Journal of the Mechanics and Physics of Solids 52 (2004) 2587 – 2616

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Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals Y.J. Wei, L. Anand∗ Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 1-310, Cambridge, MA 02139 4307, USA Received 16 September 2003; accepted 5 April 2004

Abstract In order to model the e4ects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic–plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations re7ect the macroscopic experimentally observed tensile stress–strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress–strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress–strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension. ? 2004 Elsevier Ltd. All rights reserved. Keywords: A. Crystal plasticity; B. Interface failure; C. Finite elements

∗

Corresponding author. Tel.: +1-617-253-1635; fax: +1-617-258-8742. E-mail address: [email protected] (L. Anand).

0022-5096/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2004.04.006

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1. Introduction It is well known that in polycrystalline metals, a substantial increase in strength and hardness can be obtained by reducing the grain size to the nanometer scale (cf., e.g., Gleiter, 1989; Suryanarayana, 1995; McFadden et al., 1999; Jeong et al., 2001; Shuh et al., 2002; Lu et al., 2000). These attributes have generated considerable interest in the use of nanocrystalline (nc) metallic materials (grain sizes less than ≈100 nm), for a wide variety of structural applications. The hardness, sti4ness, strength and ductility of some nc-fcc metals (e.g., Cu, Ni), as measured by microindentation and simple tension experiments, have been reported in the recent literature (e.g., Nieman et al., 1991; Sanders et al., 1997; Ebrahami et al., 1998, 1999; Legros et al., 2000; Lu et al., 2001; Torre et al., 2002). Typically, relative to their microcrystalline counterparts, nanocrystalline metals exhibit a high yield strength and a high strain-hardening rate leading to a very high tensile strength, but at the expense of a much reduced tensile ductility. The limited ductility is of major concern. For example, while the ultimate tensile strength levels approach ≈1500 MPa in electro deposited nanocrystalline Ni, the ductility that can be obtained in this material is generally low and usually does not exceed ≈3–5%, Torre et al. (2002). The reduced ductility at room temperature is observed in many, if not most, of the nanocrystalline metals synthesised using existing methods, which include (a) gas-phase condensation of particulates and consolidation, (b) mechanical alloying and compaction, (c) severe plastic deformation, and (d) electrodeposition. The reduced ductility is believed to be mainly due to pre-existing 7aws, such as impurities and porosity introduced during the processing of nanocrystalline materials. Nanocrystalline metals produced by electrodeposition methods are expected to have a fully-dense as-deposited structure, but as mentioned above, this material also shows a limited ductility (Torre et al., 2002). Kumar et al. (2003) have recently reported on an extensive set of experiments that they conducted to observe the deformation mechanisms in electro-deposited nanocrystalline Ni both after and during deformation using transmission electron microscopy (TEM). Brie7y, they concluded • That the grain interiors of their as-received electro-deposited nanocrystalline Ni (average grain size of 30 nm) were clean and (almost) devoid of dislocations, and that the grain boundaries did not contain any amorphous layers or second-phase particles, or nano-scale voids; Fig. 1. • Observations of specimens which were Frst deformed by compression, rolling or nano-indentation, and then examined by TEM indicated that while isolated dislocations, infrequent dislocation networks, and some dislocation debris were seen within grains, the density of dislocations visible in the specimens after the deformation could not account for the macroscopically imposed plastic strains; Fig. 2 shows the microstructure of nc-Ni after 4% strain in compression. Of course, post-mortem TEM observations do not tell us anything about how many dislocations have passed and subsequently been annihilated (at grain-boundaries, opposite signed dislocations, etc.) in the specimen. • Observations of tensile specimens which contained perforations in the center of their gage sections and which were incrementally strained in discrete steps in-situ in a

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Fig. 1. As-received electrodeposited nc-Ni: (a) Bright-Feld TEM image showing nanocrystalline grain structure. (b) Image showing a clean atomically faceted grain boundary (K.S. Kumar, personal communication).

transmission electron microscope, provided evidence of nucleation and growth of grain-boundary cracks and triple junction voids ahead of a growing crack (which had nucleated from the perforation), as well as substantial dislocation activity in the interiors of the grains adjacent to the crack front; Fig. 3. That is, the nucleation and growth of grain-boundary slip and separation, together with dislocation-based plasticity in the interior of grains adjacent to the propagating crack, are the dominant mechanisms of inelastic deformation in the “process-zone” associated with the tip of a crack in nanocrystalline Ni. As a caution, it is worth noting that in-situ deformation of a perforated thin foil specimen in a TEM is not always representative of mechanical response of a bulk specimen. Atomistic computer simulations of nanocrystalline metals (particularly for Ni and Cu) have also been widely reported in the recent literature (cf., e.g., El-Sherik and Erb, 1995; Schiotz et al., 1998, 1999; Swygenhoven et al., 1998, 1999a,b, 2002; Yamakov et al., 2001; Derlet and Swygenhoven, 2002; Swygenhoven, 2004). These atomistic simulations collectively show that as the grain size decreases, intragranular plasticity driven by dislocation mechanisms becomes more di
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Fig. 2. Microstructure of nc-Ni after 4% plastic strain in compression: (a) Bright-Feld TEM image showing a few dislocations within a grain. (b) A possible crack at a grain-boundary triple junction. (c) Substantially dislocation-free grains and possibly a low-angle grain boundary. (d) Clean grain boundaries with no evidence of residual dislocation debris left after the deformation (K.S. Kumar, personal communication).

inelastic deformation in the grain interiors of nanocrystalline materials. Again, we note that since atomistic calculations involve only a limited number of grains and nearly instantaneous (in a matter of picoseconds) loading, they only provide a qualitative understanding of the real mechanical response of materials at laboratory length and time-scales. Thus, from the physical experiments and atomistic simulations reported in the literature, it is clear that grain-boundary-related slip and separation phenomena begin to play an important role in the overall inelastic response of a polycrystalline material when the grain-size decreases to diameters under ≈100 nm, and dislocation activity within the grain interiors becomes more diKcult. To the best of our knowledge, the only numerical modeling study that includes grain boundary phenomena at the continuum level (that is, not molecular dynamics)

