R0 ¼ 20 nm, the difference in rm is still considerable. This difference is mainly due to the difference in the surface energy itself – the term r02rcy in Eq. (2.21).
Fig. 3.4. Comparison of the loading–deformation responses with and without the consideration of the pre-history plastic flow.
Fig. 3.5. The size effect caused by the change in the void radius, R0 : loading–deformation curves.
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We have also studied the dependence on the ratio increasing
R1 , R0
the deformation
r0 R0
R1 R0
when R0 is fixed, as shown in Figs. 3.7 and 3.8. From Fig. 3.7, with
corresponding to the peak load increases, while the post-peak drop becomes slower. From
Fig. 3.8, the Von Mises effective strain is the highest with the minimum value of R1 R0
R1 . R0
One issue deserves pointing out: when
¼ 20, among the spherical shell, only a small region around the void has entered the plastic state, as shown in Fig. 3.9: in
the region 7R0 < R < 20R0 , there is ee < ey , meaning that only elastic deformation happens. Therefore, the consideration of the elastic effective strain is very important. Huang et al. [32] and Gao and Huang [31] also emphasize that elastic deformation is essential in the analysis of cavitation instability.
Fig. 3.6. The size effect caused by the change in the void radius, R0 : distributions of the dimensionless effective strain when the deformation reaches r0 ¼ 1:5. R0
Fig. 3.7. The size effect caused by the change in the geometrical ratio,
R1 : R0
loading–deformation curves.
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Fig. 3.8. The size effect caused by the change in the geometrical ratio, RR10 : distributions of the dimensionless effective strain when the deformation reaches r0 ¼ 1:5. R0
Fig. 3.9. Distribution of the dimensionless effective strain in the specimen with
R1 R0
¼ 20 when the deformation reaches
r0 R0
¼ 1:5.
3.4. Size-dependent growth of nanovoids ð1Þ
ð2Þ
For demonstration purposes, consider two spherical shells with different void radii respectively R0 and R0 . These two specimens are under the same remote hydrostatic tensile stress, rm . For stage 1,
2c ðiÞ
R0
ry
¼
Z e
e
R
ðiÞ 1
fI 3
R
ðiÞ 0
eRðiÞ
e2eRðiÞ 1
deRðiÞ
ð3:4Þ
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and for stage 2,
Z
rmðiÞ 2c ¼ ry r0ðiÞ ry
e ðiÞ r
1
sign
eelast þ eðiÞ r RðiÞ
e ðiÞ r
f e
e
RðiÞ ; r ðiÞ
II
3
e2erðiÞ 1
derðiÞ ;
ð3:5Þ
0
where the superscript ‘‘ðiÞ’’ means the size number, say, i ¼ 1; 2. Eqs. (3.4) and (3.5) imply that when erð1Þ ¼ erð2Þ , say 0
ð1Þ m
ð2Þ m
ð1Þ R0
0
ð1Þ
r0
ð1Þ
R0
ð2Þ
¼
r0
ð2Þ
R0
,
ð2Þ R0 .
the external loadings are generally different, namely r – r because – In other words, the growth of nanovoids is intrinsically size-dependent. Zhang et al. [19] also make this observation but without explanation. The size dependence destroys the shell approximation for classic porous plasticity such as the Gursontype model [33]. The shell approximation is valid when we adopt the following formula to describe the void growth:
r1 ¼ ry
Z
0
f II ð0; erðiÞ Þ 3
e2erðiÞ 1
e ðiÞ r
derðiÞ :
ð3:6Þ
0
When void 1 and void 2 are under the same external stress
Z
0
e ð1Þ r
f II ð0; erð1Þ Þ 3
e2erð1Þ 1
derð1Þ ¼
Z
0
e ð2Þ r
0
f II ð0; erð2Þ Þ 3
e2erð2Þ 1
derð2Þ ;
r1 , we have ð3:7Þ
0
immediately leading to
erð1Þ ¼ erð2Þ : 0
ð3:8Þ
0
With the help of the definition ð1Þ
r0
ð1Þ R0
er ¼ 2 ln
r , we obtain R
ð2Þ
¼
r0
ð2Þ
R0
ð3:9Þ
:
The above Eqs. (3.6)–(3.9) can be taken as the theoretical background for the shell approximation. However, the current theory, Eqs. (3.4) and (3.5), does not satisfy Eq. (3.9). The surface effect is stronger for smaller voids. The surface related terms in both Eqs. (2.8) and (2.21) are inversely proportional to void radius, leading to the inclusion of the void size itself instead of dimensionless Rr00 into the constitutive response. The pre-plastic-hardening is also larger. Nevertheless, the Gurson model can still be used for some particular cases. For example, when the uni-sized nanovoids are periodically distributed in the matrix, we can choose one nanovoid as a representative cell, and reformulate the Gurson model based on the present theoretical derivation. Actually, this strategy has already been adopted by some investigators, e.g. [34]. But, for nanoporous media with a considerable scattering in nanovoid sizes, we have not been able to see an easy way to extend the Gurson model. 3.5. Tension–compression asymmetry The present model can predict the asymmetric properties of tension and compression. Such an asymmetry has been discussed by many other investigators, for example, [19]. However, in the following the instability in void shrinkage is firstly explored. For tension, the complete description is composed of Eq. (2.8) for stage 1 and Eq. (2.21) for the tensile stage 2. For compression, the formulation includes Eq. (2.8) for stage 1, and the following equation for void-shrinkage stage 2:
rm 2c ¼ ry r 0 ry |ffl{zffl} a
Z
er1
compress
ðeR ; er Þ der 3 e2er 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f II
ð3:10Þ
er0
b
with the flow rule compress f II ð R; r Þ
e e
8 eR þer < ey N ¼ : eR þ er ey
when
eR þ er 6 ey ;
when
eR þ er > ey ;
ð3:11Þ
obtained by following the similar procedure in Section 2.2 and letting dr0 < 0 instead. Fig. 3.10 shows the responses during both expansion and shrinkage processes of voids with different radii. The following trends are observed. (1) Within the neighborhood around Rr00 ¼ 1, the slopes of the expansion curves are larger than the shrinkage counterpart, and this disparity increases with decreasing initial void size R0 . This means that void shrinkage can be implemented under lower stress levels than can void expansion. (2) When voids shrink to some degree, voids can continue to be driven to shrink further under much lower stress levels. (3) Finally, unstable shrinkage happens; that is, voids
J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
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Fig. 3.10. Loading–deformation curves during shrinkage/expansion of voids with different radii.
