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The fundamental equations and some special solutions of contact mechanics for solid fractal surfaces are proposed in this paper. Firstly, as for fractal solid surface, the probability distribution equation of height without pressure, the probability distribution equation of vertical displacement with pressure and the probability distribution equation of contact stress are established. Secondly, considering the adhesion force, a special solution is given for the probability distribution equation of contact stress when the fractal index is 2. Finally, when the area ratio of multi-scale contact is 0.5, the influence of different derivative orders on the results is discussed. In this paper, a complete analytical framework is given for the static multi-scale contact mechanics. [ABSTRACT FROM AUTHOR]

In this paper, an exact elasticity solution of functionally graded material (FGM) rectangular plate under cylindrical bending is presented. The formulation is based on the displacement approach in which 3D elasticity equations are reduced to 2-D equations using plane-strain conditions. In the present formulation, an exact elasticity solution is feasible due to the consideration of simply supported boundary conditions with applied loads expressed in harmonic forms. Functionally graded material (FGM) infinite rectangular plate subjected to realistic transverse normal loads are analyzed. Solutions have been presented for FGM plate with various aspect ratios and gradation factors. Variation of displacements and stresses through the thickness has been investigated for the FGM plate subjected to single sinusoidal load, uniformly distributed load, hydrostatic load, and strip load. Validation of results is presented using reference results in the literature. The extensive results presented in this paper can be served as benchmark solutions for the assessment of improved plate theories. [ABSTRACT FROM AUTHOR]

This paper deals with one-dimensional axisymmetric quasi-static coupled magnetothermoelastic problems subjected to magnetic and vapor fields. Laplace transform and finite difference methods are used to analyze the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions for a transient and steady state. It is demonstrated that the computational procedures established in this paper are capable of solving the generalized magnetothermoelasticity problem of a hollow cylinder with nonhomogeneous layers. [ABSTRACT FROM AUTHOR]

This paper investigates Mei symmetry and new adiabatic invariants of the disturbed non-material volumes. A infinitesimal transformation group and the infinitesimal transformations vectors of generators are proposed. The definition of Mei symmetry and the determining equation for the systems are presented. The perturbation to the Mei symmetry and new adiabatic invariants for non-material volumes is employed, two types of Mei adiabatic invariant induced by Mei symmetrical perturbation are obtained. Two theorems on new adiabatic invariants are given, and the corresponding deductions about new exact invariants are derived. An example is given to illustrate the application of the method, the corresponding adiabatic invariants are obtained. The example is verified numerically, and it proofs that the theoretical derivation is correct. [ABSTRACT FROM AUTHOR]

The application of piezoelectric actuators in smart structures is a rapidly developing field, particularly in aerospace environments. Given the significant impact of thermal effects in aerospace applications, the study of thermopiezoelectricity has gained attention. This phenomenon accounts for the thermal field in addition to the mechanical and electrical fields. Consequently, coupling phenomena among these three fields, including pyroelectric and electrocaloric effects, must be considered. This paper examines how these coupling effects influence the performance of piezoelectric bender actuators in normal operation and under varying external environments. This analysis is conducted through a custom-written finite element code which takes the three fully coupled field equations of thermopiezoelectricity into account. Then, the temperature dependence of the piezoelectric strain coefficients is included into the developed code in a numerically efficient manner using a pre-computation approach. The effect of temperature-dependent material properties is investigated via a case study of a stratospheric balloon flight where the actuator is used as a lens positioning element and subjected to significant temperature variations. [ABSTRACT FROM AUTHOR]

