Scalar Potential Functions Research Articles

A Viscoplastic Constitutive Theory for Monolithic Ceramics—I

This paper, which is the first of two in a series, provides an overview of a viscoplastic constitutive model that accounts for time-dependent material deformation (e.g., creep, stress relaxation, etc.) in monolithic ceramics. Using continuum principles of engineering mechanics, the complete theory is derived from a scalar dissipative potential function first proposed by Robinson (1978), and later utilized by Duffy (1988). Derivations based on a flow potential function provide an assurance that the inelastic boundary value problem is well posed, and solutions obtained are unique. The specific formulation used here for the threshold function (a component of the flow potential function) was originally proposed by Willam and Warnke (1975) in order to formulate constitutive equations for time-independent classical plasticity behavior observed in cement and unreinforced concrete. Here constitutive equations formulated for the flow law (strain rate) and evolutionary law employ stress invariants to define the functional dependence on the Cauchy stress and a tensorial state variable. This particular formulation of the viscoplastic model exhibits a sensitivity to hydrostatic stress, and allows different behavior in tension and compression.

An Axisymmetric Inclusion in One of Two Perfectly Bonded Dissimilar Elastic Half-Spaces

This paper introduces an effective method for the solution of axisymmetric elastic inclusions in bonded dissimilar half-spaces. The method leads to new concise expressions for the elastic fields in terms of only two scalar Papkovich potential functions. A host of earlier solutions, which were obtained using a range of alternative techniques, involving surface stress relaxation and Hankel transforms, are shown to be subsumed in the present result.

Analysis technique for interaction of high-frequency rhombic inducer field with cracks in metals

In the nondestructive evaluation (NDE) involving the ac field measurement (ACFM), a current carrying structure is required to induce the eddy current in the work-piece and a probe to sample the field. Due to its flat profile, slender shape, and other advantages, the rhombic wire loop is a suitable inducer for developing linear flexible arrays for the ACFM inspection of large surfaces of ferrous and nonferrous metals. This paper introduces an analysis technique for the evaluation of the interaction of the field of the rhombic inducer carrying a high-frequency current, with long surface cracks of uniform depth in flat metal plates. The technique is accurate and very efficient computationally. It uses the two-dimensional Fourier transform together with a special boundary condition at the metal surface. The boundary condition takes into account the thin-skin nature of the eddy current in the metal as well as the flux leakage at the crack mouth. The analysis technique benefits from the use of scalar potential functions and can be extended to simply or multiply connected wire loops. Also, it is applicable to high-frequency (thin-skin) eddy current problems. Using the analysis technique, the tangential field below a rhombic inducer along its long diagonal when the inducer is located above the surface of aluminum and steel is given in the presence and absence of a crack. This field was found to have a nonuniform phase distribution. Near a crack, the phase change is significant, even for shallow cracks. The role of the nonuniform phase in the detection sensitivity is addressed. Also, simulated ACFM crack responses using an inducer with a linear probe attached along the long diagonal are presented and discussed. To support the validity of the analysis technique, experimental results obtained for some of the simulations are also reported. In addition to its application in predicting crack responses, the technique can be used for model-based inversion of crack signals.

Micromechanics and homogenization of inelastic composite materials with growing cracks

A homogenization scheme is employed to derive the effective constitutive equations of an elastoplastic composite system with growing damage. The homogenization procedure followed herein is based on the thermodynamics of dissipative media. It is shown that when damage consists of sharps microcracks the macroscopic constitutive behavior is that of a so-called generalized standard material. The latter is a general dissipative medium whose constitutive equations are completely characterized by a single scalar convex potential function of the chosen state variables and whose evolution is completely characterized by a single convex dissipation potential function of the thermodynamic forces conjugate to the chosen internal state variables. The analysis presented is valid under the assumption that the evolution of the representative volume element at hand is unique and stable. The results of the theoretical analysis are then employed for formulating an approximate method for practically deriving the macroscopic constitutive equations. Computer software development for the application of said method is currently ongoing. A simple example of the numerical results obtained so far is presented.

Finite Difference Calculus Invariant Structure of a Class of Algorithms for the Nonlinear Klein–Gordon Equation

In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some numerical experiments are presented to illustrate the conservation property of the proposed algorithms.

