Accurate modeling of collagen molecules including their stiffness is essential for our understanding of mechanics of collagen fibers and tissues where these fibers play a prominent role. Studies of mechanical properties of collagen molecules employing various experimental methods and molecular dynamics (MD) simulations yield a broad range of values of the modulus of elasticity. The effect of nonlocal elasticity on the molecule stiffness derived from experiments and simulations is assessed in this brief. The estimate of the correction accounting for the nonlocal effect utilizes the exact solution of the nonlocal elasticity theory for one-dimensional elastic bars. It is demonstrated that the effect of nonlocal elasticity on the stiffness of collagen molecules can be neglected.
This paper studies the influence of road camber on the stability of single-track road vehicles. Road camber changes the magnitude and direction of the tire force and moment vectors relative to the wheels, as well as the combined-force limit one might obtain from the road tires. Camber-induced changes in the tire force and moment systems have knock-on consequences for the vehicle's stability. In order to study camber-induced stability trends for a range of machine speeds and roll angles, we study the machine dynamics as the vehicle travels over the surface of a right circular cone. Conical road surfaces allow the machine to operate at a constant steady-state speed, a constant roll angle and a constant road camber angle. The results show that at low speed both the weave- and wobble-mode stability is at a maximum when the machine is perpendicular to the road surface. This trend is reversed at high speed, since the weave- and wobble-mode damping is minimized by running conditions in which the wheels are orthogonal to the road. As a result, positive camber, which is often introduced by road builders to aid drainage and enhance the friction limit of four-wheeled vehicle tires, might be detrimental to the stability of two-wheeled machines.
Shows that in the use of Lie groups for the study of the relative
motion of rigid bodies some assumptions are not explicitly stated. A
commutation diagram is shown which points out the “reference
problem” and its simplification to the usual Lie group approach
under certain conditions which are made explicit
An anisotropic constitutive model based on crystallographic slip theory was formulated for nickel-base single-crystal superalloys. The current equations include both drag stress and back stress state variables to model the local inelastic flow. Specially designed experiments have been conducted to evaluate the existence of back stress in single crystals. The results showed that the back stress effect of reverse inelastic flow on the unloading stress is orientation-dependent, and a back stress state variable in the inelastic flow equation is necessary for predicting inelastic behavior. Model correlations and predictions of experimental data are presented for the single crystal superalloy Rene N4 at 982 C.
An experimental investigation of water flow through an abrupt circular-channel expansion is described over a Reynolds number range between 20 and 4200. The shear layer between the central jet and the reverse flow region along the wall downstream behaved differently in the various flow regimes that were observed. With increasing Reynolds number these regimes changed progressively from a laminar flow to an unstable vortex sheetlike flow and then to a more random fluctuating flow. The distance between the step and the reattachment location downstream correspondingly increased, reached a maximum, and then decreased. Of particular significance are the shear layer wave instabilities observed in the shear flow and their relationship to rettachment which apparently has not received much attention previously. Visual observations aided in understanding the results.
A criterion is proposed which provides a priori means of choosing increments in the independent variables for effecting accurate finite-difference solutions to parabolic partial-differential equations. This is accomplished by relating the increments to the thickness of the diffusion layer. In this manner, the size of the increments is related to the magnitude of the derivatives which are known to influence strongly the accuracy. The fac- that the thickness of the diffusion layer is unknown is surmounted by relating the parameters of the discrete, physical plane to the diffusion variable and to the diffusion thickness. The diffusion variable is a dimensionless coordinate which governs the diffusion process. In similar problems, the diffusion variable is identical to the independent similarity variable. The thickness of the diffusion layer in terms of the diffusion coordinate is shown to be of the same order of magnitude for a wide variety of problems. The utility of the proposed criterion is demonstrated with numerous finite-difference solutions to problems in the areas of heat conduction and boundary-layer theory.
A technique is introduced which extends the range of useful approximation of numerical inversion techniques to many cycles of an oscillatory function without requiring either the evaluation of the image function for many values of s or the computation of higher-order terms. The technique consists in reducing a given initial value problem defined over some interval into a sequence of initial value problems defined over a set of subintervals. Several numerical examples demonstrate the utility of the method.