This paper deals with the local bifurcation analysis of a nanotube with a nanostring passing through it. Eringen’s two-phase local/nonlocal model and Eringen’s differential model are employed as constitutive equations. The governing equations are derived as two nonlinear first-order systems of ordinary differential equations. Nonlinear analysis is performed by using the Lyapunov–Schmidt method. The influence of the small length scale parameter and the phase parameter on critical buckling load, type of bifurcation and post-buckling shape of the nanotube is examined for both types of constitutive equations. Depending on the values of the small length scale parameter and the phase parameter, the critical buckling load corresponding to Eringen’s two-phase local/nonlocal model can be greater or less than that corresponding to Eringen’s differential model. It is shown that for both models supercritical pitchfork bifurcation occurs. The post-buckling shapes of nanotube, obtained by numerical integration, exhibit a qualitative difference between the two models. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a compressed Euler-Bernoulli beam on a Winkler elastic foundation under the action of an external nonlinear force, which models a wind force, is studied. The beam is assumed to be long, and the lower part of its spectrum is prescribed. An asymptotic method is proposed to find the parameters of the beam, in order to have this prescribed lower part of the spectrum. All these parameters are necessary to guarantee the stability of the beam and to avoid resonances between the low frequency modes. These modes have special spatial supports that exclude a direct interaction between them. It is shown that the Galerkin system describing the time evolution can be decomposed into a system of almost independent equations which describes n independent nonlinear oscillators. Each oscillator has its own phase and frequency. It is shown that interaction between oscillators can exist only through high frequency modes. [ABSTRACT FROM AUTHOR]

Factors such as the memory effect of viscoelastic materials and the influence of microcantilever deflection still pose challenges in the mechanical modeling, mathematical solution, and material parameter identification for AFM indentation-relaxation experiments to characterize the viscoelastic properties of cells. This paper aims to provide a rational mechanical model for interpreting the detection signals in AFM indentation-relaxation experiments. Considering the contribution of microcantilever deflection under the Euler beam assumption to the viscoelastic indentation force, the Lee-Radok’s model was updated. Combining a piecewise integration method and a time-domain differential method, we derived the implicit/explicit governing equations of the probe-cell viscoelastic indentation forces under ramp-hold protocol. After building the finite element (FE) model, we examined the effect of microcantilever deflection on the acquired data in FE simulation results and the updated model predictions, and compared our updated model with relevant experiments. Then, we proposed a new two-stage (TS) approach for parameter extraction and compared the differences between this new approach and the classic single-stage (SS) approach, thus the strategy for suppressing parameter extraction error was presented. The results validate that the transient modulus extracting by the classical SS approach is essentially an equivalent transient modulus dependent on the ramp loading history, which incurs a divergence in identification of cellular viscoelastic parameter; whereas the new TS approach is valid at a broader range of loading conditions due to explicitly reflecting the dependence of viscoelastic material parameters on ramp loading history. These conclusions provide a theoretical foundation and reference for the advancement of AFM-based detection techniques for cellular mechanical properties. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a relativistic generalization of noncentral force, and it is a prototypical example of relativistic curl force. We study reductions of relativistic noncentral forces using two new variables, angular momentum and integral torque, which lead to relativistic generalization of the Emden–Fowler equation. We also formulate the relativistic generalization of Lane–Emden and Thomas–Fermi equations using fractional angular momentum and torque. [ABSTRACT FROM AUTHOR]

The development of the generalized probability density evolution equation (GDEE) provides a new direction for analyzing stochastic dynamical systems. In particular, it offers an effective solution for nonlinear problems. In this paper, a multiresolution adaptive grid method (MAGM) is proposed for the numerical solution of the GDEE. In this method, the size of multiresolution adaptive grids, which are constructed based on the second-generation wavelet transform, can be dynamically adjusted according to the change of the probability density function (PDF). Besides, the MAGM, which combines the spatial discretization scheme WENO-Z and the temporal discretization scheme third-order TVD Runge–Kutta, ensures the accuracy of the PDF. The method's advantages and accuracy are verified by analyzing a single-degree-of-freedom system, a two-dimensional frame structure, and a three-dimensional frame structure. The adaptive grids can capture the local singularity of PDF and provide more details for local mutation areas. Moreover, the algorithm configures fewer grid points in the region with high regularity to speed up the calculation efficiency. [ABSTRACT FROM AUTHOR]