Incremental stresses in loaded orthotropic circular membrane tubes—I. Theory

Starting from the equations of general elastic nonlinear membrane theory in intrinsic form, the equations governing the incremental state of stress in an orthotropic circular membrane tube are derived and discussed. The tube is initially subjected to uniform internal pressure and to longitudinal extension, which lead to large homogeneous deformation. Then some changes in loading and/or geometry are considered, e.g. an additional load is applied, the shape of the boundary is changed or a slit is formed in the membrane. These changes are regarded as small perturbations on the initial homogeneous state of stress. The general form as well as some simplified forms of the equations are presented, leading finally to a set of two equations in terms of two scalar potential functions. Two different variational formulations for the problem are also presented, each of which may serve as a basis for numerical treatment.

Regular boundary integral solution with dual and complementary variational formulations applied to two‐dimensional Laplace problems

AbstractRegular boundary integral elements are employed for the dual and complementary variational formulations of Laplace problems. The problems are defined only on the boundary as usual, but as in the manner of charge simulation method (CSM), the source terms are arranged outside the domain so that the singular integral can be avoided. The Laplace problems are analysed and the usefulness of the formulation is discussed in comparison with direct BEM. The scalar potential and electric displacement vector potential functions are used in the dual and complementary formulations, with which the upper and lower bounds of the system energy or the capacitance can also accurately be evaluated.

Multipolar dielectrophoretic force calculation

The conventional theory of dielectrophoresis, based on the approximation that only the dipole is induced, fails to predict observed behavior for a particle situated near a null or in a highly non-uniform field because of the neglect of induced higher-order moments. A recently published equivalent multipolar moment theory, while it enables calculation of these higher-order DEP force contributions, is not easily interpreted or used because it is expressed in terms of a Legendre function expansion of the field. The new formulation of the multipolar DEP theory provided here expresses the net force as the gradient of a series of scalar electromechanical potential functions. This result, valuable in its own right because it enables calculation of the higher-order DEP force terms without reliance upon Legendre function expansions, also leads to a generalized formalism for the complete ponderomotive force on a particle using multipolar moment tensors. The tensor form, readily recognized as a generalization of the original dipole-based DEP theory, provides clear physical insight about the magnitude and the direction of the net force on an uncharged particle in a non-uniform electric field.

Overview no. 117 The structure of constitutive laws for the sintering of fine grained materials

This paper examines the general structure of constitutive laws for the sintering of fine grained materials for situations where power-law creep and grain-boundary diffusion are the dominant mechanisms of deformation and densification. The description of grain-boundary diffusion accepts an interface reaction. Constitutive laws for general multiaxial stress states are expressed in terms of scalar potential functions. Bounds to the potential are obtained which take into account the coupling between power-law creep and diffusion.

RCS Computation of Coax-Loaded Microstrip Patch Antennas of Arbitrary Shape

Abstract A space-domain mixed-potential integral equation approach is applied in conjunction with the method of moments to compute the radar cross-section (RCS) of coax-loaded microslrip patch antennas having arbitrary or irregular shapes. The effects of Lhe substrate—which may be electrically thick and may consist of any number of planar, possibly uniaxially anisotropic dielectric layers, backed by a ground plane—are rigorously incorporated in the analysis by means of the vector and scalar potential Green's functions. The latter are expressed in terms of the voltages and currents on transmission line analogs of the layered medium, associated with TM and TE partial fields. The current distribution on the microstrip patch is approximated using vector basis functions defined over triangular elements and the coax probe current is expanded in terms of piecewise-lincar subdomain basis functions. A simple probe-to-patch attachment mode, compatible with the triangular element model of the microstrip patch, is used to enforce current continuity at the junction, and the coax aperture is modeled by a magnetic enrrent frill. The far zone fields are found by the stationary phase method, and are expressed in terms of the Fourier-transformed basis functions and the transmission line voltages and currents evaluated at the stationary phase point value of the transverse wavenumber. Computed RCS results are presented for several loaded and unloaded microstrip patch antennas of various shapes and are shown to be in agreement with published measured data and with computed results obtained by specialized techniques, which—unlike the method presented here—are not easily extendable to arbitrary shapes.

