Stability and Imperfections in Quasistatic Viscoplastic Solutions

1990 ◽  
Vol 43 (5S) ◽  
pp. S251-S255 ◽  
Author(s):  
T. Belytschko ◽  
B. Moran ◽  
M. Kulkarni
Keyword(s):  
Strain Field ◽  
Strain Softening ◽  
Fourier Spectrum ◽  
Shear Bands ◽  
Closed Form Solution ◽  
Step Function ◽  
Form Solution ◽  
Viscoplastic Material ◽  
Strain Fields ◽  
The Stability

The effect of imperfections on the structure of shear bands in strain-softening viscoplasticity is studied via a closed form solution. The stability of various solutions is then examined by varying the data through imperfections. It is shown that a step-function imperfection, such as commonly used in finite element solutions, leads to a step-function shear strain field, which is an unstable solution. Arbitrary C0 and C1 imperfections lead to C0 and C1 strain fields, respectively. Fourier analyses show that the imperfection scales the response of the viscoplastic material: the Fourier spectrum of the strain field is strongly influenced by the Fourier spectrum of the imperfection.

On the Crucial Role of Imperfections in Quasi-static Viscoplastic Solutions

1991 ◽  
Vol 58 (3) ◽  
pp. 658-665 ◽  
Author(s):  
T. Belytschko ◽  
B. Moran ◽  
M. Kulkarni
Keyword(s):  
Shear Band ◽  
Strain Softening ◽  
Shear Bands ◽  
Step Function ◽  
Viscoplastic Material ◽  
One Dimensional ◽  
Initial Imperfections ◽  
Dimensional Boundary ◽  
The Stability

The stability and structure of shear bands and how they relate to initial imperfections is studied within the framework of a one-dimensional boundary value problem. It is shown that in strain-softening viscoplasticity the structure of the band depends on the structure of the imperfection. A Fourier analysis shows that the width of the shear band depends directly on the width of the imperfection, suggesting that the imperfection scales the response of the viscoplastic material. For continuously differentiable imperfections, the shear band is continuously differentiable, whereas when the imperfection is C° at the maximum, the shear band is C°, and cusp-shaped. For step function imperfections, the shear band is shown to be a step function, but it is shown that this solution is unstable.

Stability Analysis of Mechanical Seals With Two Flexibly Mounted Rotors

1998 ◽  
Vol 120 (2) ◽  
pp. 145-151 ◽  
Author(s):  
J. Wileman ◽  
I. Green
Keyword(s):  
Dynamic Properties ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fluid Film ◽  
Mechanical Seals ◽  
Stability Threshold ◽  
Seal Design ◽  
The Stability ◽  
Gyroscopic Effects

Dynamic stability is investigated for a mechanical seal configuration in which both seal elements are flexibly mounted to independently rotating shafts. The analysis is applicable to systems with both counterrotating and corotating shafts. The fluid film effects are modeled using rotor dynamic coefficients, and the equations of motion are presented including the dynamic properties of the flexible support. A closed-form solution for the stability criteria is presented for the simplifled case in which the support damping is neglected. A method is presented for obtaining the stability threshold of the general case, including support damping. This method allows instant determination of the stability threshold for a fully-defined seal design. A parametric study of an example seal is presented to illustrate the method and to examine the effects of various parameters in the seal design upon the stability threshold. The fluid film properties in the example seal are shown to affect stability much more than the support properties. Rotors having the form of short disks are shown to benefit from gyroscopic effects which give them a larger range of stable operating speeds than long rotors. For seals with one long rotor, counterrotating operation is shown to be superior because the increased fluid stiffness transfers restoring moments from the short rotor to the long.