A noncentral force is a prototypical example of nonconservative position dependent force, known as curl force; this terminology was first proposed by Berry and Shukla in [1]. In this paper, we extend the construction of a noncentral force on the Euclidean plane to constant curvature spaces. It is known that the angular momentum is not conserved in a noncentral setting; we take angular momentum and integral torque to be two independent coordinates and study two different reductions of noncentral forces using these two new variables. These lead to the curvature-dependent generalized Emden–Fowler and generalized Lane–Emden equations. These reduce to the standard form of the Emden–Fowler and Lane–Emden equations when the curvature vanishes. We compute all the generalized Emden–Fowler equations based on polynomial noncentral forces. Finally, we also give a brief outline of our construction for nonpolynomial forces. [ABSTRACT FROM AUTHOR]

In this paper, a numerical solution strategy is proposed for studying the large deformations of rectangular plates made of hyperelastic materials in the compressible and nearly incompressible regimes. The plate is considered to be Mindlin-type, and material nonlinearities are captured based on the Neo-Hookean model. Based on the Euler–Lagrange description, the governing equations are derived using the minimum total potential energy principle. The tensor form of equations is replaced by a novel matrix–vector format for the computational aims. In the solution strategy, based on the variational differential quadrature technique, a new numerical approach is proposed by which the discretized governing equations are directly obtained through introducing differential and integral matrix operators. Fast convergence rate, computational efficiency and simple implementation are advantages of this approach. The results are first validated with available data in the literature. Selected numerical results are then presented to investigate the nonlinear bending behavior of hyperelastic plates under various types of boundary conditions in the compressible and nearly incompressible regimes. The results reveal that the developed approach has a good performance to address the large deformation problem of hyperelastic plates in both regimes. [ABSTRACT FROM AUTHOR]

In this paper, a locally-defined stabilized adaptive explicit approach is considered to analyse generalized coupled thermo-mechanical models. In this sense, a modified central difference method is applied, which performs adapting itself along the solution process, considering the properties and results of the model, as well as the relations between the adopted temporal and spatial discretizations. The proposed technique enables stabilized decoupled analyses, allowing each "phase" of the coupled model to be handled separately, without considering stability restrictions for their time solutions, providing a very versatile and efficient methodology. In addition, the new approach is designed as a single-solve framework based on reduced systems of equations, which further greatly improves the efficiency of the technique. The new method enables adaptive algorithmic dissipation in the higher modes, and it is highly accurate, simple to implement and entirely automated, requiring no decision or expertise from the user. Numerical results are presented at the end of the manuscript, illustrating the performance and effectiveness of the new approach. [ABSTRACT FROM AUTHOR]

A numerical algorithm that can quantitatively solve transcendental equation sets with multiple independent variables in the complex number domain is presented in this paper. A strict mathematical proof of solving the unary transcendental equation in the real domain is given in detail to show the process of the algorithm, which can be generalized to the equations with two or more variables. The complex variable concerned in this method is treated as two independent variables in real domain that represent the real part and the imaginary part of the variable, respectively. The criterion for choosing the dimension of scanning elements is given to overcome the difficulty in solving the general multivariate equation. Finally, the algorithm is applied in the problem of plotting 3-D dispersion curves in complex wave number and real frequency domains. Three examples are used to illustrate the correctness and effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The paper is concerned with the limiting classical model which may be approached by a micropolar plasticity theory proposed previously. The aim is to investigate the limiting classical model in comparison with some standard classical plasticity models. The essential differences to such models consist in the constitutive equations governing kinematic hardening. As shown in the paper, this causes different responses mainly in the so-called second-order effects. [ABSTRACT FROM AUTHOR]

The major objective of this paper is to clarify the effect of constitutive laws on bulk forming design based on the ideal flow theory. The latter theory is in general applicable for perfectly/plastic materials. However, its kinematics equations constitute a closed-form system, which are valid for any incompressible materials, therefore enabling us to extend design solutions based on the perfectly/plastic constitutive law to more realistic laws with rate sensitive hardening behavior. In the present paper, several constitutive laws commonly accepted for the modeling of cold and hot metal forming processes are considered and the effect of these laws on one particular plane strain design is demonstrated. The closed form solution obtained describes a nontrivial nonsteady ideal process. The design solutions based on the ideal flow theory are not unique. To achieve the uniqueness, the criterion that the plastic work required to deform the initial shape of a given class of shapes into a prescribed final shape attains its minimum is adopted. Comparison with a non-ideal process is also made. [ABSTRACT FROM AUTHOR]