A simple and accurate complex image interpretation of vertical antennas present in contiguous dielectric half-spaces

An efficient technique is developed to solve the important problem of vertical antennas located above and penetrating the interface between contiguous dielectric half-spaces. It begins by deriving the spectral expressions of the vector and scalar potential Green functions in terms of the TM wave spectral transmission coefficient. The complex image technique is used to derive simple, accurate, and rapidly convergent expressions of the full-wave spatial Green functions. The spatial Green functions are used in the moment method to model vertical thin-wire antennas residing above the penetrating the interface separating the contiguous dielectric half-spaces with and without loss. >

Magnetic Field Solutions of E-Polarization Induction Problems.

An E-polarization problem in geo-electromagnetic induction would not normally be solved in terms of the two components of the magnetic field since it is much simpler to use the single component of the electric field which serves as a scalar potential function. When one is dealing with the two-dimensional limit at infinity of a three-dimensional structure, however, and when a finite difference solution of the full three-dimensional model is obtained in terms of the magnetic vector, then the unorthodox form of the problem emerges quite naturally on the boundaries of the three-dimensional grid. It is not difficult to write down finite difference equations in terms of the two magnetic components and the resistivity, but uncertainties arise when assigning resistivity values to the nodes because normally in two-dimensional E-polarization problems, the nodal values are defined as weighted averages of conductivities which are not the same as the reciprocals of the averaged resistivities. An alternative procedure, which avoids this ambiguity in the numerical model, is to obtain the finite difference equations by integrating over rectangular domains at each node. This approach leads to two 9-point equations in the magnetic components (rather than the usual single 5-point equation for the electric component) whose solution was initially found to be numerically unstable. A method of circumventing this instability is described and numerical solutions of E-polarization problems obtained using the magnetic components as variables are compared with solutions found by traditional finite difference methods.

Coaxial electromagnetic launcher calculations using FE-BE method and hybrid potentials

A hybrid method combining finite and boundary elements (the FE-BE method) is presented to analyze the transient electromagnetic/mechanical behavior of coaxial induction launchers (i.e., the coilgun). The corresponding initial/boundary value problem is formulated in terms of the hybrid potentials, which mixes the vector and scalar magnetic potential functions. The problem is assumed to be axisymmetric and the forcing currents are circumferential. The thermal effect due to ohmic losses is considered during the launching processes. Simulations of a single-stage induction launcher and of a five-stage launcher are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The use of a relaxation method to calculate the 3D magnetic field contribution of an iron yoke

A computational procedure has been developed for calculating the 3-D field produced by an axisymmetric iron yoke of high permeability in the presence of a system of conductors. The procedure is particularly applicable to the end regions of multipole magnets of the sort used in particle accelerators. The field produced by the conductors is calculated using the Biot-Savart law. If one solves the appropriate individual differential equations for specified scalar potential functions throughout the iron-free region, with the proper applied boundary condition for the scalar potential of each harmonic number, one will achieve upon summation the appropriate potential function to describe the field contribution of the surrounding high-permeability iron.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of microstrip resonators of arbitrary shape

A space-domain approach based on a mixed-potential integral equation formulation is developed for efficient computation of complex resonant frequencies of laterally open microstrip-pitch resonators of arbitrary shape. The effects of the substrate-which may consist of any number of planar, possibly uniaxially anisotropic, dielectric layers-are rigorously incorporated in the formulation by means of the vector and scalar potential Green's functions. The current distribution on the conducting patch is approximated in terms of vector basis functions defined over triangular elements. Computed resonant frequencies, quality factors, modal currents, and far-field radiation patterns are presented for several microstrip resonators. For patches of simple, regular shapes, the results are in agreement with published data obtained by specialized techniques, which, unlike the method presented here, are not easily extendible to arbitrary shapes. >

Inductance of perfectly conducting foils including spiral inductors

A numerical method for calculating the lumped inductance parameters of perfectly conducting foils (i.e. current sheets) is presented. A quasi-static analysis for computing the inductance for foils arbitrarily shaped in three dimensions is described. The vector current distribution on the structure is solved in terms of a scalar current potential function. The method of moments is utilized to solve the integral equation. Numerical results are also presented. The strength of this technique is that a bound on the numerical accuracy can be provided. The relative error provides not only a self-consistency check, but also the accuracy with which the numerical values have been computed. Foil structures which can be approximated accurately with a triangular patch model and whose terminal edge current distributions are known can be analyzed with this method. >

An analytical solution for the field of a hysteresis motor based on complex permeability

