Effective Yield Criterion Accounting for Microvoid Coalescence

2013 ◽  
Vol 81 (3) ◽  
Author(s):  
A. Amine Benzerga ◽  
Jean-Baptiste Leblond
Keyword(s):  
Boundary Conditions ◽  
Limit Analysis ◽  
Yield Criterion ◽  
Closed Form Solution ◽  
Yield Function ◽  
Form Solution ◽  
Homogenization Theory ◽  
Porous Solids ◽  
Microvoid Coalescence ◽  
Effective Yield

An effective yield function is derived for a porous ductile solid near a state of failure by microvoid coalescence. Homogenization theory combined with limit analysis are used to that end. A cylindrical cell is taken to contain a coaxial cylindrical void of finite height. Plastic flow in the intervoid matrix is described by J2 theory while regions above and below the void remain rigid. Velocity boundary conditions are employed which are compatible with an overall uniaxial straining for the cell, a postlocalization kinematics that is ubiquitous during the coalescence of neighboring microvoids in rate-independent solids. Such boundary conditions are not of the uniform strain rate kind, as is the case for Gursonlike models. A similar limit analysis problem for a square-prismatic cell containing a square-prismatic void was posed long ago (Thomason, P. F., 1985, “Three-Dimensional Models for the Plastic Limit–Loads at Incipient Failure of the Intervoid Matrix in Ductile Porous Solids,” Acta Metallurgica, 33, pp. 1079–1085). However, to date a closed-form solution to this problem has been lacking. Instead, an empirical expression of the yield function proposed therein has been widely used in the literature. The fully analytical expression derived here is intended to be used concurrently with a Gursonlike yield function in numerical simulations of ductile fracture.

Nonmodal Solution of Spherical Shells With Cutouts Excited by High-Frequency Axisymmetric Forces

1970 ◽  
Vol 37 (4) ◽  
pp. 977-983 ◽  
Author(s):  
M. C. Junger
Keyword(s):  
Boundary Conditions ◽  
High Frequency ◽  
Near Field ◽  
Closed Form Solution ◽  
Radial Force ◽  
Form Solution ◽  
Mode Shapes ◽  
High Frequency Response ◽  
Circular Cutout ◽  
Line Loads

A closed-form solution is obtained for the high-frequency response of a thin spherical shell embodying a circular cutout and excited axisymmetrically by a concentrated radial force. The solution is constructed by combining the shell response to the radial exciting force with its response to radial, tangential, and moment line loads applied along the cutout boundary, these line loads being selected to match the boundary conditions. Concise expressions for the shell response are obtained by applying the Sommerfeld-Watson transformation to the slowly converging high-frequency modal series which is thereby reduced to only two terms, viz., an exponentially decaying near-field and a standing or propagating-wave field. These two terms are in the nature of the creeping waves commonly used to formulate electromagnetic or acoustic diffracted wave fields in the short-wavelength limit. The method is illustrated for the simple case of a circular cutout with a clamped boundary, but lends itself to more complicated boundary conditions, viz., intersecting shells or wave guides. The natural frequencies and mode shapes are found from a single, characteristic equation involving trigonometric functions.

scholarly journals A Novel Closed-Form Solution for Circular Openings in Generalized Hoek-Brown Media

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Qing Xiang Meng ◽  
Wei Wang
Keyword(s):  
Rock Mass ◽  
Closed Form ◽  
Hydrostatic Stress ◽  
Yield Criterion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Preliminary Design ◽  
Exact Integration ◽  
Sensitive Factor ◽  
Brittle Rock Mass

A novel closed-form solution is presented in this paper for the estimation of displacements around circular openings in a brittle rock mass subject to a hydrostatic stress field. The rock mass is assumed to be elastic-brittle-plastic media governed by the generalized Hoek-Brown yield criterion. The present closed-form solution was validated by employing the existing analytical solutions. Results of several example cases are analyzed to show that, with the simplified assumption, a novel closed-form solution is derived and found to be in an excellent agreement with those obtained by using the exact integration method with mathematical software. Parametric sensitivity analysis is carried out and the parameterartends to be the sensitive factor. As a closed-form solution that does not require transformation technique and the use of any numerical method, this work can provide a better choice in the preliminary design for circular opening.

A Closed Form Solution of Dual-Phase Lag Heat Conduction Problem With Time Periodic Boundary Conditions

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Pranay Biswas ◽  
Suneet Singh ◽  
Hitesh Bindra
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Closed Form ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fourier Series Expansion ◽  
Phase Lag ◽  
Dual Phase ◽  
Time Periodic

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.

