Heat and mass transfer of a multiatomic gas in a cylindrical capillary with arbitrary Knudsen numbers

Processes of heat and mass transfer of a multiatomic gas in a cylindrical channel of circular cross section with arbitrary Knudsen numbers are considered on the basis of a model kinetic equation, taking account of the excitation of rotational and vibrational degrees of freedom of the molecules.Notation Kn Knudsen number - f, f^tr total and translational Eucken factors - R_o capillary radius - m molecular mass - k Boltzmann's constant - n, T numerical density and temperature of gas - v_i i-th component of the molecular velocity - h_ij perturbation function - E_i ^(r), e_j ^(v) energy of the i-th rotational and j-th vibrational levels - E_o ^(r), E_o ^(v) equilibrium values of the rotational and vibrational energy - P_i ^(r), P_i ^(v) probability of rotational and vibrational states of energy E _i ^r and E _j ^v - , logarithmic pressure and temperature gradients - T_o mean gas temperature - R rarefaction parameter of gas - C _V ^r , C _V ^v contributions of rotational and vibrational degrees of freedom of the molecule to the specific heat at constant volume - U macroscopic gas velocity - q^(t), q^(r), q^(v) components of the heat flux density due to translational, rotational, and vibrational degrees of freedom of the molecules - P, pressure and dynamic viscosity of the gas - l free path length of molecules - up velocity of Poiseuille flow - u_T rate of thermal creep - cross-sectional area of capillary - I_n, I_q numerical and heat fluxes averaged over the channel cross section - universal index characterizing the thermomolecular pressure difference - ^t, ^r, ^v thermal conductivities due to translational, rotational, and vibrational degrees of freedom of the molecules - mass density of the gas - D_rr, D_vv diffusion coefficients of rotationally and vibrationally excited molecules among the unexcited molecules - Z_r rotational collisional number Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 71–82, July, 1984.     

A mixed finite element formulation for shells through a reduced Reissner's principle in elastodynamics

Summary The use of Mixed models based in Reissner's principle in statics has been found to lead to some desirable simplifications in Finite Element formulations, in particular in plates and shells. Reduced formulations of Reissner's principle such as the one used by Prato have proved to be even more successful. In this paper, a reduction similar to that of Prato is attempted on a mixed elastodynamic variational principle by Karnopp.

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Eine gemischte finite Elemente-Formulierung für Schalen durch ein reduziertes Reissnersches Prinzip der Elastodynamik

Zusammenfassung Die Verwendung von gemischten Modellen basiert auf Reissners Prinzip der Statik führt zu erwünschten Vereinfachungen bei der Formulierung von finiten Elementen im speziellen bei Untersuchungen von Platten und Schalen. Reduzierungen des Reissnerschen Prinzips, wie sie von Prato angewendet worden sind, haben sich sogar als noch erfolgreicher erwiesen. In dieser Untersuchung wird eine Reduktion, ähnlich der von Prato, für ein gemischtes elastodynamisches Variationsprinzip nach Karnopp, vorgenommen.

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Symbol Table A Domain of integration of the Functional. Also area of the triangle - b Second fundamental form of the shell middle surface - C ^ijkl Elastic Constants - E ₁,E ₁ ^* Strain Energy and Co-Energy density - e _ij Elastic strain tensor - f _i Body force density tensor - I _ks Karnopp's functional, specialized to shells - I _ksc Contracted Karnopp's functional, specialized to shells - i, j, k Index 1, 2, 3 - K ₁,K ₁ ^* Kinetic Energy and Co-Energy density - K ^* Kinetic co-energy density for shell - m Moment tensor defined at the mid-surface - n In-plane stress tensor defined at the middle surface - n Qualifier for the boundary normal - p ,p ³ Boundary forces - Prescribed boundary forces - p Shear force tensor defined at the mid-surface - R Position vector of a point in the volume of the shell - r Position vector of a point on the mid-surface - r _i Net impulse density tensor - S _u Portion of the boundary where displacements are preseribed - S Portion of the boundary where forces are prescribed - s Qualifier for the direction tangent to the boundary - t Time variable - t ^ij Stress tensor - u ,u ₃ Mid-surface displacements - Mid-surface velocities - V Volume - v _i Displacement tensor - , Indices. Range 1, 2 - Shear strain tensor for the middle surface - Variation operator - Mid-surface strain tensor - Mid-surface curvature strain tensor - Direction cosine tensor for boundary normal - Mid-surface rotation tensor - Mid-surface angular velocity tensor - _M Strain energy density - _M * Strain co-energy density - _B * Bending strain co-energy density - _TS ^* Transverse shear strain co-energy density - | Covariant differentiation with respect tox , etc - Partial differentiation with respect tox , etc - .(dot) Time differentiation - -(bar) Prescribed quantities     

