On the Plastic Strains in Slabs With Cutouts

1953 ◽  
Vol 20 (2) ◽  
pp. 183-188
Author(s):  
P. G. Hodge
Keyword(s):  
Maximum Shear Stress ◽  
Yield Condition ◽  
Slab Thickness ◽  
Tresca Yield Condition ◽  
Plastic Potential ◽  
Plastic Domain ◽  
Leading Term ◽  
Elastic Slab ◽  
Plastic Slab ◽  
Set Up

Abstract The complete elastic-plastic problem is set up for a circular slab with a central circular cutout, subjected to uniform external tension. The analysis is carried out under the assumption of generalized plane stress but with possibly finite deformations. The material is assumed to be isotropic, homogeneous, and incompressible, to yield according to the Tresca yield condition, and to satisfy Hooke’s law in the elastic domain and the plastic potential law in the plastic domain. The equations are solved by a perturbation method based on the ratio of the maximum shear stress to the shear modulus, in which each of the significant quantities, stress, displacement, and slab thickness, is expanded in a power series in this ratio. A similar technique is employed to solve the fully plastic flow problem. It is shown that for cutout radii greater than 1 per cent of the radius of the slab, the “first approximations” obtained by neglecting all but the leading term in each series are satisfactory, up to loads at which the slab becomes wholly plastic. The ratio of the maximum strain in the just fully plastic slab to that in the completely elastic slab is computed. It is found that this ratio is less than about 6 if the cutout radius is at least 1 per cent of the radius of the slab. Some observations are advanced on the case where the cutout is very small compared to the slab.

Dynamic Plastic Response of Cylindrical Shells Under Gaussian Impulse

1980 ◽  
Vol 24 (01) ◽  
pp. 24-30
Author(s):  
S. Anantha Ramu ◽  
K. J. Iyengar
Keyword(s):  
Cylindrical Shells ◽  
Yield Condition ◽  
Longitudinal Axis ◽  
Plastic Behavior ◽  
Plastic Response ◽  
Marine Structures ◽  
Impulsive Load ◽  
Tresca Yield Condition ◽  
Impulsive Loads

The determination of the inelastic response of cylindrical shells under general impulsive loads is of relevance to marine structures such as submarines, in analyzing their slamming damages. The present study is concerned with the plastic response of a simply supported cylindrical shell under a general axisymmetric impulsive load. The impulsive load is assumed to impart an axisymmetric velocity to the shell, with a Gaussian distribution along the longitudinal axis of the shell. A simplified Tresca yield condition is used. The shell response is determined for various forms of impulses ranging from a concentrated impulse to a uniform impulse over the entire length of the shell. Conclusions about the influence of geometry of the shell and the spatial distribution of impulse on the plastic behavior of cylindrical shells are presented.

Limit Analysis for Combined Edge and Pressure Loading on a Cylindrical Shell

1971 ◽  
Vol 93 (4) ◽  
pp. 998-1006
Author(s):  
H. S. Ho ◽  
D. P. Updike
Keyword(s):  
Cylindrical Shell ◽  
Velocity Field ◽  
Axial Force ◽  
Shear Force ◽  
Circular Cylindrical Shell ◽  
Yield Condition ◽  
External Pressure ◽  
Tresca Yield Condition ◽  
Stress States ◽  
Range Of Values

Equations describing the stress field and velocity field occurring in a circular cylindrical shell at plastic collapse are derived corresponding to stress states lying on each face of a yield surface for a uniform shell of material obeying the Tresca yield condition. They are then applied to the case of a shell under combined axisymmetric loadings (moment, shear force, and axial force) at one end and uniform internal or external pressure on the lateral surface. For a sufficiently long shell, complete solutions are obtained for a fixed far end, and for a certain range of values of axial force and pressure, they are obtained for a free far end. All the solutions are represented by either closed form or by quadratures. It is shown that in many cases the radial velocity field is proportional to the shear force.

