Certain basic issues in finite-deformation continuum plasticity

Summary I consider the kinematical implications embedded in the decomposition of the deformation gradient F into two constituents, F^* (say, elastic) and F^p (say, plastic), in the form F=F^* F^p. Whatever the origin, nature, and physical meaning of these three deformation gradients, the decomposition can always be recast in the form F==V^* Q U^p, where V^* and U^p are stretch tensors, respectively associated with F^* and F^p, and Q is a unique rotation tensor. It then follows that the (elastic) spin associated with V^* and the (plastic) spin associated with U^p are fully defined by their corresponding deformation rates. I give explicit expressions defining elastic and plastic spins in terms of the elastic and plastic deformation rates, respectively. I show that, if the total velocity gradient and either the elastic or the plastic (but not both) deformation rate tensor, are given, then all the remaining kinematical rate quantities can be calculated explicitly, with no constitutive or other assumptions. I give explicit expressions for the rate of change of V^*, Q, and U^p, from which these quantities can be incrementally but directly obtained. In particular, I give explicit expressions for the elastic spin, plastic spin, and the rigid spin. Constitutive implications of these purely kinematical results are briefly discussed in connection with rate-independent and rate-dependent phenomenological elastoplasticity, and, particularly, kinematical hardening.

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Sommario Si considerano le implicazioni cinematiche insite nella decomposizione del gradiente di deformazione F in due componenti, F^* (per esempio, la parte elastica) ed P^p (per esempio, la parte plastica), nella forma F=F^*F^p. Qualunque sia l'origine, la natura e il significato fisico di questi tre gradienti, la decomposizione si può sempre riscrivere nella forma F=U^* Q U^p, in cui V^* ed U^p sono tensori di deformazione associati rispettivamente con F^* ed F^p, e Q è un tensore di rotazione univocamente definito. Ne segue che lo spin (elastico) associato con U^* e lo spin (plastico) associato con U^p sono completamente definiti dalle corrispondenti derivate delle deformazioni: di questa dipendenza si forniscono le espressioni esplicite. Si mostra quindi che, se sono dati il gradiente della velocità totale e il tensore delle deformazioni elastiche o quello delle deformazioni plastiche (ma non entrambi), tutte le altre quantità cinematiche possono essere calcolate esplicitamente, senza alcuna ulteriore ipotesi, neanche sulla legge costitutiva. Si forniscono ancora espressioni per i gradienti di V^*, Q ed U^p, dalle quali queste quantità si possono ottenere direttamente, per incrementi successivi: in particolare, si danno espressioni esplicite per lo spin elastico, plastico e rigido. Si discutono brevemente le implicazioni costitutive di questi risultati puramente cinematici, in connessione con vari aspetti fenomenologici dell'elasto-plasticità, e in particolare l'incrudimento cinematico.

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Invited Lecture presented at New Developments in Structural Mechanichs, International Meeting in memory of Professor Manfredi Romano; Catania, Italy, 4–6 July 1990.     

Phase evolution of （K,Na）NbO3 powder prepared by high temperature mixing under hydrothermal conditions

（Na, K）NbO3 （KNN） powders were successfully prepared by high temperature mixing method （HTMM） under hydrothermal conditions to study the effect of reaction time on the formation of KNN for three K＋/（K＋ ＋Na＋） ratios of 0.6, 0.7 and 0.8. The products were characterized by X-ray diffraction （XRD）, scanning electron microscope （SEM）, transmission electron microscopy （TEM） and selected area electron diffraction （SAED）, to show the change of phase and morphology of the as-prepared particles with the K＋/（K＋ ＋ Na＋） molar ratio in the solution. Pure Na-rich KNN monoclinic phase and pure K-rich KNN orthorhombic phase could be obtained quickly after mixing the solutions at high temperature when the K＋/（K＋ ＋Na＋） molar ratio was either 0.6 or 0.8. When the K＋/（K＋ ＋Na＋） molar ratio was 0.7, however, the K-rich KNN orthorhombic phase grain formed first, followed by the Na-rich KNN monoclinic phase grain, with the two phases coexisting in the final product.     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

