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     

Tangent modulus tensor in plasticity under finite strain

Summary The tangent modulus tensor, denoted as , plays a central role in finite element simulation of nonlinear applications such as metalforming. Using Kronecker product notation, compact expressions for have been derived in Refs. [1]–[3] for hyperelastic materials with reference to the Lagrangian configuration. In the current investigation, the corresponding expression is derived for materials experiencing finite strain due to plastic flow, starting from yield and flow relations referred to the current configuration. Issues posed by the decomposition into elastic and plastic strains and by the objective stress flux are addressed. Associated and non-associated models are accommodated, as is plastic incompressibility. A constitutive inequality with uniqueness implications is formulated which extends the condition for stability in the small to finite strain. Modifications of are presented which accommodate kinematic hardening. As an illustration, is presented for finite torsion of a shaft, comprised of a steel described by a von Mises yield function with isotropic hardening.Notation B strain displacement matrix - C=F ^T F Green strain tensor - compliance matrix - D=(L+L ^T )/2 deformation rate tensor - D fourth order tangent modulus tensor - tangent modulus tensor (second order) - d VEC(D) - e VEC() - E Eulerian pseudostrain - F, F _e ,F _p Helmholtz free energy - F=x/X deformation gradient tensor - f consistent force vector - residual function - G strain displacement matrix - h history vector - h time interval - H function arising in tangent modulus tensor - I, I ₉ identity tensor - i VEC(I) - k ₀,k ₁ parameters of yield function - K _g geometric stiffness matrix - K _T tangent stiffness matrix - k _k kinematic hardening coefficient - J Jacobian matrix - L=v/x velocity gradient tensor - m unit normal vector to yield surface - M strain-displacement matrix - N shape function matrix - n unit normal vector to deformed surface - n ₀ unit normal vector to undeformed surface - n unit normal vector to potential surface - r, R, R ₀ radial coordinate - s VEC() - S deformed surface - S ₀ undeformed surface - t time - t, t ₀ traction - t VEC() - VEC( ) - t VEC() - t _r reference stress interior to the yield surface - t t–t _r - T kinematic hardening modulus matrix - u=x–X displacement vector - U permutation matrix - v=x/t particle velocity - V deformed volume - V ₀ undeformed volume - X position vector of a given particle in the undeformed configuration - x(X,t) position vector in the deformed configuration - z, Z axial coordinate - vector of nodal displacements - =(F ^T F–I)/2 Lagrangian strain tensor - history parameter scalar - , azimuthal coordinate - elastic bulk modulus - flow rule coefficient - twisting rate coefficient - elastic shear modulus - iterate - Second Piola-Kirchhoff stress - Cauchy stress - Truesdell stress flux - deviatoric Cauchy stress - Y, Y yield function - residual function - plastic potential - X, Xe, Xp second order tangent modulus tensors in current configuration - X, Xe, Xp second order tangent modulus tensors in undeformed configuration - (.) variational operator - VEC(.) vectorization operator - TEN(.) Kronecker operator - tr(.) trace - Kronecker product     

Die Herleitung von Schalengleichungen über ein erweitertes Variationsprinzip bei Berücksichtigung der Querschubverzerrungen

Ohne ZusammenfassungBezeichnungen L Bezugsgrößen für dimensionslose Koordinaten - L charakteristische Schalenabmessung - t Schalendicke - Schalenparameter - körperfeste, krummlinige, dimensionslose Koordinaten der Schalenmittelfläche - Dimensionslose Koordinate in Richtung der Schalennormalen - i, j,...=1,2,3 Indizierung des dreidimensionalen Euklidischen Raumes - ,,...=1,2 Indizierung des zweidimensionalen Riemannschen Raumes - (...), Partielle Differentiation nach der Koordinate - (...), Kovariante Differentiation für Tensorkomponenten des zweidimensionalen Raumes nach der Koordinate - (...)| Kovariante Differentiation für Tensorkomponenten des dreidimensionalen Raumes nach der Koordinate - Variationssymbol - a ,a ₃ Basisvektoren der Schalenmittelfläche - V Verschiebungsvektor - U ,U ₃ Verschiebungskomponenten des Schalenraumes - v ,w,w ,W Verschiebungskomponenten der Schalenmittelfläche - Verhältnis der Metriktensoren des Schalenraumes und der Schalenmittelfläche - _ik Verzerrungstensor des Raumes - _{(, )}, Symmetrische Verzerrungstensoren der Schalenmittelfläche - _{[, ]} Antimetrischer Term des Verzerrungsmaßes - ^, Spannungstensor - n ,m ,q Tensorkomponenten der Schnittgrößenvektoren - p ,p,c Tensorielle Lastkomponenten     

