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A constitutive model for powder-processed nanocrystalline metals
Acta Materialia 55 (2007) 921–931 www.actamat-journals.com
A constitutive model for powder-processed nanocrystalline metals Yujie Wei, Lallit Anand
*
Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 1-310C, 77 Massachusetts Avenue, Cambridge, MA 02139, United States Received 14 February 2006; received in revised form 9 September 2006; accepted 11 September 2006 Available online 29 November 2006
Abstract Significant developments in the processing of nanocrystalline metals from powder precursors have occurred in recent years, and materials are beginning to be produced which exhibit not only high strength, but also a reasonable amount of ductility. In order to develop a simulation capability for the engineering design of structural components made from such materials, it is necessary to formulate continuum-level constitutive equations which faithfully model the complicated experimentally observed pressure- and strain-rate-sensitive response of these materials. From a macroscopic point of view, powder-consolidated nanocrystalline metals are similar to cohesive granular materials. Guided by this similarity, we have developed a phenomenological large-deformation, isotropic, rate-dependent plasticity model, in which the plastic flow is taken to be pressure-dependent, plastically dilatant and non-normal. We have implemented this model in a finite-element program. We have conducted compression, tension and notched-bar tension experiments on a nanocrystalline magnesium-based alloy (nc-Mg), and the constitutive parameters for this material appearing in the model are estimated from these experiments. Using our numerical finite-element-based capability, we show that our constitutive model may be used with reasonable accuracy to simulate the response of the nc-Mg in some representative boundary-value problems, which include micro-indentation and three-point bending. Based on data available in the literature, we have also explored the applicability of the model to powder-consolidated nc-Cu. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocrystalline metals; Powder-consolidated; Constitutive equations; Finite elements
1. Introduction Nanostructured metals with grain sizes typically less than 100 nm have been shown to exhibit high strength and hardness, but they also exhibit limited ductility due to difficulties in synthesis, especially for materials produced from powders by various consolidation techniques. Significant developments in the processing of such materials have occurred in recent years, and materials are beginning to be produced which exhibit not only high strength, but also a reasonable amount of ductility. Such materials, when produced in sufficient quantities, may therefore actually become viable for use in structural applications (cf. e.g., Ma [1] and Newberry et al. [2]).
*
Corresponding author. Tel.: +1 617 253 1635; fax: +1 617 258 8742. E-mail address: [email protected] (L. Anand).
Of particular note is the recent work reported by Youssef et al. [3] and Cheng et al. [4] for the processing of copper, in which the authors report achieving nominally dense samples with a narrow grain-size distribution in the nanocrystalline range by a special in situ processing technique that involves a two-step high-energy ball-milling process, first at liquid nitrogen temperature and subsequently at room temperature. Their nanocrystalline copper-based alloy (nc-Cu) shows strength levels greater than 650 MPa, and a tensile elongation to failure of at least 6%. Further, the nc-Cu shows a strain rate sensitivity mð o ln rf =o ln _ Þ 0:026, which represents a fourfold increase over the strain rate sensitivity of conventional coarse-grained Cu (m 0.006). Similarly impressive properties have also been recently reported for powder-consolidated nanocrystalline Mgbased materials by Lu et al. [5]. These authors have developed nanostructured Mg alloys of nominal composition
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.09.014
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
a
b 400
True Stress (MPa)
Mg–5%Al–x%Nd, with x = 0.5%, 1% and 5% by weight. A typical processing schedule used by these authors is to first mechanically alloy the Mg, Al and Nd powders. The mechanically alloyed powders are then cold-compacted into cylinders of 35 mm diameter, and sintered at 400 C or 500 C in a vacuum furnace for 2 hours. The sintered cylinders are then extruded at 400 C with an extrusion ratio of 25:1. For additional processing details see [5]. Samples of their nc-Mg–5%Al–1%Nd alloy were kindly provided to us by Prof. Li Lu of the National University of Singapore, and we conducted transmission electron microscopy (TEM), X-ray pole-figure measurements and some standard tension and compression experiments on their material. Fig. 1(a) shows a typical TEM bright-field image of their nc-Mg. Fig. 1(b) shows true stress–strain curves in tension and compression at a strain rate of 104 s1. The strength level is approximately 400 MPa. Note that there is a small tension–compression asymmetry, with the compressive curve being slightly higher than the tensile curve; also, while the tensile ductility is at a respectable level of 9%, the ductility in compression is substantially higher, approaching levels greater than 35% at low strain rates. Further, the material shows strain-softening in both tension and compression. Fig. 1(c) shows true stress–strain curves from compression tests on the same material at strain rates of 102, 103 and 104 s1. Note the pronounced strainrate sensitivity exhibited by the material; the nc-Mg has a strain-rate sensitivity of m 0.07, which represents a sevenfold increase over that of a conventional coarse-grained magnesium alloy, AZ31B (m 0.01). The ductility in compression is seen to decrease as the strain rate increases, but it is still greater than 10% at a strain rate of 102 s1. Finally, note that the high values of specific stiffness (E/q) and specific strength (ry/q) of nc-Mg, relative to that of powder-consolidated nanocrystalline Cu and electrodeposited Ni (cf. Table 1), makes the Mg-based nanocrystalline material potentially very attractive for lightweight structural applications. Concerning the strain-softening of the nc-Mg alloy observed in Fig. 1(b), pole figures from our crystallographic texture evolution experiments on this material, shown in Fig. 2(a) and (b), show that there is very little texture evolution after compressive strain levels of 15%. However, as shown in a representative TEM micrograph of the deformed specimen in Fig. 2(c), there is evidence of distributed internal damage due to grain-boundary decohesion in the material. Based on these experimental results, we conclude that the strain-softening is due not to texture evolution, but to distributed internal grainboundary damage, which is to be expected in powderconsolidated materials. With the emerging capabilities to produce nanocrystalline materials which exhibit not only high strength but also a reasonable amount of ductility, it is of interest to formulate continuum-level constitutive models which, when implemented in finite-element solution procedures, may be used in the design of structural components made from
Tension Compression
200
0
0
5
10
15
True Strain (%)
c
True Stress (MPa)
922
600
400
200
Compression, 0.01/s Compression, 0.001/s Compression, 0.0001/s 0
0
10
20
30
40
50
True Strain (%) Fig. 1. (a) TEM bright-field image of nc-Mg; material supplied by Lu et al. [5]. (b) Stress–strain curves for nc-Mg in tension and compression at a strain rate of 104 s1. (c) Stress–strain curves in compression at strain rates of 102 s1, 103 s1 and 104 s1.
such materials. It is the purpose of this paper to report on such a model. From a macroscopic point of view, powderconsolidated metals (nanocrystalline or otherwise) are similar to cohesive granular materials. Guided by this similarity, the constitutive model that we have formulated is a large-deformation, isotropic, rate-dependent elasto-plasticity model in which the plastic flow is taken to be pressuredependent, plastically dilatant and non-normal. Our theory
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Table 1 Specific stiffness and strength of some nanocrystalline metals Material
q (Mg/m3)
E (GPa)
ry (MPa)
E/q
ry/q
Grain size (nm)
nc-Mg [5] nc-Cu [4] nc-Ni [6]
1.8 8.9 8.7
43 107 192
400 700 1150
25 14 28
222 80 130
90 45 28
Fig. 2. (a) Initial crystallographic texture for the nc-Mg as represented by pole figures for the {0 0 0 2}, {1 0 1 0} and {1 1 2 0} planes. (b) Pole figures after 15% compressive strain. Note that there is very little difference in the pole figures after deformation relative to those before, indicating that there is little twinning or lattice re-rotation due to slip after 15% strain. (c) TEM micrograph of specimen after 15% compression showing decohesion at grain boundaries.
may be viewed as a rate-dependent generalization of the rate-independent theories of Rudnicki and Rice [7] and Anand [8]. The plan of this paper is as follows. We develop our constitutive model in Section 2. We have implemented the constitutive model in the commercial finite-element
program ABAQUS/Explicit [9] by writing a user-material subroutine. In Section 3 we use data from our compression, tension and notched-bar tension experiments on nc-Mg to estimate the material parameters appearing in the model for this material.
