Large deformation theory of coupled thermoplasticity including kinematic hardening

Summary Thermoplasticity is a topic central to important applications such as metalforming, ballistics and welding. The current investigation introduces a thermoplastic constitutive model accommodating the difficult issues of finite strain and kinematic hardening. Two potential functions are used. One is interpreted as the Helmholtz free energy. Its reversible portion describes elastic behavior, while its irreversible portion describes kinematic hardening. The second potential function describes dissipative effects and arises directly from the entropy production inequality. It is shown that the dissipation potential can be interpreted as a yield function. With two simplifying assumptions, the formulation leads to a simple energy equation, which is used to derive a rate variational principle. Together with the Principle of Virtual Work in rate form, finite element equations governing coupled thermal and mechanical effects are presented. Using a uniqueness argument, an inequality is derived which is interpreted as a finite strain thermoplastic counterpart to the classical inequality for stability in the small. A simple example is introduced using a von Mises yield function with linear kinematic hardening, linear isotropic hardening and linear thermal softening.Symbols D rate of deformation tensor - d VEC(D) - F deformation gradient tensor - h heat generation per unit mass - L velocity gradient tensor - q heat flux vector - workless internal variable - Lagrangian strain - e VEC() - E quasi-Eulerian strain - entropy - internal energy per unit mass - Helmholtz free energy - Cauchy stress tensor - Truesdell stress flux tensor - t VEC() - yield function - First Piola Kirchhoff stress - Second Piola Kirchhoff stress - s VEC() - s ^* backstress, center of the yield surface - Kronecker product symbol - VEC vectorization operator - tr(.) trace - DEV deviator of a tensor - TEN22 Kronecker tensor operator     

Completely hierarchical p-version curved shell element for laminated composite plates and shells

This paper presents a new completely hierarchical three dimensional curved shell finite element formulation for linear static analysis of laminated composite plates and shells. The element displacement approximation can be of arbitrary polynomial orders p , p and p in the , , and directions thereby permitting strains of at least (p –1), (p –1) and (p –1) order. The element approximation functions as well as the nodal variables are hierarchical. The element formulation ensures C ⁰ continuity. The lamina properties are incorporated by numerically integrating the element stiffness matrix for each lamina. The formulation has no restriction on either the number of laminas or the layup pattern of the laminas. The geometry of the laminated shell element is described by the coordinates of the nodes lying on the middle surface of the element and the lamina thicknesses at each node. The element formulation is equally effective for very thin as well as very thick laminated plates and curved shells. The results obtained from the present formulation are compared with those available in the literature as well as available analytical solutions.     

Boojums on the fermi surface in3He-A and Hamiltonian dynamics of the orbital momentum

Boojums (nodes) on the Fermi surface of³He-A lead to peculiar behavior of the intrinsic orbital momentum L in this superfluid. AtT=0, L has the form L=(/2m ₃)l(–C ₀), whereC ₀ is the prefactor of the anomalous term in the supercurrent l(l rot l). From the algebra of the hydrodynamical Poisson brackets and from microscopic theory it follows thatC ₀ is a dynamical invariant, (/t)C ₀=0. The closed system of nonlinear hydrodynamic equations of³He-A atT=0 is constructed in the framework of the Poisson brackets scheme.     

Hierarchical three dimensional curved beam element based on p-version

This paper presents a three node curved three dimensional beam element for linear static analysis where the element displacement approximation in the axial () and transverse directions ( and ) can be of arbitrary polynomial orders p , p and p . This is accomplished by, first constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators in , and directions using Lagrange interpolating polynomials and then taking the products (also called tensor product) of these hierarchical one dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three dimensional beam element were hierarchical. The formulation guarantees C ⁰ continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element all six components of the stress and strain tensor are ratained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors representing the nodal cross-sections. The results obtained from the present formulation are compared with analytical solutions (when available) and the h-models using isoparametric three dimensional solid elements. The formulation is equally effective for very slender as well as deep beams since no assumptions are made regarding such conditions during the formulation.     

