Scalar Potential Functions Research Articles

Transient response of a thermoelastic half‐space to mechanical and thermal buried sources

With the aid of a complete set of two scalar potential functions, the transient responses of an isotropic thermoelastic half‐space subjected to time dependent tractions and heat flux applied to a finite patch at an arbitrary depth below a free surface are derived. Using the displacements‐ and temperature‐potential function relationships, the coupled equations of motion and energy equation are uncoupled, resulting in two (6th and 2nd order) partial differential equations in the cylindrical coordinate system, which are solved with the aid of Fourier series expansion and joint Hankel‐Laplace integral transforms. The solutions are also investigated in details for tractions varying with time in terms of a Heaviside step function and heat flux as a Dirac delta function, which may be used as a kernel in any integral based method for more complicated thermoelastodynamic initial‐boundary value problems. Due to the complexity of the integrands involved in the general case, the integrals cannot be resolved analytically and thus an appropriate numerical algorithm is used for the inversion of the Laplace and Hankel integral transforms. To demonstrate the pattern of deformations as well as the distribution of change of temperature at the free surface of the half‐space, numerical evaluations for these functions are presented for an isotropic material.

Coupled thermoviscoelastodynamic Green's functions for bi‐material half‐space

By virtue of the representations of displacements, stresses, and temperature fields in terms of two scalar potential functions and the use of correspondence principle, an analytical derivation of fundamental Green's functions for bi‐material half‐space composed of a transversely isotropic thermo‐elastic layer and an isotropic thermo‐visco‐elastic half‐space affected by finite surface or interfacial sources is presented. With the aid of the potential function relationships, the coupled equations of motion and energy equation in both the half‐space and the layer are uncoupled and solved with the aid of Fourier series and Hankel integral transforms. Responses of the medium are derived in the form of improper line integrals related to Hankel inversion transforms. To show the effects of anisotropy and viscoelasticity on the propagation of coupled thermoviscoelastic waves, the derived integrals for displacements, stresses, and temperature Green's functions are evaluated by a numerical scheme.

Brans-Dicke theory and the emergence ofΛCDMmodel

The dynamics of the Brans-Dicke theory with a scalar field potential function is investigated. We show that the system with a barotropic matter content can be reduced to an autonomous three-dimensional dynamical system. For an arbitrary potential function we found the values of the Brans-Dicke parameter for which a global attractor in the phase space representing de Sitter state exists. Using linearized solutions in the vicinity of this critical point we show that the evolution of the Universe mimics the $\Lambda$CDM model. From the recent Planck satellite data, we obtain constraints on the variability of the effective gravitational coupling constant as well as the lower limit of the mass of the Brans-Dicke scalar field at the de Sitter state.

The wave-function description of the electromagnetic field

For an arbitrary electromagnetic field, we define a prepotential S, which is a complex-valued function of spacetime. The prepotential is a modification of the two scalar potential functions introduced by E. T. Whittaker. The prepotential is Lorentz covariant under a spin half representation. For a moving charge and any observer, we obtain a complex dimensionless scalar. The prepotential is a function of this dimensionless scalar. The prepotential S of an arbitrary electromagnetic field is described as an integral over the charges generating the field. The Faraday vector at each point may be derived from S by a convolution of the differential operator with the alpha matrices of Dirac. Some explicit examples will be calculated. We also present the Maxwell equations for the prepotential.

Thermoelastodynamics with Scalar Potential Functions

AbstractA linear thermoelastic isotropic material is considered. A complete solution in terms of three scalar potential functions for the coupled displacement-temperature equations of motion and heat equation is presented, where the governing equations for the potential functions are the wave, heat, or a repeated wave-heat equation. The completeness theorem is proven based on a retarded Newtonian potential function, existence of solutions for the repeated wave equation and heat equation, and perturbation theory. A connection is also made between a complete solution already existing in the literature and the solution introduced in this paper, which in itself is another method to prove the completeness. If no heat source exists, the number of potential functions is reduced to two. In some circumstances, the number of potential functions is reduced to only one, and the required conditions for this case are discussed. As a special case, the torsionless and rotationally symmetric configuration with respect to ...

Analytical Solution of Coupled Thermoelastic Axisymmetric Transient Waves in a Transversely Isotropic Half-Space

A half-space containing transversely isotropic thermoelastic material with a depth-wise axis of material symmetry is considered to be under the effects of axisymmetric transient surface thermal and forced excitations. With the use of a new scalar potential function, the coupled equations of motion and energy equation are uncoupled, and the governing equation for the potential function, is solved with the use of Hankel and Laplace integral transforms. As a result, the displacements and temperature are represented in the form of improper double integrals. The solutions are also investigated in detail for surface traction and thermal pulses varying with time as Heaviside step function. It is also shown that the derived solutions degenerate to the results given in the literature for isotropic materials. Some numerical evaluations for displacement and temperature functions for two different transversely isotropic materials with different degree of anisotropy are presented to portray the dependency of response on the thermal properties as well as the degree of anisotropy of the medium.