Fig. 3.11. Comparison of loading–deformation curves with and without surface effects during void shrinkage/expansion.
can shrink spontaneously. To maintain the equilibrium, considerable external tensile stress, rm > 0, should be applied. (4) When R0 ¼ 5 nm, the shrinkage is unstable from the very beginning of the shrinkage process, indicating that the initial configuration, R is an unstable equilibrium state along the direction of void shrinkage. It is notable that such a shrinkage instability cannot be predicted if the surface energy is not considered. As shown in Fig. 3.11, when c ¼ 0, the compressive stress should continue to increase to drive the void to shrink; however, when c ¼ 2 N m1 , the compressive stress first increases until a maximum compressive stress is reached and then begins decreasing until it becomes a tensile stress. The above tension–compression asymmetry can be easily explained as follows (refer to Section 3.2 and Fig. 3.3). In Fig. 3.3, the void expansion corresponds to the unloading–reloading routine, B ! C ! B ! D; whereas the void shrinkage
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J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
corresponds to the routine B ! D, the continuous plastic evolution following the ending state of stage 1. In Eq. (3.10), we 1 have term a > 0, term b > 0. Nevertheless, term a is proportional to r1 0 , say a / r 0 . Therefore, an extremely small r 0 can lead to a b > 0, i.e. tensile hydrostatic stress rm . Such an instability may be one hidden cause for the pop-in events observed in nanoindentation tests where the indenter suddenly enters deeper into the material without any additional force being applied [35]. 4. Conclusions The evolution of nanovoids in an imcompressible plastic material is described analytically with consideration of the surface energy in the framework of continuum plastic mechanics. The method described here to understand the growth mechanism in nanovoids is a two-stage procedure. In stage 1, namely the process from the fictitious stress-free configuration to the initial configuration can produce considerable pre-history plastic flow. For the case of void expansion, such a pre-history deformation causes the response under the external loading become an intrinsic unloading–reloading process. While void shrinkage becomes much easier to happen due to the pre-history plastic deformation and surface energy effects, and shrinkage instability can be expected at least under the current parameter settings and serves as a potential cause for the pop-in phenomenon found in the indentation tests [36]. Nanovoid growth is strongly size-dependent, leading to a necessity to include void sizes in constitutive continuum models. Acknowledgements This work was funded by the Jiangsu University and Jiangsu Provincial Specially-Appointed Professor grants, Jiangsu Science Fund for Youth No. BK20140520, Jiangsu Provincial ‘‘Shuang Chuang’’ grant and the open fund from the state key laboratory of refractories and metallurgy (Wuhan University of Science and Technology). References [1] Z. Suo, Evolving material structures of small feature sizes, Int. J. Solids Struct. 37 (2000) 367–378. [2] B. Lautrup, Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World, CRC, 2009. [3] J. Wang, Z. Huang, H. Duan, S. Yu, X. Feng, G. Wang, W. Zhang, T. Wang, Surface stress effect in mechanics of nanostructured materials, Acta Mech. Solida Sin. 24 (2011) 52–82. [4] N. Cabeo, Ferrara, Philosophia Magnetica (1629). [5] J. Segner, De figuris superficierum fluidarum, Commentarii Societ. Regiae Scientiarum Gottingensis 1 (1751) 301–372. [6] R. Shuttleworth, The surface tension of solids, Proc. Phys. Soc. A 63 (1950) 444. [7] L. Zhou, H. Huang, Are surfaces elastically softer or stiffer?, Appl Phys. Lett. 84 (2004) 1940. [8] Y. Tang, Z. Zheng, M. Xia, Y. Bai, Mechanisms underlying two kinds of surface effects on elastic constants, Acta Mech. Solida Sin. 22 (2009) 605–622. [9] L. Vitos, A. Ruban, H. Skriver, J. Kollar, The surface energy of metals, Surf. Sci. 411 (1998) 186–202. [10] J. Diao, K. Gall, M. Dunn, J. Zimmerman, Atomistic simulations of the yielding of gold nanowires, Acta Mater. 54 (2006) 643–653. [11] X. Chen, H. Ma, L. Liang, Y. Wei, A surface energy model and application to mechanical behavior analysis of single crystals at sub-micron scale, Comput. Mater. Sci. 46 (2009) 723–727. [12] F. Cajori, A History of Mathematics, Chelsea Pub Co., 1991. [13] J. Gibbs, Scientific Papers of J. Willard Gibbs: Thermodynamics, vol. 1, Longmans, Green and Co., 1906. [14] Y. Huang, S. Qu, K. Hwang, M. Li, H. Gao, A conventional theory of mechanism-based strain gradient plasticity, Int. J. Plast. 20 (2004) 753–782. [15] F. Yang, Size-dependent effective modulus of elastic composite materials: spherical nanocavities at dilute concentrations, J. Appl. Phys. 95 (2004) 3516–3520. [16] H. Duan, J. Wang, B. Karihaloo, Z. Huang, Nanoporous materials can be made stiffer than non-porous counterparts by surface modification, Acta Mater. 54 (2006) 2983–2990. [17] L. He, Z. Li, Impact of surface stress on stress concentration, Int. J. Solids Struct. 43 (2006) 6208–6219. [18] G. Wei, S. Yu, G. Huang, Finite element characterization of the size-dependent mechanical behaviour in nanosystems, Nanotechnology 17 (2006) 1118. [19] W. Zhang, T. Wang, X. Chen, Effect of surface/interface stress on the plastic deformation of nanoporous materials and nanocomposites, Int. J. Plast. 