超强钢18NiC250在不同加载速率下的断裂韧性

为了揭示国产超强钢18NiC250的强度、断裂韧性随加载速率的变化规律,利用电子万能试验机和Hopkinson压杆,测试其在0.001～2 000 s-1的塑性流动应力应变曲线及在10-1~106 MPam1/2/s的断裂韧性,同时对断裂破坏机理进行了微观分析。结果表明:该材料的强度对加载速率不敏感,即流动应力基本保持在1.9 GPa;而断裂韧性很敏感,当加载速率由10-1 MPam1/2/s增大到106 MPam1/2/s时,断裂韧性降低了38.2%,断裂模式由韧窝断裂转变为解理断裂。     

A field model interpretation of crack initiation and growth behavior in ferroelectric ceramics: change of poling direction and boundary condition

Strain energy density expressions are obtained from a field model that can qualitatively exhibit how the electrical and mechanical disturbances would affect the crack growth behavior in ferroelectric ceramics. Simplification is achieved by considering only three material constants to account for elastic, piezoelectric and dielectric effects. Cross interaction of electric field (or displacement) with mechanical stress (or strain) is identified with the piezoelectric effect; it occurs only when the pole is aligned normal to the crack. Switching of the pole axis by 90° and 180° is examined for possible connection with domain switching. Opposing crack growth behavior can be obtained when the specification of mechanical stress σ^∞ and electric field E^∞ or (σ^∞,E^∞) is replaced by strain ε^∞ and electric displacement D^∞ or (ε^∞,D^∞). Mixed conditions (σ^∞,D^∞) and (ε^∞,E^∞) are also considered. In general, crack growth is found to be larger when compared to that without the application of electric disturbances. This includes both the electric field and displacement. For the eight possible boundary conditions, crack growth retardation is identified only with (E_y^∞,σ_y^∞) for negative E_y^∞ and (D_y^∞,ε_y^∞) for positive D_y^∞ while the mechanical conditions σ_y^∞ or ε_y^∞ are not changed. Suitable combinations of the elastic, piezoelectric and dielectric material constants could also be made to suppress crack growth.     

Dehydration characteristics and mathematical modelling of lemon slices drying undergoing oven treatment

In this study, the effect of drying temperature on drying behaviour and mass transfer parameters of lemon slices was investigated. The drying experiments were conducted in a laboratory air ventilated oven dryer at temperatures of 50, 60 and 75 °C. It was observed that the drying temperature affected the drying time and drying rate significantly. Drying rate curves revealed that the process at the temperature levels taken place in the falling rate period entirely. The usefulness of eight thin layer models to simulate the drying kinetics was evaluated and the Midilli and Kucuk model showed the best fit to experimental drying curves. The effective moisture diffusivity was determined on the basis of Fick’s second law and obtained to be 1.62 × 10^?11, 3.25 × 10^?11 and 8.11 × 10^?11 m² s^?1 for the temperatures of 50, 60 and 75 °C, respectively. The activation energy and Arrhenius constant were calculated to be 60.08 kJ mol^?1 and 0.08511 m² s^?1, respectively. The average value of convective mass transfer coefficient for the drying temperatures of 50, 60 and 75 °C was calculated to be 5.71 × 10^?7, 1.62 × 10^?6 and 2.53 × 10^?6 m s^?1, respectively.     

Determination of dynamic yield strengths with embedded manganin gages in plate-impact and long-rod experiments

The dynamic yield strengths of three steels were determined at strain rates of about 10³ s⁻¹ and 10⁶ s⁻¹. The measurements at 10³ s⁻¹ were obtained by a new technique based on measurements of large amplitude elastic waves in long bars struck by rigid flyer plates. Embedded manganin gages were used to measure stress, and the gage records were long enough to observe subsequent reverberations between the bar free end and the plastically deformed impact end. The measurements at 10⁶ s⁻¹ were made with a slightly modified version of a conventional flyer-plate impact configuration. The data are combined with static results to show the behavior of these steels at strain rates of 10⁻³ s⁻¹ to 10⁶ s⁻¹.     

