SPH-based simulation of granular collapse on an inclined bed

The paper presents a numerical model for simulating a granular flow and its deposition on an inclined bed. A granular material is described as an elastic–plastic continuum and its constitutive law, namely Hooke's law, is discretized on the basis of the Smoothed Particle Hydrodynamics (SPH) method. In the equation of motion, however, the artificial viscosity, which is widely used in SPH, is not applied. The diffusive term derived from Hooke's law is introduced with a diffusion coefficient that varies depending on the stress and strain rate based on the Drucker–Prager yield function. The model is verified and validated through two numerical tests. It is shown that the basic elastic–perfectly plastic characteristics are reproduced with a simple shearing test. The effects of the diffusion coefficient and spatial resolution are investigated to show the validity of the model. In the simulation of the gravitational collapse of a granular column on an inclined bed, the performance of the model from the final deposition profile, the time history of the front position of the granular flow, the maximum runout distance, and the velocity profile are investigated for several cases of basal inclinations. The calculated results show good agreement with the experimental results.     

The extension and oscillation of a non-Hooke's law spring

We derive a wave equation for small-amplitude, undamped, extensional oscillation of a spring-mass system consisting of a mass suspended on a spring governed by a quadratic force-extension relationship. We justify this quadratic model using a Taylor series expansion of the general elasticity equations for a helical spring. Transformation of the equation of motion of the spring leads to a separable wave equation with the spacial component being a transformation of Bessel's equation. The model is successful in predicting static extension and period of oscillation of a helical wire spring for which the wave equation based on Hooke's law is inadequate.     

Analytical modelling of thermal stresses in anisotropic composites

This paper represents a continuation of the author's previous work which deals with an analytical model of thermal stresses which originate during a cooling process of an anisotropic solid elastic continuum. This continuum consists of anisotropic spherical particles which are periodically distributed in an anisotropic infinite matrix. The infinite matrix is imaginarily divided into identical cubic cells with central particles. This multi-particle–matrix system represents a model system which is applicable to two-component materials of the precipitate–matrix type. The thermal stresses, which originate due to different thermal expansion coefficients of components of the model system, are determined within the cubic cell. The analytical modelling results from fundamental equations of continuum mechanics for solid elastic continuum (Cauchy's, compatibility and equilibrium equations, Hooke's law). This paper presents suitable mathematical procedures which are applied to the fundamental equations. These mathematical procedures lead to such final formulae for the thermal stresses which are relatively simple in comparison with the final formulae presented in the author's previous work which are extremely extensive. Using these new final formulae, the numerical determination of the thermal stresses in real two-component materials with anisotropic components is not time-consuming.     

Optimal design of bars under nonuniform tension with respect to mixed creep rupture time

The first attempt of finding of optimal shape for bars in presence of body forces with respect to mixed creep rupture is made. For given volume of the bar, distribution of initial cross-section, ensuring the longest life-time to mixed rupture is sought. The finite strain theory and physical law in form of Norton's law generalized for true stresses and logarithmic strains are applied. Using the method of parametric optimization, the best of linear and quadratic functions describing the initial shape of the bar are found. The shape of initial strength is corrected in a way leading to longer life-time. Results of both approaches are compared.     

An explicit formulation of a three-dimensional material damping model with transverse isotropy

A constitutive three-dimensional (3D) damping model is derived for transversely isotropic material symmetry, using the augmented Hooke's law [Intl. J. Solids Struct. 32 (1995) 2835] as a starting point. The proposed material model is tested numerically, via finite-element techniques, on a laminate structure built from stacked aluminium and Plexiglas plates. Effective 3D transversely isotropic material properties are given in terms of homogeneous material damping functions in connection with homogenised elastic laminate properties. Comparisons made between the results from the elastic (undamped) eigenvalue problem of the detailed (layerwise) model of the laminate and the effective 3D elastic model show that the homogenised model is reasonably accurate, in terms of predicted elastic eigenfrequencies for the first 20 modes. The dynamic homogenisation process, with damping included, is evaluated in terms of forced vibration response for the laminate structure, using effective transversely isotropic frequency dependent material properties. The dynamic 3D effective homogeneous material model is found to simulate very closely the detailed model in the studied frequency interval for the numerical test case.     