Initial Post-Buckling and Growth of a Circular Delamination Bridged by Nonlinear Fibers

2000 ◽  
Vol 67 (4) ◽  
pp. 777-784 ◽  
Author(s):  
S. Li ◽  
J. Nie ◽  
J. Qian ◽  
Y. Huang ◽  
Y. Hu
Keyword(s):  
Fracture Toughness ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Dynamic Effect ◽  
Form Solution ◽  
Delamination Growth ◽  
Nonlinear Fibers ◽  
Post Buckling ◽  
Bridging Model ◽  
The Stability

Axisymmetric buckling, initial post-buckling and growth of a circular delamination bridged by nonlinear fibers in three-dimensional composites are studied by a perturbation method. The through-thickness fibers are assumed to provide nonlinear restoring traction resisting the deflection of the delaminated layer. A closed-form solution for the central deflection of the delamination due to on applied compressive stress during initial post-buckling is obtained. In addition, some simple formulas for calculating the strain energy release rate and the mixed mode stress intensity ratio (i.e., Mode II versus Mode I) at the delamination crack tip are also established. Some interesting conclusions arising directly from the perturbation solutions are drawn. These include: (1) initial post-buckling behavior of a circular delamination is unstable for a softening bridging model; this may result in initial delamination growth for some materials with lower fracture toughness when the delamination buckles rather than post-buckles. However, stable growth is obtained for a hardening bridging model; (2) with an increase of the nonlinear fiber bridging parameter β¯, the residual stiffness of a three-dimensional composite structure with a circular delamination increases gradually; (3) bridging force changes the catastrophic nature of the delamination growth and increases the stability of the delamination. The range and the dynamic effect of the unstable delamination growth diminish or disappear as the bridging parameters increase; (4) for the bridged delamination, the higher the material fracture toughness, the higher the stability of the delamination growth, and the smaller the range and dynamic effect of its unstable growth. [S0021-8936(00)03203-7]

Strain fields in cracked bodies under antiplane shear for a generalised non-hardening material law

2019 ◽  
Vol 24 (10) ◽  
pp. 3125-3135
Author(s):  
M Zappalorto
Keyword(s):  
Closed Form ◽  
Large Strain ◽  
Closed Form Solution ◽  
Form Solution ◽  
Antiplane Shear ◽  
Elastic Behaviour ◽  
Strain Fields ◽  
Hardening Material ◽  
Hardening Law ◽  
Non Linear

An exact, closed form, solution is derived for the non-linear stress distribution in a cracked body under antiplane shear deformation. A generalised, non work-hardening, law is introduced to describe the material behaviour, and the stress and strain fields are derived in closed form. Such a new generalised material law includes the effect of a new parameter, a, which allows the transition from the ideally elastic behaviour (low strain regime) to the pure non-linear behaviour (large strain regime) to be modulated. A discussion is carried out on the features of the new solution and on the behaviour of stresses and strains close to and far away from the crack tip.

An Inverse Force Analysis of a Tetrahedral Three-Spring System

1995 ◽  
Vol 117 (2A) ◽  
pp. 286-291 ◽  
Author(s):  
P. Dietmaier
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
The Other ◽  
Force Analysis ◽  
Equilibrium Configurations ◽  
Spring System ◽  
The Common ◽  
The Stability ◽  
Inverse Force

A tetrahedral three-spring system under a single load has been analyzed and a closed-form solution for the equilibrium positions is given. Each of the three springs is attached at one end to a fixed pivot in space while the other three ends are linked by a common pivot. The springs are assumed to behave in a linearly elastic way. The aim of the paper at hand was to find out what the maximum number of equilibrium positions of such a system might be, and how to compute all possible equilibrium configurations if a given force is applied to the common pivot. First a symmetric and unloaded system was studied. For such a system it was shown that there may exist a maximum of 22 equilibrium configurations which may all be real. Second the general, loaded system was analyzed, revealing again a maximum of 22 real equilibrium configurations. Finally, the stability of this three-spring system was investigated. A numerical example illustrates the theoretical findings.

scholarly journals On the stability of the μ(I) rheology for granular flow

2017 ◽  
Vol 833 ◽  
pp. 302-331 ◽  
Author(s):  
J. D. Goddard ◽  
Jaesung Lee
Keyword(s):  
Linear Stability ◽  
Granular Flow ◽  
Convective Instability ◽  
Continuum Model ◽  
Shear Bands ◽  
Closed Form Solution ◽  
Static Field ◽  
Form Solution ◽  
Base Flow ◽  
Field Equations