26 (2010) 957–975. [20] B. Huo, Q. Zheng, Y. Huang, A note on the effect of surface energy and void size to void growth, Eur. J. Mech. A/Solids 18 (1999) 987–994. [21] H. Gao, Y. Huang, W. Nix, J. Hutchinson, Mechanism-based strain gradient plasticity–I. theory, J. Mech. Phys. Solids 47 (1999) 1239–1263. [22] Y. Huang, H. Gao, W. Nix, J. Hutchinson, Mechanism-based strain gradient plasticity–II. analysis, J. Mech. Phys. Solids 48 (2000) 99–128. [23] Z. Bazˇant, Scaling of dislocation-based strain-gradient plasticity, J. Mech. Phys. Solids 50 (2002) 435–448. [24] Y. Huang, F. Zhang, K. Hwang, W. Nix, G. Pharr, G. Feng, A model of size effects in nano-indentation, J. Mech. Phys. Solids 54 (2006) 1668–1686. [25] S. Brinckmann, T. Siegmund, Y. Huang, A dislocation density based strain gradient model, Int. J. Plast. 22 (2006) 1784–1797. [26] C. Reina, J. Marian, M. Ortiz, Nanovoid nucleation by vacancy aggregation and vacancy-cluster coarsening in high-purity metallic single crystals, Phys. Rev. B 84 (2011) 104117. [27] Z. Huang, J. Wang, A theory of hyperelasticity of multi-phase media with surface/interface energy effect, Acta Mech. 182 (2006) 195–210. [28] J. Mackenzie, R. Shuttleworth, A phenomenological theory of sintering, Proc. Phys. Soc. B 62 (1949) 833. [29] A. Needleman, E. Van der Giessen, Discrete dislocation and continuum descriptions of plastic flow, Mater. Sci. Eng. A-Struct. 309 (2001) 1–13. [30] D. Balint, V. Deshpande, A. Needleman, E. Van der Giessen, Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals, Int. J. Plast. 24 (2008) 2149–2172. [31] H. Gao, Y. Huang, Taylor-based nonlocal theory of plasticity, Int. J. Solids Struct. 38 (2001) 2615–2637. [32] Y. Huang, K. Hu, C. Yeh, N. Li, K. Hwang, A model study of thermal stress-induced voiding in electronic packages, J. Electron. Packag. 118 (1996) 229. [33] A. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I-Yield criteria and flow rules for porous ductile media, J. Eng. Mater-Trans. ASME 99 (1977) 2–15. [34] J. Wen, Y. Huang, K. Hwang, C. Liu, M. Li, The modified Gurson model accounting for the void size effect, Int. J. Plast. 21 (2005) 381–395. [35] R. Nowak, D. Chrobak, S. Nagao, D. Vodnick, M. Berg, A. Tukiainen, M. Pessa, An electric current spike linked to nanoscale plasticity, Nat. Nanotechnol. 4 (2009) 287–291. [36] H. Leipner, D. Lorenz, A. Zeckzer, H. Lei, P. Grau, Nanoindentation pop-in effect in semiconductors, Physica B 308 (2001) 446–449.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Analysis of surface effects on the deformation of a nanovoid in an elasto-plastic material J.X. Liu ⇑ Faculty of Civil Engineering and Mechanics, Jiangsu University, 301 Xuefu Road, Zhenjiang, Jiangsu 212013, China
a r t i c l e
i n f o
Article history: Received 10 May 2014 Received in revised form 8 March 2015 Accepted 1 April 2015 Available online 23 April 2015 Keywords: Nanovoid Surface energy Plastic deformation Size effect Tension–compression asymmetry Pre-history yielding
a b s t r a c t Surface effects, usually embodied as surface tension or surface energy, become considerable in nanostructured materials. With attention to static finite deformations, we present a theoretical analysis about the surface energy effects on the evolution of a nanovoid in a plastic material. The following improvements have been incorporated based on the previous studies: (a) the initial configuration is assumed to be the equilibrium state, which is reached by applying the surface tension onto a fictitious stress-free configuration; (b) both cases of applying compressive and tensile hydrostatic stress is discussed. This illustrative solution provides some reasonable physical interpretations of the following ‘‘unconventional’’ phenomena: (a) when the void radius is smaller than some critical value, finite plastic flow happens in the process from the fictitious stress-free configuration to the initial configuration, which will dramatically influence the subsequent responses; (b) size effects exist, i.e., the growth of nanovoids is closely related to their sizes; (c) the tension–compression asymmetry becomes stronger with decreasing the void size; (d) void shrinkage instability can arise. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Deformations of voids on the nanometric scale are studied with the consideration of surface energy, whose importance has been widely recognized [1,2]. The main characteristic of this study is to consider the prehistory elasto-plastic deformation caused by the surface energy on the void surface, during the investigation of its response under external loadings.
1.1. A brief literature review on surface effects Surface effects in the mechanics of nanostructured elements, including nanoparticles, nanowires, nanobeams, nanofilms and materials containing nanoscale inhomogeneities such as cavities, have attracted considerable research attention, e.g. [1,3]. Here, we focus on the basic concepts of surface tension and composition as related to growth of nanovoids in plastic materials. The first two concepts are surface tension and surface energy. The idea of surface tension is proposed by Cabeo [4] and is stated more explicitly by Segner [5]. Shuttleworth [6] derives the relationship between surface tension and surface energy. We adopt his definitions in this research: ‘‘The surface energy is the work necessary to form unit area of surface by a process ⇑ Tel.: +86 18851280096. E-mail address: [email protected] http://dx.doi.org/10.1016/j.apm.2015.04.007 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.
