On the elastic–viscoplastic behavior of nanocrystalline materials

A new constitutive law is introduced to quantify the macroscopic effect of grain boundary dislocation emission on the behavior of pure face center cubic nanocrystalline materials. It is postulated that an emitted dislocation ends its trajectory in the grain boundary opposite to the source causing mass transfer. Dislocation emission by grain boundary ledges, considered here as the primary grain-boundary sources, is modeled as a thermally activated mechanism and the penetration of an emitted dislocation is assimilated as a soft collision. The macroscopic behavior of the material is retrieved via the use of a secant self-consistent scheme. The material is seen as a two-phase composite where the inclusion phase represents grain cores, their behavior is driven by dislocation glide, and where the matrix phase, governed by the newly introduced dislocation emission and penetration mechanism, represents both grain boundaries and triple junctions. The long range stress field arising from the presence of grain boundaries is taken into account in the critical glide resistance stress at 0 K in the inclusion phase. The model is applied to polycrystal copper and results in pure tension and creep are compared to experiments. Good agreements between the experimental measurements and the model predictions are observed.     

Mechanics of very fine-grained nanocrystalline materials with contributions from grain interior, GB zone, and grain-boundary sliding

In this paper, we formulated an atomically-equivalent continuum model to study the viscoplastic behavior of nanocrystalline materials with special reference to the low end of grain size that is typically examined by molecular dynamic (MD) simulations. Based on the morphology disclosed in MD simulations, a two-phase composite model is construed, in which three distinct inelastic deformation mechanisms disclosed from MD simulations are incorporated to build a general micromechanics-based homogenization scheme. These three mechanisms include the dislocation-related plastic flow inside the grain interior, the uncorrelated atomic motions inside the grain-boundary region (the GB zone), and the grain-boundary sliding at the interface between the grain and GB zone. The viscoplastic behavior of the grain interior is modeled by a grain-size dependent unified constitutive equation whereas the GB zone is modeled by a size-independent unified law. The GB sliding at the interface is represented by the Newtonian flow. The development of the rate-dependent, work-hardening homogenization scheme is based on a unified approach starting from elasticity to viscoelasticity through the correspondence principle, and then from viscoelasticity to viscoplasticity through replacement of the Maxwell viscosity of the constituent phases by their respective secant viscosity. The developed theory is then applied to examine the grain size- and strain rate-dependent behavior of nanocrystalline Cu over a wide range of grain size. Within the grain-size range from 5.21 to 3.28 nm, and the strain rate range from 2.5 × 10⁸ to 1.0 × 10⁹/s, the calculated results show significant grain-size softening as well as strain-rate hardening that are in quantitative accord with MD simulations [Schiotz, J., Vegge, T., Di Tolla, F.D., Jacobsen, K.W., 1999. Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. Phys. Rev. B 60, 11971–11983]. We have also applied the theory to investigate the flow stress, strain-rate sensitivity, and activation volume over the wider grain size range from 40 nm to as low as 2 nm under these high strain rate loading, and found that the flow stress initially displays a positive slope and then a negative one in the Hall–Petch plot, that the strain-rate sensitivity first increases and then decreases, and that the activation volume first decreases and then increases. This suggests that the maximum strain rate sensitivity and the lowest activation volume do not occur at the smallest grain size but, like the maximum yield strength (or hardness), they occur at a finite grain size. These calculated results also confirm the theoretical prediction of Rodriguez and Armstrong [Rodriguez, P., Armstrong, R.W., 2006. Strength and strain rate sensitivity for hcp and fcc nanopolycrystal metals. Bull. Mater. Sci. 29, 717–720] on the basis of grain boundary weakening and the report of Trelewicz and Schuh [Trelewicz, J.R., Schuh, C.A., 2007. The Hall–Petch breakdown in nanocrystalline metals: a crossover to glass-like deformation. Acta Mater. 55, 5948–5958] on the basis of hardness tests. In general the higher yield strength, higher strain rate sensitivity, and lower activation volume on the positive side of the Hall–Petch plot are associated with the improved yield strength of the grain interior, but the opposite trends displayed on the negative side of the plot are associated with the characteristics of the GB zone which is close to the amorphous state.     

The scale effect on the yield strength of nanocrystalline materials

The microstructure of the nanocrystalline can be divided generally into two parts: grain and interface. When the grain size is about or less than 10 nm, the interface can be divided into grain boundary and triple junctions. The mechanical performance of nanocrystalline materials with complicated microstructures is greatly different from that of the coarse grain materials. In this paper, the nanocrystalline material is considered as a composite with three phases: the grain core, the grain boundaries, and the triple junction. The model analysis for nanocrystalline material deformation is established and the relationship between yield strength and grain size is obtained. The obtained result explains the inverse Hall–Petch relation.     