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Fig. 3. Microstructure of a perforated nc-Ni tensile specimen deformed in situ in an electron microscope: (a) A bright Feld image of an advancing crack. The image shows a saw-tooth crack which runs along grain boundaries, and also often across grains that have necked-down plastically. (b) A pair of “freeze-frame” images showing the microstructural evolution and progression of damage with increases in the applied displacement pulses; grain-boundary cracks and triple junction voids are indicated by arrows. These cracks and voids have grown after eight pulses. Subsequent images (not shown here) also indicate dislocation emission from the tip of crack B (K.S. Kumar, personal communication).

is that by Fu et al. (2001). These authors idealized a polycrystalline material as a two-dimensional composite of grain interiors and grain boundary layers of various thicknesses. The grain interiors were modeled using a limited form of crystal plasticity, while the grain boundary regions were modeled using an isotropic plasticity model with a more rapidly strain-hardening rate. By varying the volume fraction of grain boundaries layers with respect to that of grain interiors, they attempted to capture the e4ect of increased strength as the grain size decreases in the nanocrystalline range. In view of the relatively small regions of grain-boundary disorder in actual nanocrystalline materials, Fig. 1, the large volume fractions of grain-boundary regions assumed by Fu et al. (2001) (their Fig. 7) appear to be unrealistic. Another possible computationally tractable modeling approach to account for the combined e4ects of grain-boundary-related deformation as well as plasticity within the grains, is to couple a single-crystal plasticity constitutive model for the grain interior, with an appropriate cohesive interface constitutive model to account for grain-boundary

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sliding and separation phenomena; it is this latter approach that we shall develop in this paper. At the outset we acknowledge that a standard crystal plasticity model for the grain-interior deformation is inadequate to represent the limited amount of inelastic deformation due to emission and eventual absorption of a few partial dislocations from grain boundaries in nanocrystalline materials. However, since elastic anisotropy and crystallographic texture e4ects are still important in these materials, and since the dislocation partials are still expected to move on slip systems, the mathematical structure of a continuum crystal plasticity theory is still useful as an indicator of inelasticity within the grains. It will of course fall short in being able to accurately represent the actual details of the discrete intra-granular dislocation response of such materials. A Coble-type grain boundary di4usion creep mechanism, together with grain boundary sliding as a concurrent accommodation mechanism for the grain boundary di4usion, has been considered to be the major deformation mechanism in nanocrystalline materials by many authors. Although di4usion-controlled processes are typically activated only at high homologous temperatures, Yamakov et al. (2002) have argued that Coble-creep, which exhibits a linear dependence of the inelastic strain rate on the stress (that is a strain-rate sensitivity of one), should also be the dominating grain-boundary mechanism in nanocrystalline materials even at low homologous temperatures. It is important to note that Yamakov et al. (2002) base their conjecture on their molecular dynamics simulations in which the lowest strain rates accessible in their numerical simulations are of the order of 107 =s. However, the physical experiments of Torre et al. (2002) at low homologous temperatures in the quasi-static strain rate range of 5:5 × 10−5 –5:5 × 10−2 =s, exhibit only a very slight strain rate sensitive response for nanocrystalline nickel, their Fig. 9. While a rate-dependent constitutive equation for grain-boundary shearing rates, di4erent from that given by the Coble-creep, may be constructed, and indeed one such equation based on thermally activated shear transformation rates in grain-boundary amorphous regions has been suggested by Conrad and Narayan (2000), in the study presented here we shall approximate the grain-boundary, as well as the grain-interior mechanisms at low homologous temperatures and quasi-static strain rates to be rate-independent. We shall report on a rate-dependent model for the grain-boundary response in a future paper. Accordingly, the purpose of this paper is (a) to develop an isothermal, rateindependent elastic–plastic interface model which accounts for both reversible elastic, as well as irreversible inelastic separation-sliding deformations at the interface prior to failure; (b) to couple the interface model with an isothermal, rate-independent crystal-plasticity model for the grain interior, and (c) to use the new computational capability to study the deformation and fracture response of an aggregate of crystals. Such a computational capability is used to study the deformation and fracture response of nanocrystalline materials whose individual crystals lie in the 6 100 nm size range. Since our model will not contain an inherent length scale, it may of course also be used for polycrystals with grains in any other larger size range. The plan of this paper is as follows. In Section 2, we develop our rate-independent interface constitutive model. In Section 3, we summarize the crystal-plasticity model that we shall use in this paper. In Section 4, we Frst estimate the material parameters in the model for nanocrystalline Ni from the limited stress–strain curves available in the

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literature, and then show some representative simulations of the deformation and damage evolution in a quasi-three-dimensional columnar-grained textured polycrystalline aggregates. The e4ects of a variation of the grain-size on the macroscopic stress– strain curve, as predicted by the continuum model, are discussed in Section 5. Results from a numerical experiment showing the development of fracture in a notched nc-Ni specimen are shown in Section 6. We close in Section 7 with some Fnal remarks. 2. Interface constitutive model Cohesive interface modeling of fracture started more than 40 years ago with the work of Barenblatt (1959) and Dugdale (1960). In recent years, cohesive interface models have been widely used to numerically simulate fracture initiation and growth by the Fnite element method (e.g., Needleman, 1990; Xu and Needleman, 1994; Camacho and Ortiz, 1996), for a recent review see Hutchinson and Evans (2000). Typically, a cohesive interface is introduced in a Fnite element discretization of the problem by the use of special interface elements which obey a non-linear interface traction-separation constitutive relation which provides a phenomenological description for the complex microscopic processes that lead to the formation of new traction-free crack faces. The loss of cohesion, and thus of crack nucleation and extension, occurs by the progressive decay of interface tractions. The interface traction-separation relation usually includes a cohesive strength and cohesive work-to-fracture. Once the local strength and work-to-fracture criteria across an interface are met, decohesion occurs naturally across the interface, and traction-free cracks form and propagate along element boundaries. Although substantial progress has been made in recent years, there are still several key issues that need to be addressed in the modeling of cohesive interfaces. SpeciFcally: (i) Most previous interface models are of the non-linear reversible elastic type. For example, in Xu and Needleman (1994), with T denoting the displacement jump across the cohesive surface, and t the work-conjugate traction vector on the surface, the constitutive equation for the cohesive surfaces is expressed in terms of a potential function ’, such that the traction t = @’=@T. The speciFc form for ’ chosen by Xu and Needleman (1994) is history- and rate-independent, and this leads to a traction-displacement relation which is fully reversible. There exists a need to develop interface constitutive models which allow for inelasticity at the interface prior to failure, much in the spirit of elastic–plastic constitutive equations which govern the deformation of the bulk material. 1 (ii) While it is relatively straightforward to construct a traction-separation relation for normal separation across an interface, appropriate relations for combined opening and sliding are less well-developed. With a view towards modeling grain-boundary interface response, in what follows we develop an elastic–plastic interface model which accounts for both reversible elastic, as well irreversible inelastic separation-sliding deformations at the interface prior to failure. 1

