Identification of the model describing viscoplastic behaviour of high strength metals

Abstract

Ultrafine grained (UFG) and nanocrystalline metals (nc-metals) are studied. Experimental investigations of the behaviour of such materials under quasistatic as well as dynamic loading conditions related with microscopic observations show that in many cases the dominant mechanism of plastic strain is a multiscale development of shear deformation modes. The comprehensive discussion of these phenomena in UFG and nc-metals is given in M.A. Meyers, A. Mishra and D.J. Benson [Mechanical properties of nanocrystalline materials, Progr. Mater. Sci. 51 (2006), pp. 427–556], where it has been shown that the deformation mode of nanocrystalline materials changes as the grain size decreases into the ultrafine region. For smaller grain sizes (d < 300 nm) shear band development occurs immediately after the onset of plastic flow. Significant strain-rate dependence of the flow stress, particularly at high strain rates, was also emphasized. Our objective is to identify the parameters of Perzyna constitutive model, a new description of viscoplastic deformation, which accounts for the observed shear banding. The viscoplasticity model proposed earlier by Perzyna [Fundamental problems in viscoplasticity, Adv. Mech. 9 (1966), pp. 243–377] was extended in order to describe the shear banding contribution in Z. Nowak, P. Perzyna, R.B. Pȩcherski [Description of viscoplastic fow accounting for shear banding, Arch. Metall. Mater. 52 (2007), pp. 217–222]. The shear banding contribution function, which was introduced formerly by Pȩcherski [Modelling of large plastic deformation produced by micro-shear banding, Arch. Mech. 44 (1992), pp. 563–584] and applied in continuum plasticity accounting for shear banding in R.B. Pȩcherski [Macroscopic measure of the rate of deformation produced by micro-shear banding, Arch. Mech. 49 (1997), pp. 385–401] plays pivotal role in the viscoplasticity model. The derived constitutive equations were identified and verified with the application of experimental data provided in the article by D. Jia, K.T. Ramesh and E. Ma [Effects of nanocrystalline and ultrafne grain sizes on constitutive behavior and shear bands in iron, Acta Mat. 51 (2003), pp. 3495–3509], where quasistatic and dynamic compression tests with UFG and nanocrystalline iron specimens of a wide range of mean grain size were reported. Numerical simulation of the compression of the prismatic specimen was made by the ABAQUS FEM program with UMAT subroutine. Comparison with experimental results proved the validity of the identified parameters and the possibilities of the application of the proposed description for other high strength metals.

Keywords:

• ultra fine grained metals
• nanocrystalline metals
• viscoplastic deformation
• shear banding
• shear banding contribution function
• numerical simulation of compression test
• identification of the model parameters

AMS Subject Classifications:

• 74C20
• 74S05
• 74P99
• 74H05

1. Introduction

The subject of this study is concerned with ultrafine grained (UFG) and nanocrystalline metals (nc-metals). Experimental investigations of the behaviour of such materials under quasistatic as well as dynamic loading conditions related with microscopic observations presented in 1–3 show that in many cases the dominant mechanism of plastic strain is multiscale development of shear deformation modes – called shear banding. The comprehensive discussion of these phenomena in UFG and nc-metals is given in 2,3 where it has been shown that the deformation mode of nanocrystalline materials changes as the mean grain size decreases into the ultrafine region. For smaller grain sizes (d < 300 nm) shear band development occurs immediately after the onset of plastic flow. Significant strain-rate dependence of the flow stress, particularly at high strain rates was also emphasized.

The aim of this article is to propose a new description of viscoplastic deformation, which accounts for the observed shear banding. A viscoplasticity model with an overstress function proposed earlier by Perzyna 4–6 was extended. The theoretical description of multiscale hierarchy of shear localization modes presented by Pȩcherski 7–9 and the new concept of shear banding contribution function introduced in 7,8 and identified for polycrystalline Cu in 10,11 were applied by that. The derived constitutive equations were identified with the application of experimental data of quasistatic and dynamic compression tests, made for the UFG and nanocrystalline iron specimens of different mean grain sizes described in 3.

