A unified theory of turbulent flow

Summary A general theory of turbulent flow is applied to incompressible flow in a circular pipe. The theoretical mean velocity distribution is found to be in good agreement with experiment, but there is some discrepancy in the normal stress distribution. The available pressure drop data are used to estimate the value of the apparent wall velocity as a function of Reynolds number and roughness. It is found that the results can be represented by simple expressions which in turn imply simple expressions for the pressure drop as a function of Reynolds number and roughness. However, it has not been possible to derive these results from fundamental considerations. The basis of Reynolds analogy and the application of the theory to channel flow are also discussed.     

A unified theory of turbulent flow

Summary A critical analysis is made of the assumptions underlying Reynolds' equations for turbulent flow. It is shown that there are regions in a flow field where these assumptions break down, and it is therefore necessary to separate the flow into a turbulent core and a laminar sublayer. The importance of the boundary conditions to be imposed on the mean velocities and Reynolds stresses at the junction is emphasized as this is the way in which the effect of surface roughness enters the theory. A set of equations for calculating turbulent flows is proposed. The distinctive feature is that the turbulent stresses are represented as the difference between viscous terms with a large eddy viscosity and terms satisfying auxiliary differential equations proposed by Broszko. These terms may be associated with the free and wall turbulence respectively. The theory enables the idea of a large eddy viscosity to be applied even where the velocity gradient is large. The results obtained for specific configurations, which will be reported in detail in future papers, are previewed.     

A unified theory of turbulent flow

Summary A general theory of turbulent flow is applied to incompressible plane Couette flow. It is found that a unique formulation is not obtained because of a singularity in the equations and problems relating to the boundary conditions. Solutions are obtained for several different assumptions. The characteristic feature is a square root velocity profile for high Reynolds numbers. The logarithmic law is obtained as a divergent approximation. There are discrepancies in the available experimental data; one set agreeing with the square root form, and a second set with the logarithmic form.     

Nonlinear theory of crystalline solids. Origination of nano- and microstructures and their stability

We develop a theory of large deformations of a crystal lattice as a generalization of linear equations of acoustic and optical modes of a complex lattice composed of two interpenetrating sublattices. We suggest a principle of internal translational symmetry with respect to mutual displacements of sublattices by an integer number of periods. Based on this principle, we assume that the force of interaction of the sublattices is a nonlinear periodic function.     

A minimum principle for microstructuring in rigid-viscoplastic crystalline solids

A minimum plastic power principle is proposed for a rigid-viscoplastic crystalline domain subdivided into two sets of lath-shaped regions, called bands. The lattice orientation in each band is assumed uniform and to differ infinitesimally from that in the other band. The proposed minimum principle yields the slip activity in the bands and semi-analytical expressions for the misorientation axis and orientation of band boundaries. These band boundary characteristics are predicted for f.c.c. lattice orientations near the ideal rolling texture components. Surprisingly, it found that the predicted band boundary characteristics closely match those of microstructural features called cell block boundaries reported in the experimental literature, except when the dislocations of activated slip systems are known to interact very strongly. This suggests that except when precluded by strong dislocation interactions, continuum extremum principles may also govern microstructural characteristics.     

On the mechanics of crystalline solids

Elastic-plastic constitutive behavior of cubic crystals (and aggregates) at large pressure is investigated taking account of a thermodynamic basis for lattice straining. Particular attention is directed to quasi-static processes in which an explicit Schmid law governing active slip systems is adopted. Connections between a precise normality rule and the pressure-dependence of moduli and critical shear strengths are analyzed and their implications assessed.     

A unified thermodynamic theory of elasticity and plasticity

A thermodynamics is developed for a unified theory of elasticity and plasticity in infinitesmal strain. The constitutive equations which relate stress and strain deviators are rate type differential equations. When they satisfy a Lipschitz condition, uniqueness for the initial value problem dictates that the stress and strain will be related through elastic relations. Failure of the Lipschitz condition occurs when a von Mises yield condition is achieved: Plastic yield then occurs and the deviator relations turn into the Prandtl-Reuss equations. The plastic yield solution is stable during loading and unstable during unloading. The requirement that the solution followed during unloading be stable dictates entry into an elastic regime. Appropriate thermodynamic functions are constructed. It then appears that stress deviator (not strain deviator) is a viable state variable, and the thermodynamic relations are constructed in terms of a Gibbs function. The energy balance leads to satisfaction of the Clausius-Duhem inequality (and thus the second law of thermodynamics) in an elastic regime because it is shown that in an elastic regime entropy production is caused only by heat flux. During yield, the proper method of differentiating yields entropy production terms in addition to those arising from heat flux. These terms are positive during loading, whence it is concluded that the requirement that a stable solution be followed leads to satisfaction of the Clausius-Duhem inequality during plastic as well as elastic behavior.     