Dislocation nucleation from bicrystal interfaces and grain boundary ledges: Relationship to nanocrystalline deformation

Molecular dynamics simulations are used to evaluate the primary interface dislocation sources and to estimate both the free enthalpy of activation and the critical emission stress associated with the interfacial dislocation emission mechanism. Simulations are performed on copper to study tensile failure of a planar Σ5 {2 1 0} 53.1° interface and an interface with the same misorientation that contains a ledge. Simulations reveal that grain boundary ledges are more favorable as dislocation sources than planar regions of the interface and that their role is not limited to that of simple dislocation donors. The parameters extracted from the simulations are utilized in a two-phase composite mesoscopic model for nanocrystalline deformation that includes the effects of both dislocation emission and dislocation absorption mechanisms. A self-consistent approach based on the Eshelby solution for grains as ellipsoidal inclusions is augmented by introduction of stress concentration in the constitutive law of the matrix phase to account for more realistic grain boundary effects. Model simulations suggest that stress concentration is required in the standard continuum theory to activate the coupled grain boundary dislocation emission and absorption mechanisms when activation energy of the dislocation source is determined from atomistic calculation on grain boundaries without consideration of impurities or other extrinsic defects.     

A secant-viscosity composite model for the strain-rate sensitivity of nanocrystalline materials

In order to address the strain-rate sensitivity of nanocrystalline solids, a secant-viscosity composite model is developed in this article. The microgeometry of the composite is taken to consist of the grain-interior phase and the grain-boundary affected zone (GBAZ) as suggested by Schwaiger et al. [Schwaiger, R., Moser, B., Dao, M., Chollacoop, N., Suresh, S., 2003. Some critical experiments on the strain-rate sensitivity of nanocrystalline nickel. Acta Mater. 51, 5159–5172], while the constituent properties are modeled by a unified viscoplastic constitutive law. The drag stress of the grain interior is assumed to follow the Hall–Petch relation, but that of the GBAZ is independent of grain size, d. Then in terms of the secant viscosity of the constituent phases, the strain-rate sensitivity of the nanocrystalline solid is determined with the help of a linear viscous comparison composite and a field-fluctuation approach. To test the applicability of the developed model, it is applied to predict the strain-rate effect of a nanocrystalline Ni, and the grain-size dependence of its stress–strain relations. Our theoretical calculations indicate that the tensile strength of a nanocrystalline Ni with d = 40 nm is about five times that of a microcrystalline one with d = 10 μm under the same strain rate of , and that the nanocrystalline Ni exhibits a much stronger strain-rate effect. These predictions are found to be consistent with the experimental data of Schwaiger et al. Possible grain-size softening with further grain-size reduction such as reported in molecular dynamic simulations is also demonstrated.     

Mechanical response and modeling of fully compacted nanocrystalline iron and copper

A comprehensive study on the response of nanocrystalline iron and copper to quasi-static and dynamic loading is reported. Bulk solid nanocrystalline iron and copper specimens used in static and dynamic loading experiments were made by compaction and hot sintering of the nanocrystalline powders. The powders, with grain size 16–96 nm, were obtained by using high energy ball milling. The stress/strain response of dense nanocrystalline iron is found to be grain size and strain rate dependent. The KHL model is modified by incorporating Hall–Petch relation (i.e. yield stress dependence on grain size) and is used to represent the behavior of fully compacted nanocrystalline material. A good correlation with the experimental results is demonstrated.     

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall-Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall-Petch slope turns negative in the nano-range. It is also found that, when the Hall-Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.     

Mechanically alloyed nanocrystalline iron and copper mixture: behavior and constitutive modeling over a wide range of strain rates

A comprehensive study on the response of a nanocrystalline iron and copper mixture (80% Fe and 20% Cu) to quasi-static and dynamic loading is performed. The constitutive model developed earlier by Khan, Huang & Liang (KHL) is extended to include the responses of nanocrystalline metallic materials. The strain rate and grain size dependent behaviors of porous nanocrystalline iron-copper mixture were determined experimentally for both static and dynamic loading. A viscoplastic model is formulated by associating the modified KHL model (representing the fully dense matrix behavior), and Gurson's plastic potential which provides the yield criteria for porous material. Simulations of uniaxial compressive deformations of iron-copper mixture with different initial porosity, grain size and at a wide range of strain rate (10⁻⁴ to 10³ s⁻¹) are made. The numerical results correlate well with the experimental observations.     