Camacho and Ortiz (1996) do allow for a form of inelasticity. They consider interface constitutive equations in which the initial cohesive response is rigid, and there is a Fnite traction at which softening occurs; they also allow loading/unloading irreversibility, with linear unloading to the origin.

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Interface e1

n

e

3

e2

Fig. 4. Schematic of interface between two bodies B+ and B− .

We consider two bodies B+ and B− separated by an interface I (Fig. 4). Let {eˆ1 ; eˆ2 ; eˆ3 } be an orthonormal triad, with eˆ aligned with the normal n to the interface, and {eˆ2 ; eˆ3 } in the tangent plane at the point of the interface under consideration. Let T denote the displacement jump across the cohesive surface, and t the powerconjugate traction, such that t · T˙ gives the power per unit area of the interface in the reference conFguration. We assume that the displacement jump may be additively decomposed as T = Te + Tp ; e

(2.1)

p

where T and T , respectively, denote the elastic and plastic parts of T. Then, t · T˙ = t · T˙e + t · T˙p :

(2.2)

Let ’ denote a free-energy per unit surface area in the reference conFguration. We consider a purely mechanical theory based on the following local energy imbalance that represents the Frst two laws of thermodynamics under isothermal conditions, ˙ ’˙ 6 t · T:

(2.3)

Then, using (2.2), the Feld  = t · T˙e + t · T˙p − ’˙ ¿ 0 represents the dissipation rate per unit area.

(2.4)

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We assume that the free-energy ’ is given by ’ = ’(T ˆ e ):

(2.5)

Then, using standard arguments, (2.4) gives @’(T ˆ e) t= @Te and  = t · T˙p ¿ 0:

(2.6)

(2.7)

We are concerned with interfaces in which the elastic displacement jumps are small. For these conditions we assume a simple quadratic free-energy ’ 1 ’ = Te · KTe (2.8) 2 with K, the interface elastic sti4ness tensor, taken to be positive deFnite. In this case (2.6) gives t = KTe = K(T − Tp ):

(2.9)

We consider an interface model which is isotropic in its tangential response, and take K to be given by K = KN n ⊗ n + KT (1 − n ⊗ n);

(2.10)

with KN ¿ 0 and KT ¿ 0 normal and tangential elastic sti4ness moduli. The interface traction t may be decomposed into normal and tangential parts, tN and tT , respectively, as t = tN + tT ;

tN ≡ (n ⊗ n)t = (t · n)n ≡ tN n;

tT ≡ (1 − n ⊗ n)t = t − tN n:

(2.11)

The quantity tN represents the normal stress at the interface. We denote the magnitude of the tangential traction vector tT by √

Q ≡ tT · tT (2.12) and call it the e>ective tangential traction, or simply the shear stress. We take the elastic domain in our rate-independent elastic–plastic model to be deFned by the interior of the intersection of two convex yield surfaces. The yield functions corresponding to each surface are taken as (i) (t; s(i) ) 6 0;

i = 1; 2;

(2.13)

and henceforth we indentify the index i = 1 with a “normal” mechanism, and the index i = 2 with a “shear” mechanism. The scalar internal variable s(1) represents the deformation resistance for the normal mechanism, and s(2) represents the deformation resistance for the shear mechanism. In particular, we consider the following simple speciFc functional form for the yield functions: (1) = tN − s(1) 6 0;

(2) = Q + tN − s(2) 6 0;

(2.14)

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tN n(1)

n(2)

Φ (1) = 0

µ

s(1)

Φ(2) =0

s(2)

τ

Fig. 5. Schematic of yield surfaces for the normal and shear mechanisms.

where  represents a friction coe
with

tT : (2.17)

Q Note that since m(2) = n(2) , we have a non-normal 7ow rule for the shear response. The evolution equations for the internal variables s(i) are taken as a pair of ordinary di4erential equations 2  h(ij) (j) ; (2.18) s˙(i) = m(1) = n

and

m(2) =

j=1

where the coeKcients h(ij) denote hardening/softening moduli.

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Finally, during inelastic deformation, an active mechanism must satisfy the consistency condition (i) ˙ (i) = 0

when

(i) = 0:

(2.19) (i)

The consistency condition serves to determine the inelastic deformation rates  when inelastic deformation occurs. Straightforward calculations using (2.9) and (2.18) give  [n(i) · Km( j) + h(ij) ](j) : ˙ (i) = n(i) · KT˙ − j

For (i) ¿ 0, when i = 0 the consistency condition requires that ˙ (i) = 0. This gives the following system of linear equations for (i) ¿ 0: 2 

A(ij) (j) = b(i) ;

A(ij) = n(i) · Km( j) + h(ij) ;

˙ b(i) = n(i) · KT:

(2.20)

j=1

We assume that the matrix A is invertible, so that the (j) are uniquely determined. 2.1. Speci@c form for the evolution equations Let def

(1) =

def

(2) =

 0



t

t

0

(1) () d;

(2.21)

(2) () d;

(2.22)

deFne equivalent relative plastic displacements for the two individual mechanisms, and
def (2.23) Q = ((1) )2 + ((2) )2 deFnes a combined equivalent relative plastic displacement, where  represents a coupling parameter between the normal and shear mechanisms. A simple set of evolution equations for s(1) and s(2) which represent strain-hardening response until a critical value of Q = Qc is reached, and a softening response thereafter is Q s(1) = sˆ (1) () with