The behavior of cambered belts transiting cylindrical rollers

2021 ◽  
pp. 095440892110644
Author(s):  
Jinxin Shi ◽  
Sheng Pan ◽  
Ron E. Markum ◽  
James K. Good
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Length Variation ◽  
Lateral Deformation ◽  
Roll To Roll ◽  
Manufactured Products ◽  
Lateral Behavior ◽  
The Web

The lateral deformations of webs in roll-to-roll process machines can affect the quality of the manufactured products. Webs with simple nonuniform length variation across their width (camber) will steer toward the long side, leading to the steady state lateral deformation and hence registration. Most previous studies have focused on a cambered web in a free span between two rollers. These studies assume some displacement and slope boundary conditions are known and seek the remaining conditions that would dictate the steady state lateral deformation of the web. This article focuses on the lateral behavior of a cambered web belt transiting between two aligned rollers as the simplest case of multiple span cambered web. Dynamic simulation has been conducted to better understand the response of a cambered web under tension that has been witnessed in tests. There are no boundary conditions enforced and no steady state deformation of the cambered webs. Thus there is no closed-form solution to the lateral movement of a cambered web transits over multiple rollers. This explained why the previous research focused more on the experimental exploration without few theoretical validations. The web travels toward the long side continually from one span to the next until a web guide attempts to return the web to an acceptable lateral location in the process machine.

Effective Yielding Characterization of Ductile Matrix Composites Containing Aligned Spheroidal Particles

1999 ◽  
Author(s):  
L. Z. Sun ◽  
J. W. Ju
Keyword(s):  
Spatial Distribution ◽  
Yield Criterion ◽  
Volume Averaging ◽  
Yield Function ◽  
Matrix Composites ◽  
Axisymmetric Loading ◽  
Matrix Interaction ◽  
Effective Yield ◽  
Yield Surfaces

Abstract The effective yielding characterization of ductile matrix composites containing randomly located yet unidirectionally aligned elastic spheroidal reinforcements is investigated. The ensemble-volume averaging procedure is employed to micromechanically derive the effective yield function of composites based on the probabilistic spatial distribution of aligned spheroidal particles and the particle-matrix interaction. The proposed effective yield surfaces of composites are shown to be convex. The initial effective yield criterion of composites under axisymmetric loading is specifically discussed.

Limit Analysis of Ice Sheet Indentation

1983 ◽  
Vol 105 (3) ◽  
pp. 352-355 ◽  
Author(s):  
D. G. Karr ◽  
S. C. Das
Keyword(s):  
Limit Analysis ◽  
Plastic Material ◽  
Yield Criterion ◽  
Ice Sheet ◽  
Yield Function ◽  
Infinite Layer ◽  
Perfectly Plastic ◽  
Aspect Ratios ◽  
Yield Functions ◽  
Stress Ratios

The methods of plastic limit analysis are used to determine the indentation pressures of a flat rectangular punch on an ice sheet. The ice sheet is idealized as a semi-infinite layer of elastic-perfectly plastic material. Lower bounds are computed by application of the lower bound limit theorem. The suitability of basic yield functions are assessed based on their ability to predict failure at demonstrated ice failure stress ratios. The particular yield functions that are employed include the generalized Mohr-Coulomb (or Drucker-Prager) criterion, a modified Drucker-Prager criterion, as well as a parabolic yield criterion used previously in literature on this topic. A study of the effects on indentation pressure of varying ice strength parameters is presented. Limit analysis solutions are obtained for plane stress conditions, and thus the applicability of a particular yield function can be evaluated for a range of ice strengths for indentation problems involving high aspect ratios.

Application of Orthotropic Plate Theory to Windmill Blade Design

1981 ◽  
Vol 103 (4) ◽  
pp. 892-894 ◽  
Author(s):  
C. Rubin
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Closed Form ◽  
Orthotropic Plate ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Layered Structures ◽  
Form Solution ◽  
Blade Design ◽  
Free Boundary Conditions

The windmill blade is treated as a semi-infinite orthotropic wedge with free-free boundary conditions. A closed form solution for the deflections and stresses is obtained as a function of the loading. The loading may be quite general. Results for three different materials which are commonly used for windmill blades (aluminum, sitka spruce, and fiberglass) are obtained. Applications also include ribbed, corrugated, and layered structures. In addition, other types of boundary conditions may be used to obtain solutions to a wide variety of other orthotropic plate problems.

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