Implications of rainfall temporal resolution for soil-moisture and transpiration modeling

Dimensionless groups of parameters characterizing an ecosystem are valuable indicators for the a priori assessment of the effect of rainfall data resolution on predictions of soil moisture and transpiration. Knowledge of these dimensionless groups enables identification of appropriate levels of rainfall data resolution, when using historical rainfall directly or when using it to derive rainfall model parameters for use in models of soil–plant–climate systems. Detailed simulation studies of the soil, plant, and climate systems in Colorado and Texas, highly resolved in time and vertical space, show that historical rainfall data resolved at the daily level allow accurate prediction of soil-moisture and transpiration dynamics for smaller time resolutions. These results support inferences based on the dimensionless groups. Furthermore, no significant improvement in the prediction of soil-moisture and transpiration dynamics is attained, when representing rainfall through a more complex Neyman–Scott model rather than the simple rectangular pulses Poisson model.     

Grain size,strain rate,and temperature dependence of flow stress in ultra-fine grained and nanocrystalline Cu and Al: Synthesis,experiment, and constitutive modeling

For the first time, high quality bulk nanocrystalline (nc) fcc metals, with least amounts of imperfections, exhibiting high strength and ductility at room and different temperatures, under quasi-static and dynamic types of loading, were prepared and a comprehensive study on their post-yield mechanical properties was performed. This investigation included study of the effect of temperature on stress–strain responses of mechanically milled bulk nc Cu and Al. The samples after preparation through mechanical milling and consolidation processes were subjected to uniaxial compressive loading at quasi-static and dynamic strain rates of 10⁻² s⁻¹ and 1840–3105 s⁻¹, respectively, at temperatures ranging from 223 to 523 K. In both materials strong dependency of flow stress to temperature was observed; this dependency was rather more pronounced when the materials were tested at the quasi-static strain rate. Further, a new grain size and temperature dependent viscoplastic phenomenological constitutive equation, Khan–Liang–Farrokh (KLF) model was developed based on the Khan–Huang–Liang (KHL) constitutive equation. The model was featured to correlate different characteristic behaviors of polycrystalline materials in the plastic regime, as the result of grain refinement. In addition, the viscoplastic responses of bulk Cu and Al of different grain sizes (from sub-micron to nanometer range), and those from bulk nc Cu and Al at different strain rates (quasi-static to dynamic), recently published (21 and 22), were simulated using the newly developed equation. The results confirmed reasonable capability of the developed model to correlate a wide spectrum of the viscoplastic responses of these fcc metals.     

纳米晶铜单向拉伸变形的分子动力学模拟

纳米材料是由尺度在1－100nm的微小颗粒组成的体系，由于它具有独特的性能而备受关注。本文简要地回顾了分子动力学在纳米材料研究中的应用，并运用它模拟了平均晶粒尺寸从1．79－5．38nm的纳米晶体的力学性质。模拟结果显示：随着晶粒尺寸的减小，系统与晶粒内部的原子平均能量升高，而晶界上则有所下降；纳米晶体的弹性模量要小于普通多晶体，并随着晶粒尺寸的减小而减小；纳米晶铜的强度随着晶粒的减小而减小，显示了反常的Hall-Petch效应；纳米晶体的塑性变形主要是通过晶界滑移与运动，以及晶粒的转动来实现的；位错运动起着次要的、有限的作用；在较大的应变下（约大于5％），位错运动开始起作用；这种作用随着晶粒尺寸的增加而愈加明显。     

The Effects of Compressibility on Inhomogeneous Deformations for a Class of Almost Incompressible Isotropic Nonlinearly Elastic Materials

Experimental data for simple tension suggest that there is a power–law kinematic relationship between the stretches for large classes of slightly compressible (or almost incompressible) non-linearly elastic materials that are homogeneous and isotropic. Here we confine attention to a particular constitutive model for such materials that is of generalized Varga type. The corresponding incompressible model has been shown to be particularly tractable analytically. We examine the response of the slightly compressible material to some nonhomogeneous deformations and compare the results with those for the corresponding incompressible model. Thus the effects of slight compressibility for some basic nonhomogeneous deformations are explicitly assessed. The results are fundamental to the analytical modeling of almost incompressible hyperelastic materials and are of importance in the context of finite element methods where slight compressibility is usually introduced to avoid element locking due to the incompressibility constraint. It is also shown that even for slightly compressible materials, the volume change can be significant in certain situations.      