A closed-form solution for the electromagnetic field distribution inside a hysteresis motor is presented. The solution is based upon Maxwell's field equations, considering the case of a circumferential-flux-type machine at synchronous speed. A hysteresis loop in the shape of an inclined ellipse is adopted. The application of the complex permeability concept to space rotating vectors is explained. A new form for the general solution of Laplace's equation is used. This new form makes it possible to obtain a proper matching between the field components at the rotor outer surface when the scalar potential function is used and the hysteresis is considered. The torque equation is developed in order to allow comparison with test results. The proposed solution is simple, accurate, and rigorous. Comparison with test results shows the accuracy of the method. >

Relativistic mean field theory for nuclei

The phenomenology of relativistic mean field theory of nuclei is discussed for general scalar potential functions U( φ) with a minimum at zero scalar field φ = 0. The cubic-plus-quartic scalar self-interaction model U 34 is discussed as an important special case. We obtain general conditions on U required by the saturation properties including the incompressibility coefficient K. The effective mass M ∗ is uniquely related to the vector coupling. The pathologies of the U 34 model are discussed and modifications are proposed which cure the model when the quartic term is negative as phenomenologically required. It is argued that the equation of state (at T = 0) is effectively independent of the form of U if the parameters are adjusted to a given K, M ∗ , and that the phenomenology of finite nuclei, in particular the spin-orbit splitting in light nuclei, determines M ∗ ≈ 0.6 M essentially independently of U. The resulting (stiff) e.o.s. is then determined within narrow limits independently of the form of U.

Matter tensor in scalar–tensorial theories of gravitation

Gravitational interaction is assumed to be partially geometrized and described by 10 potential functions in the framework of a Riemannian geometry. It is also partially nongeometrized and described by an additional scalar potential function nonlinearly coupled to the energy density. The scalar potential naturally leads to a conformally Riemannian metric, enabling us to completely geometrize the gravitation and to establish a close analogy with the Brans–Dicke theory. Finally, the scalar field is endowed with the rest mass, and we suggest that it be identified with the short-range graviscalar field.

Three-dimensional magnetic engineering: The programs magnus and epilog

We present the post-processor EPILOG for the well established finite element program MAGNUS for three-dimensional magnetic engineering. MAGNUS solves problems of magnetostatics with nonlinear magnetic materials, permanent magnets and electric currents, for any 3-D geometry. The two-scalar-potentials formulation of magnetostatics used by MAGNUS combines numerical accuracy and computational efficiency, and is considered state of the art. The well known program KUBIK is used as a pre-processor to describe the geometry and finite element mesh. KUBIK is highly interactive and allows the user to effectively control all geometric details.The needs of magnetic engineers, however, go far beyond the simple availability of a mathematical solution. Once the solution has been obtained by MAGNUS in the form of a continuous magnetic scalar potential function defined at every point in the solution domain, those needs are met by EPILOG. EPILOG is command operated. Commands are independent of each other and can be used in any order, or not used at all. The purpose of each command is to use the solution for the calculation of a derived quantity or the production of a plot or table. The following derived quantities can be obtained: the magnetic energy in specific regions, the magnetic force on specified conductors in space, the magnetic torque on specified conductors, the magnetic flux across a given surface in space, the inductance of a circuit, and a variety of line integrals for specified lines in space. A useful facility is the automatic calculation of harmonic multipoles averaged along the beam direction for accelerator magnets, essential for end analysis and the integral effect of the magnetic field on the beam.Graphical facilities include color plots of the shapes of the conductors, the geometry, field lines and surfaces of constant magnetic scalar potential in specified regions of space. EPILOG produces a device independent graphical metafile, which can be seen on any device by running the Graphical Kernel System GKS or some other graphics package. Another important feature of EPILOG is its ability to produce tables of quantities such as the scalar potential, the total field, the field of conductors alone, the magnetic permeability, etc. Editing options allow the user to specify where the quantities are to be tabulated, such as on equispaced points on a line or arc in space. One useful application is the evaluation of the maximum field inside conductors for superconducting magnets.The ability of EPILOG to accurately provide the magnetic field components at any given point in space is convenient for its use as input to particle tracking programs. EPILOG can easily be interfaced to such a program. EPILOG is appropriate for complicated 3-D geometries, common in wigglers and undulators, or for the analysis of ends of bending magnets, as required in the design of synchrotron light sources.