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In Section 4 we report on the predictive capabilities of our constitutive model and computational procedures. We have carried out simulations for (i) conical micro-indentation experiments and (ii) a three-point bend experiment, and compared the predictions of certain key macroscopic features from the simulations against corresponding experimental measurements. The numerically predicted load (P) versus indentation depth (h) curves in the indentation experiments are shown to closely match those observed in the experiments. Also, the load–displacement curve in the three-point bending simulation reasonably approximates the corresponding experimentally measured curve. In Section 5 we turn our attention to the applicability of the model to powder-consolidated nc-Cu. We have used the limited published experimental data from macroscopic stress–strain tests for nc-Cu [4] to estimate the material parameters appearing in the model for this material. Using the material parameters so-determined, we carry out a three-dimensional finite-element simulation of a sheet tension specimen of this material, and numerically study the evolution of deformation patterns and localized necking in this material. In this section we also compare the results from our numerical simulations of localized necking against those observed experimentally by Cheng et al. [4] (their Fig. 8), and show that our numerical simulations satisfactorily reproduce their experimental results. We close in Section 6 with a few final remarks. 2. Constitutive model
w
free energy density per unit volume of intermediate space. The set of constitutive equations is summarized below: (1) Free energy:
^ e Þ ¼ GjEe j þ KðtrEe Þ w ¼ wðE 0 2
2
ð1Þ
with G the elastic shear modulus and K bulk modulus. In general, we expect that the elastic moduli G and K will depend on the plastic volumetric strain g; for now, because of lack of experimental data, we neglect such a dependence. (2) Equation for the stress: ~ eÞ owðE ¼ 2GEe0 þ KðtrEe Þ1 oEe (3) Flow rule: The evolution equation for Fp is
Te ¼
F_ p ¼ Dp Fp ;
Dp ¼ Dps þ Dpc ;
with
ð2Þ
Fp ðX; 0Þ ¼ 1
ð3Þ
p
where the plastic stretching D is taken to arise from two micromechanisms: a shear-dominated contribution Dps and a cavitating contribution Dpc . The plastic stretching Dps is given by 9 Dps ¼ ðDps Þ0 þ 13 ðtrDps Þ1; with > = ð4Þ ðDps Þ0 ¼ d p Np ; and > ; p p trDs ¼ bd ; where pffiffiffi d p ¼ 2Dp 0
ð5Þ
is an equivalent plastic shear strain rate We use standard notation of modern continuum mechanics. The symbols $ and Div denote the gradient and divergence with respect to the material point X in the reference configuration; grad and div denote these operators with respect to the point x=v(X, t) in the deformed configuration; a superposed dot denotes the material time-derivative. Throughout, we write Fe1 = (Fe)1, Fp> = (Fp)>, etc. We write sym A, skw A, A0, and sym0 A, respectively, for the symmetric, skew, deviatoric, and symmetric–deviatoric parts of a tensor A. Also, the inner product of tensors A andpBffiffiffiffiffiffiffiffiffiffiffi is denoted by A:B, and the magnitude of A by jAj ¼ A : A. Our constitutive equations relate the following basic fields: x = v(X, t) motion F = $v, J = detF > 0 deformation gradient F = FeFp multiplicative elastic–plastic decomposition Fp, Jp = detFp > 0 plastic distortion g = lnJp plastic volumetric strain Fe, Je = detFe > 0 elastic distortion Fe = ReUe polar decomposition of Fe P Ue ¼ 3a¼1 kea ra ra spectral decomposition of Ue P Ee ¼ 3a¼1 ðlnkea Þra ra logarithmic elastic strain T, T = T> Cauchy stress Te = Re>(JeT)Re stress conjugate to elastic strain Ee
Np ¼
Te0 2s
ð6Þ
is a direction of deviatoric plastic flow given in terms of the stress deviator Te0 , with 1 s ¼ pffiffiffi Te0 ð7Þ 2 an equivalent shear stress. The equivalent plastic shear strain rate dp is specified by a power-law constitutive function 1=m s p d ¼ d0 P0 ð8Þ c þ lp where 1 p ¼ trTe 3
ð9Þ
is mean-normal pressure; c > 0 and l P 0 are internal variables called the cohesion and internal friction, respectively; also d0 is a reference plastic shear strain rate, and m > 0 is a strain-rate sensitivity parameter. For dp > 0, the flow Eq. (8) may be inverted to read p m d s ¼ fc þ lpg ð10Þ d0 thus, the limit m ! 0 corresponds to the rate-independent limit.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
925
The term l p accounts for pressure-sensitivity of plastic flow. The quantity b in Eq. (4)3 is a shear-induced plastic dilatancy function, which is assumed to depend on the plastic volumetric strain g = ln Jp. As a simple specific form for the dilatancy function, we assume that g b ¼ g0 1 ð11Þ g
The evolution of the cohesion c is in general a function of g and Dp. This is modelled by introducing the following phenomenological function for the evolution of c: 9 3
p P > i c a c c_ ¼ h0 1 c sign 1 c d hc d c ; with > > > = i¼1 p n > ¼ c dd 0 þ bðg gÞ; and initial value c > > > ; cð0Þ ¼ c0 :
where g0 and g* are positive-valued material parameters; in this case the shear-induced dilatancy vanishes, b = 0, when g = g*. (Rate-independent versions of our flow rule (with Dpc ¼ 0) are broadly used to model the plastic response of granular materials in which pressure sensitivity and plastic dilatancy are quite pronounced. We emphasize that we do not assume that l = b, which would correspond to an associated/normality flow rule.) Next, recalling the spectral representation of the stress:
ð16Þ
Te ¼
3 X
ri^ei ^ei ;
r1 P r2 P r3
ð12Þ
i¼1
the constitutive equation for Dpc , representing the cavitating micromechanism, is taken as 9 3 P > p ðiÞ > Dc ¼ d c ð^ei ^ei Þ; with > > > i¼1 > > 8 n = o1=m < ri rth ð13Þ d 0 rcr if ri > rth ; > ðiÞ dc ¼ > > : > 0 if ri 6 rth ; > > > ; rcr ¼ c1 þ c2 p > 0; where c1 and c2 are material parameters. In this model the contribution d ðiÞ ei ^ei Þ to Dpc is zero if c ð^ the principal stress ri is less than or equal to a positive threshold value rth. When ri is greater than rth, the value of (ri rth) relative to the value of a parameter ^cr ðpÞ, which depends on the hydrostatic tension rcr ¼ r (p), controls the magnitude of d ðiÞ c . The value of rcr is taken to decrease linearly as the hydrostatic tension ðpÞ increases. For simplicity, the reference strain rate d0 and the rate-sensitivity parameter m in Eq. (13) are taken to be the same as that in Eq. (8). (4) Evolution equations for g, c and l: For the cohesive powder-consolidated nanocrystalline materials under consideration, we take the internal friction l to be a positive constant (there is not enough experimental information available for powder-consolidated nanocrystalline materials to be more definitive about the evolution of l at this time). l constant > 0:
ð14Þ
By definition g = lnJp, and therefore g_ ¼ trDp . Thus, using Eq. (4) and Eq. (13), we have 3 X g d ðiÞ gð0Þ ¼ 0 ð15Þ g_ ¼ g0 1 d p þ c ; g i¼1 with d ðiÞ c given in Eq. (13)2.
