Some Exact and Approximate Solutions for the Modified von Mises Yield Criterion

1988 ◽  
Vol 55 (2) ◽  
pp. 260-266 ◽  
Author(s):  
J. H. Lee
Keyword(s):  
Lower Bounds ◽  
Yield Criterion ◽  
Approximate Solutions ◽  
Upper And Lower Bounds ◽  
Plasticity Model ◽  
Plastic Deformations ◽  
Volume Increase ◽  
Strength Differential ◽  
Von Mises Yield Criterion ◽  
Von Mises

The effects of Strength Differential (SD) and plastic compressibility for materials obeying the modified von Mises yield criterion were exemplified by solving two boundary-value problems. The assumptions of associated plasticity (leading to maximum plastic volume increase) and nonassociated plasticity (leading to zero plastic volume increase) were used for comparative studies on the effects of plastic compressibility. The solutions for compression processes showed that SD effects increased the pressure at initial yielding and at failure, as well as increased the capacity of the materials to withstand plastic deformations. The opposite was true for tension processes. For associated and nonassociated plasticity, upper and lower bounds for stresses and strains for load and stroke-controlled situations were indicated. The results also showed unrealistic restrictions on the Poisson’s ratio and C/T for nonassociated plasticity under certain conditions. Hence, plastic volume increase, although small, should be incorporated into a more realistic plasticity model.

Limit pressure for a spherical vessel with a circumferential slot at its junction with a radial cylindrical protruding branch

1987 ◽  
Vol 22 (4) ◽  
pp. 215-227 ◽  
Author(s):  
M Robinson ◽  
C S Lim ◽  
R Kitching
Keyword(s):  
Spherical Shell ◽  
Lower Bounds ◽  
Yield Criterion ◽  
Spherical Vessel ◽  
Non Linear Programming ◽  
Limit Pressure ◽  
Wide Range ◽  
Cylindrical Branch ◽  
Von Mises ◽  
Plastic Limit Pressure

One of the requirements of the two criteria method of safety assessment of a pressure vessel with a defect is an estimate of the plastic limit pressure. Here the defect is in a spherical shell close to its junction with a protruding radial cylindrical branch. The defect is assumed to be an axisymmetric circumferential slot of uniform depth on the outer surface of the shell. Lower bounds to the limit pressure are calculated for a wide range of geometries. The material is assumed to obey the von Mises yield criterion and a non-linear programming method is used to give optimum lower bounds. Data is supplied for spherical shell radius to thickness ratios from 25 to 100, nozzle radius to vessel radius ratios from 0 to 0.4, nozzle to vessel thickness ratios from 0.25 to 1.0 and ligament thickness to vessel thicknesses (ligament efficiencies) of 0 to 1. Slot widths vary from the significant to the infinitesimal, where it becomes a crack. Vessels of some proportions were shown to have their limit pressures reduced only a little by very low ligament efficiencies.

Upper and Lower Bounds for the Parameters of One-Dimensional Theories for Sandwich Fracture Specimens

2020 ◽  
Vol 88 (3) ◽  
Author(s):  
Roberta Massabò
Keyword(s):  
Crack Length ◽  
Lower Bounds ◽  
Beam Theory ◽  
Approximate Solutions ◽  
Upper And Lower Bounds ◽  
Model Parameters ◽  
One Dimensional ◽  
Limit Value ◽  
Elastic Mismatch ◽  
Fracture Specimens

Abstract Upper and lower bounds for the parameters of one-dimensional theories used in the analysis of sandwich fracture specimens are derived by matching the energy release rate with two-dimensional elasticity solutions. The theory of a beam on an elastic foundation and modified beam theory are considered. Bounds are derived analytically for foundation modulus and crack length correction in single cantilever beam (SCB) sandwich specimens and verified using accurate finite element results and experimental data from the literature. Foundation modulus and crack length correction depend on the elastic mismatch between face sheets and core and are independent of the core thickness if this is above a limit value, which also depends on the elastic mismatch. The results in this paper clarify conflicting results in the literature, explain the approximate solutions, and highlight their limitations. The bounds of the model parameters can be applied directly to specimens satisfying specific geometrical/material ratios, which are given in the paper, or used to support and validate numerical calculations and define asymptotic limits.

Design Equation for Minimum Required Thickness of a Cylindrical Shell Subject to Internal Pressure Based on Von Mises Criterion

2019 ◽  
Author(s):  
James Lu ◽  
Barry Millet ◽  
Kenneth Kirkpatrick ◽  
Bryan Mosher
Keyword(s):  
Cylindrical Shell ◽  
Internal Pressure ◽  
Yield Criterion ◽  
Design Equation ◽  
Shell Wall ◽  
Section Viii ◽  
Von Mises Yield Criterion ◽  
Von Mises ◽  
Division 2 ◽  
Shell Wall Thickness