 (1)

Q = sˆ ()

Q sˆ(1) hard ()

if Q 6 Qc ;

sˆ(1) Q soft ()

if Q ¿ Qc and s(1) ¿ 0:

(2.24)

Similarly, Q s(2) = sˆ (2) () with

 (2)

Q = sˆ ()

sˆ(2) Q hard ()

if Q 6 Qc ;

sˆ(2) Q soft ()

if Q ¿ Qc and s(2) ¿ 0:

(2.25)

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In this case s˙(1) =

@sˆ (1) ˙ 1 @sˆ (1) (1) (1) Q = (  + (2) (2) ); @Q Q @Q

(2.26)

s˙(2) =

@sˆ (2) ˙ 1 @sˆ (2) (1) (1) Q = (  + (2) (2) ); @Q Q @Q

(2.27)

so that h(11) = h(22) =

(1) @sˆ (1) ; Q @Q

h(12) =

(2) @sˆ (1) ; Q @Q

h(21) =

(1) @sˆ (2) ; Q @Q

(2) @sˆ (2) : Q @Q

(2.28)

A simple explicit constitutive time-integration procedure for the rate-independent interface model is summarized in Appendix A. The model, using this time-integration procedure has been implemented in the Fnite element program ABAQUS/Explicit (ABAQUS, 2002), by writing a USER INTERFACE subroutine. 3. Constitutive model for single-crystals We shall employ the (now classical) single-crystal plasticity theory (cf., e.g., Taylor, 1938; Mandel, 1965; Teodosiu, 1970; Rice, 1971; Mandel, 1972; Hill, 1965; Teodosiu and Sidoro4, 1976; Asaro, 1983; Asaro and Needleman, 1985; Bronkhorst et al., 1992). A rate-independent version of the single-crystal plasticity model is summarized below. Single-crystal plasticity is based on the hypothesis that plastic 7ow takes place through slip on prescribed slip systems  = 1; 2; : : : ; N , with each system  deFned by a slip direction s0 and a slip-plane normal m0 , where s0 · m0 = 0;

|s0 |; |m0 | = 1;

s0 ; m0 = constant:

(3.1)

For a material with slip systems  = 1; 2; : : : ; N deFned by orthonormal vector pairs (m0 ; s0 ), the constitutive equations relate the following basic Felds: , F, T, Fp ,  = {s1 ; s2 ; : : : ; sN }, Fe = FFp−1 ,
Ce = Fe Fe , 1 Ee = (Ce − 1), 2 Te = (det Fe )Fe−1 TFe− ,

J = det F ¿ 0, det Fp = 1, s ¿ 0, det Fe ¿ 0,

free energy density per unit referential volume, deformation gradient, Cauchy stress, plastic part of the deformation gradient, slip resistances, elastic part of the deformation gradient, elastic right Cauchy–Green strain, elastic strain, stress conjugate to Ee .

The constitutive theory, intended to characterize small elastic strains, is summarized below.

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The free energy is given by (Ee ) = 12 Ee · C[Ee ];

(3.2)

where C is the elasticity tensor. The elastic stress–strain relation has the form Te = C[Ee ];

(3.3)

while the resolved stress on the th slip system is given by

 = s0 · (Ce Te )m0 :

(3.4)

The condition for slip on the th slip system is taken as  = |  | − s 6 0;

(3.5)



where s is the slip system deformation resistance. The ?ow rule is deFned by an evolution equation for Fp of the form N   s0 ⊗ m0 ;  ¿ 0; and   = 0; F˙ p = Lp Fp ; Lp = =1

Fp (0) = 1;

(3.6) 

The evolution equations for the internal variables s are N  h ()| |; s (0) = s0 : s˙ =

(3.7)

=1

During plastic 7ow the following consistency conditions must be satisFed:  ˙  = 0 if  = 0:

(3.8)

The consistency conditions serve to determine the shearing rates  ¿ 0 on the slip systems. To complete the constitutive model for a given ductile single crystal, the material parameters/functions that need to be speciFed are the slip-systems (m0 ; s0 ) and their initial orientations, as well as {C; s0 ; h }: A constitutive time-integration procedure for the rate-independent crystal plasticity model has been previously detailed in Anand and Kothari (1996), and for brevity, is not repeated here. The single-crystal plasticity model using this time-integration procedure has been implemented in the Fnite element program ABAQUS/Explicit (ABAQUS, 2002) by writing a USER MATERIAL subroutine. 4. Application to nanocrystalline nickel Experimentally measured stress–strain curves in simple tension from specimens of electrodeposited nanocrystalline nickel with grain sizes ranging from 15 to 40 nm from di4erent groups show a highly nonlinear stress–strain response up to an ultimate tensile strength between ≈1:3–1:7 GPa, and Fnal fracture at strain levels ranging from

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Fig. 6. (a) Initial microstructure represented by 50 columnar grains. (b) Finite-element mesh. (c) Experimental (111) pole Fgure of as-received electrodeposited nc-nickel from Xiao et al. (2001); pole Fgure projection direction is normal to the plane of the sheet specimen. (d) (111) pole Fgures corresponding to the grain orientations used in the polycrystal simulation.

≈2:5–5% (e.g., Wang et al., 1997; Yin and Whang, 2001; Xiao et al., 2001; Torre et al., 2002). We shall use this representative experimental information to estimate the material parameters for the grain interiors as well as the grain boundaries by judiciously adjusting the values of the material parameters in our constitutive model to approximately match these stress–strain curves. We recognize that such a procedure for material parameter estimation is not unique. However, the values for the material parameters, so estimated, serve reasonably well the primary objective of this paper, which is to qualitatively study the competition between the grain interior and grain boundary deformation modes in nanocrystalline fcc materials. For reasons of computational eKciency, in our simulations we have used a quasi-three-dimensional polycrystalline aggregate consisting of a collection of columnar grains. 2 An aggregate consisting of 50 such grains is shown in Fig. 6a. This initial microstructure was generated by Frst using a (more or less) standard Voronoi 2

The electrodeposited nanocrystalline nickel does possess an approximately columnar grain structure; see Fig. 1c of Kumar et al. (2003). However, the grain structure with only one grain through the thickness used in our calculations does not quite approach the actual columnar grain structure. The numerical grain structure is more suitable for modeling @ne-grained polycrystalline thin @lms which show a similar macroscopic stress –strain response.