Chemical characteristics and Pb isotopic compositions of PM2.5 in Nanchang,China

In mid-September 2013, PM_2.5 samples were collected at six sites in Nanchang, Jiangxi Province, China, to quantify nine water-soluble ions (Ca²⁺, Mg²⁺, K⁺, Na⁺, NH₄⁺, SO₄²⁻, Cl⁻, F⁻, NO₃⁻), 29 trace elements (Ba, Zn, Pb, Ni, Mo, Cr, Cu, Sr, Sb, Rb, Cd, Bi, Zr, V, Ga, Li, Y, Nb, W, Cs, Tl, Sc, Co, U, Hf, In, Re, Be, and Ta), and to characterize Pb isotopic ratios (²⁰⁷Pb/²⁰⁶Pb, ²⁰⁸Pb/²⁰⁶Pb, and ²⁰⁷Pb/²⁰⁴Pb) for identifying the main source(s) of Pb. The results showed that the average daily PM_2.5 concentration (53.16 ± 24.17) μg/m³ was within the secondary level of the Chinese ambient air quality standard. The combined concentrations of SO₄²⁻, NH₄⁺, and NO₃⁻ to total measured water-soluble ion concentrations in PM_2.5 ranged from 79.40% to 95.18%, indicating that anthropogenic sources were significant. Coal combustion and vehicle emissions were both contributors to PM_2.5 based on the NO₃⁻/SO₄²⁻ ratios. Wushu School experienced the lowest concentrations of PM_2.5 and most trace elements among the six sampling sites. Enrichment factor results showed that Tl, Cr, In, Cu, Zn, Pb, Bi, Ni, Sb, and Cd in PM_2.5 were affected by anthropogenic activities. Cluster analysis suggested that Cd, Sb, Pb, Re, Zn, Bi, Cs, Tl, Ga, and In were possibly related to coal combustion and vehicle exhaust, while Ni, Nb, Cr, and Mo may have originated from metal smelting. Pb isotopic tracing showed that coal dust, cement dust, road dust and construction dust were the major Pb sources in PM_2.5 in Nanchang. Combined, these sources contributed an average of 72.51% of the Pb measured, while vehicle exhaust accounted for 27.49% of Pb based on results from a binary Pb isotope mixed model.     

Asymptotic Analysis for a Mixed Boundary-Value Contact Problem

The variational solution of the nonlinear Signorini contact problem determines also the active contact zone Γ ^c . If the latter is known, then the elastic field is a solution of a linear mixed boundary value problem in which on Γ ^c the normal displacement and tangential traction are given, while on the non-contact part the total traction is zero. Such mixed boundary conditions in general generate singularities of the solution's stress field at the points P ^{( k )} where the boundary conditions change. For smooth data, however, the variational solution of the Signorini contact problem actually belongs to H ²(Ω)², which implies the disappearance of these singularities, i.e., that the corresponding stress intensity factors vanish. This paper is devoted to the characterization of the active contact zone Γ ^c by the vanishing stress intensity factors including their sensitivity with respect to varying Γ ^c for two-dimensional problems provided that Γ ^c consists of a finite number of intervals. We use the method of asymptotic expansions and derive an explicit formula for the sensitivity, which is rigorously justified by employing weighted Sobolev spaces with detached asymptotics. These results can be used to determine the points P ^{( k )} with a corresponding Newton iteration. Accepted July 6, 2000?Published online January 22, 2001     

Effect of the use of the balance and gradient methods as a result of experimental investigations of natural convection action with regard to the conception and construction of measuring apparatus

Measurement apparatus designed and constructed according to conceptions of the authors, enabled a more precise calculation of the heat transfer coefficient with the balance and gradient methods. Construction and use of the apparatus and devices are described below, results of experimental investigations for horizontal and vertical, isothermal, flat plates obtained independently with the balance and gradient methods, are also presented. The following equations were found:Nu=0.612 · (Ra)^1/4 10⁴ ≦Ra ≦ 10⁸ for vertical platesNu=0.766 · (Ra)^1/5 10⁴ ≦Ra ≦ 10⁷ Nu=0.173 · (Ra)^1/3 10⁵ ≦Ra≦ 10⁸ for horizontal plates. On the basis of the results obtained from both these methods, differences of natural convection acting from vertical and horizontal plates are discussed. The usefulness of the balance and gradient methods have been considered for qualitative and quantitative investigations of heat transfer by natural convection.     

Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.     