Abstract Design equation (4.3.1) for the minimum required thickness of a cylindrical shell subjected to internal pressure in Part 4 “design by rule (DBR)” of the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 [1] is based on the Tresca Yield Criterion, while design by analysis (DBA) in Part 5 of the Division 2 Code is based on the von Mises Yield Criterion. According to ASME PTB-1 “ASME Section VIII – Division 2 Criteria and Commentary”, the difference in results is about 15% due to use of the two different criteria. Although the von Mises Yield Criterion will result in a shell wall thickness less than that from Tresca Yield Criterion, Part 4 (DBR) of ASME Division 2 adopts the latter for a more convenient design equation. To use the von Mises Criterion in lieu of Tresca to reduce shell wall thickness, one has to follow DBA rules in Part 5 of Division 2, which typically requires detailed numeric analysis performed by experienced stress analysts. This paper proposes a simple design equation for the minimum required thickness of a cylindrical shell subjected to internal pressure based on the von Mises Yield Criterion. The equation is suitable for both thin and thick cylindrical shells. Calculation results from the equation are validated by results from limit load analyses in accordance with Part 5 of ASME Division 2 Code.

scholarly journals On Hybrid Type Nonlinear Fractional Integrodifferential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 984
Author(s):  
Faten H. Damag ◽  
Adem Kılıçman ◽  
Awsan T. Al-Arioi
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Lower Bounds ◽  
Fixed Point Theorems ◽  
Approximate Solutions ◽  
Integrodifferential Equations ◽  
Upper And Lower Bounds ◽  
Hybrid Type ◽  
Fractional Equations

In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated.

Percentiles of von Mises Stress from Combined Random Vibration and Static Loading by Approximate Noncentral Chi Square Distribution

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Patrick. A Tibbits
Keyword(s):  
Lower Bounds ◽  
Cumulative Distribution ◽  
Von Mises Stress ◽  
Upper And Lower Bounds ◽  
Linear Structures ◽  
Chi Square ◽  
Mean Values ◽  
Random Loads ◽  
Nonzero Mean ◽  
Von Mises

Firstly, a calculation for percentiles of von Mises stress in linear structures subjected to Gaussian random loads is extended to the case of Gaussian random loads having nonzero mean values, i.e.,the inclusion of static loads. The development is restricted to the case of plane stress. The method includes calculation of a given percentile of von Mises stress to any desired accuracy, a rapid estimate of the percentile, and upper and lower bounds on the von Mises stress. The calculation expands the cumulative distribution function of the von Mises stress as a series of noncentral chi-square distributions. Summation of a sufficient number of terms of the series calculates the percentile to the desired accuracy. The rapid estimate of the percentile interpolates the distribution of the von Mises stress in a small number of inverse noncentral chi-square2 distribution functions. The upper and lower bounds on the percentiles take advantage of the noncentral chi-square distribution of summations of normally distributed stress components. Second and third calculation methods arise from approximations of the distribution of quadratic forms of noncentral normal variables, or equivalently, linear combinations of noncentral chi-square variables. These methods provide rapid estimates of percentiles of von Mises stress in linear structures under random loads having nonzero mean values. The accuracy and computational efficiency of the methods are reviewed and compared. The methods are expected to have wide application in design of and prognostics for components subjected to constant structural loads coupled with random loading arising from vibrations caused by wind, waves, seismic events, engines, turbulence, acoustic noise, etc.

Forming Limit Analysis of Sheet Metals Based on a Generalized Deformation Theory

2003 ◽  
Vol 125 (3) ◽  
pp. 260-265 ◽  
Author(s):  
C. L. Chow ◽  
M. Jie ◽  
S. J. Hu
Keyword(s):  
Strain Hardening ◽  
Deformation Theory ◽  
Yield Criterion ◽  
Strain Ratio ◽  
Yield Function ◽  
Forming Limit ◽  
Sheet Metals ◽  
Theory Of Plasticity ◽  
Von Mises Yield Criterion ◽  
Von Mises

This paper presents the development of a generalized method to predict forming limits of sheet metals. The vertex theory, which was developed by Sto¨ren and Rice (1975) and recently simplified by Zhu, Weinmann and Chandra (2001), is employed in the analysis to characterize the localized necking (or localized bifurcation) mechanism in elastoplastic materials. The plastic anisotropy of materials is considered. A generalized deformation theory of plasticity is proposed. The theory considers Hosford’s high-order yield criterion (1979), Hill’s quadratic yield criterion and the von Mises yield criterion. For the von Mises yield criterion, the generalized deformation theory reduces to the conventional deformation theory of plasticity, i.e., the J2-theory. Under proportional loading condition, the direction of localized band is known to vary with the loading path at the negative strain ratio region or the left hand side (LHS) of forming limit diagrams (FLDs). On the other hand, the localized band is assumed to be always perpendicular to the major strain at the positive strain ratio region or the right hand side (RHS) of FLDs. Analytical expressions for critical tangential modulus are derived for both LHS and RHS of FLDs. For a given strain hardening rule, the limit strains can be calculated and consequently the FLD is determined. Especially, when assuming power-law strain hardening, the limit strains can be explicitly given on both sides of FLD. Whatever form of a yield criterion is adopted, the LHS of the FLD always coincides with that given by Hill’s zero-extension criterion. However, at the RHS of FLD, the forming limit depends largely on the order of a chosen yield function. Typically, a higher order yield function leads to a lower limit strain. The theoretical result of this study is compared with those reported by earlier researchers for Al 2028 and Al 6111-T4 (Grafand Hosford, 1993; Chow et al., 1997).

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