The material constants used in the coupled evolution Eqs. (15) and (16) to describe the rate-dependent strain-hardening or softening response of the material are: fg0 ; g ; h0 ; a; c0 ; c; n; b; hc g; when d ic ¼ 0, the quantity c* in Eq. (16) represents a saturation value of c which depends on g and dp: c_ is positive for c < c* which corresponds to strain-hardening, and negative for c > c* which corresponds to strain-softening. (5) Damage and failure due to cavitation: We use the following simple criteria to model damage and failure of nanocrystalline materials due to the cavitation mechanism. Let _ p ¼ jDpc j define a cavitating strain rate, and Z t p def ¼ ð17Þ _ p ðnÞdn 0
define a volumetric plastic strain. Then, as a simple model for damage by inelastic cavitation, we introduce a damage variable D defined by 8 0 if p 6 pcr ; > > < p p ð cr Þ def if pcr < p < pf ; ð18Þ D¼ pf pcr Þ ð > > : p 1 if p P f ; whose value is zero for p less than a critical value pcr , and thereafter D ¼ ðp pcr Þ=ðpf pcr Þ, so that the damage variable evolves linearly towards a value of unity as p evolves to a failure value pf . Correspondingly, the rcr in Eq. (13) is decreased linearly towards a value of zero rcr ¼ fc1 þ c2 pg ð1 DÞ: Further, to account for damage to the elastic properties, the elastic shear and bulk moduli G and K are replaced by G · (1 D) and K · (1 D) when damage occurs. As D approaches unity, the material is deemed to have failed by cavitation, and it is removed from the finite element calculation. The damage model outlined above is quite rudimentary, and much work needs to be done to develop more realistic models for the transition from shear plasticity, to damage and final fracture. We have implemented our constitutive model in the finite-element computer program ABAQUS/Explicit [9] by writing a user material subroutine. We apply this numerical capability to model the response of nc-Mg and nc-Cu in the sections below.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
In this section we apply our model to describe the deformation behavior of nc-Mg. Recall that the material parameters in our model are (i) the elastic shear and bulk moduli (G, K); (ii) the parameters {d0, m, l} in the flow rule; (iii) the parameters fg0 ; g ; h0 ; a; c0 ; c; n; b; hc g in the evolution equations for c and g; and (iv) frth ; c1 ; c2 ; pc ; pf g in the evolution of cavitation resistance in Eq. (13) and the associated damage defined in Eq. (18). We recognize that our phenomenological model contains numerous material parameters, and an unambiguous determination of these material parameters would require a large number of independent experiments of different types which accentuate a particular type of material response. The nanocrystalline material available to us was limited in quantity, and we have not conducted an extensive set of experiments to estimate/determine the material parameters. However, we have conducted two independent sets of experiments: (i) compression and tension experiments (in which the hydrostatic tension is relatively low), to estimate the material parameters {d0, m, l} in the flow rule for Dps , and fg0 ; g ; h0 ; a; c0 ; c; n; bg in the evolution equations for c and g; and (ii) notched-bar tension experiments to estimate the parameters frth ; c1 ; c2 ; pc ; pf ; hc g associated with the cavitation mechanism.
a
600 500
True Stress (MPa)
3. Application to nc-Mg
400 300
0
b
We started with simple compression experiments and tension experiments where the maximum hydrostatic tensile stresses are relatively small, and hence cavitation failure is not expected to contribute significantly. In this case, we used a single ABAQUS/C3D8R element to conduct numerical simulations in compression and tension, to estimate the material parameters by curve-fitting numerically calculated stress–strain curves to our experimentally measured true stress–strain curves, Fig. 1. The results of our curve-fitting procedure are shown in Fig. 3(a) and (b), and the material parameters obtained by our heuristic curve-fitting procedure (see below) are listed in Table 2. A heuristic curve-fitting procedure: The elastic shear and bulk moduli (G, K) were taken as their counterparts for bulk coarse-grained magnesium. The referential strain rate d0 was taken to be the lowest strain rate in the experiments, d0 = 0.0001 s1. The parameters c0 and c, were estimated by fitting the stress–strain curve at the referential strain rate: c0 controls the initial strength and c is the stabilized flow stress under shearing when g* = g; the values obtained from pffiffiffi the compression stress–strain curves are divided by 3 to approximate fc0 ; cg in shear. The parameters {b, g0, g*} control the strain-softening due to shear-induced dilatancy; the factor b(g* g) con-
0
10
20 30 True Strain (%)
40
50
600 500 400 300
Experiment, 0.01/s Simulation Experiment, 0.001/s Simulation Experiment, 0.0001/s Simulation
200 100 0
3.1. Compression and tension experiments
Experiment, Compression Simulation Experiment, Tension Simulation
200 100
True Stress (MPa)
926
0
10
20 30 True Strain (%)
40
50
Fig. 3. (a) Comparison of numerically calculated true stress–strain curves in tension and compression at a strain rate of 104 s1 against corresponding experimentally measured curves for nc-Mg. (b) Comparison of numerically calculated true stress–strain curves in compression at strain rates of 102, 103 and 104 s1 against corresponding experimentally measured curves for nc-Mg.
Table 2 Estimated material constants for nc-Mg G = 17 GPa d0 = 0.0001 s1 h0 = 6.4 GPa n = 0.04 c1 = 2.5 GPa rth = 0 GPa
K = 50 GPa m = 0.03 a = 1.2 b = 36 GPa c2 = 11 hc = 0 GPa
l = 0.03 c0 = 200.0 MPa g* = 1.0 · 103 pc = 0.06
c = 187.0 MPa g0 = 0.015 pf = 0.1
trols the difference between the peak stress and the fully developed flow stress on a stress–strain curve, and g0 controls the initial rate of strain-softening. The friction coefficient l was estimated by comparing the difference in flow strength between tensile and compressive stress–strain curves at the slowest strain rate. The rate-sensitivity parameters m and n in the flow and evolution equations were estimated from the compression stress–strain curves at different strain rates.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
3.2. Notched bar tension test To estimate the material parameters appearing in the cavitation/damage equations we conducted notched-bar tensile experiments. The geometry of the notched tensile specimen is shown in Fig. 4(a). A representative load–displacement curve from such an experiment is shown in Fig. 4(b). A finite element mesh representing one-quarter of the axisymmetric specimen is shown in Fig. 4(c). The load–displacement curve from the finite element simulation using the parameters estimated from the tension and compression experiments, and with the cavitation mechanism suppressed, is shown as in Fig. 4(b) (dotted line, labelled no failure); the match between the simulation and experiment, until just before strong softening and fracture, is quite satisfactory. The parameters fc1 ; c2 ; pc ; pf ; rth ; hc g, which control evolution of cavitation damage, were then adjusted to approx-
927
imately match the failure part of the experimental load– displacement curve. The material parameters associated with cavitation mechanism, so estimated, are also listed in Table 2. (There is not enough experimental information to estimate values for rth and hc; for simplicity we set rth = hc = 0.) The fitted curve with the cavitation mechanism taken into account (dot–dash, labelled cavitation failure) is seen to reproduce the complete experimental load–displacement curve with reasonable accuracy. Fig. 4(d) shows a contour plot of the hydrostatic pressure at the stage marked by x on the load–displacement curve for the case labelled cavitation failure; the contour has a minimum value of 200 MPa (maximum hydrostatic tension) in the light region ahead of the notch-root; as expected, damage due to cavitation initiates in this region. A contour plot of the cavitation strain p at the end of the simulation is shown in Fig. 4(e); this plot confirms that
Fig. 4. (a) Geometry of the tensile notched-bar. (b) Load–displacement curves at a cross-head velocity of 0.0005 mm s1 from simulations and experiment. (c) Finite element mesh for one-quarter of the specimen. (d) Pressure contours corresponding to the peak load, labelled as x on the load–displacement curve for the case with cavitation failure mechanism. The minimum pressure is 200 MPa (maximum hydrostatic tension) in the light region ahead of the notch root. (e) Contour plot of the accumulated cavitation strain p at the end of the calculation, a minimum of 0 (white region) and a maximum of 0.1 (dark region). The irregular boundary at the bottom of this figure is due to the elements removed after cavitation failure.
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Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
damage initiates inside the material, away from the notchroot. The irregular boundary at the bottom of Fig. 4(e) is due to the removed elements after failure.
measured P–h curves are shown and compared against the numerically simulated curves in Fig. 5(c). (Approximately ten indentation experiments were carried out at each loading rate to ensure the repeatability of the P–h curves.) The numerically simulated P–h curves agree reasonably well with the corresponding experimental curves.