Exact solutions of the generalized Navier– Stokes equations for benchmarking

The generalized Navier– Stokes equations for incompressible viscous flows through isotropic granular porous medium are studied. Some analytical classic solutions of the Navier– Stokes equations are generalized to the case of the considered equations. Obtained solutions of generalized equations reduce to classic ones as porosity effect disappears. Average velocity of generalized solutions is calculated and evaluated in two limiting regimes of flow. In the shallow conduit, the generalized flow rate approximates the free (without porous medium) flow rate and in the case of removed boundaries this approaches Darcy's law. The use of the derived exact solutions for benchmarking purposes is described. Copyright © 2002 John Wiley & Sons, Ltd.     

Linear limit stresses of some photoelastic and mechanical model materials

Certain laws of similarity must be observed in structural-model analyses. In this paper, one aspect of model similarity—that of linearity—is examined quite extensively. Most model analyses assume that both prototype and model materials obey Hooke's law. But the plastics often used for structural or photoelastic models are viscoelastic or photoviscoelastic. The stress-strain and stress-birefringence relations are time dependent and may be nonlinear. Through careful calibration of model materials and proper design of model tests, potential errors due to the time dependence of material properties can usually be avoided. If the results of the test are to be interpreted conveniently and accurately, the stresses in the model material must be within the linear range. This range is limited and time dependent for most plastics. The linear range may extend only to stresses considerably below the ultimate or fracture strength of the material. Hence, analyses based don linearity may be in error if the initial stresses are too high and/or if given stresses are sustained too long before desired information is collected. The stresses which limit the linear range, called linear limit stresses, were determined for both stress-strain and stress-birefringence relations for four commonly used plastics: CR-39 (Cast Optics Co.), PS-1 and PS-2 (Photolastic, Inc.) and P6-K (B.A.S.F., Germany). A graphical presentation of the time-dependent photoelastic and mechanical properties is employed. It was concluded that linear limit stresses for birefringence are approximately equal to those based on strain and can therefore be used to establish, within reasonable bounds of accuracy, the linear range of behavior of the material.     

Anisotropy of elastic properties of materials

Papers dealing with the generalized Hooke’s law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin’s approach disclosing the structure of the generalized Hooke’s law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 131–151, November–December, 2008.     

The Hanging Rope of Minimum Elongation for a Nonlinear Stress-Strain Relation

We consider the problem of determining the shape that minimizes the elongation of a rope that hangs vertically under its own weight and an applied force, subject to either a constraint of fixed total mass or fixed total volume. The constitutive function for the rope is given by a nonlinear stress-strain relation and the mass-density function of the rope can be variable. For the case of fixed total mass we show that the problem can be explicitly solved in terms of the mass density function, applied force, and constitutive function. In the special case where the mass-density function is constant, we show that the optimal cross-sectional area of the rope is as that for a linear stress-strain relation (Hooke's Law). For the total fixed volume problem, we use the implicit function theorem to show the existence of a branch of solutions depending on the parameter representing the acceleration of gravity. This local branch of solutions is extended globally using degree theoretic techniques.     

Extension of Noether's theorem to constrained non-conservative dynamical systems

A method based on a differential variational principle is developed in order to extend Noether's theorem to constrained non-conservative dynamical systems. The result is applied to generate constants of the motion for a generic example of a non-linear, dissipative dynamical system with time-varying coefficients represented by the Emden equation. The converse of Noether's theorem, whereby the symmetries of the system are determined from the knowledge of the Lagrangian and a first integral is also considered for both the Emden equation, and that of the damped harmonic oscillator. It is further shown that the presence of ideal constraints (whether holonomic or non-holonomic) does not affect the statement of Noether's theorem. The constraints affect the Jacobi energy integral, however, because they enter into consideration through real work instead of virtual work. It is shown that the Jacobi integral is conserved provided that: (a) the Lagrangian is explicitly independent of time, (b) the real power of the generalized forces not derivable from a potential vanish, (c) the holonomic constraints are explicitly independent of time, (d) the non-holonomic constraints are linear and homogeneous in the generalized velocities.     

On the direct measurement of very large strain at high strain rates

When dynamic plastic strain exceeds 4-percent deformation in completely annealed polycrystalline aluminum, difficulties in the optical measurment of strain occur because of changes in the diffuse-ambient-background light arising from the growth of a mottled surface, or “organe peel.” This paper describes how the diffraction-grating technique may be modified to measure dynamic plastic strain for very large strain at high strain rates in the presence of changing light intensity. The experimental results obtained show that the strain-rate-independent finite-amplitude wave theory, governed by the present writer's generalized, linearly temperature-dependent parabolic stress-strain law, still applies.     