Universality and the scaling laws in the superfluid phase transition of3He-4He mixtures

From the second-sound velocityU ₂ near the superfluid transition point, the superfluid densities in³He-⁴He mixtures, _s (X) and _s (), were deduced along the paths of constant³He concentrationX and of constant chemical potential difference of³He and⁴He. The following critical exponents of _s are determined: (a) =XX for _s (X) in the(X, T) plane,(b) X for _s (X) in the(, T) plane, and(c) for _s () in the(, T) plane. It is found that and _X change by about 4–6% relative to with increasing³He concentration up toX=0.4 and by 8–10% up toX=0.53. It seems that, belowX=0.53, universality hold for . Values of have been found to be in good agreement with the critical exponent of _s in pure⁴He under constant pressure. The values of and _X forX0.53 are also found to be consistent with the scaling relations in the (,T) plane of³He-⁴He mixture.Work performed in part while at the Electrotechnical Laboratory.     

Spatial structure of the superconducting phase in heavy-fermion systems and anomalies in the excitation spectrum

A degenerate superconducting phase in a metal with strong spin-orbital coupling may have a domain structure at low temperatures. The domain walls (of width =v_F/₀) carry a finite density of states N(0)mk _f/² per unit volume, which in turn results in the presence of a linear term T in the electronic specific heatC _e (T). With the help of the Atiyah-Singer theorem, this can be proved for a wide class of domain structures.     

Diffusion-controlled evaporation of a moving droplet

The problem of the quasisteady diffusion-controlled evaporation of a droplet moving in a viscous medium is solved numerically. The relative values of the droplet radius and velocity are determined, along with the total droplet evaporation time and the mass-transfer characteristics of the droplet and the medium.Notation R droplet radius - t time - D diffusion coefficient - density - C_s mass concentration of saturated vapor - C mass concentration of vapor far from the droplet - C=(C–C)/ (C_s–C) dimensionless vapor concentration - (r, , ) spherical coordinates - Re=2RU/ Reynolds number - Sc=/D Schmidt number - Sh Sherwood number - T=tD/R² dimensionless time - C_x coefficient of fluid friction - m mass of the droplet - T total droplet evaporation time - ratio of dynamic viscosities inside and outside the droplet - Q masstransfer intensity factor Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 739–744, October, 1980.     

Ultrasonic study of the temperature and pressure dependences of the elastic properties of β-silicon nitride ceramic

Ultrasonic wave velocity measurements have been used to determine the elastic stiffness moduli and related elastic properties of high-purity, dense -Si₃N₄ ceramic samples as functions of temperature in the range 150–295 K and hydrostatic pressure up to 0.2 GPa at room temperature. Due to its covalently bonded, rigid structural framework -Si₃N₄ is an elastically stiff material; the elastic stiffness moduli of the ceramic at 295 K are: C _L = 396 GPa, = 119 GPa, B ^S = 238 GPa, E = 306 GPa, Poisson's ratio = 0.285. The longitudinal elastic stiffness C _L increases with decreasing temperature and shows a knee at about 235 K; the decrease in slope below the knee indicates mode softening. The shear elastic stiffness shows mode softening which results in a plateau centred at about 235 K and an anomalous decrease with further reduction in temperature. The hydrostatic-pressure derivatives of elastic stiffnesses at 295 K are (C _L/P) _P=0 = 4.5 ± 0.1, (B ^S/P) _P=0 = 4.3 ± 0.1 and (/P) _P=0 = 0.17 ± 0.02 (pressure < 0.12 GPa). An interesting feature of the nonlinear acoustic behaviour of this ceramic is that, in the pressure range above 0.12 GPa, the values obtained for (/P) _P=0 and the shear mode Grüneisen parameter (_S) are small and negative, indicating acoustic-mode softening under these higher pressures. Both the anomalous temperature and pressure dependences of the shear elastic stiffness indicate incipient lattice shear instability. The shear _S(=0.005) is much smaller than the longitudinal _L(=1.18) accounting for the thermal Grüneisen parameter ^th(=1.09): since the acoustic Debye temperature _D(=923 ± 5 K) is so high, the shear modes play an important role in acoustic phonon population at room temperature. Hence knowledge of the elastic and nonlinear acoustic properties sheds light on the thermal properties of ceramic -Si₃N₄.     

