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.     

Strain-rate history effects and microstructural aspects in lithium fluoride and aluminium single crystals after dynamic loading

Single crystals of LiF and Al are deformed in shear at a number of constant strain-rates in the range 10^–4 to 1600 s^–1. These constant rate tests are supplemented by a series of jump tests in which a sharp increment in strain rate is imposed during the quasi-static straining. Dislocation arrangements are observed by etch-pits technique for LiF crystals and by TEM for Al crystals. It is shown that cell sizes vary inversely with flow stress and strain-rate sensitivity.     

Thermomechanical model of elastic–plastic materials with defect structures

A complete thermomechanical model for elastic–plastic materials that account for the interaction of defects in structures is proposed. The characteristics of affine-metric space can be used as thermodynamic variables for constructing the model. Applications to describe the pattern of fracture for underground mines are made.     

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.     

Nonlinear analysis of beam–column joints using softened truss model

The aim of this study is to expand the application of the nonlinear softened truss model for membrane elements on beam–column joints. The softened truss model employs three equations for equilibrium, three for compatibility and four equations for the constitutive laws of materials. The constitutive equations for both the concrete and steel are based on the actually observed stress–strain relationships. The model has three important attributes. The first is the nonlinear association of stress and strain. The second, and conceivably more noteworthy, is the softening of concrete in compression due to tensile strains in the perpendicular direction. The third is that the influence of the concrete tensile stresses between cracks on the average stress–strain relationship for reinforcing steel and the influence of orthogonal tensile stresses on the compression stress–strain relationship for concrete can be considered in the model. For beam–column joints, one of the most important factors influencing the behaviour is certainly the bond conditions of the beam bars. In this study, the softened truss model is expanded to take into account the influence of this important factor into account. In the revised version of the model, full strain compatibility does not exist between the steel reinforcement and the surrounding concrete and thus the factors influencing the bond-slip between concrete and reinforcement is adequately considered. The improved softened truss model is applied on 51 exterior beam–column joint tests. It is apparent from the results that the revised model gives very accurate predictions of the shear strength of joints and is an improvement on the existing version of the model proposed by Hsu.     

Scale-dependent constitutive relations and the role of scale on nominal properties

Size effects in strength and fracture energy of heterogeneous materials is considered within a context of scale-dependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a one-dimensional solid, constitutive relations which include scale as a parameter are derived from a ‘background’ gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stress–strain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stress–strain diagram integrated along the length of the considered one-dimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scale-dependent constitutive relations are calibrated from experimental data on concrete specimens.     

Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys

The results from a systematic study of the response of a Ti–6Al–4V alloy under quasi-static and dynamic loading, at different strain rates and temperatures, are presented. The correlations and predictions using modified Khan–Huang–Liang (KHL) viscoplastic constitutive model are compared with those from Johnson–Cook (JC) model and experimental observations for this strain rate and temperature-dependent material. Overall, KHL model correlations and predictions are shown to be much closer to the observed responses, than the corresponding JC model predictions and correlations. Similar trend has been demonstrated for other titanium alloys using published experimental data [Mech. Mater. 33(8) (2001) 425; J. Mech. Phys. Solids 47(5) (1999) 1157].     

Spatio-temporal characteristics of the Portevin–Le Châtelier effect in austenitic steel with twinning induced plasticity

An experimental investigation of spatio-temporal characteristics of the Portevin–Le Châtelier (PLC) effect in austenitic steel with twinning induced plasticity (TWIP) is presented. Post-processing of high resolution digital images captured from specimens in quasi-static, room temperature tensile tests was conducted with a digital image correlation (DIC) method. This provided direct measurement of strain fields during all stages of the tests. Variable rate digital image capture, enabled with a custom image acquisition algorithm, guaranteed a suitable number of images recorded during serrations in load–time records. Nucleation, propagation, and morphology of individual PLC bands in both straight gage and tapered specimens were quantified with strain rate contours computed with a backward differentiation scheme. Time histories of strain evolution in the PLC band wakes were extracted from cumulative strain contours. Of the three types of PLC bands, only the continuously propagating Type A bands were observed. Band nucleation, which occurred at serration crests in flow curves derived from the DIC results, was not limited to regions of geometry-induced stress concentrations. Due to its importance in finite element springback predictions and to support theoretical model development of inelastic behavior in TWIP steel, we measured Young’s modulus variation with strain in periodic loading–unloading tests. Implications of the experimental results for theoretical modeling of the PLC effect in TWIP steel are discussed.     