Extensions of Kronecker product algebra with applications in continuum and computational mechanics

Summary Kronecker product algebra is widely applied in control theory. However, it does not appear to have been commonly applied to continuum and computational mechanics (CCM). In broad terms the goal of the current investigation is to extend Kronecker product algebra so that it can be broadly applied to CCM. Many CCM quantities, such as the tangent compliance tensor in finite strain plasticity, are very elaborate or difficult to derive when expressed in terms of tensor indicial or conventional matrix notation. However, as shown in the current article, with some extensions Kronecker product algebra can be used to derive compact expressions for such quantities. In the following, Kronecker product algebra is reviewed and there are given several extensions, and applications of the extensions are presented in continuum mechanics, computational mechanics and dynamics. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker operations on block matrices are introduced. Kronecker product algebra is extended to third and fourth order tensors. The tensorial nature of Kronecker products of tensors is established. A compact expression is given for the differential of an isotropic function of a second-order tensor. The extensions are used to derive compact expressions in continuum mechanics, for example the transformation relating the tangent compliance tensor in finite strain plasticity in undeformed to that in deformed coordinates. A compact expression is obtained in the nonlinear finite element method for the tangent stiffness matrix in undeformed coordinates, including the effect of boundary conditions prescribed in the current configuration. The aforementioned differential is used to derive the tangent modulus tensor in hyperelastic materials whose strain energy density is a function of stretch ratios. Finally, block operations are used to derive a simple asymptotic stability criterion for a damped linear mechanical system in which the constituent matrices appear explicitly.Appendix: Notation A, Â, a_ij matrix, second-order tensor - a, a_i vector, VEC (A) - a vector - â scalar - B, b_ij matrix, second-order tensor - b, b_i VEC (B) - b vector - block permutation matrix - C, C, c_ijkl fourth-order tensor - right Cauchy Green strain tensor - c VEC () - c_i eigenvalues of - C _a,C _b third-order tensors - C ₁,C ₂,C ₃ outer product functions - D deformation rate tensor - D damping matrix - d VEC (D) - E boundary stiffness matrix - e VEC () - Eulerian strain - F isotropic tensor-valued function ofA - deformation gradient tensor - f VEC - f scalar valued counterpart ofF - G coordinate transformation tensor - G strain-displacement matrix - g VEC (G) - g consistent force vector - H dynamic system matrix - h_n lowest eigenvalue ofH - I,I _n,I ₉ identity matrix/tensor - i VEC (I) - I₁ TRACE () - I index - i index - J determinant of - J matrix relating d to da - J index - j index - J matrix relating d to da - K,K _T,K _b,K stiffness matrices - K index - k index - L velocity gradient tensor - L index - l index - M mass matrix - strain-displacement matrix - M matrix arising from Ogden model - M index - m index - N shape function matrix - N matrix arising from Ogden model - n, n₀ exterior normal vectors - n n² - N index - n index - P, dynamic system matrix - p VEC (P) - p_n eigenvalue ofP - Q rotation tensor - R,r _ij tensor used with outer products - r rank, index - r VEC (R) - S,s _ij tensor used with outer products - s VEC (S) - S, S₀ surface area - S matrix diagonalizingA - s index - T unitary matrix - t₀, t traction - t VEC () - VEC ( ) - t time - T time interval - U _n ,U ₉,U _M permutation matrices - u displacement vector - V matrix appearing in linear system stability criterion - V projection matrix - W,W _i multipliers d - w, w₁ vectors - strain energy function - X undeformed position - x deformed position - Y,Y coordinate system - y, y_j vectors - z, z_j vectors - _j eigenvalue ofA - _j eigenvalue ofB - , ₁, ₂ nodal displacement vectors - _j eigenvalue of - matrix - , ₁, ₂, _a, _b diagonal matrices - _ij entries of the Kronecker tensor (I) - Lagrangian strain - _ijk permutation tensor - _i coefficient of Ogden model - parameter in linear system stability criterion - _i eigenvalue - Lamé coefficient - matrix - surface area factor - Lamé coefficient - _j eigenvalue ofA - _i coefficient of Ogden model - matrix - second Piola-Kirchhoff stress - Cauchy stress - Truesdell stress flux - matrix/tensor - matrix/tensor - matrix/tensor - TEN22 (C) - d rotation vector - d rotation tensor - TRACE(.) trace of a matrix - left Kronecker function - right Kronecker function - VEC(.) vectorization operator - VECB(.) block vectorization operator - TEN22(.) tensor operator - TEN12(.) tensor operator - TEN21(.) tensor operator - _x, _y divergence operator - d(.) differential operator - (.) variational operator - (right) Kronecker product - Kronecker sum - Kronecker difference - block Kronecker product - left Kronecker product - AsB block Kronecker sum ofA andB - AdB block Kronecker difference ofA and      