The solutions developed in this paper are based on the finite Hankel-Fourier-Laplace transform. The Love-Timoshenko shell equations are reduced through the application of the Fourier-Laplace transforms to the axial space and time variables. The equation of motion for the acoustic fluid contained within the shell are reduced through the application of the finite Hankel-Fourier-Laplace transform to the radial, axial spatial, and time variables. The boundary conditions at the fixed end of the semi-infinite shell are met through the application of loadings symmetric with respect to the origin and the application of a ring loading at the origin which is chosen so as to make the radial deflection there zero. Expressions are found for the transforms of the axial, tangential, and radial shell displacements, the axial, tangential, and radial fluid velocities, and the radiated fluid pressure. Numerical inversion of the Fourier-Laplace transform is accomplished in terms of a series of ultraspherical polynomials and Gauss-Hermite or Gauss-Laguerre quadrature.
In this paper an adhesively bonded lap joint is analyzed by assuming that the adherends are elastic and the adhesive is linearly viscoelastic. After formulating the general problem a specific example for two identical adherends bonded through a three parameter viscoelastic solid adhesive is considered. The standard Laplace transform technique is used to solve the problem. The stress distribution in the adhesive layer is calculated for three different external loads namely, membrane loading, bending, and transverse shear loading. The results indicate that the peak value of the normal stress in the adhesive is not only consistently higher than the corresponding shear stress but also decays slower.
The effect of a standing shock wave on the static and dynamic aeroelastic stability of a flexible panel is investigated using a linear structural and aerodynamic theoretical model. It is found that the shock is generally stabilizing. The lowest critical dynamic pressures are associated with shock positions downstream from the panel, where the panel is uninfluenced by the shock.
The purpose is to outline a computational approach to spatial dynamics of mechanical systems that substantially enlarges the scope of consideration to include flexible bodies, feedback control, hydraulics, and related interdisciplinary effects. Design sensitivity analysis and optimization is the ultimate goal. The approach to computer generation and solution of the system dynamic equations and graphical methods for creating animations as output is outlined.
The purpose of this research is to develop a tool for the mechanical analysis of nickel-base single-crystal superalloys, specifically Rene N4, used in gas turbine engine components. This objective is achieved by developing a rate-dependent anisotropic constitutive model and implementing it in a nonlinear three-dimensional finite-element code. The constitutive model is developed from metallurgical concepts utilizing a crystallographic approach. An extension of Schmid's law is combined with the Bodner-Partom equations to model the inelastic tension/compression asymmetry and orientation-dependence in octahedral slip. Schmid's law is used to approximate the inelastic response of the material in cube slip. The constitutive equations model the tensile behavior, creep response and strain-rate sensitivity of the single-crystal superalloys. Methods for deriving the material constants from standard tests are also discussed. The model is implemented in a finite-element code, and the computed and experimental results are compared for several orientations and loading conditions.
The plane motion of an articulated pipe made of two segments is examined and the flow velocity at which flutter manifests itself is sought. The pressure in the reservoir feeding the pipe is kept constant. In contrast to previous works, the flow velocity is not taken as a prescribed parameter of the system but is left to follow the laws of motion. This approach requires a nonlinear formulation of the problem and the equations of motion are solved using Krylov-Bogoliubov's method. A graph of the amplitude of the limit cycles, as a function of the fluid-system mass ratio, is presented and conclusions are drawn as to the necessity of considering nonlinearities in the analysis.
This paper develops closed-form solutions for the extension of twisted wire ropes with fibrous cores which are subjected to axial forces as well as axial moments. The analytical results are compared with the corresponding numerical results obtained by Costello and Phillips. A close agreement between the two establishes validity of the analytical solutions. Finally, an expression for the effective rigidity modulus of wire ropes with fibrous core is obtained in terms of the helix angle and the number of helical wires in the rope for each of the two end conditions.
Turbulent swirling flows in stationary cylindrical ducts were investigated analytically using Taylor’s modified vorticity transport theory and von Karman’s similarity hypothesis extended to consider a three-dimensional fluctuating velocity field. The resulting similarity conditions were used to formulate the expression for eddy diffusivity in the entire flow field except in a small region near th e pipe wall where a mixing-length expression analogous to that assumed by Prandtl for parallel flow in channels was used. The swirl equation was solved numerically using a constant that was obtained indirectly from an experiment by Taylor, and the analytical results were compared with two different sets of experimental measurements. In both cases, the agreement between experiment and analysis was satisfactory. Some discrepancies appeared when the flow field was predominantly irrotational or in solid-body rotation: This might have been expected since, for these situations, some of the similarity conditions were indeterminate or infinite.
This note presents closed-form solutions for axisymmetrical and axially unsymmetrical buckling of angle-ply laminated circular cylindrical shells under axial compression. The axisymmetrical and axially unsymmetrical buckling stress are found to be different from each other, and the best lamination angles which give the highest buckling stress are obtained.