准静态压缩应力-应变曲线测量方法的探索

本文介绍了用应变片直接测量材料的准静态应力-应变曲线的试验研究。在MTS810材料试验机上分别对93W、G50、砂浆等几种材料进行了准静态压缩试验。由于仪器的系统误差不能由MTSCod规准确的得到材料的真实的弹性变形。为此在试件的中部贴应变片得到材料的弹性变形,塑性变形仍旧由MTSCod规记录,从而得到试样的真实的应变,准确获得准静态压缩应力-应变曲线。试验结果表明:在试件中部贴应变片的方法能够准确得到该材料的杨氏模量;在试件两端垫块的刀口上安置Cod规可得到材料的应力应变曲线。两者的合理组合即可得到准确而完整的准静态压缩应力-应变曲线。试验中还发现准静态实验中试件的断面加工不平,偏心压缩等都会影响E的准确测量。     

Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem

A systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales is developed. The proposed computational framework is applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary value problem on the macroscale is derived by using the double scale asymptotic analysis in space and time. It is shown that an extra history-dependent long-term memory term introduced by the homogenization process in space and time can be obtained by solving a first order initial value problem. This is in contrast to the long-term memory term obtained by the classical spatial homogenization, which requires solutions of the initial-boundary value problem in the unit cell domain. The validity limits of the proposed spatial–temporal homogenized solution are established. Numerical example shows a good agreement between the proposed model and the reference solution obtained by using a finite element mesh with element size comparable to that of material heterogeneity.     

New insights on free energies and Saint-Venant’s principle in viscoelasticity

Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll’s definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation.In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space–time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space–time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy.Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case.The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.     

The mathematical simulation of various condensation regimes in turbulent isobaric jets

Mathematical models are considered and calculations made for flows in turbulent isobaric steam—air jets in the presence of condensation of the water vapor they contain. The models consist of gasdynamic equations for a turbulent jet, equations for a differential two-parameter model of turbulence, thermodynamic relations, and kinetic equations. A study is made of steam—air jets in a regime of condensation in equilibrium, when the flow region is broken down into zones of frozen flow and flow in equilibrium, described by the equations for a turbulent jet with the use of the traditional thermodynamic relations and of the thermodynamic relations for condensation in equilibrium. An analysis is made of the influence of pulsating motion on the kinetic parameters: rate of nucleation, the critical size of the nuclei, and rate of growth of the drops. It is shown that the rate of nucleation, determined from a quasilaminar averaging model, is several orders of magnitude less than the mean value obtained by averaging using the density distribution of the passive admixture concentration probability. A numerical study is made of the heterogeneous condensation in turbulent jets on extraneous particles entering from the nozzle. Kinetic equations are written down for the case when the rate of growth of the drops does not depend on their radius. A study is made of the dynamics of the transition of heterogeneous condensation from disequilibrium to equilibriumTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–67, January–February, 1985.     

Passive effects of rare-earth permanent magnets on flexible conductive structures

The paper presents a theoretical and experimental study of vibrating structures where paramagnetic or diamagnetic systems interact with rare-earth passive magnets.The theoretical model of the system is focused on the damping properties of permanent magnets and on their interactions with the dynamic behaviour of an Euler–Bernoulli beam. In particular, the magnetic model is based on the analogy of the equivalent currents method in a quasi-static open-circuit-type configuration and it is used to determine the influence of eddy currents on the dynamic behaviour of conducting material structures. The magnetic effects are characterised by a viscous-type damping and by a stiffening dynamic effect of the structure, called “phantom effect”.The authors present the experimental outcomes for uniform cantilever clamped-free beams of different kinds of paramagnetic or diamagnetic conducting materials. It appears that the system frequency response can be modified by the presence of a pair of concordant or discordant permanent magnets of high residual induction settled at the free end.Through the comparison between theoretical and experimental results, the paper demonstrates the validity of the model, that is able to describe both the above mentioned effect of dynamic stiffening of the structure and the considerable localised damping properties in paramagnetic or diamagnetic materials having low electric resistivity.     