Thermomechanic behavior of a polymer upon its formation in a crystallization process: Theoretical principles,hypotheses, and mathematical model

A model of thermomechanic behavior of a polymer upon its formation in a crystallization process is proposed. Based on methods of nonequilibrium thermodynamics governing relationships are obtained which make it possible to establish the dependence of the final degree of crystallicity of the material on the history of the crystallization process and to explain the mechanism of formation of the remanent stresses in a polymer article.Notation u translation vector - v velocity vector - acceleration vector - absolute temperature - density - c specific heat capacity - deformation tensor - strain tensor - specific enthropy - U ^* internal energy - z specific free enthalpy - _i internal parameters of state - t time - q heat flux vector - matrix of heat conduction coefficients - W ^* energy dissipation - F vector of mass forces - the 4th rank tensor of elastic pliabilities - matrix of heat expansion coefficients - tensor of contribution of structural variations to deformation - function of equilibrium value ^* - p mean pressure - deviator of the tensor of deformations - spherical part of the deformation tensor - deviator of the tensor of stresses - K volume modulus - unity tensor - Q enthalpy of the crystallization process - Q _eq enthalpy of the equilibrium crystallization process - _g glass transition temperature - ^*() the curve obtained in the equilibrium crystallization process - f final degree of crystallicity Institute of Mechanics of Continuous Media of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Institute of Technical Chemistry of the Ural Branch of the Russian Academy of Sciences, Perm', Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 3, pp. 479–485, May–June, 1995.     

Equations of generalized thermoelasticity of a Cosserat medium in stresses

Thermoelasticity equations in stresses are derived in this paper for a Cosserat medium taking into account the finiteness of the heat propagation velocity. A theorem is proved on the uniqueness of the solution for one of the obtained systems of such equations.Notation u displacement vector - small rotation vector - absolute temperature - ₀ initial temperature of the medium - relative deviation of the temperature from the initial value - , , , , , ,, m constants characterizing the mechanical or thermophysical properties of the medium - density - I dynamic characteristic of the medium reaction during rotation - k heat conduction coefficient - ₀ a constant characterizing the velocity of heat propagation - X external volume force vector - Y external volume moment vector - w density of the heat liberation sources distributed in the medium - E unit tensor - T force stress tensor - M moment stress tensor - nonsymmetric strain tensor - bending-torsion tensor - s entropy referred to unit volume - V volume occupied by the body - surface bounding the body - (T)_ki, (M)_ki components of the tensorsT andM - q thermal flux vector Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 482–488, March, 1981.     