Strongly nonlinear theory of nanostructure formation owing to elastic and nonelastic strains in crystalline solids

We develop an essentially nonlinear theory of elastic and nonelastic microstrains resulting in the formation of nanostructures. Using the model of mutually penetrating lattices, we generalize the well-known theory of acoustic and optical vibrations to the case of nonlinear interaction between sublattices. This permits treating the sublattice interaction forces as periodic (for example, sinusoidal) functions of the relative displacement of the sublattices. We obtain equations for the macroscopic and microscopic displacement fields containing two characteristic scales of the nanostructure. We find a number of their solutions describing the effects of decrease in the potential interatomic barriers in the external stress field and the formation of defects and domain nanostructure as a result of bifurcation transitions. We prove their stability.     

Nonlinear theory of localized and periodic waves in solids undergoing major rearrangements of their crystalline structure

This article analyses the propagation of nonlinear periodic and localized waves. It examines crystals whose lattice consists of two periodic sub-lattices. Arbitrary large displacements of sub-lattices u are assumed. This theory takes into account the additional element of translational symmetry. The relative displacement in a sub-lattice for one period (and even for a whole number of periods) does not alter the structure of the whole complex lattice. This means that its energy does not vary under such a relatively rigid translation of sub-lattices and should represent the periodic function of micro-displacement. The energy also depends on the gradients of macroscopic displacement describing alterations in the elementary cells of a crystal. The variational equations of macro- and micro-displacements are shown to be a nonlinear generalization of the well-known linear equations of acoustic and optical modes of Karman, Born, and Huang Kun. Exact solutions to these equations are obtained in the one-dimensional case—localized and periodic. Criteria are established for their mutual transmutations.     

Wave dispersion in gradient elastic solids and structures: A unified treatment

Analytical wave propagation studies in gradient elastic solids and structures are presented. These solids and structures involve an infinite space, a simple axial bar, a Bernoulli–Euler flexural beam and a Kirchhoff flexural plate. In all cases wave dispersion is observed as a result of introducing microstructural effects into the classical elastic material behavior through a simple gradient elasticity theory involving both micro-elastic and micro-inertia characteristics. It is observed that the micro-elastic characteristics are not enough for resulting in realistic dispersion curves and that the micro-inertia characteristics are needed in addition for that purpose for all the cases of solids and structures considered here. It is further observed that there exist similarities between the shear and rotary inertia corrections in the governing equations of motion for bars, beams and plates and the additions of micro-elastic (gradient elastic) and micro-inertia terms in the classical elastic material behavior in order to have wave dispersion in the above structures.     

A unified intrinsic functional expansion theory for solitary waves

A new theory is developed here for evaluating solitary waves on water, with results of high accuracy uniformly valid for waves of all heights, from the highest wave with a corner crest of 120°down to very low ones of diminishing height. Solutions are sought for the Euler model by employing a unified expansion of the logarithmic hodograph in terms of a set of intrinsic component functions analytically determined to represent all the intrinsic properties of the wave entity from the wave crest to its outskirts. The unknown coefficients in the expansion are determined by minimization of the mean-square error of the solution, with the minimization optimized so as to take as few terms as needed to attain results as high in accuracy as attainable. In this regard, Stokes‘s formula, F^2μπ= tanμπ, relating the wave speed (the Froude number F) and the logarithmic decremen t# of its wave field in the outskirt, is generalized to establish a new criterion requiring (for minimizing solution error) the functional expansion to contain a finite power series in M terms of Stokes‘s basic term (singular in #), such that 2Mμ is just somewhat beyond unity, i.e. 2Mμ≌ 1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio α= a/h, especially about α≌0.01, at which M = 10by the criterion.In this pursuit, the class of dwarf solitary waves, defined for waves with ≌≤ 0.01, is discovered as a group of problems more challenging than even the highest wave. For the highest wave, a new solution is determined here to give the maximum height αhst = 0.8331990, and speed Fhst = 1.290890, accurate to the last significant figure, which seems to be a new record.     

