Constitutive theories for nonlocal micropolar continua with implicity and with multiple interactions

In this paper the constitutive theory for nonlocal micropolar continua which was proposed by A. C. Eringen is extended to the cases for nonlocal micropolar continua with implicity and with multiple interactions. Here nonlocal micropolar thermoelastic solids with implicity and with multiple interactions are cited as instances to illustrate the procedure for the establishment of their constitutive theories as well as two relevant theorems concerning the constitutive theories for those solids are given.     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

Dispersion Equation for Water Waves with Vorticity and Stokes Waves on Flows with Counter-Currents

The two-dimensional free-boundary problem of steady periodic waves with vorticity is considered for water of finite depth. We investigate how flows with small-amplitude Stokes waves on the free surface bifurcate from a horizontal parallel shear flow in which counter-currents may be present. Two bifurcation mechanisms are described: one for waves with fixed Bernoulli’s constant, and the other for waves with fixed wavelength. In both cases the corresponding dispersion equations serve for defining wavelengths from which Stokes waves bifurcate. Necessary and sufficient conditions for the existence of roots of these equations are obtained. Two particular vorticity distributions are considered in order to illustrate the general results.     

Rheological models with microstructural constraints

Rheological models of complex fluids with a physically restricted microstructure are analyzed to obtain general classes of dynamical evolution equations for these materials. These classes insure that the appropriate mathematical constraints, associated with each type of physical restriction, are consistently incorporated into the corresponding model development. Describing the microstructure of the complex fluid with a second-rank tensor variable, a general class of dynamical evolution equations is derived for three physically meaningful constraints associated with constancy of the invariants of this microstructural tensor. The physical rationale for each of these constraints is discussed, and a corresponding set of constrained dynamical evolution equations is derived in general terms. Abdellatif Ait-Kadi passed away suddenly during the course of this research. The surviving authors express their gratitude to Abdellatif for our many hours of productive work and companionship.     

Simple Thermomechanical Materials with Memory

This paper presents a framework for formulating frame-free constitutive laws for simple materials accounting for thermal effects. It is shown how memory can be modeled within this framework and fading memory is given as an example. The restrictions that the second law of thermodynamics on the constitutive laws for materials with fading memory are found using the Coleman?CNoll procedure. It is shown that materials with fading memory, when undergoing a process that is long and slow enough can be approximated in some sense. Finally, equilibrium states are considered and under weak assumptions the familiar relations from thermostatics are obtained.     

On hyperelasticity with internal constraints

By requiring the constitutive equation for the specific internal energy to be such that energy is balanced for all motions compatible with the internal constraint, we are able to infer the exitence and the direction of the reactive stress as well as the usual stress relation for the active stress. In contrast with previous work along this line, our analysis avoids the Lagrange multiplier formalism, and we need not assume that the internal energy response function is extendable off the constraint manifold.     

Percolation with a limiting gradient in a medium with random inhomogeneities

The coverage of a medium by percolation and the effective permeability of a medium with stagnant zones are determined. It is shown that effective permeability is a function of external conditions, particularly the average pressure gradient. Three-, two-, and one-dimensional flows are discussed. The theory of overshoots of random functions and fields beyond a prescribed level [1, 2] is used for the investigation. Overshoots of elements of the percolation field in media with random inhomogeneities are studied. Overshoots of energy being dissipated in a volume are discussed in particular; this permits an approximate determination of the coverage of an inhomogeneous porous medium by migration during percolation with a limiting gradient, i.e., in the case of formation of stagnant zones chaotically disseminated in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–165, September–October, 1970.The authors thank V. M. Entov for discussing the article and useful comments.     

Interpolation with boundary condition using bivariate quadric splines connected with triangular partition

In this paper, we study the relations between the existance uniqueness of interpolation with boundary condition using bivariate quadric splines connected with triangular partition and positions of interpolating points. After proving the existence uniqueness of interpolation problems at center, vertex and partial center, we give the construction methods for these three interpolatiny functions.     

On laminar bouudary layers with suction

In this paper, we obtain the asymptotic solution of the general equation for laminar boundary-layer flows with suction. Formulae for calculating the displacement thickness, momentum thickness, an’d skin friction are then derived. Furthermore, the problem of determining the separation point is dealt with. Finally, as a numerical example, we compute certain characteristic boundary-layer parameters for the case of uniform flow over flat plate with constant suction. Our numerical results obtained are in good agreement with those of Iglisch.     