Analysis of strain localization in strain-softening hyperelastic materials,using assumed stress hybrid elements

Newly developed assumed stress finite elements, based on a mixed variational principle which includes unsymmetric stress, rotation (drilling degrees of freedom), pressure, and displacement as variables, are presented. The elements are capable of handling geometrically nonlinear as well as materially nonlinear two dimensional problems, with and without volume constraints. As an application of the elements, strain localization problems are investigated in incompressible materials which have strain softening elastic constitutive relations. It is found that the arclength method, in conjunction with the Newton Raphson procedure, plays a crucial role in dealing with problems of this kind to pass through the limit load and bifurcation points in the solution paths. The numerical examples demonstrate that the present numerical procedures capture the formation of shear bands successfully and the results are in good agreement with analytical solutions.List of Symbols u displacement - R rotation - U right stretch tensor - r^* Biot stress tensor - t first Piola Kirchhoff stress tensor - Cauchy stress tensor - I identity tensor - F deformation gradient I+(u)^T - ab a _ib_jg ⁱ g ^j =dyad - a·b a _ibⁱ=dot product - A·b A _ijb^j g ⁱ - A·B A _ikB _inf.j ^supk. g ⁱ g ^j - A:B A _ijB^ij - v velocityu - W spin tensorR - D rate of stretch r - r^* UL rate of r^* - t UL rate of t - _n Kronecker's delta - - J det F=det{I+(u)^T} - symm (A) 1/2(A+A ^T ) - skew (A) 1/2(A-A ^T ) - trace (A) A _inf.i ^supi. This research is supported by the Office of Naval Research. The first author wishes to express his appreciation to Dr. H. Murakawa, Dr. E. F. Punch, Mr. A. Cazzani, and Dr. H. Okada for fruitful discussions on the subject     

Relative diffusion resistance in drying wheat

An apparatus is described for examining various methods of convective drying.Notation tan=N_I drying rate in the first period - tan =(dW^c/d)_II drying rate in the second period - drying time - W _e ^c equilibrium water content - W^c water content of grain on dry mass - N^*=(1/N_I)(dW^c/d) dimensionless drying rate - T_sur surface temperature - T_a ambient temperature - T_w wet-bulb temperature - A,, experimental coefficients Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 31, No. 5, pp. 839–843, November, 1976.     

Stress relaxation in hot-drawn low density polyethylene

The tensile stress relaxation behaviour of hot-drawn low density polyethylene, (LDPE), has been investigated at room temperature at various draw ratios. The drawing was performed at 85° C. The main result was an increase in relaxation rate in the draw direction, especially at low draw ratios when compared to the relaxation behaviour of the isotropic material. This is attributed to a lowering of the internal stress. The position of the relaxation curves along the log time axis was also changed as a result of the drawing, corresponding to a shift to shorter times. The activation volume, , varied with the initial effective stress ₀ ^* according to ₀ ^* 10kT, where ₀ ^* =₀ — _i, is the difference between the applied initial stress, ₀, and the internal stress _i. This result supports earlier findings relating to similarities in the stress relaxation behaviour of different solids.     