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of division; the surface tension is the tangential stress (force per unit length) in the surface layer; this stress must be balanced either by external forces or by volume stresses in the body’’. Zhou and Huang [7] and Tang et al. [8] argue that the surface tension depends on the competition between electron redistribution and the lower coordination on surfaces. For simplicity, the surface tension and surface energy are assumed to be constant in the present analysis, and commonly denoted by c, which has values 0–5 N m1 for most metals as shown by Vitos et al. [9]. Another important concept, the atomic structure of the surface, also deserves attention. From an atomistic viewpoint, it is actually a surface layer within a few atomic spacings of the outmost surface atom layer [6]. For example, molecular dynamics (MD) simulations were used by Diao et al. [10] to model the free relaxation process of a gold nanowire from a fictitious unrelaxed configuration to a physically realistic configuration. The outmost two layers of atoms decrease upon relaxation. The formation of such a layer is driven by unequilibrium interatom forces arising when a new surface is made, for instance, by cutting operation. Furthermore, plastic deformation in nanostructured elements under external loadings will generally change the density and number of the atoms at a surface [3]. Processes of surface formation and deformation are complicated and usually associated with atomic rearrangement. Nevertheless, this complexity adds no difficulty to the current study because surface tension (energy) is in principle a continuum-type concept. The value of surface energy is largely dependent on the atomistic spacing in the surface layer. For metals, finite deformations are mainly achieved by dislocations instead of changes in atomistic spacing. Thus, we assume the constant atomistic spacing and therefore the constant surface energy in the finite deformation processes in metals. The last concept is the specific surface area (SSA), namely the ratio of the surface area to the volume of the bulk [11]. The high SSA value of nanostructured materials is generally considered to be the cause for their mechanical behaviors, which can be very different from the behaviors of the macro-bulk. This conclusion is true for nanofilms, nanowires, nanobeams and nanoparticles, but is not true for nanovoids because the SSA value is close enough to zero when a nanovoid is embedded in infinite bulk materials. The unique characteristic of nanovoids is that the bulk is outside of the void surface, unlike in nanoparticles in which the bulk is surrounded by its surface. Instead of revisiting detailed studies on effects of the SSA in the fields of materials science and mechanics [3], here we reflect its association to Torricelli’s trumpet, an ‘‘unthinkable’’ idea first observed by Evangelista Torricelli, a student of Galileo [12]. This trumpet has a finite volume, V ¼ p, but an infinite surface area, S ¼ 2p ln xj1 1 ¼ 1. This can be only mathematically true. Actually, this infinite surface cannot be manufactured in the real physical world. If we consider that everything is composed of atoms, the maximum surface area will reach 10
¼ 20pln 10 if the atom radius is equal to 1010 m. By the way, there is a continuous increase in SSA along S ¼ 2p ln xj10 1 the length of the trumpet and it can be a potential idea for making a functionally graded material. The purpose of mentioning the Torricelli’s trumpet is to show the atomic composition nature of materials, because it becomes crucial for accurate descriptions of material composition especially at the nanometer level. The above reflection on Torricelli’s trumpet indicates that mathematics and physics do not always coincide with each other, and should be a note on the Gibbs idealization [13] describing the surface layer as a ‘‘mathematical surface’’ with a zero thickness, as well as the feasibility of continuum theories, which is questioned by some researchers such as Huang et al. [14]. Although the present study is conducted within the continuum framework, the validity of continuum concepts deserve a great awareness. Nanovoids with the surface effects have been studied intensively in the elastic regime. Yang [15] discusses the effective bulk and shear moduli of heterogeneous materials containing spherical nanovoids at dilute concentrations. Duan et al. [16] discover that nanoporous materials can be made stiffer than non-porous counterparts by surface modification. He and Li [17] examine the effect of surface stress on stress concentration near a spherical void in an elastic medium in the framework of continuum surface elasticity. Unfortunately, these theoretical works are confined to systems of simple geometry, therefore numerical simulation techniques have also been widely applied to more complex geometries. For example, Wei et al. [18] study the size-dependent mechanical properties of nanostructures by developing a kind of surface element to take into account the surface elastic effect. Chen et al. [11] propose a continuum model by introducing the surface energy to the total potential energy, then implement the model using the finite element method. To date, plastic deformation with the inclusion of nanovoids has been studied much less compared with the elastic response, mainly because both surface atom rearrangements and atom exchange can occur during finite plastic deformations. Some efforts in this area have been made within the framework of continuum mechanics. Within the small deformation regime, Zhang et al. [19] study the plastic deformation of nanoporous materials and nanocomposites with consideration of both the surface residual stress and the surface elasticity. Huo et al. [20] assume a constant surface energy and study its effects on the growth of a spherical shell or a thick cylindrical column, concluding that for typical metals, the surface effect is negligible for voids larger than 100 nm, but it becomes significant when the void size is on the order of 10 nm. Considering the size effect due to the plastic strain gradient which becomes important from hundreds of microns to hundreds of nanometers, Gao et al. [21] and Huang et al. [22] propose the mechanism-based strain gradient plasticity and then use this theory to solve the void growth problem. Nevertheless, the strain gradient effect is not possible to increase unlimitedly with decreasing the characteristic size of plastic deformation. For example, Bazˇant [23] addresses that the questionable small-scale asymptotic behavior can exist. That is why the concept of maximum allowable dislocation density has been later mentioned [24,25]. Based on this fact, we are not going to consider the gradient effect because we argue that the surface effect is more dominant than the gradient effect when the void size is smaller than 20 nm. We admit this ignorance makes the result somehow less accurate, but we would say that the bottom line is that the tendencies found in this study is qualitatively
J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