4. Predictive capabilities of the constitutive model 4.1. Micro-indentation experiments at different loading rates Next, in order to check the predictive capabilities of our constitutive model in the absence of cavitation failure, we carried out axisymmetric finite-element simulations of micro-indentation experiments with a conical indenter (half-cone angle of 70.3). The undeformed finite-element mesh is shown in Fig. 5(a). The simulations were conducted at three different loading rates: 0.5, 5 and 50 mN s1. The load (P) versus indentation depth (h) curves at these different loading rates are shown in Fig. 5(b). As expected, because of the intrinsic strain-rate sensitivity of the nc-Mg, the indentation P–h curves also show significant loading rate-sensitivity. Corresponding micro-indentation experiments were performed on the nc-Mg, and the experimentally
a 19.7
4.2. Three-point bending experiment In order to check the predictive capability of the model under situations where the cavitation mechanism is also operative, a three-point bending experiment was conducted. The bending geometry is sketched in Fig. 6(a); a corresponding finite element simulation of this bending experiment was carried out. The predicted load–displacement curve is shown and compared against the experimental result in Fig. 6(b), and (c) from the finite element simulation shows the cracked configuration of the body just before final fracture; the numerical prediction approximates the experimental result.
b
o
Force, P (N)
0.6 50 mN/s 5 mN/s 0.5 mN/s
0.4
0.2
0
0
1
2 3 Depth, h (μm)
4
4
5
5
c 0.6 Exp., 50 mN/s Force, P (N)
Model Exp., 5 mN/s
0.4
Model Exp., 0.5 mN/s Model
0.2
0 0
1
2 3 Depth, h ( μ m)
Fig. 5. (a) Finite element mesh used in axisymmetric conical indentation simulations. (b) Numerically predicted P–h curves for nc-Mg at loading rates of 0.5, 5 and 50 mN s1. (c) Comparison of numerically calculated P–h curves against corresponding experimental measurements.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
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Fig. 6. (a) Schematic of a three-point bending experiment; the out-of-plane thickness is also 4 mm. The bottom rolls are fixed and the top roll is moved downwards at a velocity of 0.0001 mm s1. (b) Comparison of load–displacement curves from simulation and experiment. (c) The deformed geometry from the finite element simulation showing the crack configuration just before final fracture.
5. Application to nc-Cu In this section we apply our constitutive model to describe the response of the powder-consolidated nc-Cu reported by Cheng et al. [4]. Fig. 7(a) shows a typical TEM bright-field image of their nc-Cu, and Fig. 7(b) shows engineering stress–strain curves from tension tests on one of their materials at strain rates of 102, 103 and 104 s1. There is very limited experimental data reported for this material, and this limits our ability to estimate the various material parameters appearing in our constitutive model. Cheng et al. [4] do not report results from corresponding compression experiments, so we neglect any pressure sensitivity of plastic flow and set l = 0. Further, since there are no reported experiments to show whether the mechanical behavior of this material is influenced by hydrostatic tension, we also suppress the cavitation mechanism and associated failure in simulating the mechanical response of this material. Since the true stress–strain curves for the nc-Cu of [4] do not exhibit any softening until very near final failure (cf. their Fig. 4(b)), our constitutive model may be further considerably simplified by also neglecting the effects of the plastic volumetric strain and its evolution. Specifically,
the plastic dilatancy part of the flow rule may be set to zero with b = 0, so that g_ ¼ 0. The evolution equation for the cohesion is then taken in the simple form: h c a i c_ ¼ h0 1 d p ; with cð0Þ ¼ c0 : ð19Þ c Thus, in the simplified model, the material parameters are the elastic shear and bulk moduli (G, K); the parameters {d0, m} in the flow rule; and the parameters {h0, a, c0, c*} in the evolution Eq. (16) for c. Cheng et al. [4] conducted their experiments on dogbone-shaped sheet tensile specimens which were electrodischarge-machined from their consolidated nc-Cu. A finite-element mesh of such a sheet tensile specimen is shown in Fig. 8(a). Using this three-dimensional specimen geometry and the implementation of our constitutive model in ABAQUS/Explicit [9], we conducted numerical tension tests and adjusted the material parameters in our model to obtain a curve-fit to the engineering stress–strain curves of [4]. The curve-fit is shown in Fig. 8(b); the fit at all three strain rates for major portions of the engineering stress–strain curves is reasonable. Since the abbreviated model considered in this section is intended to represent only the inelastic deformation of these materials and does not contain provisions to model failure, it clearly cannot
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Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
a
Engineering Stress (MPa)
b
800
600
200
0
Fig. 7. From Cheng et al. [4]: (a) TEM bright-field image of nc-Cu obtained after a two-stage high-energy ball-milling process. (b) Tensile engineering stress–strain curves at strain rates of 102 s1, 103 s1 and 104 s1.
Experiment, 0.01/s Simulation Experiment, 0.001/s Simulation Experiment, 0.0001/s Simulation
400
0
5 10 Engineering Strain (%)
15
Fig. 8. (a) A finite element mesh of an undeformed sheet tension specimen. (b) Comparison of numerically calculated engineering stress– strain curves, using the mesh shown in (a) and the material properties listed in Table 3 for nc-Cu, against experimentally measured curves at three different strain rates; experimental data from Cheng et al. [4].
Table 3 Estimated material parameters for nc-Cu
capture the final failure process observed in the experiments. The material parameters used to obtain the curvefit shown in Fig. 8(b) are listed in Table 3. Contour plots for the equivalent plastic shear strain: Z cp ¼ d p dt; keyed to points b, c and d on the engineering stress–strain curve at a strain rate of 104 s1, Fig. 9(a), are shown in Fig. 9(b)–(d). The numerical simulation shows the classical diffuse-necking that eventually gives way to intense localized-necking, which occurs at an angle of 55 to the tensile axis. A photograph of the corresponding experimentally measured localized-neck reported by [4] (their Fig. 8(b)) is shown in Fig. 9(e). The qualitative similarity between the numerical simulation and experimental observation is very encouraging.
G = 45 GPa h0 = 45 GPa
K = 140 GPa a = 1.2
d0 = 0.0001 s1 c0 = 330 MPa
m = 0.026 c* = 395 MPa
6. Concluding remarks In order to model the macroscopic response of powderconsolidated nanocrystalline metals, we have outlined a constitutive framework for the finite deformation elastic– viscoplastic response of these materials. With the limited amount of nc-Mg available to us, we have conducted a few select experiments to estimate the material parameters appearing in our model, and to check its predictive capabilities. The results of our study show that this model, when suitably calibrated and numerically implemented, may be used with reasonable accuracy to simulate the deformation
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However, both nanocrystalline materials and the continuum-level constitutive models of the type discussed in this paper are in their infancy. The materials themselves are typically available only in limited quantities, and are expensive. This limits the ability of researchers to conduct sophisticated multi-axial experiments in order to calibrate the material parameters appearing in the complex models of the type under consideration, and also to conduct corresponding validation studies. Our study represents only a first effort towards a robust constitutive theory for this class of materials. Much work remains to be done to refine the experimental database and corresponding constitutive and failure theories for this important emerging class of materials.
Acknowledgements This work was supported by the Defense University Research Initiative on NanoTechnology (DURINT) on Damage- and Failure-Resistant Nanocrystalline and Interfacial Materials, which is funded at the Massachusetts Institute of Technology (MIT) by the Office of Naval Research under Grant N00014-01-1-0808.
References
Fig. 9. (a) Numerically calculated and experimentally measured engineering stress strain curve for nc-Cu at a strain rate of 104 s1. (b)–(d) Contour plots of equivalent plastic shear strain at three different points, b,c and d, keyed to the stress–strain curve in (a). (e) A photograph of the experimentally measured localized neck reported by Cheng et al. [4].
behavior of nanocrystalline materials, which exhibit not only high strength but also a reasonable amount of ductility.