A microstructure continuum approach to electromagneto-elastic conductors

A micromorphic continuum model of a deformable electromagnetic conductor is established introducing microdensities of bound and free charges. The conductive part of electric current consists of contributions due to free charges and microdeformation. Beside the conservation of charge, we derive suitable evolution equations for electric multipoles which are exploited to obtain the macroscopic form of Maxwell’s equations. A constitutive model for electromagneto-elastic conductors is considered which allows for a natural characterization of perfect conductors independently on the form of the constitutive equation for the conduction current. A generalized Ohm’s law is also derived for not ideal conductors which accounts for relaxation effects. The consequences of the linearized Ohm’s law on the classic magnetic transport equation are shown.     

A simplified model for nonlinear dynamic analysis of steel column subjected to impact

This paper presents a new simplified model of the nonlinear dynamic behavior of a steel column subjected to impact loading. In this model, the impacted column, which undergoes large displacement, consists of two rigid bars connected by generalized elastic–plastic hinges where the deformation of the entire steel column as well as the connections is concentrated. The effect of the rest of the structure on the column is modeled by an elastic spring and a point masse both attached to the top end of the column which is also loaded by a compressive force. The plastification of the hinges follows the normality rule with a yield surface that accounts for the interaction between M and N. The latter is described by a super-elliptic yield surface that allows ones to consider a wide range of convex yield criterion by simply varying the roundness factor that affects the shape of the limit surface. By including these features, the model captures both geometry and material nonlinearities. Both the flow rule and the equations of motion are integrated using the midpoint scheme that conserves energy. The non-smooth nature of impact is considered by writing the equations of motion of colliding masses using differential measures. Contact conditions are written in terms of velocity and combined with Newton's law to provide the constitutive law describing interactions between masses during impact. Numerical applications show that the model is able to capture the behavior of a column subjected to impact.     

Optimal Design of Rotating Disks in Creep

Abstract

Minimum-weight design of axially-symmetric rotating disks in a state of stationary creep is determined for a prescribed value of the creep velocity at the outer edge of the disk. The constitutive equations used for stationary creep are Norton's law generalized to multiaxial states of stress based on von Mises' criterion and associated flow rule. The resulting nonlinear optimization problem is solved iteratively using a series expansion to approximate the thickness variation of the disk, In each iteration step the nonlinear creep equations are solved for the stresses and a linearized perturbation problem is solved for the stress gradients. The optimization procedure is used to determine the optimal shape of a solid rotating disk carrying a uniform traction at the outer edge, and this result is compared with the corresponding disk of uniform strength. Variations in the optimal shape due to a central hole and due to temperature distributions are illustrated by some examples.     

An analytical approach to determine conventional S−N curves from accelerated-fatigue data

This paper describes an analytical method of obtaining conventional S?N curves from the accelerated-fatigue tests, namely the generalized Prot accelerated-fatigue-testing technique in which the stress amplitude increases linearly with respect to cycle. Miner's cumulative-damage theory was applied and an expression for the sum of a series of natural numbers raised to a certain nonintegral power was developed to achieve this. The agreement between analytical prediction and experimental verification is quite reasonable.     

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love's thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the generalized differential quadrature (GDQ) method. Hill's method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.     

EFFECTS OF POISSON’S RATIO ON SCALING LAW IN HERTZIAN FRACTURE

In this paper the Auerbach＇s scaling law of Hertzian fracture induced by a spherical indenter pressing on a brittle solid is studied. In the analysis, the singular integral equation method is used to analyze the fracture behavior of the Hertzian contact problem. The results show that the Auerbach＇s constant sensitively depends on the Poisson＇s ratio, and the effective Auerbach＇s domain is also determined for a given value of the Poisson＇s ratio.     

Large deflections of nonlinearly elastic non-prismatic cantilever beams made from materials obeying the generalized Ludwick constitutive law

This paper studies large deflections of nonlinearly elastic cantilever beams made from materials obeying the generalized Ludwick constitutive law. An exact moment-curvature formula which can be applied to study arbitrarily loaded and supported beams of rectangular cross-sections is developed. Several advantages of the generalized Ludwick’s model are illustrated. Numerical examples considered in this materially and geometrically nonlinear analysis clearly indicate rich nonlinear behavior of the beams.