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reasonable. Recently, the same issue is revisited in the appendix of the work by Reina et al. [26], providing a relation between surface energy, external hydrostatic pressure and initial yield strength within the elastic range. 1.2. Challenges Our attempt is to address the following challenges based on the existing investigation of nanovoids. Challenge No. 1: what can happen during the free relaxation process driven by surface tension? Even before any external loading is applied, nanostructured materials have already deformed to balance the surface tension. Therefore, Huang and Wang [27] and Zhang et al. [19] emphasize that it is necessary to construct a nonstress configuration that they call ‘‘a fictitious stress-free configuration’’ based on which stress tensors are calculated within the infinitesimal deformation range. Plastic deformation possibly arises during free relaxation. Some proofs are listed as follows. Firstly, Mackenzie and Shuttleworth [28] study the calculation of the decrease in the radius of a spherical pore, surrounded by a shell of incompressible but shearable material, when a surface-tension induced pressure is applied inside the pore. They solve the viscous dissipation problem by analogy with the theory of plasticity. Diao et al. [10] employ atomistic simulation and explore that a h 1 0 0 i gold nanowire yields under surface stresses when the cross-sectional area is less than 2.45 nm 2.45 nm. Zhang et al. [19] also predict the possible surface-tension-induced yielding using their proposed nonlinear micromechanics theory. Although the pre-existing deformation field is generally considerable, it is extremely complicated to determine it exactly, because it is closely related to manufacturing techniques associated with chemical and/or thermal processes. Therefore, we will adopt the approximating strategy proposed by Huang and Wang [27], who implement the free relaxation process by fixing the material properties and applying surface forces onto the ‘‘fictitious stress-free configuration’’. Challenge No. 2: Is the isotropic continuum theory too simplistic to describe the surface effects?. Surface tension/energy is a continuum-level concept, which can become meaningless with extremely small sizes. Huang et al. [14] argue that discrete dislocation models should be employed rather than the continuum plasticity theory especially at the nanometer level. Needleman and Van der Giessen [29] and Balint et al. [30] address the same concern. Molecular dynamics simulations are an alternative for obtaining accurate results. Another fact is that surface energy depends on the lattice orientation of the surface [9]. Nevertheless, Zhang et al. [19] demonstrate that the isotropic continuum theory still gives plausible predictions if the void radius is on the order of several nanometers and above. We emphasize that our current analysis cannot be extended to too small a size without jeopardizing the rationality of continuum theory. 5 nm is the minimum void radius in Huo et al. [20], and that in Zhang et al. [19] is 2 nm. In the present study, results will be provided when the void radius is no less than 2 nm. 1.3. Present purpose We investigate surface effects on the plastic deformation of a spherical void in a plastic medium. As an effort to make a further step forward based on previous investigations, the following factors are considered simultaneously. 1. Pre-history yielding. 2. Finite deformation. 3. Nanovoid shrinkage and expansion. When compared with the work of Huo et al. [20], we further consider the first one of above factors and the shrinkage response. In consequence, we point out that the void expanding process under external loading is intrinsically an unloading-loading process, as well as tension–compression asymmetry and the shrinkage instability. While the research in Zhang et al. [19] is within the small deformation framework, we show that plastic deformations due to surface effects can be considerably large at least under the current material parameter settings. The effect of surface elasticity is neglected in this study. The same simplification is adopted by Huo et al. [20]. Zhang et al. [19] include such an effect into their micromechanical model, but conclude that the surface elasticity effect is weak. 1.4. The paper structure The remainder of this paper is organized as follows. In Section 2, the theoretical formulation is presented within the continuum plasticity framework for both the free relaxation process and the subsequent process under external hydrostatic pressure. These solutions are analyzed using a series of illustrative diagrams in Section 3. The paper concludes in Section 4. 2. Finite plastic deformation of a nanovoid in a plastic material Consider an isotropic, homogeneous and incompressible elasto-plastic shell that sustains uniform hydrostatic tensile traction, rm . As shown in Fig. 2.1, the initial inner radius is R0 , while the outer radius is R1 . This problem has a physical
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Fig. 2.1. A spherical void in a finite sphere with R0 and R1 the inner and outer radii in the fictitious stress-free configuration R, and R0 and R1 are the radii in the initial configuration R, while r 0 and r 1 are the radii in the external load-driven deformed configuration r.
background for studying the void growth [20]. The surface tension on the outer shell surface is zero because it is not a true material surface but an imaginary one in the original problem, as pointed out by Huo et al. [20]. If the surface effect is considered, any physical experiment is conducted on specimens that are initially stressed, say, due to the self-driven reconstruction in order to form the surfaces. Therefore, as shown in Fig. 2.1, there arise three configurations: the fictitious stress-free configuration, R, the observed initial configuration, R, and the final configuration, r, under external loadings and surface effects. Correspondingly, this study divides the whole deformation process into two stages: stage 1 and stage 2 (refer to Fig. 2.1). In this section, we consider the cases in which the void expands during stage 2, say, r > R. Cases of void shrinkage (r < R) are studied in Section 3. 2.1. Stage 1: from a fictitious stress-free state, R, to the initial configuration, R During this stage, the loading is related to the surface energy. That is, the traction on the ‘‘bulk surface’’ attached to the void surface is given by
Ps ¼
2c : R0
ð2:1Þ
Even though later the void surface-related virtual work is calculated directly via c instead of P s , Eq. (2.1) provides a more descriptive picture of how the void is driven to shrink. For simplicity, we assume the Poisson’s ratio is equal to 0.5. Such an incompressibility condition can be expressed in the form
R3 R30 ¼ R3 R30 ;
ð2:2Þ
where the spherical surface with radius R in the fictitious configuration can be mapped to the surface with radius R in the initial configuration, as shown in Fig. 2.1. Eq. (2.2) immediately gives the radial and tangential logarithmic strain components, eR and eT , as follows
eR ¼ 2eT ¼ 2 ln
R ; R
ð2:3Þ
where subscripts ‘‘R’’ and ‘‘T’’ mean variables during stage 1, while subscripts ‘‘r’’ and ‘‘t’’ represent variables during stage 2 as will be shown in Section 2.2. In this study, we have adopted the logarithmic strain and the corresponding work-conjugate as the strain and stress measurements, therefore, the present formulation is robust for finite deformations. To set up the equilibrium equation according to the virtual work principle, assume an infinitesimal variation:
dR0 < 0:
ð2:4Þ
The positive work related to the surface shrinkage is assumed to be completely transferred into the deformation energy of the material, leading to
Ps 4pR20 dR0 ¼
Z
R1
ðrR deR þ 2rT deT Þ4pR2 dR ¼
R0
where Ps is given by Eq. (2.1).