[1] Ma E. J Met 2006;49(April):53. [2] Newberry AP, Nutt SR, Lavernia EJ. J Met 2006;56(April): 61. [3] Youssef KM, Scattergood RT, Murty KL, Koch CC. Appl Phys Lett 2004;85:929. [4] Cheng S, Ma E, Wang YM, Kecskes LJ, Youssef KM, Koch CC, et al. Acta Mater 2005;53:1521. [5] Lu L, Lai MO, Yan C, Ye L. Rev Adv Mater Sci 2004;6:28. [6] Torre FD, Van Swygenhoven H, Victoria M. Acta Mater 2002;50:3957. [7] Rudnicki JW, Rice JR. J Mech Phys Solids 1975;23:371. [8] Anand L. J Appl Mech 1980;47:439. [9] HKS, Inc. Abaqus Reference Manuals, Pawtucket RI; 2005.
A constitutive model for powder-processed nanocrystalline metals Yujie Wei, Lallit Anand
*
Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 1-310C, 77 Massachusetts Avenue, Cambridge, MA 02139, United States Received 14 February 2006; received in revised form 9 September 2006; accepted 11 September 2006 Available online 29 November 2006
Abstract Significant developments in the processing of nanocrystalline metals from powder precursors have occurred in recent years, and materials are beginning to be produced which exhibit not only high strength, but also a reasonable amount of ductility. In order to develop a simulation capability for the engineering design of structural components made from such materials, it is necessary to formulate continuum-level constitutive equations which faithfully model the complicated experimentally observed pressure- and strain-rate-sensitive response of these materials. From a macroscopic point of view, powder-consolidated nanocrystalline metals are similar to cohesive granular materials. Guided by this similarity, we have developed a phenomenological large-deformation, isotropic, rate-dependent plasticity model, in which the plastic flow is taken to be pressure-dependent, plastically dilatant and non-normal. We have implemented this model in a finite-element program. We have conducted compression, tension and notched-bar tension experiments on a nanocrystalline magnesium-based alloy (nc-Mg), and the constitutive parameters for this material appearing in the model are estimated from these experiments. Using our numerical finite-element-based capability, we show that our constitutive model may be used with reasonable accuracy to simulate the response of the nc-Mg in some representative boundary-value problems, which include micro-indentation and three-point bending. Based on data available in the literature, we have also explored the applicability of the model to powder-consolidated nc-Cu. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanocrystalline metals; Powder-consolidated; Constitutive equations; Finite elements
1. Introduction Nanostructured metals with grain sizes typically less than 100 nm have been shown to exhibit high strength and hardness, but they also exhibit limited ductility due to difficulties in synthesis, especially for materials produced from powders by various consolidation techniques. Significant developments in the processing of such materials have occurred in recent years, and materials are beginning to be produced which exhibit not only high strength, but also a reasonable amount of ductility. Such materials, when produced in sufficient quantities, may therefore actually become viable for use in structural applications (cf. e.g., Ma [1] and Newberry et al. [2]).
*
Corresponding author. Tel.: +1 617 253 1635; fax: +1 617 258 8742. E-mail address: [email protected] (L. Anand).
Of particular note is the recent work reported by Youssef et al. [3] and Cheng et al. [4] for the processing of copper, in which the authors report achieving nominally dense samples with a narrow grain-size distribution in the nanocrystalline range by a special in situ processing technique that involves a two-step high-energy ball-milling process, first at liquid nitrogen temperature and subsequently at room temperature. Their nanocrystalline copper-based alloy (nc-Cu) shows strength levels greater than 650 MPa, and a tensile elongation to failure of at least 6%. Further, the nc-Cu shows a strain rate sensitivity mð o ln rf =o ln _ Þ 0:026, which represents a fourfold increase over the strain rate sensitivity of conventional coarse-grained Cu (m 0.006). Similarly impressive properties have also been recently reported for powder-consolidated nanocrystalline Mgbased materials by Lu et al. [5]. These authors have developed nanostructured Mg alloys of nominal composition
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.09.014
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
a
b 400
True Stress (MPa)
Mg–5%Al–x%Nd, with x = 0.5%, 1% and 5% by weight. A typical processing schedule used by these authors is to first mechanically alloy the Mg, Al and Nd powders. The mechanically alloyed powders are then cold-compacted into cylinders of 35 mm diameter, and sintered at 400 C or 500 C in a vacuum furnace for 2 hours. The sintered cylinders are then extruded at 400 C with an extrusion ratio of 25:1. For additional processing details see [5]. Samples of their nc-Mg–5%Al–1%Nd alloy were kindly provided to us by Prof. Li Lu of the National University of Singapore, and we conducted transmission electron microscopy (TEM), X-ray pole-figure measurements and some standard tension and compression experiments on their material. Fig. 1(a) shows a typical TEM bright-field image of their nc-Mg. Fig. 1(b) shows true stress–strain curves in tension and compression at a strain rate of 104 s1. The strength level is approximately 400 MPa. Note that there is a small tension–compression asymmetry, with the compressive curve being slightly higher than the tensile curve; also, while the tensile ductility is at a respectable level of 9%, the ductility in compression is substantially higher, approaching levels greater than 35% at low strain rates. Further, the material shows strain-softening in both tension and compression. Fig. 1(c) shows true stress–strain curves from compression tests on the same material at strain rates of 102, 103 and 104 s1. Note the pronounced strainrate sensitivity exhibited by the material; the nc-Mg has a strain-rate sensitivity of m 0.07, which represents a sevenfold increase over that of a conventional coarse-grained magnesium alloy, AZ31B (m 0.01). The ductility in compression is seen to decrease as the strain rate increases, but it is still greater than 10% at a strain rate of 102 s1. Finally, note that the high values of specific stiffness (E/q) and specific strength (ry/q) of nc-Mg, relative to that of powder-consolidated nanocrystalline Cu and electrodeposited Ni (cf. Table 1), makes the Mg-based nanocrystalline material potentially very attractive for lightweight structural applications. Concerning the strain-softening of the nc-Mg alloy observed in Fig. 1(b), pole figures from our crystallographic texture evolution experiments on this material, shown in Fig. 2(a) and (b), show that there is very little texture evolution after compressive strain levels of 15%. However, as shown in a representative TEM micrograph of the deformed specimen in Fig. 2(c), there is evidence of distributed internal damage due to grain-boundary decohesion in the material. Based on these experimental results, we conclude that the strain-softening is due not to texture evolution, but to distributed internal grainboundary damage, which is to be expected in powderconsolidated materials. With the emerging capabilities to produce nanocrystalline materials which exhibit not only high strength but also a reasonable amount of ductility, it is of interest to formulate continuum-level constitutive models which, when implemented in finite-element solution procedures, may be used in the design of structural components made from
Tension Compression
200
0
0
5
10
15
True Strain (%)
c
True Stress (MPa)
922
600
400
200
Compression, 0.01/s Compression, 0.001/s Compression, 0.0001/s 0
0
10
20
30
40
50
True Strain (%) Fig. 1. (a) TEM bright-field image of nc-Mg; material supplied by Lu et al. [5]. (b) Stress–strain curves for nc-Mg in tension and compression at a strain rate of 104 s1. (c) Stress–strain curves in compression at strain rates of 102 s1, 103 s1 and 104 s1.
such materials. It is the purpose of this paper to report on such a model. From a macroscopic point of view, powderconsolidated metals (nanocrystalline or otherwise) are similar to cohesive granular materials. Guided by this similarity, the constitutive model that we have formulated is a large-deformation, isotropic, rate-dependent elasto-plasticity model in which the plastic flow is taken to be pressuredependent, plastically dilatant and non-normal. Our theory
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Table 1 Specific stiffness and strength of some nanocrystalline metals Material
q (Mg/m3)
E (GPa)
ry (MPa)
E/q
ry/q
Grain size (nm)
nc-Mg [5] nc-Cu [4] nc-Ni [6]
1.8 8.9 8.7
43 107 192
400 700 1150
25 14 28
222 80 130
90 45 28
Fig. 2. (a) Initial crystallographic texture for the nc-Mg as represented by pole figures for the {0 0 0 2}, {1 0 1 0} and {1 1 2 0} planes. (b) Pole figures after 15% compressive strain. Note that there is very little difference in the pole figures after deformation relative to those before, indicating that there is little twinning or lattice re-rotation due to slip after 15% strain. (c) TEM micrograph of specimen after 15% compression showing decohesion at grain boundaries.
may be viewed as a rate-dependent generalization of the rate-independent theories of Rudnicki and Rice [7] and Anand [8]. The plan of this paper is as follows. We develop our constitutive model in Section 2. We have implemented the constitutive model in the commercial finite-element
program ABAQUS/Explicit [9] by writing a user-material subroutine. In Section 3 we use data from our compression, tension and notched-bar tension experiments on nc-Mg to estimate the material parameters appearing in the model for this material.