Flow around two tandem square cylinders near a plane wall

An experimental study has been conducted to investigate the flow around two identical square cylinders in tandem arrangement and placed near a plane wall at a Reynolds number of 6,300. The inter-cylinder spacing ratio was varied from S ^* = 0.5 to 6, and the cylinder-to-wall gap ratio from G ^* = 0.25 to 2. Totally, 42 cases were considered to systematically examine the effects of wall proximity and the mutual interference between the two cylinders in the normalized gap–spacing (G ^*–S ^*) plane. The flow fields were captured using digital particle image velocimetry, in conjunction with measurements of the fluid forces (drag and lift) acting on the downstream cylinder using a piezoelectric load cell. The results show that the flow is highly dependent on the combined values of G ^* and S ^*. Categories relating to G ^* could be broadly classified as small-gap regime (G ^* < 0.5) at which periodic vortex shedding from the cylinders is suppressed, intermediate-gap regime (0.5 < G ^* < 1) where vortex shedding occurs but is under the influence of the wall proximity, and large-gap regime (G ^* > 1) where the wall effects become negligible. Similarly, the flow interference between the two cylinders can be divided into three basic categories as a function of S ^*, namely, shielding regime at S ^* < 1, reattachment regime at 1 < S ^* < 3, and impinging regime at S ^* > 3. Variations of force coefficients, amplitude spectra, Strouhal numbers, and Reynolds shear stress with G ^* and S ^* are presented to characterize the different flow regimes.     

Effect of thermal losses on the microscopic hyperbolic heat conduction model

The effects of radiative losses on the thermal behavior of thin metal films, as described by the microscopic two-step hyperbolic heat conduction model, are investigated. Different criteria, which determine the ranges within which thermal radiative losses are significant, are derived. It is found that radiative losses from the electron gas are significant in thin films having [(C_R e_e^4/₃ T_￥ ⁴ )/(k_e^1/₃ L^2/₃ G)] 3 4.6 ×10⁷{{C_R \epsilon _e^{{4 \over 3}} T_\infty ^4 } \over {k_e^{{1 \over 3}} L^{{2 \over 3}} G}}\geq 4.6 \times 10^7 for /_o > 4 and F_F < 1 and [(C_R e_e^3/₂ T_￥ ^9/₂)/(k_e^1/₂ L^1/₂ G)] 3 7.4 ×10¹⁰{{C_R \epsilon _e^{{3 \over 2}} T_\infty ^{{9 \over 2}}} \over {k_e^{{1 \over 2}} L^{{1 \over 2}} G}}\geq 7.4 \times 10^{10} for /_o < 4 and F_F > 1.     

Precursor ionization ahead of strong shock waves in argon

The mechanism of precursor ionization ahead of strong shock waves has been studied in a low density shock tube. The experimental results are illustrated with Arrhenius plots with kink points dividing them into two parts with apparent activation energy ratio 1:2, namely with the values 7.7 eV and 15.3 eV, and varying with first and third power of the density respectively. A model is proposed to interpret the facts where the process taking place in the precursor region, is a two step photo ionization accompanied with the drift flow effect of the gas relative to the shock wave or the ionization recombination effect according to whether the shock speed and initial density are low enough. The product of the A-A collision excitation cross section coefficientS ^* multiplied by the radiation cross sectionQ ^* of ArgonS ^*×Q ^*=1×10⁻³⁶ (cm⁴eV⁻¹) and the three body recombination coefficient of Argon at room temperaturek _ra =1×10⁻²⁴ (cm⁻⁶s⁻¹). The project supported by the National Natural Science Foundation of China     

A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-orderaccuracy is proposed for solving three-dimensional parabolic equations. The stabilitycondition is r＝△t/△x² ＝△t/△y²＝△t/△z²≤1/4, and the trumcation error is O(△t²+△x⁴).     

The mechanisms governing the activation of dislocation sources in aluminum at different strain rates

This article examines the time to activate Frank–Read sources in response to macroscopic strain rates ranging from 10¹ s⁻¹ to 10¹⁰ s⁻¹ in aluminium under athermal conditions. We develop analytical models of the bowing of a pinned dislocation segment as well as numerical simulations of three dimensional dislocation dynamics. We find that the strain rate has a direct influence on both the activation time and the source strength of Frank–Read sources at strain rates up to 10⁶ s⁻¹, and the source strength increases in almost direct proportion to the strain rate. This contributes to the increase in the yield stress of materials at these strain rates. Above 10⁶ s⁻¹, the speed of the bowing segments reaches values that exceed the domain of validity of the linear viscous drag law, and the drag law is modified to account for inertial effects on the motion of the dislocation. As a result the activation times of Frank–Read sources reach a finite limit at strain rates greater than 10⁸ s⁻¹, suggesting that Frank–Read sources are unable to operate before homogeneous nucleation relaxes elastic stresses at the higher strain rates of shock loading. Elastodynamic calculations are carried out to compare the contributions of Frank–Read sources and homogeneous nucleation of dislocations to plastic relaxation. We find that at strain rates of 5×10⁷ s⁻¹ homogeneous nucleation becomes the dominant generation mechanism.     