Numerical method for solving heat-conduction problems for two-dimensional bodies of complex shape

A finite-difference scheme is described for a curvilinear orthogonal net which permits the use of a single algorithm for calculating bodies of various shapes.Notation x, y independent variables - u, v orthogonal coordinates - F(w)=F(u + iv) function of a complex variable - g(u,v)= F(w)/w Jacobian of transformation from (u,v) to (x,y) - thermal conductivity - c volumetric heat capacity - Q heat release per unit volume - T temperature - f value of temperature on boundary of region - time - L, L₁, L₂ differential operators - (u,v) solution of differential problem in canonical region - ^j, ₁ ^j , ₂ ^j , t^J, t ₁ ^j , t ₂ ^j network functions in canonical region - ^j, t^*j solutions of difference problems using rectangular and orthogonal nets respectively - {u_i, v_k} rectangular net in canonical region G - {x_i,k, y_i, k} orthogonal net in given region G^* - u_i, v_k dimensions of cell of rectangular net - u_i,v _i,k dimensions of cell of orthogonal net - h, maximum dimension of cell for rectangular and orthogonal nets respectively - ₁, ₂, difference operators for rectangular and orthogonal nets - A, B, C, D, A^*, B^*, C^*, D^* coefficients of difference scheme for rectangular net - D, Ã, B coefficients of difference scheme for orthogonal net Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 503–509, March, 1981.     

Thermal diffusivity of inhomogeneous systems

The possibility of analyzing the nonsteady temperature fields of inhomogeneous systems using the quasi-homogeneous-body model is investigated.Notation t, t_I, t_i temperature of quasi-homogeneous body inhomogeneous system, and i-th component of system - a, , c thermal diffusivity and conductivity and volume specific heat of quasi-homogeneous body - a_{i i}, c_i same quantities for the i-th component - q heat flux - S, V system surface and volume - x, y coordinates - macrodimension of system - dimensionless temperature Fo=a/² - Bi=/ Fourier and Biot numbers - N number of plates - =h/ ratio of micro- and macrodimensions - _V, volumeaveraged and mean-square error of dimensionless-temperature determination - time - m_i i-th component concentration Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 126–133, July, 1980.     

Flow visualization glass-ceramic: Preliminary experimental and modelling results

A glass-ceramic material was developed to act as a flow visualization material. Preliminary experiments indicate that aperiodic, thermally induced, convective flows can be sustained at normal processing conditions. These flows and the stress and temperature gradients induced are most likely responsible for the anomalous behaviour seen in these materials and the difficulties encountered in their development and in their production on industrial and experimental scales. A simple model describing the dynamics of variable-viscosity fluids was developed and was shown to be in qualitative agreement with more sophisticated models as well as with experimental results. The model was shown to simulate the dependence of the critical Rayleigh number for the onset of convection on the viscous properties of the fluid at low T, and also to simulate quenching behaviour when the temperature differences were high.Nomenclature C _p Heat capacity - D, E, F Expansion coefficients - H Height of the roll cell - Pr Prandtl number - R _a Rayleigh number - R _c Critical Rayleigh number for the onset of convection in a constant-viscosity fluid - S Dimensionless stream function - T Temperature - T _m Mean temperature - T ₀ Bottom surface temperature - T _r Reference temperature - a Aspect ratio of cell - g Acceleration due to gravity - k Thermal conductivity - k ₁ Function related to ²v/T ² - k ₂ Function related to ⁴v/T ⁴ - r Rayleigh number ratioR _a/R _c - t Time - w Dimensionless vertical coordinate - w _m Mean cell height - x Horizontal coordinate - y Dimensionless horizontal coordinate - z Vertical coordinate - , Constants - _t Thermal expansion coefficient - Constant in viscosity function - T Temperature difference between top and bottom surfaces - _i Viscosity coefficients - Kinematic viscosity - _m Mean kinematic viscosity - Dimensionless kinematic viscosity - Thermal diffusivity - Non-linear temperature function - Dimensionless non-linear temperature function - _o - Stream function - Dimensionless time - Eigenvalues     

Negatively isotone optimal policies for random walk type Markov decision processes

Summary This paper considers a random walk type Markov decision process in which the state spaceI is an integer subset of IR ^m , and the action spaceK is independent ofi I. The natural order, overI, and a quasi order,, overK, is assumed, together with aconditional convexity assumption on the returns {r _i ^k }, and certain other assumptions about these rewards and the transition probabilities in relationship to the orders and.A negatively isotone policy is one for whichi i(i))(i) (i.e.(i) (i) or(i) i)). It is shown that, under specified conditions, a negatively isotone optimal policy exists. Some consideration is given to computational implications in particular relationship to Howard's policy space method.