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construction in two-dimensions, and then extruding it to obtain the third direction. 3 After determining the vertices of each grain and its neighbors, the geometry is imported into in ABAQUS/CAE (ABAQUS, 2002) for generating the Fnite element mesh, Fig. 6b, and specifying the single crystal constitutive model described in Section 3 for the grain interiors, as well as the interaction between neighboring grains using the interface model described in Section 2. Nanocrystalline nickel produced by electrodeposition typically possesses a strong crystallographic texture. Fig. 6c shows the (111) pole Fgure for such a material, Xiao et al. (2001); the pole Fgure projection direction is normal to the plane of the sheet specimen. In our numerical simulations the lattices of the individual grains were assigned a set of orientations that approximate this initial texture, Fig. 6d. For the grain interiors, the anisotropic elasticity tensor C for cubic materials is speciFed in terms of three standard sti4ness parameters, C11 , C12 and C44 . The values of the elastic parameters for nickel are taken as (Simmons and Wang, 1971): C11 = 247 GPa;

C12 = 147 GPa;

C44 = 125 GPa:

For fcc crystals, crystallographic slip is assumed to occur on the twelve {111} 1 1 0 slip systems. The components of the slip-plane normals m0 and slip directions s0 with respect to an orthonormal basis associated with the crystal lattice are listed in Table 1. The interface sti4nesses, KN and KT in the normal and tangential directions, are estimated as KN ≈E=g and KT ≈G=g where E and G are, respectively, nominal values of isotropic polycrystalline Young’s modulus and shear modulus, and g is an estimate of a “grain-boundary thickness.” Our continuum model has no inherent length scale. However, in this paper we use a scale in which the average grains in our @nite element mesh shown in Figs. 6a and b are taken to be ≈30 nm, and assume that the grain-boundary thickness is g≈1 nm. For nickel, using representative values of E and G and g = 1 nm gives an estimate of KN = 200 GPa=nm;

KT = 76 GPa=nm:

(4.1)

In the simulations to follow, we take the resistance of an interface in tangential direction to be independent of the normal stress on that interface, i.e.,  = 0. Referring back to Eqs. (2.24) and (2.25), we assume that the hardening portion of the interface traction–separation curves can be Ft to the following special form of the evolution for the two resistances s(i) :  a(i) s(i) (i) (i) ; Q˙ with initial values s(i) (0) = s0(i) ; s˙hard = h0 1 − ∗(i) s for

Q 6 Qc ;

(4.2)

with s∗(i) ¿ s(i) . 3

Instead of using a totally random set of seeds to generate the voronoi tesselation, which occasionally results in extremely small grains, we numerically imposed a minimum distance between the possible “random seeds.” This helps to increase the quality of the microstructural geometry for Fnite element computations.

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Table 1 Components of slip plane normals m0 and slip directions s0 in the crystal basis for the twelve slip systems of an fcc crystal. m0



s0

1

1 √ 3

1 √ 3

1 √ 3

1 √ 2

1 −√ 2

0

2

1 √ 3

1 √ 3

1 √ 3

1 −√ 2

0

1 √ 2

3

1 √ 3

1 √ 3

1 √ 3

0

1 √ 2

1 −√ 2

4

1 −√ 3

1 √ 3

1 √ 3

1 √ 2

0

1 √ 2

5

1 −√ 3

1 √ 3

1 √ 3

1 −√ 2

1 −√ 2

0

6

1 −√ 3

1 √ 3

1 √ 3

0

1 √ 2

1 −√ 2

7

1 √ 3

1 −√ 3

1 √ 3

1 −√ 2

0

1 √ 2

8

1 √ 3

1 −√ 3

1 √ 3

0

1 −√ 2

1 −√ 2

9

1 √ 3

1 −√ 3

1 √ 3

1 √ 2

1 √ 2

0

10

1 −√ 3

1 −√ 3

1 √ 3

1 −√ 2

1 √ 2

0

11

1 −√ 3

1 −√ 3

1 √ 3

1 √ 2

0

1 √ 2

12

1 −√ 3

1 −√ 3

1 √ 3

0

1 −√ 2

1 −√ 2

Recall that we have deFned the equivalent relative plastic displacement by
Q = ((1) )2 + ((2) )2 ; we assume a value for the coupling parameter =0:25, and (guided by our assumption of a grain-boundary thickness g=1 nm) a value of Qc =1 nm when the interface starts to soften in any combination of tension or shear. Thus, in pure tension Qc = (1) c = 1 nm, while in pure shear (2) c = 2 nm. In our numerical simulations we allow for a softening branch to the interface traction-separation response, by assuming a simple linear softening from Qc to a failure value Qfail : (i) (i) ˙ s˙soft = −hsoft Q

for

Qc ¡ Q 6 Qfail :

(4.3)

Recall that molecular dynamics simulations show that when grain-boundary deformation cannot be accommodated due to geometric restrictions, local stress concentrations develop to cause the emission of a few partial dislocations from grain boundaries, and

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Table 2 Parameters for inelastic response of interface (i)

i = 1, Tension i = 2, Shear

(i)

(i)

s0 (MPa)

h0 (GPa/nm)

s∗(i) (MPa)

a(i)

Qc (nm)

Qfail (nm)

hsoft (GPa/nm)