Bounding-surface plasticity: Stress space versus strain space

Summary A bounding-surface plasticity model is formulated in stress space in a general enough manner to accommodate a considerable range of hardening mechanisms. Conditions are then established under which this formulation can be made equivalent to its strain-space analogue. Special cases of the hardening law are discussed next, followed by a new criterion to ensure nesting. Finally, correlations with experimental data are investigated.Notation (a) centre of the stress-space (strain-space) loading surface; i.e., backstress (backstrain) - ^* (a ^*) centre of the stress-space (strain-space) bounding surface - (a ) target toward which the centre of the stress-space (strain-space) loading surface moves under purely image-point hardening - (b) parameter to describe how close the loading surface is to nesting with the bounding surface in stress (strain) space; see (H10) - (c) elastic compliance (stiffness) tensor - (d) parameter to describe how close the stress (strain) lies to its image point on the bounding surface; see (H10) - (D) generalised plastic modulus (plastic compliance); see (1) - function expressing the dependence of the generalised plastic modulus on (plastic complianceD ond) - ^* (D ^*) analogue to (D) for the bounding surface - function expressing the dependence of ^* on (D ^* ond) - () strain (stress) - ' (') deviatoric strain (stress) - ^P ( ^R ) plastic strain (stress relaxation); see Fig. 1 - () image point on the bounding surface corresponding to the current strain (stress) - _iso (f _iso) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change of radius; i.e., fraction of isotropic hardening in the stress-space theory - _kin (f _kin) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change in the backstress (backstrain); i.e., fraction of kinematic hardening in the stress-space theory - _nor (f _nor) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - _ima (f _ima) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - function relating _iso to , , and (f _iso tob,d, andl) - function relating _kin to , , and (f _kin onb,d, andl) - function relating _nor to , , and (f _nor onb,d, andl) - function relating _ima to , , and (f _ima onb,d, andl) - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change of radius - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change in the centre - function relating _iso ^* to (f _iso ^* tod) - function relating _kin ^* to (f _kin ^* tod) - (l) parameter to describe the full extent of plastic loading up to the present, giving the arc length of plastic strain (stress relaxation) trajectory; see (H10) - function relating the direction for image-point translation of the loading surface to various other tensorial directions associated with the current state; see (H5). With 6 Figures     

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     

Decreasing of the initial shear modulus with increasing axial strain explained by means of a plastic-hypoelastic model

Summary The purpose of this work is to examine in detail the possibility to explain the decreasing of the initial shear modulus with increasing axial strain, observed first by Feigen, by means of the plastic-hypoelastic stress-strain relation suggested by Lehmann and by the author of the present paper.Notations _ij components of the infinitesimal strain tensor dilatation - strain deviator - _ij components of the stress tensor - spherical part of the stress tensor - stress deviator - ²= _{ij ij} second invariant of the stress deviator - = ₃₃ axial strain - e= ₁₃ shear component of the strain tensor - =2 ₁₃ shear strain - = ₃₃ axial stress - s= ₁₃ shear stress - T _ij components of Cauchy's stress tensor - F _ij components of the deformation gradient - L _ij components of the velocity gradient (Eulerian coordinates) - components of the rate of deformation tensor - components of the spin tensor - components of the rate of deformations deviator - components of Cauchy's stress deviator - T=T ₃₃ axial Cauchy's stress With 7 Figures     

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     

Order parameters and energies of analytic and singular vortex lines in rotating3He-A