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Zusammenfassung Wir betrachten einen Markovschen Entscheidungsprozeß vom random walk Typ. Der ZustandsraumI sei eine Teilmenge des IR^m, wobeii I ganzzahlige Komponenten habe. Die MengeK der zulässigen Aktionen ini I sei unabhängig voni I. Sei die natürliche Ordnung aufI und sei eine Quasiordnung aufK. Die Erträge {r _i ^k }seienbedingt konvex, darüberhinaus seien weitere Voraussetzungen über diese Erträge und die Übergangswahrscheinlichkeiten in Bezug auf die Ordnungen und erfüllt. Eine Politik heißt negativ isoton, falls ausi i folgti(i) (d. h.(i) (i) oder(i)(i)). Wir zeigen, daß unter gewissen Voraussetzungen einenegativ isotone optimale Politik existiert: Auch diskutieren wir einige Folgerungen für die Numerik, insbesondere hinsichtlich Howards Politikiteration.

     

Internally Consistent Thermodynamic Description of Three-Component Liquid-Metal Systems at High Temperatures in the Entire Region of Concentration Triangle

A method is suggested for the investigation of the thermodynamic properties of ternary liquid-metal alloys at high temperatures in the entire region of concentration triangle. The method is demonstrated for a Na–K–Cs ternary system. Data are obtained for the enthalpy and Gibbs energy of formation of alloy in the temperature range of 200 T 1200 K and concentration range of 0 x _{i (j, k)} 1. The results reveal a very fine effect associated with the temperature rise, namely, the inversion of excess partial Gibbs energy G¯ _i ^*= RTln _i ( _i is the activity coefficient of the liquid component) and the change of sign of deviation of partial pressure, as well as of total pressure, from the respective values in accordance with Raoult's law. The obtained results may be used to interpret the available literature data on independent measurements of the saturation pressure.     

Determination of the local angular radiation coefficients in certain two-body systems

A method is shown and formulas are derived by which local angular radiation coefficients can be determined in certain two-body systems where the configuration is arbitrary but one of the bodies is either a cylinder or a rectangular plate.Notation _int radiation vector of body 1 - _E ^int intrinsic radiation intensity of body 1 - _x, _y, _z components of the geometrical radiation vector along rectangular coordinates - r₀=x²+z² shortest distance from point M(x, y, z) to linear radiator - ₀ , ₀ ^' angles subtending the two segments of the linear radiator from point M(x, y, z) on area element 2 of irradiated surface - l length of the cylinders - x, y, z space coordinates of point M Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 6, pp. 1080–1088, June, 1972.     