Comparison of experimental data with results of some drying models for regularly shaped products

This paper presents an experimental and theoretical investigation of drying of moist slab, cylinder and spherical products to study dimensionless moisture content distributions and their comparisons. Experimental study includes the measurement of the moisture content distributions of slab and cylindrical carrot, slab and cylindrical pumpkin and spherical blueberry during drying at various temperatures (e.g., 30, 40, 50 and 60°C) at specific constant velocity (U = 1 m/s) and the relative humidity φ = 30%. In theoretical analysis, two moisture transfer models are used to determine drying process parameters (e.g., drying coefficient and lag factor) and moisture transfer parameters (e.g., moisture diffusivity and moisture transfer coefficient), and to calculate the dimensionless moisture content distributions. The calculated results are then compared with the experimental moisture data. A considerably high agreement is obtained between the calculations and experimental measurements for the cases considered. The effective diffusivity values were evaluated between 0.741 × 10⁻⁵ and 5.981 × 10⁻⁵ m²/h for slab products, 0.818 × 10⁻⁵ and 6.287 × 10⁻⁵ m²/h for cylindrical products and 1.213 × 10⁻⁷ and 7.589 × 10⁻⁷ m²/h spherical products using the Model-I and 0.316 × 10⁻⁵–5.072 × 10⁻⁵ m²/h for slab products, 0.580 × 10⁻⁵–9.587 × 10⁻⁵ m²/h for cylindrical products and 1.408 × 10⁻⁷–13.913 × 10⁻⁷ m²/h spherical products using the Model-II.     

不同展长比旗帜在均匀来流中摆动的实验研究

为研究展长比对旗帜摆动特性的影响,在低速风洞内开展了质量比M*为0.6,展长比H*为0.2～1.5的柔性旗帜在不同来流速度下的失稳实验.利用高速摄影机和图像处理技术分析了无量纲速度U*和展长比H*对旗帜运动学特征(频率和振幅)的影响;利用自行研制的天平测量了旗帜受到的阻力,首次给出了旗帜的动力学特征.结果发现,保持展长...     

Seasonal variations and sources of mass and chemical composition for PM10 aerosol in Hangzhou,China

Aerosol observation was conducted for four seasons from September 2001 to August 2002 at five sampling sites in Hangzhou, South China, on PM10 mass, 22 elements （Na, Mg, AI, Si, P, S, K, Ca, Ti, V, Cr, Mn, Fe, Ni, Cu, Zn, As. Se, Br, Cd, Ba, and Pb）, 5 major ions （F^-, Cl^ , NO3^-, SO4^2- , and NH4^＋）, and organic and elemental carbon （OC and EC）, showing that PM10 mass ranged from 46.7 to 270.8 μg/m^3, with an annual average of 119.2 μg/m^3. Na, AI, Si, S, K, Ca, and Fe were the most abundant elements in PM10, most of S being in the form of SO4^2- . SO4^2-, NO3^-, and NH4^＋ were the major ions, which contributed to about 20% of the PM10 mass. The mean seasonal concentrations for SO4^2- , averaged over all sites, were found to be 18.0, 18.5, 24,Z and 21.4 μg/m^3, for spring, summer, autumn, and winter, respectively, while the corresponding loadings for NO3^- were 7.2, 4.7, 7.1, and 11.2 μg/m^3, and for NH4^＋ were 6.0, 5.9, 8.2, and 9.3 μg/m^3, in the form mostly of NH4NO3 in spring, autumn, and winter, and mostly of （NH4）2SO4 in summer. The low NO3^-/SO4^2- ratio found indicates coal combustion as the major source throughout the year. The mean annual concentrations of OC and EC in PM10 were found to be 21.4, and 4.1 μg/m^3, respectively. Material balance calculation indicated that fugitive dust, the secondary aerosol, and carbonaceous matter were the most abundant species in PM10 for the four seasons, as is characteristic for cities in South China.     

Global structure of a kind of predator-prey system

In this paper we prove a theorem, theorem 2, on nonexistence of closed trajectory for a general predator-prey system. Then, using this theorem and another theorem on existence and uniqueness of limit cycle for predator-prey system, we complete the investigation of a concrete model of predator-prey system (?)=γx(1-x/K)-yx~n/(a+x~2) (?)=y(μx~n/(a+x~2)-D)(n=1,2)under the conditions of all kinds of parameters.