Boussinesq–Flamant problem in gradient elasticity with surface energy

The strain gradient elasticity theory with surface energy is applied to Boussinesq–Flamant problem. The solution for the vertical displacements at the surface of half space due to the surface normal line load is presented. The theory includes both volumetric and surface energy terms. Boussinesq–Flamant problem in the strain gradient elasticity is solved by means of Fourier transform. The results obtained show that the vertical displacements of half space in the gradient elasticity are some different from that in the classical elasticity and the effects of the strain gradient parameters (material characteristic lengths) on the vertical displacements do exist.     

Analysis of worm-like locomotion driven by the sine-squared strain wave in a linear viscous medium

In this paper, a worm-like locomotion in a linear resistive medium is studied to achieve controlled shape changes of the worm-like body by choosing a kind of driving with low energy expended and high-velocity locomotion in certain condition. To this end, we first develop the full dynamic model of the system under consideration to obtain the mean velocity related to friction coefficient, wave speed, linear density, body length and wave width. Correspondingly, a quasi-static model is also given from which the velocity can be expressed analytically. In the case of the shape change driven by the sine-squared strain wave (SSSW), it is seen that these two velocities will tend to uniformity with the friction coefficient or length of the body increasing or the wave speed decreasing when keeping the other parameters unchanged. Thus, the inertia term is ignorable for a large friction, a long body-length but a small wave-speed of the SSSW, which implies that the dynamical model can be reduced to the quasi-static one. The relative criterion is approximately given. As a result, the corresponding quasi-static model is employed to consider two typical drives, namely, the SSSW and the square strain wave (SSW). The result shows the shape change driven by the SSSW has an advantage in both the mean velocity and the average energy expended over that by the SSW when the necessary condition is satisfied. The analytical results are verified by numerical simulation.     

A unified plasticity model for cyclic behaviour of clay and sand

This paper presents the development and an experimental evaluation of a simple unified bounding surface plasticity theory for modelling the stress–strain behaviour of sand and clay under both drained and undrained cyclic loading conditions. The model concerned is called CASM-c, which is based on the unified critical state model CASM developed by Yu [Yu, H.S., 1995. A unified critical state model for clay and sand. Civil Engineering Research Report No. 112.08.1995. University of Newcastle, NSW 2308, Australia; Yu, H.S., 1998. CASM: a unified state parameter model for clay and sand. International Journal of Numerical and Analytical Methods in Geomechanics 22, 621–653]. CASM is a relatively simple model as it only requires seven model constants, five of which are the same as those used in the modified Cam-clay model. All these constants have clear physical meanings and may be easily determined from the results of triaxial tests. A key advantage of CASM over many other existing critical state models lies on its simplicity and unified nature as it can model the behaviour of both clay and sand.The extension of the model CASM presented in this paper consists of adopting the bounding surface plasticity theory and treating the reloading and unloading processes differently when calculating the hardening modulus. As a result, a smooth transition of stiffness and gradual accumulation of permanent strain and/or pore pressure in unload–reload cycles as well as the hysteretic behaviour can be reproduced. The results of model simulations show an encouraging agreement with experimental data from triaxial tests subjected to both one-way and two-way cyclic loading conditions.     

Heterogeneity of Plastic Flow of Zirconium Alloys with a Parabolic Law of Strain Hardening

The specific features of plastic–strain macrolocalization at the stage of the parabolic law of strain hardening in samples from industrial zirconium–based alloys are considered. It is shown that in predeformed blanks, zones with a different character of plastic–strain localization are formed. It is also shown that the strain–localization macropattern can be used as a characteristic of the susceptibility of a material to further plastic form–changing, for example, upon tube rolling. The sign of fracture of alloys upon plastic deformation is revealed. The scale effect in the formation of localizedplastic–flow zones is shown and studied.     