Frequency Domain Analysis of an Axisymmetric Thermoelastic Transversely Isotropic Half-Space

By virtue of a new complete scalar potential function, an analytical formulation is presented to determine displacements, stresses, and temperature in an axisymmetric linear thermoelastic transversely isotropic half-space affected by time harmonic surface axisymmetric vertical traction and/or surface heat flux. With the use of only one scalar potential function in a cylindrical coordinate system, the coupled partial differential equations in thermoelasticity are transformed to a sixth-order partial differential equation governing the potential function. Then, by using the Hankel integral transforms, a sixth-order ordinary differential equation is received, which after solving, results in the displacements, stresses, and temperature fields. To prove the validation of the analytical solution presented in this paper, the solutions are reduced to the case of elastodynamics in transversely isotropic half-space and in a quasi-static thermoelastic isotropic half-space, where exact agreements with the solutions reported in the literature are achieved. Because of complexity of the integrands, numerical evaluations of the integrals involved in this paper are carried out using a suitable quadrature scheme by Mathematica software. To show the accuracy and efficiency of the numerical algorithm, the solution of this study is compared with the results of an existing elastodynamic transversely isotropic half-space, where an excellent agreement is achieved.

Interaction of rocking vibration of a buried rigid circular disc and a two-layer transversely isotropic half-space

A linear elastic layer of finite thickness bonded on a half-space each containing a transversely isotropic material is considered. A rigid circular disc attached to the interface of these two domains is considered to be affected by a rocking vibration of constant amplitude. With the aid of a scalar potential function, the equations of motion in each domain are solved using Fourier series and Hankel integral transforms. Because of the involved integral transforms, the mixed boundary value problem is changed to dual integral equations; which are reduced to Fredholm integral equations. Because of the complex integrand function existing in the dynamic case, analytical solution cannot be given in general. However, a closed-form solution is introduced for the static case, which itself degenerates to the solution for an isotropic case existing in the literature. The contact stress in between the rigid disc and the surrounding media, and the related impedance function are analytically determined in the static case. With the help of contour integration, the governing Fredholm integral equations are numerically evaluated in the dynamic case. The dynamic contact pressure and the impedance function are numerically evaluated in a general dynamic case. The shape induced singularity in the contact pressure is investigated in detail. Some numerical evaluations are given for different transversely isotropic materials to show the effect of anisotropy.

Dynamic Green's Functions of an Axisymmetric Thermoelastic Half-Space by a Method of Potentials

AbstractWith the aid of a new complete scalar potential function, an analytical formulation for thermoelastic Green's functions of an axisymmetric linear elastic isotropic half-space is presented within the theory of Biot's coupled thermoelasticity. By using the potential function, the governing equations of coupled thermoelasticity are uncoupled into a sixth-order partial differential equation in a cylindrical coordinate system. Then, by using Hankel integral transforms to suppress the radial variable, a sixth-order ordinary differential equation is received. By solving this equation and considering boundary conditions, displacements, stresses, and temperature are derived in the Hankel integral transformed domain. By applying the theorem of inverse Hankel transforms, the solution is obtained generally for arbitrary surface time-harmonic traction and heat distribution. Subsequently, point-load Green's functions for the displacements, temperature, and stresses are given in the form of some improper line in...

Reflection and refraction of attenuated waves at boundary of elastic solid and porous solid saturated with two immiscible viscous fluids

The propagation of elastic waves is studied in a porous solid saturated with two immiscible viscous fluids. The propagation of three longitudinal waves is represented through three scalar potential functions. The lone transverse wave is presented by a vector potential function. The displacements of particles in different phases of the aggregate are defined in terms of these potential functions. It is shown that there exist three longitudinal waves and one transverse wave. The phenomena of reflection and refraction due to longitudinal and transverse waves at a plane interface between an elastic solid half-space and a porous solid half-space saturated with two immiscible viscous fluids are investigated. For the presence of viscosity in pore-fluids, the waves refracted to the porous medium attenuate in the direction normal to the interface. The ratios of the amplitudes of the reflected and refracted waves to that of the incident wave are calculated as a non- singular system of linear algebraic equations. These amplitude ratios are used to further calculate the shares of different scattered waves in the energy of the incident wave. The modulus of the amplitude and the energy ratios with the angle of incidence are computed for a particular numerical model. The conservation of the energy across the interface is verified. The effects of variations in non-wet saturation of pores and frequencies on the energy partition are depicted graphically and discussed.