An isothermal theory of anisotropic rods

Summary To provide an isothermal, deterministic theory of anisotropic rods is the primary objective of this paper. Our starting point is the 3-D linear theory of micropolar elastodynamics. First, the governing equations of the theory are established by the use of a suitable averaging procedure together with a separation of variables solution for kinematic variables. Next, without making the usual definiteness assumption for the strain energy density, a dynamic uniqueness theorem is constructed for the solutions of the governing equations. Logarithmic convexity arguments are then used to enumerate a set of conditions sufficient for uniqueness. The theory includes the effects of warping and shearing deformations, and in fact, it incorporates as many higher order effects as deemed necessary in any special case. Also, the application of the theory is illustrated in a sample example.Nomenclature Euclidean 3-space - X_k, X₁z, X a system of right-handed Cartesian coordinates in, rod axis, lateral coordinates; k=1, (=2, 3) - , , a regular region of space in , its closure and boundary surface - L _d,L complementary subsets of, on which deformations and stresses are prescribed, respectively;L _d L =L,L _d L = 0 - L length of rod - A area of cross-section of rod - C a Jordan curve which boundsA - V, L entire volume of rod and its boundary surface - L ₁,L _t lateral surface of rod, surface portion ofL ₁ on which stresses are prescribed - A _r,L ₁ right and left faces of rod - dv, dA, ds element of volume and area onA, and line element alongC - n_i,v _i unit exterior normal vectors toL andC - t time - t_kl, m_kl components of stress and couple stress tensors - f_i, l_i body force and body couple vectors per unit volume - , J_kl mass density, components of microinertia tensor - u_i, _i displacement and microrotation vectors - prescribed steady temperature increment - _kl, _klm components of Kronecker's delta and alternating tensor - t_i, m_i stress and couple stress vectors - _kl, e_kl components of strain and infinitesimal strain tensors - C_klmn, D_klmn components of isothermal elastic stiffnesses - B_kl thermal coefficients of material - , Lamé's elasticity constants - , , , elastic moduli of microisotropic continuum - B coefficient of linear thermal expansion - d_i, d_kl u_i or _i, _kl or e_kl and/or _kl - T _kl ^(m,n) , M _kl ^(m,n) components of stress and couple stress resultants of order (m, n) - T _k ^(m,n) M _k ^(m,n) components of stress and couple stress vector resultants of order (m, n) - d _k ^(m,n) , d _kl ^(m,n) components of deformation (u _k ^(m,n) , _k ^(m,n) ) and strain ( _kl ^(m,n) , e _kl ^(m,n) , _kl ^(m,n) ) of order (m, n) - F _i ^(m,n) , L _i ^(m,n) body force and body couple resultants of order (m, n) - U _i ^(m,n) , _i ^(m,n) displacement and microrotation resultants of order (m, n) - P _i ^(m,n) , Q _i ^(m,n) effective loads of order (m, n) - N , M, , P, Q, w, T ₁₁ ^(0,0) ,M ₁₁ ^(0,0) , P ₁ ^(0,0) , Q ₁ ^(0,0) , U ₁ ^(0,0) , ₁ ^(0,0) - v₀ rod velocity, (E/)^1/2 - K, W kinetic and potential energy densities - V, U kinetic and potential energies per unit length of rod - total energy of rod - C^(m,n) functions with derivatives of order up to and including (m) and (n) with respect to space coordinates and time, respectively - G (t) logarithmic convexity function - ()^. time differentiation, /t () - () partial differentiation with respect to the axial coordinate, /z () - E,v Young's modulus, Poisson's ratio     

Transient heat conduction in hollow spheres with a moving inner boundary

The finite integral transform method is used to obtain the solution of unsteady heat conduction problems for a hollow sphere with a moving internal boundary and various boundary conditions at the outer surface. For the solution of the problems of interest integral transform formulas are presented with kernels (16), (20), and (24) and the corresponding inversion formulas (18), (22), (26), (29) and characteristic equations (17), (21), (25), (28), (31), (33).Nomenclature a, thermal diffusivity and conductivity - t temperature of phase transformation - density - heat transfer coefficient - Q total quantity of heat passing through inner boundary - F latent heat of phase transformation - Fo(1,)=a/R ₁ ² , Fo(i,)=/r _i ² , Fo(i, _i)=a _i/r _i ² Fourier numbers - Bi₂=R₂/ Biot number     

Thermal expansion of bitumen-mineral compositions

The article presents the results of investigations of the thermal expansion of bitumen-mineral compositions and their components of different origin, carried out with newly developed dilatometers.Notation P_c index of the property of the composition - P_m, P_n, P_i indices of the properties of the components of the composition - V_m, V_n, V_i volumes of the components of the compositions - V_c volume of the composition - V_f volume of the filler - V_b volume of the bituminous binder - V_p volume of pores - _c, _f, _b thermal coefficients of volume expansion of composition, filler, and bituminous binder - K_b, K_f volumetric moduli of elasticity of bituminous binder and filler - G_b shear modulus of bituminous binder - T _g ^b , T _g ^c glass-transition temperatures of bituminous binder and composition - T ₁ ^b , T_2/b temperatures of beginning and end of the glass transition of bitumen - T ₁ ^c , T ₂ ^c temperatures of beginning and end of the glass transition of composition - T _F ^c fluidity temperature of the composition - _f, _b linear thermal expansion coefficients of filler and bituminous binder - _1e, _2e, _3e linear thermal expansion coefficients of composition at temperatures above T _F ^c , in the interval T _F ^c -T _g ^c , and below T _g ^c , respectively, found experimentally - _1t ^M , _2t ^M , _3t ^M the same, found theoretically by the equation for a mixture - _1t ^c , _2t ^c , _3t ^c the same, found theoretically by Kerner's equation - k₂, k₃ coefficients taking into account the effect of the filler surface on the thermal expansion of the bituminous binder in the temperature interval T_F/c-T_g/c and below T _g ^c Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 886–893, November, 1979.     