Analytic three-dimensional thermoelasticity solutions are presented for the thermal buckling of multilayered angle-ply composite plates with temperature-dependent thermoelastic properties. Both the critical temperatures and the sensitivity derivatives are computed. The sensitivity derivatives measure the sensitivity of the buckling response to variations in the different lamination and material parameters of the plate. The plates are assumed to have rectangular geometry and an antisymmetric lamination with respect to the middle plane. The temperature is assumed to be independent of the surface coordinates, but has an arbitrary symmetric variation through the thickness of the plate. The prebuckling deformations are accounted for. Numerical results are presented, for plates subjected to uniform temperature increase, showing the effects of temperature-dependent material properties on the prebuckling stresses, critical temperatures, and their sensitivity derivatives.
A study of the mechanics and failure modes of delamination initiated from a surface flaw in angle-ply fiber-reinforced composites is presented. The analysis employs a hybrid-stress finite-element method including a crack-tip singular element with its field variables expressed by Muskhelishvili's complex stress functions. Solutions are obtained for the delaminated composites with various laminate parameters. The results elucidate unique and important characteristics of delamination crack-tip response and interlaminar stress transfer mechanisms. Of particular interest are the mixed-mode stress-intensity factors associated with the delamination crack. The influence of ply orientation on KI and KII and their effects on subsequent crack extension are discussed.
A procedure for deriving nondimensional parameters and equations for bifurcation buckling of anisotropic shallow shells subjected to combined loads is presented. First, the Donnell-Mushtari- Vlasov equations governing buckling of symmetrically laminated doubly curved thin elastic shallow shells are presented. Then, the rationale used to perform the nondimensionalization of the buckling equations is presented, and fundamental parameters are identified that represent measures of the shell orthotropy and anisotropy. In addition, nondimensional curvature parameters are identified that are analogues of the well-known Batdorf Z parameter for isotropic shells, and analogues of Dunnell's and Batdorf's shell buckling equations are presented. Selected results are presented for shear buckling of balanced symmetric laminated shells that illustrate the usefulness of the nondimensional parameters.
The stresses and displacements induced in anisotropic plates by short-duration impact line forces are calculated in this paper. The anisotropy is related to the lay-up angles of the fibers of a laminated fiber composite plate. A modification of Mindlin's theory of crystal plates is used. The plate responses are calculated for various fiber lay-up angles and inclination of the line load to the axes of symmetry. Use is made of the computational method of the fast Fourier transform.
The hybrid stress finite element method is used to study the effects of filament angle and orthotropicity parameter on the vibration and flutter characteristics of cantilevered anisotropic plates. The results indicate a generally strong lack of monotonic dependence on filament angle. Also, the natural frequency in certain cases involving the first few modes, can become relatively insensitive to both filament angle and orthotropicity parameter for a range of filament angle beyond 70 deg. Values of critical dynamic pressure are obtained by a modal approach, in which the mode shapes are obtained by the hybrid stress method. Convergence of the modal method is rather rapid for the configurations analyzed, and a comparison of the method with an exact solution for the case of an isotropic simply supported plate shows that use of six modes gives excellent agreement.
Beam theory plays an important role in structural analysis. The basic assumption is that initially plane sections remain plane after deformation, neglecting out-of-plane warpings. Predictions based on these assumptions are accurate for slender, solid, cross-sectional beams made out of isotropic materials. The beam theory derived in this paper from variational principles is based on the sole kinematic assumption that each section is infinitely rigid in its own plane, but free to warp out of plane. After a short review of the Bernoulli and Saint-Venant approaches to beam theory, a set of orthonormal eigenwarpings is derived. Improved solutions can be obtained by expanding the axial displacements or axial stress distribution in series of eigenwarpings and using energy principles to derive the governing equations. The improved Saint-Venant approach leads to fast converging solutions and accurate results are obtained considering only a few eigenwarping terms.
The propagation of surface acoustic waves in a rotating anisotropic crystal
is studied. The crystal is monoclinic and cut along a plane containing the
normal to the symmetry plane; this normal is also the axis of rotation. The
secular equation is obtained explicitly using the "method of the polarization
vector", and it shows that the wave is dispersive and decelerates with
increasing rotation rate. The case of orthorhombic symmetry is also treated.
The surface wave speed is computed for 12 monoclinic and 8 rhombic crystals,
and for a large range of the rotation rate/wave frequency ratio.