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In Section 4 we report on the predictive capabilities of our constitutive model and computational procedures. We have carried out simulations for (i) conical micro-indentation experiments and (ii) a three-point bend experiment, and compared the predictions of certain key macroscopic features from the simulations against corresponding experimental measurements. The numerically predicted load (P) versus indentation depth (h) curves in the indentation experiments are shown to closely match those observed in the experiments. Also, the load–displacement curve in the three-point bending simulation reasonably approximates the corresponding experimentally measured curve. In Section 5 we turn our attention to the applicability of the model to powder-consolidated nc-Cu. We have used the limited published experimental data from macroscopic stress–strain tests for nc-Cu [4] to estimate the material parameters appearing in the model for this material. Using the material parameters so-determined, we carry out a three-dimensional finite-element simulation of a sheet tension specimen of this material, and numerically study the evolution of deformation patterns and localized necking in this material. In this section we also compare the results from our numerical simulations of localized necking against those observed experimentally by Cheng et al. [4] (their Fig. 8), and show that our numerical simulations satisfactorily reproduce their experimental results. We close in Section 6 with a few final remarks. 2. Constitutive model
w
free energy density per unit volume of intermediate space. The set of constitutive equations is summarized below: (1) Free energy:
^ e Þ ¼ GjEe j þ KðtrEe Þ w ¼ wðE 0 2
2
ð1Þ
with G the elastic shear modulus and K bulk modulus. In general, we expect that the elastic moduli G and K will depend on the plastic volumetric strain g; for now, because of lack of experimental data, we neglect such a dependence. (2) Equation for the stress: ~ eÞ owðE ¼ 2GEe0 þ KðtrEe Þ1 oEe (3) Flow rule: The evolution equation for Fp is
Te ¼
F_ p ¼ Dp Fp ;
Dp ¼ Dps þ Dpc ;
with
ð2Þ
Fp ðX; 0Þ ¼ 1
ð3Þ
p
where the plastic stretching D is taken to arise from two micromechanisms: a shear-dominated contribution Dps and a cavitating contribution Dpc . The plastic stretching Dps is given by 9 Dps ¼ ðDps Þ0 þ 13 ðtrDps Þ1; with > = ð4Þ ðDps Þ0 ¼ d p Np ; and > ; p p trDs ¼ bd ; where pffiffiffi d p ¼ 2Dp 0
ð5Þ
is an equivalent plastic shear strain rate We use standard notation of modern continuum mechanics. The symbols $ and Div denote the gradient and divergence with respect to the material point X in the reference configuration; grad and div denote these operators with respect to the point x=v(X, t) in the deformed configuration; a superposed dot denotes the material time-derivative. Throughout, we write Fe1 = (Fe)1, Fp> = (Fp)>, etc. We write sym A, skw A, A0, and sym0 A, respectively, for the symmetric, skew, deviatoric, and symmetric–deviatoric parts of a tensor A. Also, the inner product of tensors A andpBffiffiffiffiffiffiffiffiffiffiffi is denoted by A:B, and the magnitude of A by jAj ¼ A : A. Our constitutive equations relate the following basic fields: x = v(X, t) motion F = $v, J = detF > 0 deformation gradient F = FeFp multiplicative elastic–plastic decomposition Fp, Jp = detFp > 0 plastic distortion g = lnJp plastic volumetric strain Fe, Je = detFe > 0 elastic distortion Fe = ReUe polar decomposition of Fe P Ue ¼ 3a¼1 kea ra ra spectral decomposition of Ue P Ee ¼ 3a¼1 ðlnkea Þra ra logarithmic elastic strain T, T = T> Cauchy stress Te = Re>(JeT)Re stress conjugate to elastic strain Ee
Np ¼
Te0 2s
ð6Þ
is a direction of deviatoric plastic flow given in terms of the stress deviator Te0 , with 1 s ¼ pffiffiffi Te0 ð7Þ 2 an equivalent shear stress. The equivalent plastic shear strain rate dp is specified by a power-law constitutive function 1=m s p d ¼ d0 P0 ð8Þ c þ lp where 1 p ¼ trTe 3
ð9Þ
is mean-normal pressure; c > 0 and l P 0 are internal variables called the cohesion and internal friction, respectively; also d0 is a reference plastic shear strain rate, and m > 0 is a strain-rate sensitivity parameter. For dp > 0, the flow Eq. (8) may be inverted to read p m d s ¼ fc þ lpg ð10Þ d0 thus, the limit m ! 0 corresponds to the rate-independent limit.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
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The term l p accounts for pressure-sensitivity of plastic flow. The quantity b in Eq. (4)3 is a shear-induced plastic dilatancy function, which is assumed to depend on the plastic volumetric strain g = ln Jp. As a simple specific form for the dilatancy function, we assume that g b ¼ g0 1 ð11Þ g
The evolution of the cohesion c is in general a function of g and Dp. This is modelled by introducing the following phenomenological function for the evolution of c: 9 3
p P > i c a c c_ ¼ h0 1 c sign 1 c d hc d c ; with > > > = i¼1 p n > ¼ c dd 0 þ bðg gÞ; and initial value c > > > ; cð0Þ ¼ c0 :
where g0 and g* are positive-valued material parameters; in this case the shear-induced dilatancy vanishes, b = 0, when g = g*. (Rate-independent versions of our flow rule (with Dpc ¼ 0) are broadly used to model the plastic response of granular materials in which pressure sensitivity and plastic dilatancy are quite pronounced. We emphasize that we do not assume that l = b, which would correspond to an associated/normality flow rule.) Next, recalling the spectral representation of the stress:
ð16Þ
Te ¼
3 X
ri^ei ^ei ;
r1 P r2 P r3
ð12Þ
i¼1
the constitutive equation for Dpc , representing the cavitating micromechanism, is taken as 9 3 P > p ðiÞ > Dc ¼ d c ð^ei ^ei Þ; with > > > i¼1 > > 8 n = o1=m < ri rth ð13Þ d 0 rcr if ri > rth ; > ðiÞ dc ¼ > > : > 0 if ri 6 rth ; > > > ; rcr ¼ c1 þ c2 p > 0; where c1 and c2 are material parameters. In this model the contribution d ðiÞ ei ^ei Þ to Dpc is zero if c ð^ the principal stress ri is less than or equal to a positive threshold value rth. When ri is greater than rth, the value of (ri rth) relative to the value of a parameter ^cr ðpÞ, which depends on the hydrostatic tension rcr ¼ r (p), controls the magnitude of d ðiÞ c . The value of rcr is taken to decrease linearly as the hydrostatic tension ðpÞ increases. For simplicity, the reference strain rate d0 and the rate-sensitivity parameter m in Eq. (13) are taken to be the same as that in Eq. (8). (4) Evolution equations for g, c and l: For the cohesive powder-consolidated nanocrystalline materials under consideration, we take the internal friction l to be a positive constant (there is not enough experimental information available for powder-consolidated nanocrystalline materials to be more definitive about the evolution of l at this time). l constant > 0:
ð14Þ
By definition g = lnJp, and therefore g_ ¼ trDp . Thus, using Eq. (4) and Eq. (13), we have 3 X g d ðiÞ gð0Þ ¼ 0 ð15Þ g_ ¼ g0 1 d p þ c ; g i¼1 with d ðiÞ c given in Eq. (13)2.