Micromechanics of plastic deformation and phase transformation in a three-phase TRIP-assisted advanced high strength steel: Experiments and modeling

The micromechanics of plastic deformation and phase transformation in a three-phase advanced high strength steel are analyzed both experimentally and by microstructure-based simulations. The steel examined is a three-phase (ferrite, martensite and retained austenite) quenched and partitioned sheet steel with a tensile strength of ~980 MPa. The macroscopic flow behavior and the volume fraction of martensite resulting from the austenite–martensite transformation during deformation were measured. In addition, micropillar compression specimens were extracted from the individual ferrite grains and the martensite particles, and using a flat-punch nanoindenter, stress–strain curves were obtained. Finite element simulations idealize the microstructure as a composite that contains ferrite, martensite and retained austenite. All three phases are discretely modeled using appropriate crystal plasticity based constitutive relations. Material parameters for ferrite and martensite are determined by fitting numerical predictions to the micropillar data. The constitutive relation for retained austenite takes into account contributions to the strain rate from the austenite–martensite transformation, as well as slip in both the untransformed austenite and product martensite. Parameters for the retained austenite are then determined by fitting the predicted flow stress and transformed austenite volume fraction in a 3D microstructure to experimental measurements. Simulations are used to probe the role of the retained austenite in controlling the strain hardening behavior as well as internal stress and strain distributions in the microstructure.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.     

Rheological characterization and constitutive modeling of bread dough

Bread dough (a flour–water system) has been rheologically characterized using a parallel-plate, an extensional, and a capillary rheometer at room temperature. Based on the linear and nonlinear viscoelastic and viscoplastic data, two constitutive equations have been applied, namely a viscoplastic Herschel–Bulkley model and a viscoelastoplastic K–BKZ model with a yield stress. For cases where time effects are unimportant, the viscoplastic Herschel–Bulkley model can be used. For cases where transient effects are important, it is more appropriate to use the K-BKZ model with the addition of a yield stress. Finally, the wall slip behavior of dough was studied in capillary flow, and an appropriate slip law was formulated. These models characterize the rheological behavior of bread dough and constitute the basic ingredients for flow simulation of dough processing, such as extrusion, calendering, and rolling.     

Multi-level modeling of woven glass/epoxy composite for multilayer printed circuit board applications

The objective of this paper is to develop a hybrid homogenization method to predict the elastic properties of a common woven glass/epoxy composite substrate for multilayer circuit board applications. Comprehensive high resolution 3D finite element (FE) models of a quarter of the repeated unit cell (RUC) for the woven glass/epoxy composite were developed based on different micromechanical schemes. . Specifically, four different micromechanics schemes were investigated: self-consistent, Mori–Tanaka, three-phase approach and composite cylinder assemblage (CCA). The element based strain concentration matrices were determined and used to obtain the homogenized woven glass/epoxy composite properties via a specially developed MATLAB code. Attention was further devoted to the predictions of the homogenized elastic moduli of the multilayer printed circuit board (PCB). The results from our simulations, based on Mori–Tanaka and CCA, are in good agreement with existing experimental results, indicating that the newly proposed homogenization scheme can be used as a design tool to predict the overall properties of woven composite materials typically used in multilayer PCB applications.     

三轴应力状态下混凝土的一种新强度准则

提出了一种新的三轴应力状态下的混凝土强度准则,它的理论基础是最小耗能原理.对不同的混凝土材料和不同的受力状态,该文给出的强度准则与实验值均吻合较好.     

A Variational Principle for the Revised Goodman–Cowin Theory with Internal Length

In the present study a variational principle is proposed for the revised Goodman–Cowin theory with internal length for cohesionless granular materials (Fang et al. in Continuum Mech Thermodyn in press). The balance equations of the internal variables employed in the theory in equilibrium states, the equilibrium expressions of the constitutive variables and the corresponding natural boundary conditions are derived by use of the proposed variational principle for both cases of compressible and incompressible grains. It is demonstrated that the derived results coincide with those obtained by use of the thermodynamic analysis. The current work serves as a supplementary variational verification of the constitutive theory proposed in Fang et al. (in Continuum Mech Thermodyn in press).     

On the yield functions in Lagrangian stress space of elastic–plastic materials with internal constraints

In the present paper it is shown that the elastic range in the second Piola–Kirchhoff stress space can be chosen in a hyperplane which is through the origin of Lagrangian stress space and perpendicular to the normal of the constraint manifold at the plastic configuration, if the determinate stress response of the elastic–plastic material with simple internal constraints with some condition is correctly chosen, otherwise, it is in general in a hypersurface and the normal flow rule by Il yushin’s postulate will have an indeterminate part. The choice of determinate stress response is probable because of its indeterminacy. Therefore the yield function should be a function of the second Piola–Kirchhoff stress lying in the hyperplane so that it is more simple and the back stress as the geometric center of the elastic range in general is inside the elastic range. Finally some examples are concerned. The project supported by the National Natural Science Foundation of China (10272055).