We present the expressions of the generalized Ginzburg-Landau (GL) theory for the free energy and the supercurrent in terms of thed vector, the magnetic fieldH, and operators containing the spatial gradient and the rotation. These expressions are then specialized to the Anderson-Brinkman-Morel (ABM) state. We consider eight single-vortex lines of cylindrical symmetry and radiusR=[2m/]^–1/2: the Mermin-Ho vortex, a second analytic vortex, and six singular vortices, i.e., the orbital and radial disgyrations, the orbital and radial phase vortices, and two axial phase vortices. These eight vortex states are determined by solving the Euler-Lagrange equations whose solutions minimize the GL free energy functional. For increasing field, the core radius of the texture of the Mermin-Ho vortex tends to a limiting value, while the core radius of the texture goes to zero. The gap of the singular vortices behaves liker forr 0, where ranges between and . The energy of the radial disgyration becomes lower than that of the Mermin-Ho vortex for fieldsH6.5H*=6.5×25 G (atT=0.99T _c and forR=10L*=60 µm, or=2.9 rad/sec). ForR 2 _T ( _T is the GL coherence length) or _c2 (upper critical rotation speed), the energies of the singular vortices become lower than the energies of the analytic vortices. This is in agreement with the exact result of Schopohl for a vortex lattice at _c 2. Finally, we calculate the correction of order (1 -T/T _c ) to the GL gap for the axial phase vortex.     

Flow of a generalized second grade fluid between heated plates

Summary We examine the fully developed flow of a generalized fluid of second grade between heated parallel plates, due to a pressure gradient along the plate. The constant coefficient of shear viscosity of a fluid of second grade is replaced by a shear dependent viscosity with an exponentm. If the normal stress coefficients are set equal to zero, this model reduces to the standard power-law model. We obtain the solution for the case when the temperature changes only in the direction normal to the plates for the two most commonly used viscosity models, i.e. (i) when the viscosity does not depend on temperature, and (ii) when the viscosity is an exponentially decaying function of temperature.

List of symbols

Alphanumeric A ₁,A ₂ Kinematical tensor - b Body force - C Dimensionless parameter related to the pressure gradient - h Separation between the plates - L Velocity gradient - m Power-law index - M Constant appearing in the Reynolds viscosity model - p Pressure field - Modified pressure field - q Heat flux vector - r Radiant heating - T Cauchy's stress tensor - l Unit tensor - v Velocity vector - V Characteristic velocity - x Axis along the plate - y Axis perpendicular to the plate Greek ₁, ₂ Normal stress coefficient - Specific internal energy - Dimensionless parameter related to the viscous dissipation - Conservative body force field - Specific entropy - Thermal conductivity - Coefficient of viscosity - ₀ Reference viscosity - Second invariant of the stretching tensor - Temperature - ₁ Temperature of the lower plate - ₂ Temperature of the upper plate - Density - Specific Helmholtz free energy Operators div Divergence - grad Gradient - tr Trace     

Axi-symmetric natural frequencies and response of a spinning liquid column under strong surface tension

Summary The response of a solidly rotating anchored finite liquid column consisting of frictionless liquid is subjected to axial harmonic excitation. The response of the free liquid surface elevation and velocity distribution has been determined analytically in the elliptic (>2 ₀) and hyperbolic frequency range (>2 ₀). For the liquid surface displacement the response has been evaluated numerically as a function of the forcing frequency/2 ₀. In addition the first natural stuck-edge frequency has been determined and compared with the slipping case.List of symbols a radius of liquid bridge - h length of liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution in rotating liquid - Weber number - z₀ axial excitation amplitude - elliptic case (>2 ₀) - hyperbolic case (>2 ₀) - liquid density - surface tension - liquid surface displacement - acceleration potential - ₀ rotational speed - axial forcing frequency - natural frequency of rotating system - _0n natural frequency of harmonic axial response     