The material constants used in the coupled evolution Eqs. (15) and (16) to describe the rate-dependent strain-hardening or softening response of the material are: fg0 ; g ; h0 ; a; c0 ; c; n; b; hc g; when d ic ¼ 0, the quantity c* in Eq. (16) represents a saturation value of c which depends on g and dp: c_ is positive for c < c* which corresponds to strain-hardening, and negative for c > c* which corresponds to strain-softening. (5) Damage and failure due to cavitation: We use the following simple criteria to model damage and failure of nanocrystalline materials due to the cavitation mechanism. Let _ p ¼ jDpc j define a cavitating strain rate, and Z t p def ¼ ð17Þ _ p ðnÞdn 0
define a volumetric plastic strain. Then, as a simple model for damage by inelastic cavitation, we introduce a damage variable D defined by 8 0 if p 6 pcr ; > > < p p ð cr Þ def if pcr < p < pf ; ð18Þ D¼ pf pcr Þ ð > > : p 1 if p P f ; whose value is zero for p less than a critical value pcr , and thereafter D ¼ ðp pcr Þ=ðpf pcr Þ, so that the damage variable evolves linearly towards a value of unity as p evolves to a failure value pf . Correspondingly, the rcr in Eq. (13) is decreased linearly towards a value of zero rcr ¼ fc1 þ c2 pg ð1 DÞ: Further, to account for damage to the elastic properties, the elastic shear and bulk moduli G and K are replaced by G · (1 D) and K · (1 D) when damage occurs. As D approaches unity, the material is deemed to have failed by cavitation, and it is removed from the finite element calculation. The damage model outlined above is quite rudimentary, and much work needs to be done to develop more realistic models for the transition from shear plasticity, to damage and final fracture. We have implemented our constitutive model in the finite-element computer program ABAQUS/Explicit [9] by writing a user material subroutine. We apply this numerical capability to model the response of nc-Mg and nc-Cu in the sections below.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
In this section we apply our model to describe the deformation behavior of nc-Mg. Recall that the material parameters in our model are (i) the elastic shear and bulk moduli (G, K); (ii) the parameters {d0, m, l} in the flow rule; (iii) the parameters fg0 ; g ; h0 ; a; c0 ; c; n; b; hc g in the evolution equations for c and g; and (iv) frth ; c1 ; c2 ; pc ; pf g in the evolution of cavitation resistance in Eq. (13) and the associated damage defined in Eq. (18). We recognize that our phenomenological model contains numerous material parameters, and an unambiguous determination of these material parameters would require a large number of independent experiments of different types which accentuate a particular type of material response. The nanocrystalline material available to us was limited in quantity, and we have not conducted an extensive set of experiments to estimate/determine the material parameters. However, we have conducted two independent sets of experiments: (i) compression and tension experiments (in which the hydrostatic tension is relatively low), to estimate the material parameters {d0, m, l} in the flow rule for Dps , and fg0 ; g ; h0 ; a; c0 ; c; n; bg in the evolution equations for c and g; and (ii) notched-bar tension experiments to estimate the parameters frth ; c1 ; c2 ; pc ; pf ; hc g associated with the cavitation mechanism.
a
600 500
True Stress (MPa)
3. Application to nc-Mg
400 300
0
b
We started with simple compression experiments and tension experiments where the maximum hydrostatic tensile stresses are relatively small, and hence cavitation failure is not expected to contribute significantly. In this case, we used a single ABAQUS/C3D8R element to conduct numerical simulations in compression and tension, to estimate the material parameters by curve-fitting numerically calculated stress–strain curves to our experimentally measured true stress–strain curves, Fig. 1. The results of our curve-fitting procedure are shown in Fig. 3(a) and (b), and the material parameters obtained by our heuristic curve-fitting procedure (see below) are listed in Table 2. A heuristic curve-fitting procedure: The elastic shear and bulk moduli (G, K) were taken as their counterparts for bulk coarse-grained magnesium. The referential strain rate d0 was taken to be the lowest strain rate in the experiments, d0 = 0.0001 s1. The parameters c0 and c, were estimated by fitting the stress–strain curve at the referential strain rate: c0 controls the initial strength and c is the stabilized flow stress under shearing when g* = g; the values obtained from pffiffiffi the compression stress–strain curves are divided by 3 to approximate fc0 ; cg in shear. The parameters {b, g0, g*} control the strain-softening due to shear-induced dilatancy; the factor b(g* g) con-
0
10
20 30 True Strain (%)
40
50
600 500 400 300
Experiment, 0.01/s Simulation Experiment, 0.001/s Simulation Experiment, 0.0001/s Simulation
200 100 0
3.1. Compression and tension experiments
Experiment, Compression Simulation Experiment, Tension Simulation
200 100
True Stress (MPa)
926
0
10
20 30 True Strain (%)
40
50
Fig. 3. (a) Comparison of numerically calculated true stress–strain curves in tension and compression at a strain rate of 104 s1 against corresponding experimentally measured curves for nc-Mg. (b) Comparison of numerically calculated true stress–strain curves in compression at strain rates of 102, 103 and 104 s1 against corresponding experimentally measured curves for nc-Mg.
Table 2 Estimated material constants for nc-Mg G = 17 GPa d0 = 0.0001 s1 h0 = 6.4 GPa n = 0.04 c1 = 2.5 GPa rth = 0 GPa
K = 50 GPa m = 0.03 a = 1.2 b = 36 GPa c2 = 11 hc = 0 GPa
l = 0.03 c0 = 200.0 MPa g* = 1.0 · 103 pc = 0.06
c = 187.0 MPa g0 = 0.015 pf = 0.1
trols the difference between the peak stress and the fully developed flow stress on a stress–strain curve, and g0 controls the initial rate of strain-softening. The friction coefficient l was estimated by comparing the difference in flow strength between tensile and compressive stress–strain curves at the slowest strain rate. The rate-sensitivity parameters m and n in the flow and evolution equations were estimated from the compression stress–strain curves at different strain rates.
Y. Wei, L. Anand / Acta Materialia 55 (2007) 921–931
3.2. Notched bar tension test To estimate the material parameters appearing in the cavitation/damage equations we conducted notched-bar tensile experiments. The geometry of the notched tensile specimen is shown in Fig. 4(a). A representative load–displacement curve from such an experiment is shown in Fig. 4(b). A finite element mesh representing one-quarter of the axisymmetric specimen is shown in Fig. 4(c). The load–displacement curve from the finite element simulation using the parameters estimated from the tension and compression experiments, and with the cavitation mechanism suppressed, is shown as in Fig. 4(b) (dotted line, labelled no failure); the match between the simulation and experiment, until just before strong softening and fracture, is quite satisfactory. The parameters fc1 ; c2 ; pc ; pf ; rth ; hc g, which control evolution of cavitation damage, were then adjusted to approx-
927
imately match the failure part of the experimental load– displacement curve. The material parameters associated with cavitation mechanism, so estimated, are also listed in Table 2. (There is not enough experimental information to estimate values for rth and hc; for simplicity we set rth = hc = 0.) The fitted curve with the cavitation mechanism taken into account (dot–dash, labelled cavitation failure) is seen to reproduce the complete experimental load–displacement curve with reasonable accuracy. Fig. 4(d) shows a contour plot of the hydrostatic pressure at the stage marked by x on the load–displacement curve for the case labelled cavitation failure; the contour has a minimum value of 200 MPa (maximum hydrostatic tension) in the light region ahead of the notch-root; as expected, damage due to cavitation initiates in this region. A contour plot of the cavitation strain p at the end of the simulation is shown in Fig. 4(e); this plot confirms that
Fig. 4. (a) Geometry of the tensile notched-bar. (b) Load–displacement curves at a cross-head velocity of 0.0005 mm s1 from simulations and experiment. (c) Finite element mesh for one-quarter of the specimen. (d) Pressure contours corresponding to the peak load, labelled as x on the load–displacement curve for the case with cavitation failure mechanism. The minimum pressure is 200 MPa (maximum hydrostatic tension) in the light region ahead of the notch root. (e) Contour plot of the accumulated cavitation strain p at the end of the calculation, a minimum of 0 (white region) and a maximum of 0.1 (dark region). The irregular boundary at the bottom of this figure is due to the elements removed after cavitation failure.