The identification procedure was made under the assumption that the Huber–Mises yield condition obeys. This is a rather big oversimplification in the situations when the strength differential effect (strength asymmetry) is observed. Such an effect is observed particularly in the case of nc-metals, where the ratio of yield strengths in compression versus yield strength in tension depends on grain size and can reach 1.6 12. More detail and a comprehensive study of the strength asymmetry and related pressure sensitivity effect on yield strength in amorphous alloys is given in 13. Some attempts in the application of Coulomb–Mohr and Drucker–Prager criteria to account for the mentioned effects were also discussed by the authors in 12–14. Although our analysis is based on the Huber–Mises criterion, this article is concluded with discussion of the application of the paraboloidal yield criterion proposed originally by Burzyński 15 and rediscovered many times by others 16.

2. Physical motivation

Experimental investigations discussed, e.g. by Meyers et al. 2 show that nanocrystalline materials exhibit very high yield strength. A conventional soft metal can acquire a 10-fold increase in strength when the mean grain size approaches the nanoscale, presumably due to the grain-boundary strengthening known as the Hall–Petch effect. For example, strengths as high as 1.0 GPa in nc-copper and 2.5 GPa in nc-iron have been reported. Constitutive models for metallic materials must account for the effects of the rate of deformation. Significant strain-rate dependence of the flow stress has been observed in many pure metals, particularly at high strain rates. However, very few results have been reported on the high-strain-rate behaviour of nc- or UFG materials. The work of Jia et al. 3 on an 80 nm-Fe found little strain-rate dependence in the strain rate range from 1 × 10⁻⁴ to 3 × 10⁺³ s⁻¹. Localized deformation has been reported for nc-metals by several groups; typically this is associated with macroscopic perfect plasticity or even apparent strain softening. Localization of plastic deformation into shear bands was observed in nc-Fe–10%Cu 1 and nc-Fe 3. Malow and Koch 17 observed shear bands in nc-Fe samples after micro hardness tests. The observations reported in 2 and in the work of Jia et al. 3 indicate that there is a transition from uniform to non-uniform deformation as the grain size decreases down to the nanoscale, accompanied possibly by a reduction of strain hardening.

3. Experimental observations of deformation behaviour of nanomaterials

3.1. Quasi-static compression

In the article by Jia et al. 3, compression tests were performed to obtain full stress–strain curves over a wide range of strain rates. The typical specimen dimensions for the low and high strain rate tests were 2.2 × 2.2 × 3.5 mm (length) and 1.6 × 1.6 × 1.4 mm (length), respectively. The quasi-static compression tests at strain rates of 1 − 2 × 10⁻³ s⁻¹ were performed using a screw-driven ATS machine. From the tests it is apparent that the yield strength increases with decreasing grain size. Compared with the 20 µm-Fe, the strength of the nano-Fe (80 nm) is increased by an order of magnitude. The strain hardening rate changes with the grain size. In the range of grain size from 20 µm to 980 nm, there is no marked change in the slope of the curves. However, there is a transition from strain hardening to apparent strain softening as the grain size changes from ∼1 µm to ∼300 nm. For grain sizes below 300 nm, apparent strain softening appears at a very low plastic strain. Ductile behaviour is observed at relatively large grain sizes, the samples with mean grain sizes smaller than 200 nm fail relatively early. It was demonstrated by Meyers et al. 2 and Jia et al. 3 that for bcc metals we do not expect to have a significant effect of mean grain size on the strain hardening, and so these observations indicate that a change of deformation mechanism has occurred at the smaller grain sizes. In the work of Jia et al. 3, the measured flow stresses (at a fixed strain of 4%) and the yield strengths are observed to satisfy the well-known Hall–Petch relationship (σ_y = σ₀ + K d^−1/2).

3.2. High strain rate compression

Experimental results of Jia et al. 3 also show that the little influence of the strain rate on the strain hardening is observed, which is typical for bcc metals. The termination of the high-rate stress–strain curves for 80 nm and 138 nm grain sizes represent specimen failure rather than unloading (Figure 3). The influence of the rate of deformation on the flow stress is also investigated by Jia et al. 3 for the entire range of grain sizes and strain rates, with the flow stresses plotted corresponding to a fixed strain of 4%. Jia et al. 3 observed that the smaller grain size materials are much stronger at low rates, but show less relative strengthening at high strain rates. An approach to understand mean grain size effects on the viscoplastic deformations and on the flow stress for bcc Fe is presented in the following sections.