In this paper, we study the deformation of a clamped-clamped elastica under edge thrust and constrained inside a straight tube. The governing equations are formulated within the framework of director theory and solved by the shooting method. The deformations calculated from the elastica model are compared with those computed from small-deformation theory. For a relatively slender tube, the early stage of the deformation sequence is similar to the one obtained from small-deformation theory. They are one-point, two-point, three-point, and point-line-point contact deformations. However, some fundamental differences exist between these two theories even in this early stage of deformation. According to small-deformation theory, the constrained rod jumps from planar one-point contact to helical two-point contact suddenly. In elastica model, the planar one-point contact evolves to spatial one-point contact first and transforms to two-point contact smoothly. In small-deformation theory, the point-line-point contact is the final stage of deformation. In elastica model, on the other hand, the deformation may transform to other shapes as the edge thrust increases further. Generally speaking, the difference between small-deformation theory and elastica model grows as the radius of the constraining tube becomes larger. In the case when the ratio of the tube radius and the beam length is larger than 0.384, the constraining tube has no effect on the elastica deformation. [ABSTRACT FROM AUTHOR]

The closed-form solution for the J-integral of a single-lap joint is presented based on the stress field derived from Reissner's mixed variational principle. In an adhesive-bonded joint, loads are carried by the surface of the adherends in shear through an adhesive layer, and thus, the shear effect is important. To improve the accuracy of shear response in fracture analysis, all transverse effects of the shear and peel stresses are considered, and then the constitutive equations and the equilibrium equations are derived from the variational principle. The obtained J-integral gives additional terms on the transverse shear part of the total integral compared with the results from previous conventional analysis, and illustrative examples are provided to show the effects of the current approach. Also, the formulation proposed in this paper can deal with non-identical adherends and laminates easily. [ABSTRACT FROM AUTHOR]

Based on a complex variable boundary integral equation (CVBIE) suggested previously, this paper provides a numerical solution for the elastic inclusion problem using CVBIE. A dissimilar elastic inclusion is embedded in the infinite matrix. The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for the elastic inclusion, while the other is an exterior BVP for the matrix with notch. Both problems are connected by conventional boundary integral equations (BIEs) in complex variables. After performing discretization for the coupled BIEs, the inverse matrix technique is suggested to solve the relevant algebraic equations. Based on the properties of some integral operators, three ways for the inverse matrix technique are suggested. Several numerical examples are carried out to prove the efficiency of the suggested method. [ABSTRACT FROM AUTHOR]

Buoyancy-driven bubbles and drops are deformable and exhibit different boundaries from mobile to immobile. Owing to the complexity of these flows, the literature knows various scaling parameters for their description. This paper determines a set of parameters on the basis of the force balance equations. It identifies the Archimedes number as the leading parameter while the Reynolds number is considered a flow variable. This ordering scheme of parameters is applied to a set of new data on bubbles rising in five different silicon oils. Data representation and limiting cases come out consistently. The use of the Archimedes number leads even to a successful simple model for the normalized rise velocity. [ABSTRACT FROM AUTHOR]

In this study, based on the three-dimensional theory of elasticity, static and free vibration characteristics of continuously graded fiber-reinforced (CGFR) cylindrical shells are considered by making use of a generalized power-law distribution. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material. The CGFR cylindrical shells have a smooth variation of matrix volume fraction in the radial direction. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by a generalized differential quadrature method. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. The main contribution of this work is to illustrate useful results for a cylindrical shell continuously graded fiber reinforced in the radial direction. Finally, these results are compared with a similar discrete laminated composite cylindrical shell. [ABSTRACT FROM AUTHOR]