300.0 300.0

28.3 27.7

2400.0 2200.0

5.0 5.0

1 1

1.1 1.1

15 15

these high stresses drive the partial dislocations across the grain interiors to be absorbed in the opposite grain boundaries. With this mechanism in mind, we assume that the slip resistances for the grain interiors s are all equal to a constant s0 , interpreted as the resistance to emission of partial dislocations from grain-boundaries into the grain interiors: s  = s0 · · ·

constant slip system resistance for grain interiors: (4.4)
The value of s0 is estimated as follows. With G = (C11 − C12 )C44 =2 denoting a shear modulus, b the magnitude of the Burgers vector, and l denoting a dislocation segment length, the crystallographic resolved shear stress required to move dislocation segments that exist in well-developed networks in conventional grain-sized metals are of the order of s0 ≈Gb=l. However, if dislocations are to be conFned to the intragranular space, l can at most be of the order of the grain diameter D, that is, l = D with 0 ¡  6 1, and hence s0 ≈Gb=D. Since grain boundaries are potent sources of dislocations and since the grain interiors in nanocrystalline materials are essentially free of dislocations, we expect that ≈1, and arrive at the simple estimate Gb s0 ≈ ; (4.5) D which represents a criterion for grain-boundary emission (and subsequent absorption) of dislocations in nanocrystalline polycrystals (e.g., Asaro et al., 2003; Cheng et al., 2003). More reFned estimates based on stacking fault energies of partial dislocations may be made, but we do not go into such reFnements here (Asaro et al., 2003). For typical values of G≈80 MPa and b≈0:3 nm for Ni, and a grain size D≈30 nm, we obtain a value s0 ≈ 800 MPa

(4.6)

for a 30 nm grain-sized nc-Nickel. Using our previously listed assumptions, and the inelastic parameters for the interface given in Table 2, we obtained the curve-Ft shown in Fig. 7a. The traction–separation curves corresponding to the values of KN and KT given in (4.1) and the inelastic parameters for the interface given in Table 2 are shown in Figs. 7b and c. Fig. 8 shows contours of a measure of an equivalent plastic strain in the grains, deFned by   

 1 def p p p ˙ ˙ $Q = $Q dt; where $Q =  with %Q = (3=2)T0e · T0e ; T0e = Te − tr(Te )1 %Q 3 corresponding to four di4erent points, labelled a through d, on the stress–strain response, Fig. 8a. For clarity, the maximum value of the $Q p contours is set to 5% for

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Fig. 7. (a) Boundary conditions used in the numerical simulation, and comparison of stress–strain curve in tension from the numerical simulation against corresponding experimental data from the literature, Wang et al. (1997) and Yin and Whang (2001). Traction–separation curves used for nano-crystalline nickel: (b) in the normal direction to an interface, and (c) in the tangential direction to an interface.

all four contour plots. Since Fig. 8b shows essentially no plastic strain in the grain interiors, the nonlinearity in the macroscopic stress–strain curve up to point a is due entirely to the nonlinear elastic–plastic interface response. By point b (Fig. 8c) on the stress–strain curve, there is some plastic strain in the vicinity of a few grain-boundary triple junctions; however, the major cause for the macroscopic nonlinear stress–strain response is still the nonlinear interface response; indeed, by this stage a few of the interfaces have visibly failed. By point c (Fig. 8d), near the peak of the stress–strain curve, there is a dominant macroscopic crack traversing along several grain boundaries (top left of Fig. 8d), and substantial plastic strain within the interiors of the grains at the tip of the crack. By point d (Fig. 8e) on the stress–strain curve at least three dominant interface cracks are clearly visible, and the material has lost a large fraction of its stress-carrying capacity. The propagation of grain-boundary cracks causes a substantial amount of plastic deformation in the interior of the grains blocking the crack-path, and in the boundary regions of the grains adjacent to the crack path. As a cautionary note, a plastic strain of 1% in a typical grain size of 30 nm represents only a single dislocation with a Burgers vector of 0:3 nm moving through the

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Fig. 8. (a) Stress–strain curve from the simulation and literature. Contour plots (b) through (e) of the equivalent plastic strain in the grain interiors corresponding to di4erent macroscopic strain level keyed to stress–strain curve.

grain! Thus, as stated earlier, a standard continuum crystal-plasticity based description of the inelastic deformation of the grain interior for nanocrystalline materials is clearly inadequate, and a discrete dislocation modeling approach which recognizes the emission of partial dislocations form grain-boundaries is perhaps more appropriate. However, we emphasize that we have used a continuum crystal-plasticity theory only as an indicator of inelastic deformation within the grains, and not as an accurate representation of the actual discrete dislocation response of nanocrystalline materials. To emphasize this point, and to further understand the origin of the nonlinear stress–strain response of the nanocrystalline nickel during tension, we numerically suppressed any plastic deformation in the grain interiors, so that the grain interiors deform elastically, and the only source of inelastic deformation is due to grain-boundary sliding and separation. Fig. 9 compares the simulated stress–strain responses when the grain interiors deform elastically, against the response where the grain interiors are allowed to deform plastically with slip resistances set at s = 800 MPa. From this Fgure it is clear that up to

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Fig. 9. (a) Comparison of stress–strain curves in tension from a numerical simulation which uses a model in which the grain interiors that deform only elastically, against results from another simulation in which the grain interiors may also deform plastically. Corresponding contours of equivalent plastic strain plotted on the deformed geometry at strain levels approximately corresponding to point c on the stress–strain curves are shown in (b) for elastically deforming grain interiors, and (c) for grain interiors which may also deform plastically.

point b, the macroscopic stress–strain is dominated by the non-linear grain-boundary response. From b to c, the local stress levels are high enough to cause some amount of plastic deformation in some of the grain interiors and soften the overall stress– strain response. Thus, the dominant non-linearity in the stress–strain response is due to the grain-boundary related relaxation phenomena, and a simpler modeling approach for this grain size range would be to treat the grain interiors as purely elastic. However, in the next section, where we explore grain-size e4ects, we shall continue to use our continuum crystal-plasticity model as an indicator of the extent of possible inelastic deformation of the grain interiors.

5. Grain-size e&ect The e4ects of a variation of the grain-size on the macroscopic stress–strain curve, as predicted by our continuum model, are shown in Fig. 10. In these simulations, the role of grain-size on the macroscopic response of nanocrystalline nickel was modelled in the following fashion. The material parameters for the grain-boundary response were @xed once they were calibrated against the experimental data by using the 50-grain mesh representing an average grain size of 30 nm with a slip resistance of 800 MPa, Fig. 7. Then, the number of grains in the same macroscopic volume as that of the mesh with 50 grains, were changed; four additional new meshes with 12, 28, 110, and 236

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Fig. 10. (a) Stress–strain curves for di4erent grain sizes. (b) Peak strength versus grain size. Plots (c) through (g) show contours of the equivalent plastic strain in the grain interiors corresponding to the strain levels right after peak stress for the 61, 40, 30, 20 and 14 nm average grain-sized microstructures.