The generating groups of geometrically uniform spherical signal sets

An [M, n] spherical signal set is a collection ofM unit-norm vectors in the Euclideann-dimensional space ⁿ . Itsconfiguration matrix C is the matrix of the scalar products between pairs of vectors. isgeometrically uniform if, given any two vectors x _i , x _j there exists an isometry that transforms x _i to x _j while leaving invariant. Agenerating group of is a group of isometries of ⁿ that transform any given vector of into each of the vectors in while leaving invariant. In this paper we characterize the configuration matrix of a geometrically uniform spherical signal set and we show how its generating groups can be obtained.Parts of an earlier version of this paper were presented at the IEEE International Symposium on Information Theory, Ithaca, NY, 1977     

Spin hydrodynamic equations with external disturbances and suitable Green's functions for superfluid3He-B. New onsager relations

The spin hydrodynamic equations for superfluid³He-B have been obtained for the case of external, time-dependent fields. On the basis of a microscopic approach, expressions are found for additional terms in equations containing these fields. Considering the linear response of the system to the switching on of external fields, formulas are found for suitable Green's functions (magnetization-magnetization, rotation-rotation, magnetization-rotation, rotation-magnetization). The rotation-rotation Green's function has the 1/q ² singularity characteristic of superfluid systems. Connections between Green's functions lead to relations among kinetic coefficients v, ₁, and ₂. It is also shown that there is a conserved quantityQ ^(B) = div v _s ^(B) that describes sources or magnetic type charges (monopoles) of the superfluid velocity v _s ^(B) . Comparison with the phenomenological approach suggests thatQ ^(B) is proportional to a pseudoscalar giving the projection of the spin density onto the vector describing the axis of rotation.     

Ultrasonic determination of the elastic and nonlinear acoustic properties of transition-metal carbide ceramics: TiC and TaC

Pulse-echo overlap measurements of ultrasonic wave velocity have been used to determine the elastic stiffness moduli and related elastic properties of ceramic transition-metal carbides TiC and TaC as functions of temperature in the range 135–295 K and hydrostatic pressure up to 0.2 GPa at room temperature. The carbon concentration of each ceramic has been determined using an oxidation method: the carbon-to-metal atomic ratios are both 0.98. In general, the values determined for the adiabatic bulk modulus (B ^S), shear stiffness (), Young's modulus (E), Poisson's ratio () and acoustic Debye temperature (_D) for the TiC and TaC ceramics agree well with the experimental values determined previously. The temperature dependences of the longitudinal stiffness (C _L) and shear stiffness measured for both ceramics show normal behaviour and can be approximated by a conventional model for vibrational anharmonicity. Both the bulk and Young's moduli of the ceramics increase with decreasing temperature and do not show any unusual effects. The results of measurements of the effects of hydrostatic pressure on the ultrasonic wave velocity have been used to determine the hydrostatic pressure derivatives of elastic stiffnesses and the acoustic-mode Grüneisen parameters. The values determined at 295 K for the hydrostatic pressure derivatives (C _L/P)_{P = 0}, (/P)_{P = 0} and (B ^S/P)_{P = 0} for TiC and TaC ceramics are positive and typical for a stiff solid. The adiabatic bulk modulus B ^S and its hydrostatic pressure derivative (B ^S/P)_{P = 0} of TiC are in good agreement with the results of recent high pressure X-ray diffraction measurements and theoretical calculations. The longitudinal (_L), shear (_S) and mean (^el) acoustic-mode Grüneisen parameters of TiC and TaC ceramics are positive: the zone-centre acoustic phonons stiffen under pressure. The shear _S is much smaller than the longitudinal _L. The relatively larger values estimated for the thermal Grüneisen parameter ^th in comparison to ^el for the TiC and TaC ceramics indicate that the optical phonons have larger Grüneisen parameters. Hence knowledge of the elastic and nonlinear acoustic properties sheds light on the thermal properties of ceramic TiC and TaC.     

Superconducting state in M3C60: Strong vs. weak coupling

The superconducting state in the doped fullerenes is due to strong coupling (e.g.,2.1 for Rb₃C₆₀) to low-frequency intramolecular modes _L 250 cm^–1 (^21/2). The analysis is based on an equation describingT _c for any strength of the coupling and on recent isotope effect and NMR data.     