3.3. Phenomenology of shear bands

The deformation mode of nc- and UFG materials changes dramatically as the grain size is decreased into the UFG range. In the 20 µm-Fe and 980 nm-Fe, the compressive deformations were uniform at all strain rates and no shear bands were evident under either the optical microscope or SEM. However, for all smaller grain sizes (d < 300 nm) shear band development was observed in 3 to occur immediately after the onset of plastic deformation, correlating to the observed change in apparent strain hardening at those grain sizes. Shear bands were observed during both quasistatic and high-rate deformations for these grain sizes.

It was demonstrated by Meyers et al. 2 that additional shear bands appear with increasing strain and that the newly generated shear bands have similar orientations (in the four possible shearing planes for these cuboidal specimens). Large numbers of shear bands are observed, rather than a single dominant band that lead to failure. Shear bands have been observed by Wei et al. 1,18 in both low and high strain-rate tests. In 3 it was observed that under dynamic loading, conventional polycrystalline iron did not exhibit localized deformation. Shear band populations were observed in all specimens with grain sizes d < 300 nm. It can be concluded that the shear bands play an important role in plastic deformation of UFG Fe.

4. Constitutive modelling

4.1. Viscoplasticity model of Perzyna

If the dependence on strain rate comes into play, the associated viscoplasticity flow law can be applied. (cf 4,5): (1) where D^vp is the rate of viscoplastic deformation, τ′ denotes deviatoric Kirchhoff stress and is the viscoplastic shear strain rate, while k is the corresponding quasistatic yield shear strength.

For Huber–Mises yield condition: J₂ − k = 0, , (2) and (3) where σ ′ is the deviatoric Cauchy stress and k is the shear yield strength while and D denote material constants. The shear strain rate (2) is controlled by an overstress function to be specified for a particular material. In our case the power-like overstress function is assumed.

4.2. Multiscale system of shear bands contribution in plastic flow

The description of a multiscale system of shear bands contribution in plastic flow was given in 8,9. The rate of inelastic deformation was assumed in the form (4)

The scalar shear banding contribution function was defined: (5) where A is a symmetric second order tensor, e.g. or D^vp, while is the rate of viscoplastic deformation produced by dislocation mediated crystallographic slips and denotes the rate of viscoplastic deformation produced by shear banding. As discussed in 7, among many possible realizations of shear banding, one can single out the group of processes characterizing with the same contribution of two symmetric shear banding systems . In the case of proportional loading paths, the total viscoplastic shear strain rate can be expressed as follows: (6) and due to , we have (7) what leads, according to (2), to the following constitutive relation for the viscoplastic shear strain rate controlled by the discussed mechanisms of crystallographic slip and shear banding: (8) Inverting (8) gives the relation for the dynamic yield condition (9)

For the compression test considered in 3, the following specification of quasi-static yield strength can be proposed: (10) where and ϵ^vp is the equivalent plastic strain . The symbols A(d) and B(d) denote the values which are dependent on mean grain diameter d, e.g. according to Hall–Petch relation. A is the quasi-static initial yield strength, B(d)(ϵ^vp)ⁿ corresponds to a plastic hardening function and n corresponds to plastic hardening parameter. Furthermore, the viscosity parameter and the contribution function f_SB are assumed to be dependent on the mean grain diameter d.

The shape of the contribution function f_SB is proposed, accounting to the studies in 7,11 supported by the numerical identification in 10,1 in the form of logistic function: (11) where are material parameters to be specified.

According to the discussion in 19, the effect of shear banding on yield strength can be described theoretically by means of the analysis of the plastic deformation of representative volume element, in which the mechanisms of dislocation slip and shear banding are operative. It appears that the effect of shear banding can be described by the instantaneous contribution function f_SB defined in Equation (7) on the one hand and by the volumetric contribution function on the other hand (cf. 19). For simple processes, without change of deformation path, the proportionality can be assumed for simplicity. Therefore the final dynamic yield strength in uniaxial compression σ_Yd reads: (12)

5. Identification of the model

The identification of the constitutive models parameters is obtained by an inverse method. A parametric identification program is developed, based on a conjugate gradients algorithm (cf. 20,21). Our objective is to identify the parameters of the Perzyna viscoplastic constitutive model parameters: . The identification of these constants is carried out by means of compression true stress–strain diagrams. These curves stem from the iron experimental tests performed at various strain rates and various average grain sizes d (cf. 3).