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damage initiates inside the material, away from the notchroot. The irregular boundary at the bottom of Fig. 4(e) is due to the removed elements after failure.
measured P–h curves are shown and compared against the numerically simulated curves in Fig. 5(c). (Approximately ten indentation experiments were carried out at each loading rate to ensure the repeatability of the P–h curves.) The numerically simulated P–h curves agree reasonably well with the corresponding experimental curves.
4. Predictive capabilities of the constitutive model 4.1. Micro-indentation experiments at different loading rates Next, in order to check the predictive capabilities of our constitutive model in the absence of cavitation failure, we carried out axisymmetric finite-element simulations of micro-indentation experiments with a conical indenter (half-cone angle of 70.3). The undeformed finite-element mesh is shown in Fig. 5(a). The simulations were conducted at three different loading rates: 0.5, 5 and 50 mN s1. The load (P) versus indentation depth (h) curves at these different loading rates are shown in Fig. 5(b). As expected, because of the intrinsic strain-rate sensitivity of the nc-Mg, the indentation P–h curves also show significant loading rate-sensitivity. Corresponding micro-indentation experiments were performed on the nc-Mg, and the experimentally
a 19.7
4.2. Three-point bending experiment In order to check the predictive capability of the model under situations where the cavitation mechanism is also operative, a three-point bending experiment was conducted. The bending geometry is sketched in Fig. 6(a); a corresponding finite element simulation of this bending experiment was carried out. The predicted load–displacement curve is shown and compared against the experimental result in Fig. 6(b), and (c) from the finite element simulation shows the cracked configuration of the body just before final fracture; the numerical prediction approximates the experimental result.
b
o
Force, P (N)
0.6 50 mN/s 5 mN/s 0.5 mN/s
0.4
0.2
0
0
1
2 3 Depth, h (μm)
4
4
5
5
c 0.6 Exp., 50 mN/s Force, P (N)
Model Exp., 5 mN/s
0.4
Model Exp., 0.5 mN/s Model
0.2
0 0
1
2 3 Depth, h ( μ m)
Fig. 5. (a) Finite element mesh used in axisymmetric conical indentation simulations. (b) Numerically predicted P–h curves for nc-Mg at loading rates of 0.5, 5 and 50 mN s1. (c) Comparison of numerically calculated P–h curves against corresponding experimental measurements.
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Fig. 6. (a) Schematic of a three-point bending experiment; the out-of-plane thickness is also 4 mm. The bottom rolls are fixed and the top roll is moved downwards at a velocity of 0.0001 mm s1. (b) Comparison of load–displacement curves from simulation and experiment. (c) The deformed geometry from the finite element simulation showing the crack configuration just before final fracture.
5. Application to nc-Cu In this section we apply our constitutive model to describe the response of the powder-consolidated nc-Cu reported by Cheng et al. [4]. Fig. 7(a) shows a typical TEM bright-field image of their nc-Cu, and Fig. 7(b) shows engineering stress–strain curves from tension tests on one of their materials at strain rates of 102, 103 and 104 s1. There is very limited experimental data reported for this material, and this limits our ability to estimate the various material parameters appearing in our constitutive model. Cheng et al. [4] do not report results from corresponding compression experiments, so we neglect any pressure sensitivity of plastic flow and set l = 0. Further, since there are no reported experiments to show whether the mechanical behavior of this material is influenced by hydrostatic tension, we also suppress the cavitation mechanism and associated failure in simulating the mechanical response of this material. Since the true stress–strain curves for the nc-Cu of [4] do not exhibit any softening until very near final failure (cf. their Fig. 4(b)), our constitutive model may be further considerably simplified by also neglecting the effects of the plastic volumetric strain and its evolution. Specifically,
the plastic dilatancy part of the flow rule may be set to zero with b = 0, so that g_ ¼ 0. The evolution equation for the cohesion is then taken in the simple form: h c a i c_ ¼ h0 1 d p ; with cð0Þ ¼ c0 : ð19Þ c Thus, in the simplified model, the material parameters are the elastic shear and bulk moduli (G, K); the parameters {d0, m} in the flow rule; and the parameters {h0, a, c0, c*} in the evolution Eq. (16) for c. Cheng et al. [4] conducted their experiments on dogbone-shaped sheet tensile specimens which were electrodischarge-machined from their consolidated nc-Cu. A finite-element mesh of such a sheet tensile specimen is shown in Fig. 8(a). Using this three-dimensional specimen geometry and the implementation of our constitutive model in ABAQUS/Explicit [9], we conducted numerical tension tests and adjusted the material parameters in our model to obtain a curve-fit to the engineering stress–strain curves of [4]. The curve-fit is shown in Fig. 8(b); the fit at all three strain rates for major portions of the engineering stress–strain curves is reasonable. Since the abbreviated model considered in this section is intended to represent only the inelastic deformation of these materials and does not contain provisions to model failure, it clearly cannot
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a
Engineering Stress (MPa)
b
800
600
200
0
Fig. 7. From Cheng et al. [4]: (a) TEM bright-field image of nc-Cu obtained after a two-stage high-energy ball-milling process. (b) Tensile engineering stress–strain curves at strain rates of 102 s1, 103 s1 and 104 s1.
Experiment, 0.01/s Simulation Experiment, 0.001/s Simulation Experiment, 0.0001/s Simulation
400
0
5 10 Engineering Strain (%)
15
Fig. 8. (a) A finite element mesh of an undeformed sheet tension specimen. (b) Comparison of numerically calculated engineering stress– strain curves, using the mesh shown in (a) and the material properties listed in Table 3 for nc-Cu, against experimentally measured curves at three different strain rates; experimental data from Cheng et al. [4].
Table 3 Estimated material parameters for nc-Cu
capture the final failure process observed in the experiments. The material parameters used to obtain the curvefit shown in Fig. 8(b) are listed in Table 3. Contour plots for the equivalent plastic shear strain: Z cp ¼ d p dt; keyed to points b, c and d on the engineering stress–strain curve at a strain rate of 104 s1, Fig. 9(a), are shown in Fig. 9(b)–(d). The numerical simulation shows the classical diffuse-necking that eventually gives way to intense localized-necking, which occurs at an angle of 55 to the tensile axis. A photograph of the corresponding experimentally measured localized-neck reported by [4] (their Fig. 8(b)) is shown in Fig. 9(e). The qualitative similarity between the numerical simulation and experimental observation is very encouraging.
G = 45 GPa h0 = 45 GPa
K = 140 GPa a = 1.2
d0 = 0.0001 s1 c0 = 330 MPa
m = 0.026 c* = 395 MPa
6. Concluding remarks In order to model the macroscopic response of powderconsolidated nanocrystalline metals, we have outlined a constitutive framework for the finite deformation elastic– viscoplastic response of these materials. With the limited amount of nc-Mg available to us, we have conducted a few select experiments to estimate the material parameters appearing in our model, and to check its predictive capabilities. The results of our study show that this model, when suitably calibrated and numerically implemented, may be used with reasonable accuracy to simulate the deformation
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However, both nanocrystalline materials and the continuum-level constitutive models of the type discussed in this paper are in their infancy. The materials themselves are typically available only in limited quantities, and are expensive. This limits the ability of researchers to conduct sophisticated multi-axial experiments in order to calibrate the material parameters appearing in the complex models of the type under consideration, and also to conduct corresponding validation studies. Our study represents only a first effort towards a robust constitutive theory for this class of materials. Much work remains to be done to refine the experimental database and corresponding constitutive and failure theories for this important emerging class of materials.
Acknowledgements This work was supported by the Defense University Research Initiative on NanoTechnology (DURINT) on Damage- and Failure-Resistant Nanocrystalline and Interfacial Materials, which is funded at the Massachusetts Institute of Technology (MIT) by the Office of Naval Research under Grant N00014-01-1-0808.
References
Fig. 9. (a) Numerically calculated and experimentally measured engineering stress strain curve for nc-Cu at a strain rate of 104 s1. (b)–(d) Contour plots of equivalent plastic shear strain at three different points, b,c and d, keyed to the stress–strain curve in (a). (e) A photograph of the experimentally measured localized neck reported by Cheng et al. [4].
behavior of nanocrystalline materials, which exhibit not only high strength but also a reasonable amount of ductility.
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