SU-E-J-202: Treatment Dose Assessment with the Consideration of Radiation Dose Inducible Organ Shrinkage/Deformation.

For asymmetric organ shrinkage caused by heterogeneity dose distribution, tissue elasticity based deformable organ registration cannot be directly applied to the treatment image to construct the treatment dose. A novel approach is proposed to include radiation dose inducible organ shrinkage/deformation in the consideration of treatment dose assessment. A model and algorithm were developed to include dose inducible organ shrinkage in the deformable image registration. In the model, we assume that the logarithm of tissue-element volume shrinkage ratio is proportional to the LQ survival function, and the tissue-element displacement field is the gradient of a scalar potential function. The derived Poisson equation of the potential was solved using a finite element method. Two steps registration was implemented to determine the tissue-element volume and displacement. The first is the conventional deformable image registration to determine the organ surface. The second is to map the element volume/position in the shrinking organ with using the dose inducible shrinkage model. Treatment dose was constructed by applying both the conventional deformable registration and the shrinkage model on CBCT images obtained from h&n treatment. The dose distribution in the patient parotid was evaluated. Treatment dose-volume, V30, constructed including the dose inducible parotid shrinkage was 11% higher than the one from the conventional deformable image registration and dose construction. Tissue elements in the high dose region shrink more than those in a low dose region which results in extra tissue elements in the organ move into the high dose area during the treatment course and an unfavorable dose-volume relationship in the organ. Without including dose inducible organ shrinkage, treatment dose in a shrinking organ could be underestimated. On the other hand, including this effect in the treatment evaluation and adaptive planning optimization will minimize the potential detriment of the effect.

Noether symmetric classical and quantum scalar field cosmology

We study the evolution of a two-dimensional minisuperspace cosmological model in classical and quantum levels by the Noether symmetry approach. The phase space variables turn out to correspond to the scale factor of a Friedmann–Robertson–Walker model and a scalar field with which the action of the model is augmented. It is shown that the minisuperspace of such a model is a two-dimensional manifold with a vanishing Ricci scalar. We present a coordinate transformation which cast the corresponding minisupermetric to a Minkowskian or Euclidean one according to the choices of an ordinary or phantom model for the scalar field. Then, the Noether symmetry of such a cosmological model is investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. We explicitly calculate the form of the scalar field potential functions for which such symmetries exist. For these potential functions, the exact classical and quantum solutions in the cases where the scalar field is an ordinary or a phantom one are presented and compared.

Elastodynamic Scalar Potential Functions in Cylindrical Coordinate Systems

The aim of this paper is to present scalar potential functions for three-dimensional elastodynamic problems in transversely isotropic media defined by different cylindrical coordinate systems. To do so, the standard cylindrical coordinate system, elliptical cylindrical coordinate system and parabolic cylindrical coordinate system are considered. By virtue of Poisson’s representation, Lame solution and Chadwick-Trowbridge, the completeness of such representations is addressed. The representations given here are useful in the study of wave scattering from the circular, elliptical and parabolic edges.

Surface-Wavefield Estimation From Coherent Marine Radars

One major impediment to using marine radars for real-time shipboard measurements of evolving ocean wavefields is the uncertainty in the transfer function that relates the radar cross section to sea-surface height. In this letter, a more direct approach is proposed to infer nonlinear sea-surface heights by using radial-velocity measurements from coherent marine radars. The radial velocities are initially integrated along range lines to obtain a scalar potential function. The velocity potential field is then differentiated with respect to time to yield the sea-surface height. Numerical simulations have been conducted to evaluate the sensitivity of the proposed scheme to sampling errors and noise. Under idealized conditions, the results demonstrate that the sea-surface elevation can be reliably estimated from the radial-velocity field provided that the antenna-rotation period is much smaller than a characteristic wave period.

The complex pre-potential and the Aharonov–Bohm effect

The Aharonov–Bohm (AB) effect is traditionally attributed to the affect of the electromagnetic 4-potential A, even in regions where both the electric field and the magnetic field are zero. We argue that the quantity measured by AB experiments may be the difference in values of a multiple-valued complex function, which we call a pre-potential. The pre-potential is a combination of the two scalar potential functions introduced by Whittaker. We show that any electromagnetic field can be described by such a pre-potential, and give an explicit expression for the electromagnetic field tensor through this potential. The covariance of the pre-potential is explained. A program for deriving the AB effect from the pre-potential is proposed.