5.1. Criterion of minimization

The criterion or cost function is the part of the program where the parameter vector β appears. The criterion chosen is the quadratic sum of errors. The method of least squares requires the residual sum in stress between the experimental observations and model results to be minimized. This sum is made with every experimental point. This implies that the proposed model of the grain size-dependent viscoplasticity can be completely calibrated by minimizing the residual: (13) where F(x) refers to the residual of the constitutive model and the experimental data with the number of experimental points α and denotes a vector of unknown material parameters to be determined.

Furthermore, are discrete values of the strains ϵ^vp. The symbols and denote the experimental and calculated stresses for the same strain level , N_α is the number of stress–strain data for the test with given strain rate and grain size. For our constitutive equation Equation (12) we have (14)

5.2. Gradient vector of the cost function

The gradient vector computation is performed with the central derivative formulation. The expression of the numerical gradient vector is (15)

The application of the standard derivative analytical formulation of a multi-variable function gives the theoretical gradient vector. In the case of the Perzyna model, the theoretical gradient vector is given as follows: (16)

It implies that for one material point for N_p measurements of (ϵ^vp)ⁱ and (σ_exp)ⁱ with known , the gradient vector g is given.

5.3. Termination test

As for the criterion, there are numerous formulations to stop the conjugate gradient algorithm. The termination test employed for the identification program consists of computing the following relation for each experimental point: (17) and for each parameter: (18) with β_i, the parameter under consideration, and j, the iteration number.

The termination test is satisfied when every point verifies: (19) and when every parameter verifies (20) with χ_ref, the point accuracy, η_ref, the parameters accuracy. The identification error limits are given by the Equations (17) and (19). The second set of equations (18) and (20) gives parameter stability relation.

5.4. Results of identification for quasi-static and dynamic compression tests

To evaluate the quality of the identified parameters, a simulation of compression test is performed for the prismatic sample and compared with the corresponding test. The ABAQUS FEM code was employed to realize this simulation. Simulation leads to plastic strain and strain rates in the same range than experimentation. An example of the application of the proposed constitutive description for modelling of the behaviour of polycrystalline iron under quasistatic and dynamic compression tests for experimental data of Jia et al. [3] is depicted in Figures 2 and 3, where the compressive yield strength .

In the case of quasistatic and dynamic compression, we use the following forms of Equation (14):

when d > 300 nm, f_SB = 0, (21) and when d < 300 nm, f_SB > 0, B = 0, =0.95, a = 5, D = 0.08 and n = 0 (no hardening) and (22)

Perzyna model parameters are determined for each kind of specimen with different average grain size. In each case we have started our computations assuming at the beginning a broad range of feasible parameters. For instance, the initial parameters values in cases for different average grain size: d = 20 µm, d = 980 nm, d = 268 nm, d = 138 nm and d = 80 nm for dynamic tests we have taken are presented in Table 1.

Table 1. The initial parameters values for dynamic tests.

Case
d=20 µm:	100.0	≤ A ≤	1000.0,	50.0	≤ B ≤	1000.0,	0.01	≤ n ≤	1.5
	1.0 × 10^+3		1.0 × 10^+5,	0.01	≤ D ≤	0.3,
d=980 nm:	100.0	≤ A ≤	1000.0,	100.0	≤ B ≤	1000.0,	0.01	≤ n ≤	1.5
	1.0 × 10^+4		1.0 × 10^+7,	0.01	≤ D ≤	0.3,
d=268 nm:	500.0	≤ A ≤	2000.0,	10.	≤ b ≤	100.0,
	1.0 × 10^+5		1.0 × 10^+7,	0.01	≤ D ≤	0.3
d=138 nm:	500.0	≤ A ≤	3000.0,	50.0	≤ b ≤	500.0,
	1.0 × 10^+5		1.0 × 10^+7,	0.01	≤ D ≤	0.3
d=80 nm:	1000.0	≤ A ≤	5000.0,	50.0	≤ b ≤	500.0,
	1.0 × 10^+5		1.0 × 10^+7,	0.01	≤ D ≤	0.3