Finite element method for thermomechanical response of near-incompressible elastomers

Summary The present study addresses finite element analysis of the coupled thermomechanical response of near-incompressible elastomers such as natural rubber. Of interest are applications such as seals, which often involve large deformations, nonlinear material behavior, confinement, and thermal gradients. Most published finite element analyses of elastomeric components have been limited to isothermal conditions. A basic quantity appearing in the finite element equation for elastomers is thetangent stiffness matrix. A compact expression for theisothermal tangent stiffness matrix has recently been reported by the first author, including compressible, incompressible, and near-incompressible elastomers. In the present study a compact expression is reported for the tangent stiffness matrix under coupled thermal and mechanical behavior, including pressure interpolation to accommodate near-incompressibility. The matrix is seen to have a computationally convenient structure and to serve as a Jacobian matrix in a Newton iteration scheme. The formulation makes use of a thermoelastic constitutive model recently introduced by the authors for near-incompressible elastomers. The resulting relations are illustrated using a near-incompressible thermohyperelastic counterpart of the conventional Mooney-Rivlin model. As an application, an element is formulated to model the response of a rubber rod subjected to force and heat.Notation A _i n _i /c - C Cauchy strain tensor - c VEC (C) - C ₁,C ₂ constants in Mooney-Rivlin model for elastomer - c ₂ VEC (C ²) - c _i eigenvalues ofC - c _e ,c _e , _e specific heat at constant strain - D _nl stiffness matrix due to the geometric nonlinearity - D _T ,D _T isothermal tangent modulus matrices - e VEC () - e _d VEC ( _d ) - f, f(T) thermal expansion function, =1/[1+(T-T ₀)/3] - f combined vector of nodal forces and heat fluxes - f _M consistent nodal force vector - f _T ,f _1T ,f _2T ,f _3T ,f _4T consistent heat flux vector - F deformation gradient tensor - g related tof _T - h time step - I _i invariants ofC - I 9×9 identity tensor - I identity tensor - i vectorial counterpart ofI:VEC(I) - J the Jacobian matrix in Newton iteration scheme - J determinant ofF - J _i invariants ofI ₁ ^–1/3 C - k, k(T) thermal conductivity - K tangent stiffness matrix - K _MM ,K _MT ,K _MP tangent stiffness submatrices - L,L _M ,L _P ,L _S lower triangular matrices related toLU decomposition ofK - M ₁,M ₂ strain-displacement matrices - N interpolation matrix - n surface normal vector - n _i (I _i /c) ^T - P matrix arising in theLU decomposition ofK - P the tension applied to the rubber rod - p (true) pressure - Q heat flux - q heat flux vector - r,r _M ,r _T ,r residual vectors - R,R ₁,R ₂,R ₃ matrices from thermal boundary conditions - R _s 1/2(R+R ^T ) - R _a 1/2(R–R ^T ) - R –R ₁+R ₂+R - R _s 1/2(R+R ^T ) - R _a 1/2(R-R ^T ) - s VEC() - S surface in undeformed configuration - t time - t traction referred to undeformed configuration - T, T ₀ temperature, reference temperature - T upper-shelf temperature in the surface convective relation - U upper triangular matrix inLU decompositionK - u displacement vector - v combined vector of nodal parameters - v _n value ofv at thenth time step - V volume in undeformed configuration - w strain energy density per unit undeformed volume - x position vector in deformed configuration - X position vector in undeformed configuration - volumetric thermal expansion coefficient - _c coefficient in the surface convective relation - ₁ strain-displacement matrix - _T interpolation matrix for thermal gradient: T - vector of nodal displacements - Lagrangian strain tensor - _d deviatoric Lagrangian strain tensor - interpolation function for - entropy per unit mass in the undeformed configuration - vector of nodal temperatures - þ isothermal bulk modulus - interpolation function forT - temperature-adjusted pressure,p/f ³(T) - mass density in the undeformed configuration - matrix arising inLU decomposition ofK - 2nd Piola-Kirchhoff stress - Cauchy stress tensor - , _M , _T , _c , ₀ Helmholtz free energy density function per unit mass - _i - _ij - vector of nodal values of - matrix arising in theLU decomposition ofK - near-incompressibility constraint function - internal energy density per unit mass - (·) variational operator - VEC(·) vectorization operator - symbol for Kronecker product of two tensors - tr(·) trace of a tensor - det(·) determinant of a tensor     