This paper presents the theoretical and numerical results for the plane problem of an arbitrarily shaped inclusion in an infinite isotropic matrix based on the Faber series method. The key of the method is to express the complex potentials in the arbitrary inclusion in the form of Faber series with unknown coefficients and then substitute them directly into the boundary conditions on the interface. These conditions lead to a set of linear equations containing all the unknown coefficients. Through solving these linear equations, one can obtain the complex potentials both inside the inclusion and in the matrix. Then, numerical results are presented and graphically shown for the cases of an elliptic, square, and triangle inclusions, respectively. It is found that as the stiffness of the inclusion increases, the hoop stress decreases at the rim of the inclusion, while the radial and shear stresses increase. Especially, it is also found that the stresses show the nature of intense fluctuations near the corners of the triangle inclusion, since the inclusion in this case is similar to a wedge. [ABSTRACT FROM AUTHOR]

In this paper, a new reformulation of the Navier equations of motion is introduced for solving the known three-dimensional elastostatics and elastodynamics problems. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. These equations are also invariant with respect to the choice of the coordinate system. In order to solve a three-dimensional elasticity problem based on the presented formulation, one of the three equations in terms of displacement components and the corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. In order to verify the relations, the closed-form solutions are obtained for deflection and natural frequencies of the thick rectangular plate. The numerical results are compared with available results in the literature and it can be seen that the results of the present study are identical to those of the previous works. [ABSTRACT FROM AUTHOR]

In this paper a general model for the analysis of concrete as multiphase porous material, based on the so-called Hybrid Mixture Theory, is presented. The development of the model equations, taking into account both bulk phases and interfaces of the multiphase system is described, starting from the microscopic scale. An exploration of the second law of thermodynamics is also presented: it allows defining several quantities used in the model, like capillary pressure, disjoining pressure or effective stress, and to obtain some thermodynamic restrictions imposed on the evolution equations describing the material deterioration. Then, two specific forms of the general model adapted to the case of concrete at early ages and beyond and to the case of concrete structures under fire are shown. Some numerical simulations aimed to prove the validity of the approach adopted also are presented and discussed. [ABSTRACT FROM AUTHOR]

A theoretical study of the heat transfer in the unsteady thermal boundary layer associated with the forced convection flow past a sharp, semi-infinite wedge that is started impulsively from rest and for which the constant heat flux at the walls is suddenly changed is presented in this paper. The velocity far from the wedge is given by u e ( x) = x m , where x is the coordinate measured along the wedge and m is a constant. Using appropriate non-dimensional transformations, the number of independent variables in the governing boundary-layer equations is reduced from three to two. These equations are then solved numerically for both small (initial) and large (steady-state) times. It is found that for the steady-state flow the Blasius-like solutions exist for each value of m in the range m* ≤ m < ∞, where m* is the value of the power law exponent m that corresponds to a self-similar solution profile with vanishing skin friction. [ABSTRACT FROM AUTHOR]

This paper is concerned with the linear theory of piezoelectricity for homogeneous and isotropic bodies with microstructure. First, we present the basic equations which govern the bending of thin plates. Then, we establish a uniqueness result with no definiteness assumption on constitutive coefficients. A variational characterization of solution to the boundary-initial-value problem is presented. Finally, the effects of a concentrated body force in an unbounded plate are investigated. [ABSTRACT FROM AUTHOR]

The paper deals with the prediction of coupled sway-roll-yaw motions of a ship with constant forward speed, subject to regular wave force. The governing equations derived in time domain after balancing the hydrodynamic and exciting forces are solved analytically while considering the motion variables, wave frequency and external force are complex quantities. Numerical experiments were carried out for a Panamax container ship under the action of a sinusoidal wave of periodicity 11.2 sec with the variations of speed and wave height. The effect of various parameters on motion responses and system stability is investigated. [ABSTRACT FROM AUTHOR]

The interaction of a buoyancy induced mixed convection flow and a free stream impinging at some angle of incidence on a vertical flat plate with a prescribed surface heat flux is studied in this paper. The similarity equations are numerically solved for some values of the governing parameters. It is found that the buoyancy force and non-orthogonal mechanisms act to either reinforce or oppose one another. The stagnation point (separation point) is shifted at the left or at the right of the origin and it depends upon the balance between obliqueness and thermal effects. [ABSTRACT FROM AUTHOR]