Z
R1
R0
ðrR rT ÞdeR 4pR2 dR;
ð2:5Þ
J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
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Eqs. (2.2) and (2.3) provide the above related variational quantities in the form
deR ¼ 2deT ¼ 2
R20 R3
ð2:6Þ
dR0 :
The Von Mises effective strain and stress are, respectively, written as
re ¼ jrR rT j ¼ rR rT ; ee ¼ jeR j ¼ eR :
ð2:7Þ
Finally, Eq. 2.5 becomes
2c ¼ R0 ry where
Z
eR1
eR0
eR0 ¼ 2 ln
e
f Ið RÞ 32eR
e
1
R0 R0
,
8 eR < ey f I ðeR Þ ¼ N : eR ey
deR ;
eR1 ¼ 2 ln
ð2:8Þ R1 R1
when
eR 6 ey ;
when
eR > ey ;
, and the flow rule f I ðeR Þ is written in the form
ð2:9Þ
where N is the hardening exponent. Because now the Von Mises effective strain is equal to eR , as shown in Eq. (2.7), the flow rule in Eq. (2.9) is exactly the elastic-power-law-hardening uniaxial stress–strain relation, which has been used widely, e.g. [31]. Now, Eq. (2.8) can be used to determine the void radius, R0 , in the fictitious stress-free configuration, R. Additionally, the elastic part among the total effective strain, eR , can be written as
eelast ¼ ey f I ðeR Þ; R
ð2:10Þ
which will be used in Section 2.2. 2.2. Stage 2: from the initial configuration, R, to the final deformed configuration, r The external loading, rm , is applied on the surface with radius R1 , say the outer surface of the spherical shell. Again, note that in this section only void expansion is considered, while void shrinkage under external loadings will be discussed in Section 3. For void expansion, we assume a variation
dr 0 > 0:
ð2:11Þ
Then, again according to the impressibility requirement, we have
r21 dr 1 ¼ r 20 dr0 :
ð2:12Þ
For clarity, again, subscripts ‘‘R’’ and ‘‘T’’ are for stage 1, while ‘‘r’’ and ‘‘t’’ are for stage 2. For example, eR is the radiusdirection strain during stage 1, while er is that during stage 2. It is worth noting that the variational quantities der and det will not change no matter which configuration is chosen as the reference configuration, R or R, because of the following fact
r r R r ¼ d 2 ln ; der ¼ d 2 ln ¼ d 2 ln 2 ln R R R R
ð2:13Þ
where d 2 ln Rr refers to configuration R, while d 2 ln Rr refers to R. Similarly, for the tangential component of the logarithmic strain, et , there is
r r det ¼ d ln ¼ d ln : R R
ð2:14Þ
The final expression takes the form
der ¼ 2det ¼ 2
r20 dr 0 < 0: r3
ð2:15Þ
According to the virtual work principle, there is
4prm r 21 dr 1 ¼
Z
r1
r0
ðrr rt Þder 4pr 2 dr þ 4pcdr 20 ;
ð2:16Þ
where ðrr rt Þder in the integrand of the integral on the right side denotes the variation in the strain energy density. There is a significant difference between the current study and the work by Huo et al. [20]. Huo et al. [20] say that ðrr rt Þ is negative during stage 2. Here, ðrr rt Þ is positive at the early period of stage 2, then becomes negative with
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the further expansion of the void. This is easy to understand: During stage 1, ðrR rT Þ is positive as given in Eq. (2.7). Because the stress state should evolve continuously, there should be a transition period, say from positive ðrR rT Þ to negative ðrr rt Þ. In Section 3.2, this is explained thoroughly as an intrinsic unloading–reloading process. ðrr rt Þ and the flow rule are related to each other in the form
rr rt ¼ sign eelast þ er f II ðeR ; er Þ; R ry where
ð2:17Þ
eelast is given by Eq. (2.10), and the flow rule f II ðeR ; er Þ is written as R 8 f I ðeR Þ þ eeyr > > > < er f II ðeR ; er Þ ¼ ey f I ðeR Þ > > > : eR er 2f ðe Þ N I R ey
when
when
er 2 eelast ;0 ; R when er 2 eelast max eelast ; ey ; eelast ; R R R
ð2:18Þ
er 2 1; eelast max eelast ; ey : R R
The Von Mises effective strain has to be calculated according to its original definition
(R
ee ¼
R
stage 1
jdeR j
stage 1
jdeR j þ
R R
stage 2
jder j
stage 2
jder j 2eelast R
er 2 eelast ;0 ; R when er 2 1; eelast : R
when
ð2:19Þ
With the help of Eqs. (2.3) and (2.15), we rewrite Eq. (2.19) as
(
2 ln Rr
ee ¼
er 2 eelast ;0 ; R when er 2 1; eelast : R when
2 ln rR 2eelast R R2
ð2:20Þ
Finally, Eq. (2.16) becomes
rm 2c ¼ ry r 0 ry where
er0 ¼ 2 ln
Z
er1
sign
er0
r0 R0
,
eelast þ er R
er1 ¼ 2 ln
f II ðeR ; er Þ 3
e2er 1
der ;
ð2:21Þ
r1 . R1
In order to compare Eq. (2.21) with previous studies, we provide the formula as follows: (1) Eq. (2.21) reduces to the solution from [32]:
rm ¼ ry
Z
er1
er0
f II ð0; er Þ 3
e2er 1
der
ð2:22Þ
by letting c ¼ 0 and eR ¼ 0. Here c ¼ 0 means that the surface energy is eliminated, while configuration transfer is not considered. (2) Eq. (2.21) reduces to the solution from [20]:
rm 2c ¼ þ ry r 0 ry by letting
Z
er1
er0
f II ð0; er Þ 3
e2er 1
eR ¼ 0 suggests that the R ! R
der
ð2:23Þ
eR ¼ 0.
3. Discussion and implications In the following analyses, the material parameters are set with the ranges shown in Table 1, where the magnitude of the surface energy is within the range provided by Vitos et al. [9] for metals. 3.1. Pre-history plastic deformation During stage 1, with the help of Eq. (2.8), the surface tension can produce considerable plastic flow measured by the Von Mises effective strain, ee ¼ eR . For instance, before any external loading is applied, under the parameter setting in Fig. 3.1, : : near the void surface, the Von Mises effective plastic strain is ee ¼15ey when the surface energy is c ¼ 1:0Nm1 ; ee ¼82ey when : the surface energy is c ¼ 1:5 N m1 ; ee ¼586ey when the surface energy is c ¼ 2:0 N m1 .
Table 1 Ranges of material properties. Initial void radius R0 (nm) Shell outer radius R1 (R0 ) Initial yield stress ry (MPa)
5–106 3–1 25–200
Hardening exponent N
0–0.5
Surface energy c (N m1) Initial yield strain ey
0.5–5.0 0.001–0.02
J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
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Fig. 3.1. Distribution of the dimensionless Von Mises effective strain, eeye , in relation to the radius-direction position, R, under different surface energies, c.
In Eq. (2.8), by letting eR0 ¼ ey , a critical void radius Rcrit 0 beyond which no plastic deformation will be caused by the surface tension can be expressed in the form
Rcrit 0 ¼
ry
R ecrit R1 ey
2c f I ð eR Þ 3e e 2 R 1
de
ð3:1Þ
;
where crit R1
e
2 x ¼ ln 3ey 3 x1þe2
! ð3:2Þ
R3
with x ¼ R13 . 0
crit crit R1 1 From Eq. (3.1), Rcrit 0 / c, R0 / ry . Fig. 3.2 shows the change of R0 along with the initial yield strain, ey . For every fixed R0 , crit Rcrit 0 decreases slightly with increasing ey , suggesting that the dependence of R0 on ey is almost negligible. On the other hand, R1 R1 R1 R1 when ey is fixed, Rcrit 0 decreases quickly with increasing R0 when R0 < 5, but it tends to become independent of R0 when R0 > 10.