Stress intensity factors in a fully interacting,multicracked, isotropic plate

An essential part of describing the damage state and predicting the damage growth in a multicracked plate is the accurate calculation of stress intensity factors (SIF's). Analytical derivations of these SIF's for a multicraked plate can be complex and tedious. Recent advances, however, in intelligent application of symbolic computation can overcome these difficulties and provide the means to rigorously and efficiently analyze this class of problems. Here, the symbolic algorithm required to implement the methodology for the rigorous solution of the system of singular integral equations for SIF's is presented. The special problem-oriented symbolic functions to derive the fundamental kernels are described, and the associated automatically generated FORTRAN subroutines are given. As a result, a symbolic/FORTRAN package named SYMFRAC, capable of providing accurate SIF's at each crack tip, has been developed and validated.Simple illustrative examples using SYMFRAC show the potential of the present approach for predicting the macrocrack propagation path due to existing microcracks in the vicinity of a macrocrack tip, when the influence of the microcracks' location, orientation, size, and interaction are accounted for.List of symbols offset angle between inner tips of two parallel cracks - _lr, _mz direction cosines between two local coordinate systems - _jl strain tensor - offset of notch-microcracks system with respect to Y axis - _k four roots of the characteristic equation - v Poisson's ratio - _jl ^o , _jl ^T far-field and total stress field, respectively - _XX ^o , _YY ^o , _XY ^o components of stress in global coordinate system - _j angle defining local frame orientation - , normalized real variable - (s, y) Fourier transform of Airy stress function with respect to x variable - a _j half crack length - a ₁₁, a₁₂, a₂₂ coefficients of strain-stress relationship - d normalized radial (tip) distance - f _nj auxiliary functions - k ₁, k₂ mode-I and mode-II stress intensity factors - ker _inf ^sup Fredholm kernels - p _j normal traction at crack surface - q _j shear traction at crack surface - s Fourier variable - r _j, r_kX, r_kY position vector and its components, for an origin of local frame - u, v x, y component of displacement, respectively - w weight function - x _j y_jand X, Y local and global coordinates - [A] matrix of coefficients - C' _j, C_j functions of s in Fourier space (i.e., constrants in x, y-real space) - E Young modulus - F _j(x_j, y_j) Airy stress function - G_nj(_p) discrete auxiliary function - {} loading function vector     

Heat flow model of rapid solidification with nonhomogeneous thermal contact

A heat flow model is presented of the solidification process of a thin melt layer on a heat conducting substrate. The model is based on the two-dimensional heat conduction equation, which was solved numerically. The effect of coexisting regions of good and bad thermal contact between foil and substrate is considered. The numerical results for thermal parameters of the Al-Cu eutectic alloy show considerable deviations from one-dimensional solidification models. Except for drastic differences in the magnitude of the solidification rate near the foil-substrate interface, the solidification direction deviates from being perpendicular to the substrate and large lateral temperature gradients occur. Interruption of the thermal contact may lead to back-melting effects. A new quantity, the effective diffusion length, is introduced which allows some conclusions to be drawn concerning the behaviour of the frozen microstructure during subsequent cooling.Nomenclature _i ,a _i Thermal diffusivity _i = _i /c _{i i} ,a _i = _i / ₁ - c _i Specific heat capacity - d Foil thickness - D Solid state diffusion coefficient - e_x, e_z Unit vectors - H Latent heat of fusion - h ,h Foil-substrate heat transfer coefficients - i Index: 1, melt; 2, solidified foil; 3, substrate - _i ,k _i Thermal conductivityk _i = _i / ₁ - n Normal unit vector - Nu ,Nu Nusselt numbers for regions of badNu(x,) and good thermal contact, respectivelyNu =h Nu _d / ₁,,Nu(x, )=h(x,)d/ ₁ - R Universal gas constant - , s Position of the liquid-solid interface ¯s/d=s=s _xe_x+s _ze_z - Local solidification rate /d = s =s _xe_x +s _ze_z - t Real time - T _i Temperature field - T ₀ Ambient temperature - T _f Melting temperature - u _i Dimensionless temperature fieldu _i (x, z,)=T _i (x,z,)/T _f - u ₀ Dimensionless ambient temperatureu ₀=T ₀/T _f - _i Local cooling rate within the foil _i = du _i /d - W Stefan numberW=H/c ₁ T _f - ,x Cartesian coordinate parallel to the foil-substrate interfacex= /d - ₀,x ₀ Lateral extension of foil sectionx ₀= ₀/d - ₁,x ₁ Lateral contact lengthx ₁= ₁/d - ,z Cartesian coordinate perpendicular to the foil-substrate interfacez= /d - ₀,z ₀ Substrate thicknessz ₀= ₀/d - E Activation energy of diffusion - T Initial superheat of the melt - u Dimensionless initial superheat u=T/T _f - (x) Step function - _eff Dimensionless effective diffusion length - _i Mass density - Dimensionless time=t ₁/d ² - _f, _f(x, z) Total and local dimensionless freezing time, respectively     