In this paper we describe a new model consisting of N articulated rigid bodies with open chain structure, that move in the combined gravitational field of two mutually attracting massive centers. We derive the general equations of motion of the chain and we study the conditions for the planar motion of the individual centers of mass of the links. [ABSTRACT FROM AUTHOR]

This paper firstly conducts a systematic investigation of the problem of a rigid punch indenting an anisotropic piezoelectric half-space. The Fourier transform method is employed to the mixed boundary value problem. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. For an arbitrarily shaped punch, two governing equations are derived, which can be solved numerically. In the case of transversely isotropic piezoelectric media, the two governing equations are corresponding with that given by others using potential theory. Particularly, when the punch has elliptic cross-section, and the pressure and electric displacement are given by some certain forms of polynomial functions, then the displacement and electric potential are prescribed by polynomial functions in the contact area. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals. The problem of indentation by a smooth flat punch is examined for special orthotropic piezoelectric media, and some results obtained can be degenerated to the case of transversely isotropic piezoelectric media. [ABSTRACT FROM AUTHOR]

The mode-I weight function proposed by Oore and Burns is an efficient and empirical mathematical relationship capable of estimating the Stress Intensity Factor in the presence of a generic planar flaw. In the case of penny shaped cracks or tunnel cracks the Oore-Burns equation provides the classic analytical expressions. In this paper, the authors prove analytically that the Oore-Burns equation is not the weight function for an ellipse in an infinite solid under traction loading. The absolute novelty is given by the analytic evaluation between the Oore-Burns integral and the Irwin solution in terms of an algebraic expression, which is independent from any numerical procedures for an ellipse close to a circle. [ABSTRACT FROM AUTHOR]

One means of preventing areas from being hit by avalanches is to divert the flow by appropriately constituted obstacles. Thus, there arises the question how a given avalanche flow is modified by obstructions and how the diverted flow depth and direction emerge. In this paper rapid gravity-driven dense granular flows, partly blocked by obstacles with different shapes, sizes and positions, are numerically investigated by solving the hyperbolic Savage-Hutter equations with an appropriate integration technique. The influences of the obstructions on the granular flows are graphically demonstrated and discussed for a finite mass and a steady inflow of granular material down an inclined plane, respectively. These flows are accompanied by shocks induced by both the presence of the obstacles and the transition of granular flows from an inclined surface into a horizontal run-out zone when the velocity transits from its supercritical to its subcritical state. The numerical results show that the theory is capable of capturing key qualitative features, such as shocks wave and particle-free regions. [ABSTRACT FROM AUTHOR]

When a singular surface is present in the region under consideration, the local forms of the equations of balance must be accompanied by jump relations in order to connect the different values of the discontinuous entities at the two sides of the singular surface. The classical general form of these jump relations is provided by the Kotchine relation. In the present contribution, we exemplary deal with balance of mass, momentum and kinetic energy, as well as with balance of total energy, denoted as the first law of thermodynamics, and with balance of internal energy. We first derive an extension of the Kotchine relation with respect to surface growth terms. By a surface growth term we mean an entity which equivalently describes the balance of some quantity associated with the material that is instantaneously located at or in the vicinity of the singular surface. In a more detailed modelling, a singular surface is often described by a thin shell-type region or layer of transition. Surface growth terms may represent a non-vanishing rate of change of the respective quantity, or they may characterise some sources of this quantity. As a main result of the present paper, we show that surface growth terms are needed in order to ensure consistency between the jump relations for balance of mass, momentum and kinetic energy, and between the jump relations for balance of total energy and balance of internal energy. Even when surface growth terms for mass, balance of momentum and total energy are absent, one generally must take into account surface growth terms for balance of kinetic energy and balance of internal energy. The presented results refer to both, a singular surface, as well as to a thin region of transition. The jump relations in the latter case are referred to an equivalent singular surface, replacing the region of transition. The uni-axial flow of a fluid in a diffusor is used to demonstrate that the present results may be used even in cases in which the... [ABSTRACT FROM AUTHOR]