Elastic-Plastic Analysis of Pressurized Cylindrical Shells

1987 ◽  
Vol 54 (3) ◽  
pp. 597-603 ◽  
Author(s):  
G. N. Brooks
Keyword(s):  
Shell Theory ◽  
Plastic Region ◽  
Yield Condition ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
Axisymmetric Shell ◽  
Bending Stresses ◽  
Principal Directions ◽  
The Moment ◽  
Thin Shell Theory

Plasticity in shells is often contained near the ends of a segment where the bending stresses are significant. Outside of this local neighborhood the behavior is elastic. Thus, an axisymmetric shell can be divided along its axis into a purely elastic region away from an end and the local region where plasticity is present. The moment-curvature relation in the elastic-plastic region is calculated using the Tresca yield condition. Use of the Tresca yield condition greatly simplifies this derivation because the principal directions are known. This moment-curvature relationship is “exact” in the sense that only the standard assumptions of thin shell theory are made. The solutions of the elastic and plastic regions are matched at their intersection for an efficient numerical solution. The technique is used here to study the semi-infinite clamped cylindrical shell with an internal pressure loading.

Finite Elastic-Plastic Expansion of a Cylindrical Tube

1973 ◽  
Vol 2 (4) ◽  
pp. 216-222
Author(s):  
B. Slevinsky ◽  
J. B. Haddow
Keyword(s):  
Plastic Deformation ◽  
Yield Condition ◽  
Cylindrical Tube ◽  
Flow Rule ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
End Conditions ◽  
Cylindrical Tubes ◽  
Limiting Case ◽  
Plastic Flow Rule

A numerical method for the analysis of the isothermal elastic-plastic expansion, by internal pressure, of cylindrical tubes with various end conditions is presented. The Tresca yield condition and associated plastic flow rule are assumed and both non-hardening and work-hardening tubes are considered with account being taken of finite plastic deformation. Tubes which undergo further plastic deformation on unloading are also considered. Expansion of a cylindrical cavity from zero radius in an infinite medium is considered as a limiting case.

Comparison of Slip-Line Solutions With Experiment

1956 ◽  
Vol 23 (2) ◽  
pp. 225-230
Author(s):  
E. G. Thomsen
Keyword(s):  
Plane Strain ◽  
Pure Aluminum ◽  
Maximum Shear Stress ◽  
Yield Condition ◽  
Approximate Solutions ◽  
Extrusion Process ◽  
Experimental Results ◽  
Wall Friction ◽  
Axially Symmetric ◽  
Area Reduction

Abstract Plane-strain and axially symmetric approximate solutions for a perfect plastic are compared with experimental results obtained from the extrusion of a billet of commercially pure aluminum. The extrusion process was carried out at room temperature under essentially negligible external wall friction with a reduction of area of 87.8 per cent. It is found that the plane-strain solution is in good agreement with the experimental results, if a constant k = σ̄/2 = 20,000 psi (maximum shear stress for Tresca’s criterion) is substituted in the plane-strain solution. The flow stress (σ̄ ≅ 40,000 psi) was determined from a property test for the aluminum billet, at a strain equivalent to the uniform deformation in the extrusion (i.e., same area reduction without shear deformation). It is found that the Haar and von Karman yield condition is not valid for the present case, and further that the axially symmetric solution demonstrates no real advantage over the simpler plane-strain solution in predicting the stress distribution in an axially symmetric extrusion problem.

Rigid-Plastic Response of Floating Plates

1988 ◽  
Vol 32 (03) ◽  
pp. 168-176
Author(s):  
John Anastasiadis ◽  
Paul C. Xirouchakis
Keyword(s):  
Rigid Body ◽  
Yield Condition ◽  
Body Motion ◽  
Phase Iii ◽  
Flow Rule ◽  
Rigid Plastic ◽  
Flexural Response ◽  
Tresca Yield Condition ◽  
Perfectly Plastic ◽  
R Ratio

This paper presents the exact formulation and solution for the static flexural response of a rigid perfectly plastic freely floating plate subjected to lateral axisymmetric loading. The Tresca yield condition is adopted with the associated flow rule. The plate response is divided into three phases: Initially the plate moves downward into the foundation as a rigid body (Phase I). Subsequently the plate deforms in a conical mode in addition to the rigid body motion (Phase II). At a certain value of the load a hinge-circle forms which may move as the pressure increases further (Phase III). The nature of the solution during the third phase depends upon the parameter α = a/R (ratio of radius of loaded area to the plate radius). When α = αs≅ 0.46 the hinge-circle remains stationary under increasing load. For α < αs the hinge-circle shrinks, whereas for α > αs the hinge-circle expands with increasing pressure. The application of the present results to the problem of laterally loaded floating ice plates is discussed.