3.2. The intrinsic unloading–reloading process There is a sharp transition in the stress state between stage 1 and stage 2, as briefly mentioned in SubSection 2.2. At the end of stage 1, there is ðrR rT Þ > 0; but in the late period of stage 2, ðrr rt Þ < 0. Therefore, there must be a continuous process implementing the change from ðrR rT Þ > 0 to ðrr rt Þ < 0, driven by the increasing rm . It is more convenient to explore this process in the space constructed by the Von Mises effective strain, ee , and stress, re , as shown in Fig. 3.3, which gives the re ee diagram for the material point at R ¼ 1:5R0 . Stage 1 corresponds to the O ! A ! B process. Point B is the end of stage 1, denoting the equilibrium state under the action of c. Then, the external hydrostatic stress, rm , is applied increasingly from zero, leading to a decrease in rr and an increase in rt during the B ! C ! B process. Briefly, it can be summarized as
0 6 re
8 > < rr rt ¼ 0 > : rt rr
during B ! C; at point C; during C ! B:
ð3:3Þ
Obviously, the above B ! C ! B process is just an unloading–reloading process in the re ee space. Eqs. (2.21) and (2.18) are valid for the process B ! C ! B ! D.
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Fig. 3.2. The critical void radius below which plastic flow arises in stage 1, Rcrit in relation to the initial yielding strain, ey , under changing geometrical ratio 0 R1 . R0
Fig. 3.3. Curve of the dimensionless effective stress in relation to the strain, for the material point at R ¼ 1:5 R0 in the initial configuration R.
With consideration of stage 1 and therefore the above unloading–reloading process, the loading–deformation curve is shown in Fig. 3.4 compared with the result by Huo et al. [20], say, Eq. (2.23). The following differences are seen: (1) when : the deformation indicator is Rr00 ¼ 1, the present theory, namely Eq. (2.21), gives rm ¼ 0, but (2.23) predicts rm ¼5:33ry . Eq. (2.23) fails to satisfy the condition, say rm ¼ 0 when Rr00 ¼ 1, because it is established based on an unequilibrated initial configuration. This is caused by neglecting stage 1. (2) Eq. (2.21) predicts a peak value of rm , which is much higher than that from Eq. (2.23). Pre-history plastic deformation leads to considerable hardening, which is an important contributing factor.
J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
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3.3. Influences of the void size R0 and the geometrical ratio R1 =R0 We can see a considerable size effect, i.e. the dependence on the void radius, R0 , when the geometrical ratio, RR10 , is fixed, as shown in Figs. 3.5 and 3.6. In Fig. 3.5, the curve with R0 ¼ 106 nm represents the result with a negligible surface effect. Within the range R0 2 ð1; 100Þ nm, the loading–deformation response has a strong dependence on R0 . Namely, the maximum external loading, max ðrm Þ, decreases quickly with increasing R0 . Fig. 3.6 should be understood by referring to Eqs. (2.19) and (2.20). In other words, the differences among curves in Fig. 3.6 are related to eR , i.e., the pre-history deformation. The curve corresponding to R0 ¼ 5 nm is much higher than others because its pre-history plastic flow is several hundred times of ey and the strongest. By combining Figs. 3.5 and 3.6, for the cases of R0 ¼ 10 nm and R0 ¼ 20 nm, the difference in rm is still considerable. This difference is mainly due to the difference in the surface energy itself – the term r02rcy in Eq. (2.21).
Fig. 3.4. Comparison of the loading–deformation responses with and without the consideration of the pre-history plastic flow.
Fig. 3.5. The size effect caused by the change in the void radius, R0 : loading–deformation curves.
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We have also studied the dependence on the ratio increasing
R1 , R0
the deformation
r0 R0
R1 R0
when R0 is fixed, as shown in Figs. 3.7 and 3.8. From Fig. 3.7, with
corresponding to the peak load increases, while the post-peak drop becomes slower. From
Fig. 3.8, the Von Mises effective strain is the highest with the minimum value of R1 R0
R1 . R0
One issue deserves pointing out: when
¼ 20, among the spherical shell, only a small region around the void has entered the plastic state, as shown in Fig. 3.9: in
the region 7R0 < R < 20R0 , there is ee < ey , meaning that only elastic deformation happens. Therefore, the consideration of the elastic effective strain is very important. Huang et al. [32] and Gao and Huang [31] also emphasize that elastic deformation is essential in the analysis of cavitation instability.
Fig. 3.6. The size effect caused by the change in the void radius, R0 : distributions of the dimensionless effective strain when the deformation reaches r0 ¼ 1:5. R0
Fig. 3.7. The size effect caused by the change in the geometrical ratio,
R1 : R0
loading–deformation curves.
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Fig. 3.8. The size effect caused by the change in the geometrical ratio, RR10 : distributions of the dimensionless effective strain when the deformation reaches r0 ¼ 1:5. R0
Fig. 3.9. Distribution of the dimensionless effective strain in the specimen with
R1 R0
¼ 20 when the deformation reaches
r0 R0
¼ 1:5.
3.4. Size-dependent growth of nanovoids ð1Þ
ð2Þ
For demonstration purposes, consider two spherical shells with different void radii respectively R0 and R0 . These two specimens are under the same remote hydrostatic tensile stress, rm . For stage 1,
2c ðiÞ
R0
ry
¼
Z e
e
R
ðiÞ 1
fI 3
R
ðiÞ 0
eRðiÞ
e2eRðiÞ 1
deRðiÞ
ð3:4Þ
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and for stage 2,
Z
rmðiÞ 2c ¼ ry r0ðiÞ ry
e ðiÞ r
1
sign
eelast þ eðiÞ r RðiÞ
e ðiÞ r
f e
e
RðiÞ ; r ðiÞ
II
3
e2erðiÞ 1
derðiÞ ;
ð3:5Þ
0
where the superscript ‘‘ðiÞ’’ means the size number, say, i ¼ 1; 2. Eqs. (3.4) and (3.5) imply that when erð1Þ ¼ erð2Þ , say 0
ð1Þ m
ð2Þ m
ð1Þ R0
0
ð1Þ
r0
ð1Þ
R0
ð2Þ
¼
r0
ð2Þ
R0
,
ð2Þ R0 .