Thermal transport properties in helium near the superfluid transition. I.4He in the normal phase

The thermal conductivity and the associated relaxation time to reach steady-state conditions are reported for the normal phase of several very dilute mixtures of³He in⁴He (X<4 × 10^–6) at saturated vapor pressure near T. The measurements were made over the reduced temperature range 2.5 × 10^–6<<2×10^–1, where (T–T)/T, and are representative for pure⁴He. The spacing between the cell plates was 0.147 cm. The systematic uncertainty in the conductivity data is estimated to increase from 2% for =0.2 to 4% for =3 × 10^–6. The random scatter due to finite temperature resolution increases to 7% at the smallest . The data are in agreement within the combined uncertainty with recent ones by Tam and Ahlers (cell F, spacing 0.20 cm) and with previous ones in this laboratory taken with a different plate spacing. The thermal diffusivity coefficientD _T = / C _p obtained from is found to agree within better than 15% with the calculated one using data for , the density , and the specific heatC _p . Measurements of the effective boundary resistivityR _b in the superfluid phase are described.R _b is found to depend on the thermal history of the cell when cycled up to 77 K and above. Also,R _b shows the beginning of an anomalous increase for ¦¦10^–4. The possible reasons for this anomaly are discussed, and their impact on the analysis of conductivity data in the normal phase is appraised.     

Viscoelastic and rheological behavior of concentrated colloidal suspensions

Molecular approaches are discussed to the density (), viscoeleastic (), and rheological () behavior of the viscosity(,,) of concentrated colloidal suspensions with 0.3 < < 0.6, where, is the volume fraction, the applied frequency, and ; the shear rate. These theories are based on the calculation of the pair distribution functionP ₂(r,,), wherer is the relative position of a pair of colloidal particles. The linear viscoelastic behavior(,,=0) follows from an equation forP ₂(r,,) derived from the Smoluchowski equation for small, generalized to large by introducing the spatial ordering and (cage) diffusion typical for concentrated suspensions. The rheological behavior(,,=0) follows from an equation forP ₂(r,) of a dense hard-sphere fluid derived from the Liouville equation. This leads to a hard-sphere viscosity^hs(,) which yields the colloidal one(,) by the scaling relation(,) ₀=^hs(,) _B, where ₀ is the solvent viscosity. _B is the dilute hard-sphere (Boltzmann ) viscosity and the's are appropriately scaled,(,) and(,) agree well with experiment. A unified theore for(,,) is clearly needed and pursued.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994. Boulder, Colorado, U.S.A.     

On the Universal Curve in Superplasticity Mechanics

We have carried out a series of tests (tensile, compression, torsion, and tension+torsion) of a superplastic titanium alloy VT9 at 950°C, with strain-rate intensity _e = 5 10^-4 - 1 10^-2 s ^-1 . By plotting and analyzing the stress intensity vs strain-rate intensity curves for this material, we justified the validity of the universal curve hypothesis for finite strains with an accuracy acceptable in engineering practice.     

Determining fracture toughness of polyethylene from fatigue

A newly developed fatigue method of determining the value of the J-integral at crack initiation, J _1C, was tested on single-edge-notch-tension (SENT) specimens of a high-density polyethylene (HDPE). The fatigue method is able to propagate plane-strain brittle cracks in thin specimens which more closely approximate the in-use thickness, and hence the in-use morphology, of the polymer. This newly developed fatigue method was compared to ASTM standard procedure E813 (the ASTM method), which employs thick single-edge-notch-bend (SENB) specimens. Because of the similar size and shape of the crack-tip damage zone, the J _1C values for both methods were nearly identical (J _1C=1.8 kJ m^–2, 1.4 kJ m^–2 for the ASTM method and fatigue method, respectively).     

Combined heat and mass transfer in natural convection flow from a vertical wavy surface

Summary In the present paper, effects of combined buoyancy forces from mass and thermal diffusion by natural convection flow from a vertical wavy surface have been investigated using the implicit finite difference method. Here we have focused our attention on the evolution of the surface shear stress,f(0), rate of heat transfer,g(0), and surface concentration gradient,h(0) with effect of different values of the governing parameters, such as the Schmidt number Sc ranging from 7 to 1500 which are appropriate for different species concentration in water (Pr=7.0), the amplitude of the waviness of the surface ranging from 0.0 to 0.4 and the buoyancy parameter,w, ranging from 0.0 to 1.Notation C species concentration in the boundary layer - C species concentration of the ambient fluid - C _w species concentration at the surface - D chemical molecular diffusivity - f dimensionless stream function - g acceleration due to gravity - Gr _x local modified Grashof number - N ratio of the buoyancy forces due to the temperature difference and the concentration difference - p pressure of the fluid - T temperature of the fluid in the boundary layer - T temperature of the ambient fluid - T _w temperature at the surface - u, v thex- andy-components of the velocity field - x, y axis in the direction along and normal to the plate Greek symbols thermal diffusivity - _T volumetric coefficient of thermal expansion - _C volumetric coefficient of expansion with concentration - stream function - nondimensional similarity variable - x/L - density of the ambient fluid - v kinematic coefficient of viscosity - stream function - dimensionless skin friction - fluid viscosity     