Zero-point motion and superfluid helium

We propose that He II exhibits macroscopic [ _P /N O(1)] quantum zero-point motion in momentum space, i.e., that a nonzero root-mean-square superfluid velocity exists even in an equilibrium superfluid system at rest. At absolute zero, using coherent states, we relate the uncertainty _P /N in the total momentumP (per particle) to the long-range-order (LRO) part of the phase gradient correlation function, which is proposed as an order parameter. The local equilibrium equation for the superfluid velocity potential derived by Biswas and Rama Rao yields, in the strict equilibrium limit, the equation determining this order parameter in terms of fluctuation correlations that remain to be determined. The order parameter is interaction dependent, nonzero atT=0 if (0)–₀V₀>0, and can vanish at some transition temperatureT when fluctuation terms become comparable to theT=0 value. (HereV _{0 0}, and (0) are the uniform parts of the potential, density, and chemical potential with shifted zero of energy, respectively.) A characteristic length (T), diverging atT=T , appears naturally, with its defining relation reducing to a macroscopic uncertainty relation ( _P /N)(0)=/2 atT=0. With certain assumptions it is shown that atT=0, LRO in the phase gradient correlation function is incompatible with off-diagonal long-range order (ODLRO) in the (r)(r) correlation function, and with nonzero condensate function.     

Free vibrations of a circular plate with attached concentrated mass,spring and dashpot

Summary The natural vibrations of a circular plate with attached concentrated mass, spring and dashpot have been obtained by means ofYoung's analysis [1]. The results are presented in terms of eigen-functions of the plate alone. The case of a plate carrying two masses and resting on elastic foundation has also been studied. Some particular cases have been deduced.

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Zusammenfassung Die Eigenschwingungen einer Kreisplatte mit lokal befestigter Einzelmasse, Feder und Dämpfer werden nach derYoungschen Methode [1] ermittelt. Die Ergebnisse werden als Entwicklung nach den Eigenfunktionen der reinen Plattenschwingung dargestellt. Der Fall der elastisch gebetteten Platte mit zwei Einzelmassen wird ebenfalls studiert. Einige Sonderfälle werden hergeleitet.

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Nomenclature a radius of circular plate - h plate thickness - k ₁ spring constant - k _c generalized spring constant - modulus of elastic foundation - decay constant - c dashpot strength - D , flexural rigidity of plate - E Young's modulus - v Poisson's ratio - p natural frequency of plate alone - natural frequency of composite system - w deflection mode of plate - r, cylindrical coordinates - mass density - r - (l/D)^1/4 - - - F _m ,L _m ,G _m ,M _m unknown constants With 5 Figures     

Theory of dielectric polarization of a substance. Calculation of dielectric permittivity of water,ammonia, and chlorine

A theory of dielectric polarization of a substance is developed. The theory is verified by experiment and by phenomenological relations that follow from the determination of polarization, molar polarization, and dielectric permittivity.Notation _s static dielectric permittivity - high-frequency dielectric permittivity - _s permittivity perpendicular to the acisC - _s permittivity in the direction of the axisC - a average molecular polarizability - dielectric susceptibility - _i molecular hyperpolarizability - p ⁰ constant dipole moment of a molecule - p dipole moment of a molecule in condensed state - p _add additional dipole moment of a molecule - P polarization of a substance - P _m molar polarization - k Boltzmann constant - T Kelvin temperature - t Celsius temperature - angle between the vectors and - F internal electric field strength - Û internal interaction energy, J·mol^–1 - û internal interaction energy per molecule - N ₀ Avogacro number - V ₀ molar volume - a _t total molecular polarizability - H ⁰ (H) enthalpy as a function of temperature - l(x) Langevin function - n molecular concentration Murmansk State Academy of Fish Fleet. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 5, pp. 767–773, September–October, 1995.     

On apparent properties of nonlinear heterogeneous bodies smaller than the representative volume

Summary Huet's model for overall properties of specimens smaller than the representative volume is generalized on nonlinear heterogeneous elastic materials with imperfect interfaces. A modified definition for the apparent properties of heterogeneous nonlinear elastic bodies is given. The size effect relationships are established between experimental results obtained on a big specimen and on an appropriate set of smaller specimens. Hierarchies between the apparent properties of the families of specimens of different sizes are constructed.Notation D domain occupied in space by a material body - D boundary ofD - x position of a material point at timet - n outward normal - internal interface of a heterogeneous body - [a]=a ⁺–a^– jump bracket ofa on an interface in the direction of the outward normal - stress tensor - strain tensor - displacement vector - P traction vector density on a surface - a spatial average of the variablea on a domainD - x tensor product (dyadic) - twice contracted tensor product - syma symmetric part 1/2(a+a ^T ) of the tensora - F ,F potential energy and complementary energy functionals, respectively - S ^eff,C ^eff effective compliance and modulus tensors, respectively - S ,C kinematic apparent compliance and modulus tensors, respectively - S ,C static apparent compliance and modulus tensors, respectively     