the external loadings are generally different, namely r – r because – In other words, the growth of nanovoids is intrinsically size-dependent. Zhang et al. [19] also make this observation but without explanation. The size dependence destroys the shell approximation for classic porous plasticity such as the Gursontype model [33]. The shell approximation is valid when we adopt the following formula to describe the void growth:
r1 ¼ ry
Z
0
f II ð0; erðiÞ Þ 3
e2erðiÞ 1
e ðiÞ r
derðiÞ :
ð3:6Þ
0
When void 1 and void 2 are under the same external stress
Z
0
e ð1Þ r
f II ð0; erð1Þ Þ 3
e2erð1Þ 1
derð1Þ ¼
Z
0
e ð2Þ r
0
f II ð0; erð2Þ Þ 3
e2erð2Þ 1
derð2Þ ;
r1 , we have ð3:7Þ
0
immediately leading to
erð1Þ ¼ erð2Þ : 0
ð3:8Þ
0
With the help of the definition ð1Þ
r0
ð1Þ R0
er ¼ 2 ln
r , we obtain R
ð2Þ
¼
r0
ð2Þ
R0
ð3:9Þ
:
The above Eqs. (3.6)–(3.9) can be taken as the theoretical background for the shell approximation. However, the current theory, Eqs. (3.4) and (3.5), does not satisfy Eq. (3.9). The surface effect is stronger for smaller voids. The surface related terms in both Eqs. (2.8) and (2.21) are inversely proportional to void radius, leading to the inclusion of the void size itself instead of dimensionless Rr00 into the constitutive response. The pre-plastic-hardening is also larger. Nevertheless, the Gurson model can still be used for some particular cases. For example, when the uni-sized nanovoids are periodically distributed in the matrix, we can choose one nanovoid as a representative cell, and reformulate the Gurson model based on the present theoretical derivation. Actually, this strategy has already been adopted by some investigators, e.g. [34]. But, for nanoporous media with a considerable scattering in nanovoid sizes, we have not been able to see an easy way to extend the Gurson model. 3.5. Tension–compression asymmetry The present model can predict the asymmetric properties of tension and compression. Such an asymmetry has been discussed by many other investigators, for example, [19]. However, in the following the instability in void shrinkage is firstly explored. For tension, the complete description is composed of Eq. (2.8) for stage 1 and Eq. (2.21) for the tensile stage 2. For compression, the formulation includes Eq. (2.8) for stage 1, and the following equation for void-shrinkage stage 2:
rm 2c ¼ ry r 0 ry |ffl{zffl} a
Z
er1
compress
ðeR ; er Þ der 3 e2er 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f II
ð3:10Þ
er0
b
with the flow rule compress f II ð R; r Þ
e e
8 eR þer < ey N ¼ : eR þ er ey
when
eR þ er 6 ey ;
when
eR þ er > ey ;
ð3:11Þ
obtained by following the similar procedure in Section 2.2 and letting dr0 < 0 instead. Fig. 3.10 shows the responses during both expansion and shrinkage processes of voids with different radii. The following trends are observed. (1) Within the neighborhood around Rr00 ¼ 1, the slopes of the expansion curves are larger than the shrinkage counterpart, and this disparity increases with decreasing initial void size R0 . This means that void shrinkage can be implemented under lower stress levels than can void expansion. (2) When voids shrink to some degree, voids can continue to be driven to shrink further under much lower stress levels. (3) Finally, unstable shrinkage happens; that is, voids
J.X. Liu / Applied Mathematical Modelling 39 (2015) 5091–5104
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Fig. 3.10. Loading–deformation curves during shrinkage/expansion of voids with different radii.
Fig. 3.11. Comparison of loading–deformation curves with and without surface effects during void shrinkage/expansion.
can shrink spontaneously. To maintain the equilibrium, considerable external tensile stress, rm > 0, should be applied. (4) When R0 ¼ 5 nm, the shrinkage is unstable from the very beginning of the shrinkage process, indicating that the initial configuration, R is an unstable equilibrium state along the direction of void shrinkage. It is notable that such a shrinkage instability cannot be predicted if the surface energy is not considered. As shown in Fig. 3.11, when c ¼ 0, the compressive stress should continue to increase to drive the void to shrink; however, when c ¼ 2 N m1 , the compressive stress first increases until a maximum compressive stress is reached and then begins decreasing until it becomes a tensile stress. The above tension–compression asymmetry can be easily explained as follows (refer to Section 3.2 and Fig. 3.3). In Fig. 3.3, the void expansion corresponds to the unloading–reloading routine, B ! C ! B ! D; whereas the void shrinkage
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corresponds to the routine B ! D, the continuous plastic evolution following the ending state of stage 1. In Eq. (3.10), we 1 have term a > 0, term b > 0. Nevertheless, term a is proportional to r1 0 , say a / r 0 . Therefore, an extremely small r 0 can lead to a b > 0, i.e. tensile hydrostatic stress rm . Such an instability may be one hidden cause for the pop-in events observed in nanoindentation tests where the indenter suddenly enters deeper into the material without any additional force being applied [35]. 4. Conclusions The evolution of nanovoids in an imcompressible plastic material is described analytically with consideration of the surface energy in the framework of continuum plastic mechanics. The method described here to understand the growth mechanism in nanovoids is a two-stage procedure. In stage 1, namely the process from the fictitious stress-free configuration to the initial configuration can produce considerable pre-history plastic flow. For the case of void expansion, such a pre-history deformation causes the response under the external loading become an intrinsic unloading–reloading process. While void shrinkage becomes much easier to happen due to the pre-history plastic deformation and surface energy effects, and shrinkage instability can be expected at least under the current parameter settings and serves as a potential cause for the pop-in phenomenon found in the indentation tests [36]. Nanovoid growth is strongly size-dependent, leading to a necessity to include void sizes in constitutive continuum models. Acknowledgements This work was funded by the Jiangsu University and Jiangsu Provincial Specially-Appointed Professor grants, Jiangsu Science Fund for Youth No. BK20140520, Jiangsu Provincial ‘‘Shuang Chuang’’ grant and the open fund from the state key laboratory of refractories and metallurgy (Wuhan University of Science and Technology). References [1] Z. 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