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Table 3 Average grain size and average slip resistance for the di4erent meshes in Fig. 10 Average grain size, nm

Average slip resistance, s0 , MPa

61 40 30 20 14

396 599 800 1180 1730

grains were generated. Thus, with the average grain-size of the 50-grain mesh scaled to be 30 nm, the appropriately scaled average grain sizes for the other four meshes are listed in Table 3. In our numerical calculations, the slip resistances for the individual grains in each mesh are assigned based on the estimate (4.5). Recall that the ideal shear strength for fcc metals is usually estimated as ≈G=30. We have used this upper bound, approximately 2:7 GPa for nickel, as the slip resistances for the very small grains in some of the Fner-grained materials, whenever their resistances calculated using (4.5) are higher than this maximum one. For reference, Table 3 also lists the average slip resistances for the grains of the Fve meshes that we have used in our simulations. The macroscopic stress–strain curves for meshes with di4erent grain sizes are shown in Fig. 10a. A plot of the peak strength versus grain size is shown in Fig. 10b. This curve clearly shows the interplay between the properties of the grain-boundaries and the grain interiors. As the grain size is decreased from 61 to 30 nm the overall strength increases, however a further decrease in the grain size to 14 nm results in a decrease of the strength. We emphasize that the occurrence of the peak strength at 30 nm grain size is the direct outcome of our scaling arguments and the particular values of the grain-interior and grain-boundary interface properties that we have used in our calculations. Our calculations are intended to qualitatively represent the interplay between the grain-interior strengths as the grain size changes, relative to @xed grain-boundary thickness and strength. The material parameters for the grain boundary properties were chosen to approximately match the stress–strain curve for the 30 nm grain size material, all other grain-size versus strength variations shown in Fig. 10b only conFrm qualitatively expected trends, and are not expected to match any actual experimental results. Plots in Fig. 10(c) through Fig. 10(g) show contours of the equivalent plastic strain in the grain interiors corresponding to a strain level right after the peak stress for each grain-size. Recall that the grain-boundary properties have been taken to be identical in all calculations, and the grain-interior slip resistances are taken to increase with decreasing grain size. Thus, at a relatively large grain size, as we see from the mesh with 12 grains (representing an average grain size of 61 nm) the plastic deformation in the grain interior dominates, whereas the grain-boundary deformation begins to dominate as the grain size is reduced. With a decrease in the grain-size, and hence an increase in the grain-boundary fraction per unit volume, we Fnd that the ductility also changes

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as the grain size changes. For meshes with 110 and 236 grains (grain size of 20 nm and 14 nm, respectively), Fig. 10 a shows a marked increase in ductility. The increase of ductility in this grain-size range is due entirely to the more distributed nature of grain boundary damage with decreasing grain-size. 6. Fracture in a notched specimen Finally, to numerically simulate the situation corresponding to a tension test on a perforated nano-nickel specimen (see Fig. 3), the mesh with 236 grain shown in Fig. 10(g) was adopted and rescaled to represent a mean grain-size of 30 nm. The overall dimensions are 1000 m × 1000 m × 80 m. An initial notch was generated by simply removing a few grains at right boundary (Fig. 11). Properties of the grain boundary and grain interior are taken as the same as we used for the calibration test for the 50 grain mesh (Fig. 8). The resulting macroscopic load–displacement curve is shown in Fig. 11(a). Contour plots of the equivalent plastic strain in the grain interiors corresponding to di4erent macroscopic displacement levels keyed to load-displacement curve are shown in Fig. 11(b) through (e). These Fgures also show the nucleation and propagation of the intergranular crack(s) which give rise to the macroscopic failure. A crack, labelled crack A, nucleates initially at the tip of the notch because of the high local stress concentration. Plastic deformation in the grain interiors at this stage is small, and mainly occurs in the vicinity of the newly-nucleated crack tip, Fig. 11(b). From the load–displacement we see that there is an approximate load-plateau from a to b. Microstructurally, at stage b the propagation of crack A is blocked by an unfavorably oriented grain G1, Fig. 11(c), and this results in a signiFcant plastic deformation within the interior of this grain. Upon further increase of the macroscopically-applied displacement, the local conditions between grains G2 and G3 cause the nucleation of a new crack, labelled crack B, Fig. 11(d). Upon further application of the macroscopic deformation, cracks A and B further extend toward each other, and the macroscopic load drops. The region in Fig. 11(e) surrounded by a dotted line shows that by this stage the remaining ligament between the two cracks exhibits both interface cracking, as well as substantial plastic deformation within the grain interiors—much in accord with experimental observations of Kumar et al. (2003). 7. Concluding remarks To model the e4ects of grain boundaries in polycrystalline materials we have coupled a standard crystal-plasticity model for the grain interiors with a new elastic–plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to qualitatively study the deformation and fracture response of nanocrystalline nickel in simple tension. The results from the simulations re7ect the major features of the experimentally observed stress–strain curves (Wang et al., 1997; Yin and Whang, 2001; Xiao et al., 2001; Torre et al., 2002),

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Fig. 11. Fracture in a notched-tensile specimen. (a): Load versus displacement. Contour plots in (b) through (e) show the equivalent plastic strain in the grain interiors corresponding to di4erent macroscopic displacement levels keyed to load–displacement curve.

and the dominant fracture mechanisms in this material (Kumar et al., 2003). Our simulations show that the macroscopically-observed nonlinearity in the stress–strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in

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the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress–strain response, and the overall ductility. In nanocrystalline nickel, the high yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension. Much work remains to be done to quantitatively account for (a) the grain boundary di4usional e4ects, and (b) the emission of a partial dislocations from grain boundaries observed in atomistic simulations of nanocrystalline materials. However, such e4ects appear diKcult to accommodate at this stage of the development of a computationally tractable theory such as ours, which has been formulated at the continuum-level (and not at molecular-dynamics or discrete-dislocation levels) to elucidate the dominant e>ects of the interplay between grain-boundary and grain interior deformation modes in nanocrystalline materials under low homologous temperature and quasi-static strain rate conditions. Finally, we note that our modeling approach should also be useful to represent the inelastic deformation and fracture response of other materials with very little or no intergranular plasticity, such as ceramics and rock-like materials, where the pseudo-plasticity and non-linear stress–strain curves arise primarily due to grain boundary separation and sliding.