Diffusive relaxation processes in liquid3He-4He mixtures. I. Normal phase

When a heat flux is switched on across a fluid binary mixture, steady state conditions for the temperature and mass concentration gradients T and c are reached via a diffusive transient process described by a series of terms modes involving characteristic times _n . These are determined by static and transport properties of the mixture, and by the boundary conditions. We present a complete mathematical solution for the relaxation process in a binary normal liquid layer of heightd and infinite diameter, and discuss in particular the role of the parameterA=k _T ² (/c) _T,P /TC _P,c coupling the mass and thermal diffusion. Herek _T is the thermal diffusion ratio, (/c) _T,P ^–1 is the concentration susceptibility, is the chemical potential difference between the components, andC _P,c is the specific heat. We present examples of special situations found in relaxation experiments. WhenA is small, the observable times (T) and (c) for temperature and concentration equilibration are different, but they tend to the same value asA increases. We present experimental results on four examples of liquid helium of different³He mole fractionX, and discuss these results on the basis of the preceding analysis. In the simple case for pure³He (i.e., in the absence of mass diffusion) we find the observed (T) to be in good agreement with that calculated from the thermal diffusivity. For all the investigated³He-⁴He mixtures, we observe (c) and (T) to be different whenA is small, a situation occurring at high enough temperatures. AsA increases with decreasingT, they become equal, as predicted. For the mixtures with mole fractionsX(³He)=0.510 and 0.603, we derive the mass diffusionD from the analysis of (c) and demonstrate that it diverges strongly with an exponent of about 1/3 in the critical region near the superfluid transition. As the tricritical point (T _t,X _t) is approached for the mixtureX=X _t0.675,D tends to zero with an exponent of roughly 0.4. These results are consistent with predictions and also with theD derived from sound attenuation data. We discuss the difficulties of the analysis in the regime close toT andT _t, with special emphasis on the situation created by the onset of a superfluid film along the wall of the cell forX=0.603 and 0.675.Work supported by grants from the National Science Foundation and the Research Corporation and by an A. P. Sloan fellowship to one of the authors (RPB).     

A non-hypersingular boundary integral formulation for displacement gradients in linear elasticity

Summary Based on boundary displacement and traction, a non-hypersingular boundary integral formulation is developed for the displacement gradient. At an arbitrary boundary point where the displacement field at least satisfies a Hölder condition (u _kC ^1, with >0), the displacement gradient can be calculated by the Cauchy Principal Value (CPV) integration. The hypersingularity involved in conventional formulation is circumvented by applying rigid body translation. The numerical implementation of the present formulation is illustrated, and both direct and indirect approaches are discussed. For two-dimensional problems, the coefficients involved in the direct approach are analytically derived. The stress formulation is also discussed. Finally, numerical examples are presented to validate the present formulation.     

Coupling of order parameter collective modes to spin density in3He-B

We derive a general expression for the dynamic spin susceptibility of³He-B which is valid for all magnetic fields. The coupling of real and imaginary modes by particle-hole asymmetry is taken into account. Then we calculate the contribution of the mode at frequency =2 – 1/4 ( is the effective Larmor frequency) to the transverse susceptibility. The spectral weight of this mode in magnetic resonance absorption is proportional to (/)^1/2 (–)², where and are particle-hole asymmetry parameters. From the experimental coupling strength of the real squashing mode to sound we estimate (–)²10^–4. The dynamic susceptibility satisfies the sum rules of Leggett. Finally we point out the difficulties in calculating the transverse NMR frequency of³He-B. These difficulties arise from theS _z =0 Cooper pairs and from the coupling ofJ _z =±1 modes forJ=1 andJ=2.