At the very least, the identification leads to one set of model parameters available for quasi-static strain rates and dynamic strain rates. In the numerical simulations of compression tests by ABAQUS 22, the specimen is completely modelled with 226,991 nodes and 216,004 solid elements (type C3D8R in Explicit or C3D8 in Standard). The specimen is supported by a fixed rigid wall and is impacted by a second moving one with an imposed velocity of 5 m s⁻¹. Rigid walls are chosen as infinite planes with infinite mass and finite friction (f_c = 0.005). The undeformed sample mesh and the Mises stress distribution for dynamical compression are presented in Figure 1(a) and (b).

Figure 1. (a) The mesh used in numerical simulations and (b) Mises stress distribution for dynamic compression test of iron for ϵ^vp = 0.08 and d = 268 nm.

Figure 2. True stress–true strain for quasistatic compression test for polycrystalline iron. Continuous lines represent curves obtained from viscoplasticity model accounting for shear bands according to Equation (21) for d > 300 nm and according to Equation (22) for d < 300 nm, symbols ⋄ correspond to the quasistatic experimental data for iron of purity 99.9% obtained in two-step consolidation procedure to form bulk Fe with desired grain size from 3.

Figure 3. True stress – true strain for dynamic compression test for polycrystalline iron. Continuous lines represent curves obtained from viscoplasticity model accounting for shear bands according to equations (21) for d > 300 nm and according to equation (22) for d < 300 nm, symbols ⋄ correspond to the dynamic experimental data for iron of purity 99.9% obtained in two-step consolidation procedure to form bulk Fe with desired grain size from 3.

Simulation leads to plastic strain and strain rates in the same range as experimentation. Figures 2 and 3 compare the experimental and simulated stress–strain diagrams for a 5 m s⁻¹ impact test. These results show that identification of constitutive models for nano-iron with shear bands can be performed even with a plastic strain range of about 0.1.

Finally, the following constitutive parameters are found for the quasi-static and dynamic compression tests when d < 300 nm, f_SB > 0, B = 0, a = 5,  = 0.95, D = 0.08 and n = 0 (no hardening) and for quasi-static compression and for dynamic compression (Table 2).

Table 2. The identified constitutive parameters when d < 300 nm.

	Quasi-static		Dynamic
d (nm)	A(d)(MPa)	b(d)	A(d)(MPa)	b(d)
80	2364	50.24	2087	56.49	145.0 × 10^+6
138	2048	52.85	1882	55.11	50.5 × 10^+6
268	1160	27.46	1237	30.01	50.1 × 10^+6

6. Conclusions

The proposed description of viscoplastic behaviour of high-strength metals, in particular UFG and nc-metals, can be extended, accounting for the application of more adequate yield criterion, which in the case of associated flow law, also provides the appropriate potential function. According to our recent studies 23, the yield criterion, which is adequate to strength materials and multiphase materials, e.g. metal matrix and ceramic matrix composites reinforced by particles of nano and micro size should be pressure dependent, as it was discussed in the early papers of Burzyński 15,24. Such criteria take, in the principal axes, the shape of a rotationally symmetric paraboloid for isotropic materials and ellipsoidal paraboloid for materials having orthotropic symmetry 23,25, Figure 4(a) and 4(b).

Figure 4. (a) Burzyński yield limit for plane stress approximating the results presented in 14; (b) Burzyński yield surface in the principal axes coordinates calculated for the data in plane stress given in (a).

This article treats the inverse problem for the viscoplastic solids. The problem of uniqueness of the solution of inverse boundary value problem is very crucial. There are several challenging problems in this subject.

Acknowledgements

Part of this work was completed within the framework of the Research Project NN501 0364 35 from the Ministry of Higher Education and Science of Poland.

Notes

1. The proposed specification of the contribution function f_SB is suitable for the description of proportional loading processes. More general formula, applicable for arbitrary states of deformation and loading paths, was proposed in 26.