Hysteresis loops predicted by isoenergy density theory for polycrystals. Part I: fundamentals of non-equilibrium thermal–mechanical coupling effects

When a polycrystal is stressed or strained at fifty percent of the corresponding yield value, damage will be inflicted non-homogeneously in the material due to the fact that the stress and/or strain distribution is non-uniform even if isotropy and homogeneity are assumed for the initial microstructure. This effect will be cumulated for each cycle of the load if the applied stress or strain is repeated continuously. Nucleation of microcracks can eventually lead to the propagation of a macrocrack.The process of damage accumulation in fatigue is defined to be sufficiently slow such that inhomogeneity of material behavior created by loading is a significant factor that can not be arbitrarily dismissed without a good reason. What this means specifically is that the difference of the stress and strain behavior at each point in a fatigue specimen must be accounted for in the analytical model in order to predict the correct cumulative effect. Such a requirement translates into a non-equilibrium formulation where the constitutive relations for each point and loading cycle must be determined separately. In this sense, the true problem of fatigue cannot be completely treated by the classical continuum mechanics approach that is limited to equilibrium mechanics for a closed system. Having said this, the isoenergy density theory will be applied to estimate the hysteresis loops of a hour-glass profile cylindrical bar specimen as recommended by the American Society for Testing and Materials (ASTM) for low-cycle fatigue.The work will be divided into two parts. Part I will cover the fundamentals of a non-equilibrium theory where the continuum elements are finite in size; they do not vanish in the limit. Therefore, size effects are immediately encountered as a function of time. General expressions for the rate change of volume of these elements with surface area are derived such that they can be computed from the nine displacement gradients. These elements can differ in size and must fit together without discontinuities or gaps to form the continuum. The condition of isoenergy energy density is invoked such that the size of these individual elements under large and finite deformation and rotation can be determined without loss in generality. The existence of such a space having the property of the same isoenergy density in all directions is thus proved. This enables the establishment of the one dimensional energy state with that in three dimensions without restriction, the absence of which has prevented the development of a complete non-linear theory of mechanics that can be solved in a direct fashion in contrast to the inverse method of assuming the displacement field. Illustration is provided for deriving the constitutive relation incrementally for a given location for the hour-glass specimen made of 6061-T6 aluminum. Once the specimen is loaded, each material point will follow a different stress and strain curve according to the local displacement rate. Hence, the method applies to material with non-homogeneous microstructure if their individual expressions can be assessed and fed into the computer.Part II computes for the non-equilibrium temperature and an entropy-like quantity that can be positive and negative. This implies that the system can absorb or dissipate energy with reference to the surrounding. Additional data for hysteresis loops are given for 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium. Accumulation of the local hysteresis energy per cycle is found to be the highest near the surface of the uniaxial specimen where load symmetry prevails. This is a consequence of the difference in accumulation of the energy density due to distortion in contrast to dilatation at the specimen center. This is why fatigue cracks tend to nucleate near the specimen surface, at a small distance towards the interior. Another distinct feature of fatigue is that the non-equilibrium temperature is found to oscillate about the ambient temperature while the local stress states fluctuate between tension and compression. This temperature reversal behavior is typical of non-equilibrium behavior and also occurs under monotonic loading. The space and time variations of the dissipated energy density for different materials are found to be related to the initial monotonic energy density or area under the true stress and true strain curve.What will be demonstrated is that no special consideration need to be made when applying the isoenergy density theory for analyzing the nucleation of micro and macrocracks in addition to failure of the specimen. Crack nucleation under fatigue is assumed to occur when the total hysteresis energy reaches a critical value. It is possible to establish a relation between the average hysteresis energy per cycle and the number of cycles to failure. The proposed method requires only a knowledge of the initial monotonic energy density curve for a given material. Predicted results for the fatigue of cylindrical bar specimens with hour-glass profile are given and they can be found in Part II of this work.     

Modeling of the transient processes in the channels of an EHD pump

The nonstationary process of development of weakly conducting electrohydrodynamic flow in the channel of an EHD pump with plane permeable electrodes, between which a potential difference is created from an external source, is considered in the hydraulic approximation within the framework of the model proposed in [1, 2]. It is shown that the efficiency of the EHD device can be improved if in the fluid the spatial process of ion formation is retarded, while the recombination process is intense. The effect of the flow velocity on the formation of a space charge region in the interelectrode gap is investigated. On a certain range of the problem parameters the flow induced in the channel substantially modifies the space charge distribution as a result of the blowoff of narrow diffusion electrode ion layers. This creates a nonmonotonic dependence of the fluid velocity on the applied potential difference.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 30–41, May–June, 1994.