Analytic three-dimensional elasticity solutions are presented for the stress and free vibration problems of multilayered anisotropic plates. The plates are assumed to have rectangular geometry and antisymmetric lamination with respect to the middle plane. A mixed formulation is used with the fundamental unknowns consisting of the six stress components and the three displacement components of the plate. Each of the plate variables is decomposed into symmetric and antisymmetric components in the thickness direction, and is expressed in terms of a double Fourier series in the Cartesian surface coordinates. Extensive numerical results are presented showing the effects of variation in the lamination and geometric parameters of composite plates on the importance of the transverse stress and strain components.
An elasticity solution is utilized to analyze an orthotropic fiber in an isotropic matrix under uniform thermal load. The analysis reveals that stress distributions in the fiber are singular in the radial coordinate when the radial fiber stiffness (C-rr) is greater than the hoop stiffness (C-theta-theta). Conversely, if C-rr is less than C-theta-theta the maximum stress in the composite is finite and occurs at the fiber-matrix interface. In both cases the stress distributions are radically different than those predicted assuming the fiber to be transversely isotropic (C-rr = C-theta-theta). It is also shown that fiber volume fraction greatly influences the stress distribution for transversely isotropic fibers, but has little effect on the distribution if the fibers are transversely orthotropic.
A simple approach to the design of feedback controls for damping the vibrations in large spaceborne antennas with flexible dish reflectors is proposed. The feedback controls consist of movable velocity-feedback dampers whose positions are determined by minimizing the rate of change of total vibrational energy at any time. The performance of the proposed feedback controls is studied via computer simulations.
The singular nature of the crack tip stress field in a nonhomogeneous medium having a shear modulus with a discontinuous derivative was investigated. The problem is considered for the simplest possible loading and geometry, namely the antiplane shear loading of two bonded half spaces in which the crack is perpendicular to the interface. It is shown that the square-root singularity of the crack tip stress field is unaffected by the discontinuity in the derivative of the shear modulus. The problem is solved for a finite crack and extensive results are given for the stress intensity factors.
We study the apsidal precession of a Physical Symmetrical Pendulum (Allais'
precession) as a generalization of the precession corresponding to the Ideal
Spherical Pendulum (Airy's Precession). Based on the Hamilton-Jacobi formalism
and using the technics of variation of parameters along with the averaging
method, we obtain approximate solutions, in terms of which the motion of both
systems admits a simple geometrical description. The method developed in this
paper is considerably simpler than the standard one in terms of elliptical
functions and the numerical agreement with the exact solutions is excellent. In
addition, the present procedure permits to show clearly the origin of the
Airy's and Allais' precession, as well as the effect of the spin of the
Physical Pendulum on the Allais' precession. Further, the method can be
extended to the study of the asymmetrical pendulum in which an exact solution
is not possible anymore.
The conservation laws for a plane fluid flow were simplified by the weak wave approximations valid for sonic boom-type waves and applied to afield of mesh points, utilizing the "artificial viscosity" concept for numerical stability. The numerical analysis was applied to predict the pressure history of the sonic boom wave on the window of a commercial store building in Oklahoma City which was broken during a sonic boom test in 1964. The results were compared with the results of a two-dimensional analytical method which was developed earlier by the authors and which rests on firm physical and mathematical foundations. Agreement was very good. The numerical method is not limited to plane cases but should be capable of extension to three-dimensional transient wave problems.
The plane interaction problem for a circular elastic inclusion embedded in an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Green's functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress-intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress-intensity factors are given.
A method for the analysis of the free vibrations of an arbitrary structure in terms of component modes is presented based upon the use of the normal, free-free modes of the components in a Rayleigh-Ritz analysis with the constraint or continuity conditions
The problem of subharmonic liquid response in a container subjected to vertical (axial) excitation has been solved to the first approximation by the method of perturbation, employing characteristic functions. In principle, the tank can be arbitrary. However, the computational effort required to construct the characteristic functions and- their derivatives may limit the application to tanks of relatively simple geometry such as a comparl-mentcd axisymmetric tank. As a check of the theoretical result, the simplest example for a rectangular tank is given, and the results are in good agreement Kith experiments and a third-order theory by Yarynwovych.
The axisymmetric problem of a line load acting along the axis of a semiinfinite elastic solid is solved using Hankel transforms. In this solution the line load is interpreted as a body force loading and by assuming the line load to be of the form of a Dirac delta function the solution of Mindlin's problem of a point load within the interior of the half space is obtained. Solutions of this problem presented in the literature have been obtained using semiinverse techniques whereas the solution given here is obtained in a systematic step-by-step manner.