The Temperature Distribution of a Two‐Dimensional Thermoelastic Material with Voids

AbstractIn this paper, an attempt has been made to investigate the temperature field in a two‐dimensional thermoelastic medium with voids. This takes place when it is subjected to a time‐dependent temperature source under the conditions of conduction heat transfer and interaction between the thermoelastic medium and its surroundings. Analytical expressions of displacements, stresses, temperature increment, and change in the volume fraction field for a layer of thermoelastic medium and a half‐space thermoelastic problem are derived by using the Laplace and Fourier transformation techniques. Two scalar potential functions of the displacement are introduced in the solution process. Then, the integral transform is inverted by using a numerical technique, and numerical results are illustrated graphically to analyze the distribution of the fields. Furthermore, the influence of the thermo‐void coefficient for two kinds of temperature boundaries is discussed. A comparison is also presented with the corresponding half‐space thermoelastic model without voids to ascertain the validity and the difference between these models. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; 39(7): 507–522, 2010; Published online in Wiley Online Library (wileyOnlinelibrary.com). DOI 10.1002/htj.20313

Structure formation by the fifth force: Segregation of baryons and dark matter

In this paper we present the results of $N$-body simulations with a scalar field coupled differently to cold dark matter (CDM) and baryons. The scalar field potential and coupling function are chosen such that the scalar field acquires a heavy mass in regions with high CDM density and thus behaves like a chameleon. We focus on how the existence of the scalar field affects the formation of nonlinear large-scale structure, and how the different couplings of the scalar field to baryons and CDM particles lead to different distributions and evolutions for these two matter species, both on large scales and inside virialized halos. As expected, the baryon-CDM segregation increases in regions where the fifth force is strong, and little segregation in dense regions. We also introduce an approximation method to identify the virialized halos in coupled scalar field models which takes into account the scalar field coupling and which is easy to implement numerically. It is find that the chameleon nature of the scalar field makes the internal density profiles of halos dependent on the environment in a very nontrivial way.

Three Dimensional Vibration Transmission of a Plate on a Layer of Porous Medium Due to a Moving Load

Three-dimensional (3D) transmission of vibration in an infinite elastic thin plate on a layer of poroviscoelastic medium, due to a harmonic, rectangular moving load, is investigated theoretically based on Biot’s theory. The material of the medium is idealized as a uniform, fully saturated poroviscoelastic layer on bedrock. By introducing four scalar potential functions and Helmholtz decomposition theorem, analytical solutions of stress, displacement, and pore pressure with and without thin plate are derived using Fourier transform technique. Numerical results are obtained with the help of inverse Fourier transform and are used to analyze the influence of load velocity, porosity, permeability, relative stiffness of plate versus ground, and the thickness of plate on the vibration. Furthermore, the results are compared with the available dynamic response results of a non-moving load on a layer of viscoelastic material.

Asymmetric and axisymmetric vibrations of finite transversely isotropic circular cylinders

This paper presents an approach for obtaining the exact frequency equations of axisymmetric and asymmetric free vibrations of transversely isotropic circular cylinders. The solution method is based on the three dimensional theory of linear elasticity and uses potential functions. Using this approach, the frequency spectra and vibration mode shapes are plotted for a number of transversely isotropic cylinders. The proposed approach introduces a number of merits compared to earlier approximate and exact solution methods. First, unlike numerically complicated series methods that provide approximate solutions, the proposed approach is exact. Second, combination of scalar functions employed for representing the displacement field is consistent with the physics of the problem. One scalar potential function has been considered for each component of the wave field inside the elastic cylinder. As a result, the solution is systematically divided into coupled and decoupled equations. In addition, by using this approach, there is no need to guess the final of the solution a priori. These merits make the proposed approach suitable for other vibration problems of anisotropic materials.

A new semi-analytical method analyzing the magnetic field in unbalanced magnetron sputtering system

A new method, semi-analytical method (SAM), is first applied to calculate and analyze the magnetic filed in unbalanced magnetron sputtering system, and introduced in detail. An analytic solution of the scalar magnetic potential in the system can be acquired by the SAM. Its unknown number is much less than that in the numerical method. The analytic series expression of magnetic flux density can be easily obtained directly by differentiating the scalar magnetic potential function, and it can also easily ensure the precision of solving the magnetic flux density. The comparison of results between the values measured by experiment and the values calculated by the SAM has shown correctness and effectiveness of this method. The SAM cannot only accurately describe the distribution of magnetic flux density and optimize magnetic field in unbalanced magnetron sputtering system, but also be conveniently used for the simulation about plasma distribution and thin film growth.