Component-transfer equation in column with longitudinal separation

method of integration over the transverse coordinate, a transfer equation in a separating column in obtained, taking the longitudinal enrichment mechanism into account.Notation j_r, j_z density of radial and axial diffusional flows of the heavy component - c heavy-component concentration - ¯c mean (over the cross section) concentration - c₀ concentration is feed cross section, z=L/2 - z longitudinal coordinate - v_z axial component of hydrodynamic mixture velocity - v _z ^* velocity corresponding to condition of maximum enrichment factor - current function - mixture density - p pressure, _p, _t, barodiffusional and thermodiffusional constants - T₂, T₁ temperature of hot axial and cold near-wall chamber regions - W electric power absorbed by gas-discharge plasma - V_ph phase velocity of traveling magnetic wave - dynamic viscosity - =/(K₃+ _zK₂) relative drawoff - L, R column length and radius - M ionic mass of readily ionized component - Q_ni, q_nN effective cross section for collisions of atoms of separation mixture with ions and added neutrals, respectively - D mutual diffusion factor of mixture isotopes Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 81–85, July, 1980.     

Motion of a thermal wave front in a nonlinear medium with absorption

We study the evolution of a thermal perturbation in a nonlinear medium whose thermal conductivity depends on the temperature and the temperature gradient according to a power law.Notation u temperature - k coefficient of thermal conductivity - t time - x spatial variable - x₊ a point on the thermal wave front - a ² generalized coefficient of thermal diffusivity - , , , and s parameters of the process - (x^s) Dirac delta-function - B[, ] a beta function - v(, x), (t) auxiliary functions - A, C, T_o, T_m, T*, R, r, p, and _m constants and parameters Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 4, pp. 728–731, October, 1980.     

A nonlinear composite shell element with continuous interlaminar shear stresses

A numerical model for layered composite structures based on a geometrical nonlinear shell theory is presented. The kinematic is based on a multi-director theory, thus the in-plane displacements of each layer are described by independent director vectors. Using the isoparametric apporach a finite element formulation for quadrilaterals is developed. Continuity of the interlaminar shear stresses is obtained within the nonlinear solution process. Several examples are presented to illustrate the performance of the developed numerical model.List of symbols reference surface - convected coordinates of the shell middle surface - ⁱ coordinate in thickness direction - ⁱ h thickness of layer i - X_o position vector of the reference surface - ⁱX_o position vector of midsurface of layer i - t _k orthonormal basis system in the reference configuration - ⁱ a _k orthonormal basis system of layer i - ⁱW axial vector - R_o orthonormal tensor in the reference configuration - ⁱ R orthonormal tensor of layer i - ⁱ Cauchy stress tensor - ⁱ P First Piola-Kirchhoff stress tensor - ⁱ q vector of interlaminar stresses - ⁱ n, ⁱ m vector of stress resultants and stress couple resultants - v _x components of the normal vector of boundary - ⁱ N, ⁱ Q, ⁱ M stress resultants and stress couple resultants of First Piola-Kirchhoff tensor - stress resultants and stress couple resultants of Second Piola-Kirchhoff tensor - ⁱ , ⁱ , ⁱ strains of layer i - _K transformation matrix - u_o displacement vector of layer 1 - ⁱ local rotational degrees of freedom of layer i