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     

Statistical computation of tensor of self-diffusion coefficients in the liquid-crystal nematic phase

Within the framework of the Enskog theory expressions are obtained for the tensors of the friction and self-diffusion coefficients in the ordered nematic phase of a system of nonspherical particles. Calculations of the tensor of the self-diffusion coefficients are performed; their comparison with the data of a computer experiment shows that the Enskog theory can be used to calculate the kinetic coefficients in such systems up to densities of 0.6–0.7 of the close-packing density.Notation N the number of particles in system - V volume occupied by system - a, b semiaxes of ellipsoid of revolution - =4/3ab ² particle volume - m particle mass - diameter of spherical particle equal to the ellipsoid of revolution - =N/V particle number density - density in the state of close packing - dimensionless density - = packing coefficient - T absolute temperature - k _B Boltzmann's constant - D, , tensors of the coefficients of self-diffusion, relaxation times, and friction - T ₂₁ operator of binary collisions - e₁, e₂, n unit vectors along the axes of symmetry of two particles of system and axis of orientational order - f(e) single-particle orientation distribution function - g _c (e₁, e₂, k) contact value of the two-particle correlation function - s order parameter - k unit vector of outward normal to surface of first particle Belarus State Technological University, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 5, pp. 743–749, September–October, 1995.     

Thermodynamics of the quasiequilibrial growth of crazes

By comparing the morphology and physical properties (averaged over the scale of 1 to 10m) of a crazed and uncrazed polymer, it can be concluded that crazing is a new phase development in the initially homogeneous material. The present study is based on recent work on the general thermodynamic explanation of the development of a damaged layer of material. The treatment generalizes the model of a crack-cut in mechanics. The complete system of equations for the quasiequilibrial craze growth follows from the conditions of local and global phase equilibrium, mechanical equilibrium and a kinematic condition. Constitutive equations of craze growth-equations are proposed that are between the geometric characteristics of a craze and generalized forces. It is shown that these forces, conjugated with the geometric characteristics of a craze, can be expressed through the known path independent integrals (J, L, M,). The criterion of craze growth is developed from the condition of global phase equilibrium. F Helmholtz's free energy - G Gibb's free energy (thermodynamic potential) - f density ofF - g density ofG - T absolute temperature - S density of entropy - strain tensor - components of - stress tensor - components of - _y stress along the boundary of an active zone (yield stress) - _b stress along the boundary of an inert zone - applied stress - value of at the moment of craze initiation - K stress intensity factor - C tensor of elastic moduli - C ^–1 tensor of compliance - internal tensorial product - V volume occupied by sample - V ₁ volume occupied by original material - V ₂ volume occupied by crazed material - V boundary ofV - (V) vector-function localized on V - (x) characteristic function of an area - (x) variation of(x) - (x) a finite function - tensor of alternation - components of the boundary displacement vector - l components of the vector of translation - n components of the normal to a boundary - _k components of the vector of rotation - e symmetric tensor of deviatoric deformation of an active zone - expansion of an active zone - J ⁽ⁱ⁾ ,L _k ⁽ⁱ⁾ ,M ⁽ⁱ⁾,N ⁽ⁱ⁾ partial derivatives ofG ⁽ⁱ⁾ with respect tol , _k, ande , respectively - [ ] jump of the parameter inside the brackets - thickness of a craze - 2l length of a craze - 2b length of an active zone - l _c distance between the geometrical centres of the active zone and the craze - ^* craze thickness on the boundary of an active and the inert zone - l ^* craze parameter (length dimension) - A craze parameter (dimensionless) - ^* extension of craze material