Oblique shock wave with sweep

An attached planar, oblique shock with sweep is investigated for the inviscid flow of a perfect gas. The ratio of specific heats, freestream Mach number, and wedge angle in the plane of the freestream velocity are prescribed, with the sweep angle as a free parameter. Explicit relations are provided for jump and detachment conditions. A number of trends, some non-intuitive, are discussed, e.g., the downstream Mach number may increase with sweep.     

Elastic bodies reinforced with inextensible surfaces

The constraint of in-plane rigidity is examined within the general framework of the theory of internally constrained materials. It is shown that, for in-plane rigid materials, local strain and active stress are both defined by vectorial quantities. Representation formulae for the elastic response mapping are established in the cases of transverse isotropy and maximal symmetry, compatible with the constraint manifold. The equilibrium problem for an elastic body reinforced with parallel inextensible planes is also considered. In particular, universal solutions for bodies with maximal material symmetry are determined within the class of deformations which leave rigid every reinforcing plane.     

Approximate models of salt transfer with washings of soils with water-impermeable base

On the basis of an approximate model of filtration with drainage washings of a soil layer with a water-impermeable base [1], shown schematically in Fig. 1, an investigation is made of the desalinization process of soils for several forms of the equations of salt transfer between the moving solution and the fixed phase. Neglect of the external diffusion makes it possible to reduce the investigation to the solution of one-dimensional boundary-value problems, characterizing salt transfer along fixed streamlines. Analogous problems were discussed in [2] with application to the one-dimensional filtration of wash waters in a dry soil.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 44–51, January–February, 1978.The author thanks V. N. Émikh and V. I. Pen'kovskii for their repeated and useful evaluations of the work.     

Transition sets of bifurcations of dynamical systems with two state variables with constraints

Bifurcation of periodic solutions widely exists in nonlinear dynamical systems.In this paper, categories of bifurcations of systems with two state variables with differenttypes of constraints are discussed, where some new types of transition sets are added.Additionally, the bifurcation properties of two-dimensional systems without constraintsare compared with the ones with constraints. The results obtained in this paper can beused by engineers for the choice of the structural parameters of the systems.     

Photoelastic concrete similutide with CR-39 reinforced with epoxy and glass

This investigation was undertaken to study the similitude of models for reinforced concrete beams by photoelasticity. All previous attempts of model analysis have not maintained the correct modular ratio of reinforcing rod to the beam material. The material problem of obtaining the necessary rods was solved. The material exploratory investigation revealed suitable reinforcing plastics that have opened a wide range of model analyses for reinforced concrete. The models fabricated provided a study in correlation of models and prototype. A sample of the stresses obtained with a photoelastic interferometer is given to demonstrate the practical application of the model analysis.     

Free Damage Propagation with Memory

We introduce a simple model for free damage propagation based on non-local potentials. The model is developed using a state based peridynamic formulation. The resulting evolution is shown to be well posed. At each instant of the evolution we identify the damage set. On this set the local strain has exceeded critical values either for tensile or hydrostatic strain and damage has occurred. For this model the damage set is nondecreasing with time and associated with damage variables defined at each point in the body. We show that energy balance holds for this evolution. For differentiable displacements away from the damage set we show that the nonlocal model converges to the linear elastic model.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Numerical simulation of viscous flows with free surface around realistic hull forms with transom

This paper describes a method for simulation of viscous flows with a free surface around realistic hull forms with a transom, which has been developed based on a FINFLO RANS solver with a moving mesh. A dry‐transom model is proposed and implemented for the treatment of flows off the transom. The bulk RANS flow with the artificial compressibility is solved by a cell‐centred finite volume multigrid scheme and the free surface deformed by wave motions is tracked by satisfying the kinematic and dynamic free‐surface boundary conditions on the actual location of the surface. The effects of turbulence on flows are evaluated with the Baldwin–Lomax turbulence model without a wall function. A test case is modern container ship model with a transom, the Hamburg Test Case. The calculated results are validated and they agree well with the measured results in terms of the free‐surface waves and the total resistance coefficient. Furthermore, the numerical solutions successfully captured many important features of the complicated interaction of the free surface with viscous flows around transom stern ships. In addition, the convergence performance and the grid refinement studies are also investigated. Copyright © 2001 John Wiley & Sons, Ltd.     

Splitting a Viscoelastic Orthotropic Body with a Rigid Wedge of Thickness Increasing with Time

A crack model with a constant-length prefracture zone and a critical opening criterion are used to solve the wedging problem for a linear elastic body split by a rigid wedge with thickness increasing with time. Governing equations describing the behavior of the crack during the incubation and transient stages are presented. An algorithm is proposed to solve numerically the governing equation of transient crack growth