The plane motion of two rigid, straight articulated pipes conveying fluid is examined. In contrast to previous work, the flow rate is not taken as constant, but is allowed to have small periodic oscillations about a mean value, as would be expected in a pump-driven system. It is shown that in the presence of such disturbances, both parametric and combination resonances are possible. When the system can also admit loss of stability by static buckling or by flutter, it is found that the presence of small periodic disturbances constitutes a destabilizing effect. Floquet theory and converging infinite determinant expansions are used to illustrate a basic difference between systems which lose stability by divergence and those that lose stability by flutter. An algebraic criterion is obtained for the minimum amplitude of flow-rate oscillation required for the system to be affected by the presence of small disturbances.
The assumption that volume changes associated with creep of a nonlinear viscoelastic material are only linearly dependent on the stress history is incorporated into a third-order multiple integral representation. This assumption reduces the number of independent kernel functions in the representation from 12 to 7. The traces of these independent kernels may be determined from two tension, two torsion, and one combined tension and torsion creep tests. Experiments on polyurethane are well represented by this method. The time-dependence of the kernel functions is expressed by time raised to a power with the power differing for different-order kernel functions.
The dynamic normal-load distribution across a strip that is required to maintain a plane progressive wave along its length is studied for the case where the strip is of infinite length and lies on the surface of a homogeneous isotropic elastic half space. This configuration is proposed as a preliminary idealized model for analyzing the dynamic interaction between soils and flexible foundations. The surface load distribution across the strip and the motion of the strip are related by a pair of dual integral equations. An asymptotic solution is obtained for the limiting case of small wavelength. The nature of this solution depends importantly on the propagation velocity of the strip-traveling wave in comparison with the Rayleigh wave speed, the shear wave speed and the dilatational wave speed. When the strip-traveling wave propagates faster than the Rayleigh wave speed, a pattern of trailing Rayleigh waves is shed from the strip. The limiting amplitude of the trailing waves is provided by the asymptotic solution.
The method of matched asymptotic expansions is employed to solve the singular perturbation problem of the vibrations of a rotating beam of small flexural rigidity with concentrated end masses. The problem is complicated by the appearance of the eigenvalue in the boundary conditions. Eigenfunctions and eigenvalues are developed as power series in the perturbation parameter beta to the 1/2 power, and results are given for mode shapes and eigenvalues through terms of the order of beta.
In this paper, a hybrid quasi-static atomistic simulation method at finite
temperature is developed, which combines the advantages of MD for thermal
equilibrium and atomic-scale finite element method (AFEM) for efficient
equilibration. Some temperature effects are embedded in static AFEM simulation
by applying the virtual and equivalent thermal disturbance forces extracted
from MD. Alternatively performing MD and AFEM can quickly obtain a series of
thermodynamic equilibrium configurations such that a quasi-static process is
modeled. Moreover, a stirring-accelerated MD/AFEM fast relaxation approach is
proposed, in which the atomic forces and velocities are randomly exchanged to
artificially accelerate the "slow processes" such as mechanical wave
propagation and thermal diffusion. The efficiency of the proposed methods is
demonstrated by numerical examples on single wall carbon nanotubes.
In this paper an automated algorithmic method is presented for the dynamic analysis of geared trains/transmissions. These are treated as a system of interconnected flexible bodies. The procedure developed explains the switching of constraints with time as a result of the change in the contacting areas at the gear teeth. The elastic behavior of the system is studied through the employment of three-dimensional isoparametric elements having six degrees-of-freedom at each node. The contact between the bodies is assumed at the various nodes, which could be either a line or a plane. The kinematical expressions, together with the equations of motion using Kane's method, strain energy concepts, are presented in a matrix form suitable for computer implementation. The constraint Jacobian matrices are generated automatically based on the contact information between the bodies. The concepts of the relative velocity at the contacting points at the tooth pairs and the subsequent use of the transmission ratios in the analysis is presented.
A change of variables that stabilizes numerical computations for periodic solutions of autonomous systems is derived. Computation of the period is decoupled from the rest of the problem for conservative systems of any order and for any second-order system. Numerical results are included for a second-order conservative system under a suddenly applied constant load. Near the critical load for the system, a small increment in load amplitude results in a large increase in amplitude of the response.
The line-spring model developed by Rice and Levy (1972) is used to obtain an approximate solution for a cylindrical shell containing a part-through surface crack. A Reissner type theory is used to account for the effects of the transverse shear deformations, and the stress intensity factor at the deepest penetration point of the crack is tabulated for bending and membrane loading by varying three-dimensionless length parameters of the problem formed from the shell radius, the shell thickness, the crack length, and the crack depth. The upper bounds of the stress intensity factors are provided, and qualitatively the line-spring model gives the expected results in comparison with elasticity solutions.