Acknowledgements This work was supported by the Defense University Research Initiative on NanoTechnology (DURINT) on “Damage- and Failure-Resistant Nanostructured and Interfacial Materials” which is funded at the Massachusetts Institute of Technology (MIT) by the OKce of Naval Research under grant N00014-01-1-0808. The help of Dr. O. Diard in the early stages of this work is acknowledged. The TEM pictures shown in Figures 1 through 3 were provided by K.S. Kumar. Valuable discussions with S. Suresh and K.S. Kumar are also gratefully acknowledged.

Appendix A. Summary of time-integration procedure for the interface constitutive model ˆ Let {e(0); eˆ2 (0); eˆ3 (0)} be an orthonormal triad, with eˆ1 (0) = n aligned with the normal to the interface in the reference conFguration, and {eˆ2 (0); eˆ3 (0)} be in the tangent plane at the point of the interface under consideration. Let {eˆ1 (t); eˆ2 (t); eˆ3 (t)} be the same basis in the current conFguration, with ei (t) = R(t)ei (0), where R(t) is the rotation that determines ei (t). Then, the traction and the total relative displacement t(t) =

 i

ti (t)eˆi (0);

T(t) =

 i

&i (t)eˆi (0);

(A.1)

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may be transformed into the current conFguration as  ti (t)eˆi (t); t˜(t) = R(t)t(t) =

(A.2)

i

˜ = R(t)T(t) = T(t)



&i (t)eˆi (t):

(A.3)

i

We consider that we are given 1. 2. 3. 4. 5. 6.

t˜(t) = t1 (t)eˆ1 (t) + t2 (t)eˆ2 (t) + t2 (t)eˆ3 (t), s(i) (t), (1) (t), (2) (t) [T˜ = [&1 eˆ1 (t) + [&2 eˆ2 (t) + [&3 eˆ3 (t), [t = − t, and R(t) which determines eˆi (t) = R(t)eˆi (0).

We need to calculate {t( ); s(i) ( ); (1) ( ); (2) ( )}, and march forward in time. Step 1: Calculate the trial stress at the end of the step (components with respect to eˆi (t)) t1∗ ( ) = t1 (t) + KN [&1 ;

(A.4)

t2∗ ( ) = t2 (t) + KT [&2 ;

(A.5)

t3∗ ( ) = t3 (t) + KT [&3 ;

(A.6)

t˜∗T ( ) = t2∗ ( )eˆ2 (t) + t3∗ ( )eˆ3 (t);

(A.7)

Q∗ ( ) =



t˜∗T ( ) · t˜∗T ( ) =



(t2∗ ( ))2 + (t3∗ ( ))2 :

(A.8)

Step 2: Calculate b(i) b(1) = {t1∗ ( ) − s(1) (t)}; b(2) = ( Q∗ ( ) + t1∗ ( ) − s(2) (t)):

(A.9) (A.10)

Step 3: Calculate A(ij) A(11) = [KN + h(11) (t)] ¿ 0;

(A.11)

A(12) = [h(12) (t)];

(A.12)

A(21) = [KN  + h(21) (t)];

(A.13)

A(22) = [KT + h(22) (t)] ¿ 0:

(A.14)

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Step 4: Calculate the plastic relative displacement increments 1. If b(1) ¿ 0 and b(2) 6 0 then x(1) =

b(1) ; A(11)

(A.15)

x(2) = 0:

(A.16)

2. If b(1) 6 0 and b(2) ¿ 0 then x(1) = 0; x(2) =

(A.17)

b(2) : A(22)

(A.18)

3. If b(1) ¿ 0 and b(2) ¿ 0 then Frst calculate A−1 ; recall that we have assumed that the matrix A is invertible. Then solve for the plastic strain increments  x(i) = (A−1 )(ij) b(j) : (A.19) j

Check if x(i) ¿ 0 then accept this solution. However, if x(1) ¿ 0 and x(2) ¡ 0 then x(1) =

b(1) ; A(11)

x(2) = 0

(A.20)

or if x(2) ¿ 0 and x(1) ¡ 0 then x(2) =

b(2) ; A(22)

x(1) = 0:

(A.21)

Step 5: Update the traction t1 ( ) = t1∗ ( ) − KN x(1) ;

(A.22)

( ) Q = Q∗ ( ) − KT x(2) ;

(A.23)

t˜T ( ) =

( ) Q

( ) Q t˜∗ ( ) = ∗ {t2∗ ( )eˆ2 (t) + t3∗ ( )eˆ3 (t)};

Q∗ ( ) T

Q ( )

(A.24)

t2 ( ) =

(t) Q t ∗ ( );

Q∗ ( ) 2

(A.25)

t3 ( ) =

(t) Q t ∗ (t);

Q∗ (t) 3

(A.26)

t˜( ) = t1 ( )eˆ1 (t) + t2 ( )eˆ2 ( ) + t3 ( )eˆ3 (t);

(A.27)

ˆ + t2 ( )eˆ2 (0) + t3 ( )eˆ3 (0): t( ) = R(t) t˜( ) = t1 ( )e(0)

(A.28)

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Step 6: Update the state variables s(1) ( ) = s(1) (t) + h(11) (t)x(1) + h(12) (t)x(2) ;

(A.29)

s(2) ( ) = s(2) (t) + h(21) (t)x(1) + h(22) (t)x(2) :

(A.30)

Step 7: Update (1) and (2) : (1) ( ) = (1) (t) + x(1) ;

(A.31)

(2) ( ) = (2) (t) + x(2) :

(A.32)

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