This article deals with the Hadamard instability of the so-called$\unicode[STIX]{x1D707}(I)$model of dense rapidly sheared granular flow, as reported recently by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). The present paper presents a more comprehensive study of the linear stability of planar simple shearing and pure shearing flows, with account taken of convective Kelvin wavevector stretching by the base flow. We provide a closed-form solution for the linear-stability problem and show that wavevector stretching leads to asymptotic stabilization of the non-convective instability found by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). We also explore the stabilizing effects of higher velocity gradients achieved by an enhanced-continuum model based on a dissipative analogue of the van der Waals–Cahn–Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyperstress, as the analogue of a special Korteweg stress, with surface viscosity representing the counterpart of elastic surface tension. Based on the enhanced-continuum model, we also present a model of steady shear bands and their nonlinear stability against parallel shearing. Finally, we propose a theoretical connection between the non-convective instability of Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818) and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the numerical simulation of continuum field equations involving the$\unicode[STIX]{x1D707}(I)$rheology and variants thereof.

A FAST AND ACCURATE STEP-BY-STEP SOLUTION PROCEDURE FOR DIRECT INTEGRATION

2011 ◽  
Vol 11 (03) ◽  
pp. 473-493 ◽  
Author(s):  
SHYH-RONG KUO ◽  
J. D. YAU
Keyword(s):  
Time Integration ◽  
Closed Form Solution ◽  
Solution Procedure ◽  
Small Time ◽  
Form Solution ◽  
Dynamic Problems ◽  
Sdof System ◽  
Linear Dynamic ◽  
The Stability ◽  
Stability And Accuracy

Very small time steps are usually needed in numerical computation as conventional time integration methods are used to compute the response of a structure subjected to a dynamic loading with rapid changes or load discontinuity. To overcome this drawback, this study proposed a fast, fourth-order accurate step-by-step time integration (FASSTI) algorithm that is unconditionally stable and allows larger time steps for linear dynamic problems. From the stability and accuracy analysis, it is shown that the FASSTI algorithm retains the features of unconditional stability, accuracy, and fast convergence than the Newmark method. As a first test, a closed-form solution of an excited single degree of freedom (SDOF) system is derived and used to verify the reliability of the present algorithm in solving linear dynamic problems. In the numerical examples, the accuracy and efficiency of the proposed method is demonstrated in the solution of the dynamic response of an SDOF system under a series of impulse-type forces.

On the Stability of Waves in a Thin Orthotropic Spinning Disk

1982 ◽  
Vol 49 (3) ◽  
pp. 570-572 ◽  
Author(s):  
J. L. Nowinski
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Elementary Functions ◽  
Wave Stability ◽  
Spinning Disk ◽  
Perturbation Parameters ◽  
The Stability ◽  
Temporal Perturbation ◽  
Stability Of Waves

A system of two ordinary coupled differential equations with periodic coefficients of the Mathieu type for two temporal perturbation parameters is derived. A closed-form solution of the system in terms of elementary functions is found and discussed. A condition for the wave stability involving the coefficients of anisotropy is established. Illustration involves a specific range of these coefficients.

Effects of Varying Dynamics of Flexible Workpieces in Milling Operations

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Adam K. Kiss ◽  
Daniel Bachrathy ◽  
Gabor Stepan
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Depth Of Cut ◽  
Stability Conditions ◽  
Flexible Beam ◽  
Milling Operations ◽  
Radial Depth ◽  
Distributed Transfer Function ◽  
Relative Errors ◽  
The Stability

Abstract In this study, surface error calculations and stability conditions are presented for milling operations in case of slender parts. The dynamic behavior of the flexible beam-type workpiece is modeled by means of finite element method (FEM), while the varying dynamical properties related to the feed motion as well as the material removal process are incorporated in the model. The FEM-generated direct frequency response function is verified through a closed-form solution based on the distributed transfer function method. Relative errors and convergence of the FEM are investigated based on the analytical solutions of the continuum model, from which appropriate element size and mode number can be selected for modal coordinate transformations. The pattern in the variation of the natural frequencies is explored using the analytical model in case of high radial depth of cut relative to the original cross section of the beam-like workpiece. Both the stability conditions and the resulted surface errors are predicted as a function of the tool position. The presented approach and the results are validated by laboratory tests.

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