Transient Thermal Stresses in Elasto-Plastic Discs

1969 ◽  
Vol 11 (4) ◽  
pp. 384-391 ◽  
Author(s):  
H. Odenö
Keyword(s):  
Temperature Distribution ◽  
Thermal Stresses ◽  
Plastic Material ◽  
Yield Condition ◽  
Transient Temperature ◽  
Flow Rule ◽  
Axially Symmetric ◽  
Exterior Surface ◽  
Tresca Yield Condition ◽  
Perfectly Plastic

A thin circular disc of elastic-perfectly plastic material, subjected to an axially symmetric transient temperature distribution, is treated analytically. All material parameters are assumed to be independent of the temperature. Poisson's ratio is taken to be one-half. The Tresca yield condition with associated flow rule is employed. The temperature distribution is that which appears when the outer rim surface of the disc receives a rapid temperature increase and it is solved approximately by the collocation method. The analysis shows that under certain circumstances, plastic deformation will occur in a moving annular region. This region starts to develop at the exterior surface and moves inward, while changing its width. After a certain finite time its width shrinks to zero. Except for a residual constant state of strain, the strain field is then again elastic. An application to the method of separating the ring and the shaft in a shrink-fit is carried out numerically. The residual stresses in the ring are calculated.

Quantifying Vessel Material Properties Using MRI Under Pressure Condition and MRI-Based FSI Mechanical Analysis for Human Atherosclerotic Plaques

2006 ◽  
Author(s):  
Chun Yang ◽  
Xueying Huang ◽  
Jie Zheng ◽  
Pamela K. Woodard ◽  
Dalin Tang
Keyword(s):  
Material Properties ◽  
Stokes Equations ◽  
Maximum Shear Stress ◽  
Maximum Principal Stress ◽  
Mechanical Analysis ◽  
Navier Stokes ◽  
Atherosclerotic Plaques ◽  
Navier Stokes Equations ◽  
Stenosis Severity ◽  
Set Up

Atherosclerotic plaques may rupture without warning and cause acute cardiovascular syndromes such as heart attack and stroke. Mechanical image analysis using MRI-based models with fluid-structure interactions (FSI) and MRI-determined material properties may improve the accuracy of plaque vulnerability assessment and rupture predictions. A plaque-phantom was set up to acquire plaque MR images under pressurized conditions. The 3D nonlinear modified Mooney-Rivlin (M-R) model was used to describe the material properties with parameters selected to fit the MRI data. The Navier-Stokes equations were used as the governing equations for the flow model. The fully-coupled FSI models were solved by ADINA. Our results indicate that doubling parameter values in the M-R model led to 12.5% decrease in structure maximum principal stress (Stress-P1) and 48% decrease in maximum principal strain (Strain-P1). Flow maximum shear stress (MSS) was almost unchanged. Results from a modified carotid plaque with 70% stenosis severity (by diameter) showed that Stress-P1 at the plaque throat from the wall-only model is 145% higher than that from the FSI model. MSS from a flow-only model is about 40% higher than that from the FSI model. This approach has the potential to develop non-invasive patient screening and diagnosis methods in clinical applications.

Elastic-Plastic Ring-Loaded Cylindrical Shells

1988 ◽  
Vol 55 (4) ◽  
pp. 761-766 ◽  
Author(s):  
Gregory N. Brooks
Keyword(s):  
Axisymmetric Problem ◽  
Shell Theory ◽  
Complete Solution ◽  
Yield Condition ◽  
Plastic Ring ◽  
Elastic Plastic ◽  
Tresca Yield Condition ◽  
Small Deflection ◽  
Principal Directions ◽  
Long Cylindrical Shell

The elastic-plastic solution for an infinitely long cylindrical shell with an axisymmetric ring load is presented. Except for the material nonlinearity, the standard assumptions of small deflection shell theory were made. Because the principal directions are known for the axisymmetric problem, the Tresca yield condition wasused. This made it possible to obtain closed-form expressions for the elastic-plastic, moment-curvature relations, greatly simplfying the computational task. The actual stress distribution through the thickness was used, making these relations exact. Yielding was contained near the load. Thus, for the analysis the cylinder was divided along its axis into elastic-plastic and purely elastic regions. Solutions were obtained for each region which were then matched at their intersection to give the complete solution. All results are given in dimensionless form so that they may be applied to any shell.

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