Some problems in non-linear elasticity of crystalline solids

The author reviews some results of his concerning the kinematics and the analysis of equilibria for non-linearly elastic solids whose constitutive equations have the invariance group postulated by Ericksen [6]. The last part of the paper contains an unpublished characterization of complex lattices which was in the back of the author's mind in [23], and thus provides a more satisfactory background for that paper as well as for its consequences in [24].     

A theory of pulse propagation in anisotropic elastic solids

A theory is described for the propagation of pulses in anisotropic elastic media. The pulse is initially defined by a harmonically modulated Gaussian envelope. As it propagates the pulse remains Gaussian, its spatial form characterized by a complex-valued envelope tensor. The center of the pulse follows the ray path defined by the initial velocity direction of the pulse. Relatively simple expressions are presented for the evolution of the amplitude and phase of the pulse in terms of the wave velocity, the phase slowness and unit displacement vectors. The spreading of the pulse is characterized by a spreading matrix. Explicit equations are given for this matrix in a transversely isotropic material. The rate of spreading can vary considerably, depending upon the direction of propagation. New reflected and transmitted pulses are created when a pulse strikes an interface of material discontinuity. Relations are given for the new envelope tensors in terms of the incident pulse parameters. The theory provides a convenient method to describe the evolution and change of shape of an ultrasonic pulse as it traverses a piecewise homogeneous solid. Numerical simulations are presented for pulses in a strongly anisotropic fiber reinforced composite.     

A microstructure theory of photoelastic behavior in amorphous solids

Available theories to explain the phenomenon of photomechanical behavior in solid materials below the transition temperature are qualitative in nature, and several distinct mechanisms are capable of producing a deformation birefringence. A new mechanism is proposed on the atomic level to account quantitatively for deformation birefringence in an ideal amorphous elastic solid. The material is an isotropic, statistically homogeneous, elastic medium consisting of a random spatial arrangement of heavy mass points (atom nuclei) with positive electric charge in static equilibrium with a corresponding number of continuous, spherical, negatively charged regions (electron clouds). The medium has a nonzero random initial polarization; changes in the components of the dielectric tensor at optical frequencies are computed for infinitesimal uniaxial strain using the Lorentz field approximation for isotropic media. The stress-optical constant is then computed from the associated change in refractive index, and is shown to be in good agreement with experimental values for ordinary photoelastic materials.     

A unified theory for cavity expansion in cohesive-frictional micromorphic media

This paper presents a unified theory for both cylindrical and spherical cavity expansion problems in cohesive-frictional micromorphic media. A phenomenological strain-gradient plasticity model in conjunction with a generalized Mohr–Coulomb criterion is employed to characterize the elasto-plastic behavior of the material. To solve the resultant two-point boundary-value problem (BVP) of fourth-order homogeneous ordinary differential equation (ODE) for the governing equations which is not well-conditioned in certain cases, several numerical methods are developed and are compared in terms of robustness, efficiency and accuracy. Using one of the finite difference methods that shows overall better performance, both cylindrical and spherical cavity expansion problems in micromorphic media are solved. The influences of microstructural properties on the expansion response are clearly demonstrated. Size effect during the cavity expansion is captured. The proposed theory is also applied to a revisit of the classic problem of stress concentration around a cavity in a micromorphic medium subjected to isotropic tension at infinity, for which some conclusions made in early studies are revised. The proposed theory can be useful for the interpretation of indentation tests at small scales.     

Couple stress theory for solids

By relying on the definition of admissible boundary conditions, the principle of virtual work and some kinematical considerations, we establish the skew-symmetric character of the couple-stress tensor in size-dependent continuum representations of matter. This fundamental result, which is independent of the material behavior, resolves all difficulties in developing a consistent couple stress theory. We then develop the corresponding size-dependent theory of small deformations in elastic bodies, including the energy and constitutive relations, displacement formulations, the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory. Next, we consider the more restrictive case of isotropic materials and present general solutions for two-dimensional problems based on stress functions and for problems of anti-plane deformation. Finally, we examine several boundary value problems within this consistent size-dependent theory of elasticity.