An Improved Fracture Criterion for Three-Dimensional Stress States

1976 ◽  
Vol 98 (2) ◽  
pp. 159-163 ◽  
Author(s):  
B. Paul ◽  
L. Mirandy
Keyword(s):  
Applied Stress ◽  
Stress Fracture ◽  
Fracture Criterion ◽  
Three Dimensional ◽  
Brittle Materials ◽  
Two Dimensional ◽  
Stress Concentrations ◽  
State Of Stress ◽  
Stress States ◽  
Free Material

A theory is developed to predict the onset of fracture in isotropic, brittle materials when subjected to three dimensional states of applied stress. It is assumed that fracture is precipitated by stress concentrations emanating from material flaws. The flaw model which has been adopted consists of randomly oriented, microscopic, flat triaxial ellipsoidal voids imbedded in an otherwise defect-free material. It is shown that the ensuing fracture criterion may be expressed as a parabolic Mohr’s envelope. These results are qualitatively similar to Paul’s earlier three-dimensional generalization of Griffith’s two-dimensional stress fracture criterion. To handle three-dimensional states of applied stress, Paul used an approximation based on two-dimensional elasticity to obtain the state of stress around a flat spheroid. Newly developed results for flat ellipsoidal cavaties are utilized herein to analyze the three-dimensional cavity. Pertinent effects due to Poisson’s ratio and ellipsoid geometry are reported.

Stresses at the Surface of a Flat Three-Dimensional Ellipsoidal Cavity

1976 ◽  
Vol 98 (2) ◽  
pp. 164-172 ◽  
Author(s):  
L. Mirandy ◽  
B. Paul
Keyword(s):  
Stress Intensity ◽  
Applied Stress ◽  
Surface Stress ◽  
Fracture Criterion ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Intensity Factors ◽  
Ellipsoidal Cavity ◽  
Isotropic Elastic Medium ◽  
Analytical Expressions

The stress field associated with a thin ellipsoidal cavity in an isotropic elastic medium with arbitrary tractions at distant boundaries is needed to generalize Griffith’s two-dimensional fracture criterion. Such a solution is given here. It is first formulated for a general ellipsoidal cavity having principal semiaxes a, b, and c, and then it is reduced to the specific case of a “flat” ellipsoid for which a and b are very much greater than c. An explicit solution of the general problem is possible but the results are somewhat unwieldy. The dominant terms of an asymptotic solution for small c/b, however, are shown to provide remarkably simple expressions for the stresses everywhere on the surface of the cavity. The applied normal stress parallel to the c axis and the shears lying in a plane perpendicular to it were found to produce surface stresses proportional to (b/c) × applied stress, with the amplification of other components of applied stress being negligible in comparison. Analytical expressions for the location and magnitude of the maximum surface stress are developed along with stress intensity factors for the elliptical crack (c = 0). Three dimensional effects due to ellipsoidal planform aspect ratio (b/a) and Poisson’s ratio are reported.

A Sequel to Technical Note 13: The Curved Tetrahedronal and Triangular Elements TEC and TRIC for the Matrix Displacement Method

1969 ◽  
Vol 73 (697) ◽  
pp. 55-65 ◽  
Author(s):  
J. H. Argyris ◽  
D. W. Scharpf
Keyword(s):  
Computational Analysis ◽  
Three Dimensional ◽  
Technical Note ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
Displacement Method ◽  
Stress Concentrations ◽  
The Past ◽  
The Matrix ◽  
Triangular Elements

It is by now well established that the computational analysis of significant problems in structural and continuum mechanics by the matrix displacement method often requires elements of higher sophistication than used in the past. This refers, in particular, to regions of steep stress gradients, which are frequently associated with marked changes in geometry, involving rapid variations of the radius of curvature. The philosophy underlying the idealisation of such configurations into finite elements was discussed in broad terms in ref. 1. It was emphasised that the so successful, constant strain, two-dimensional TRIM 3 and three-dimensional TET 4 elements do not, in general, prove the best choice. For this reason elements with a linear variation of strain like TRIM 6 and TET 10 were originally evolved and followed up with the quadratic strain elements TRIM 15, TRIA 4 (two-dimensional) and TET 20, TEA 8 (three-dimensional) of ref. 2. However, all these elements are characterised by straight edges and necessitate a polygonisation or polyhedrisation in the idealisation process. This may not be critical in many problems, but is sometimes of doubtful validity in the immediate neighbourhood of a curved boundary, where stress concentrations are most pronounced. To overcome this difficulty with a significant (local) increase of elements does not always yield the most economical and technically satisfactory solution. Moreover, there arises another inevitable shortcoming when dealing with TRIM and TET elements with a linear or quadratic variation of strain. Indeed, while TRIM 3 and TET 4 elements permit a very elegant extension into the realm of large displacements, this is not possible for the higher order TRIM and TET elements. This is simply due to the fact that TRIM 3 and TET 4 elements, by virtue of their specification, always remain straight under any magnitude of strain, but this is not so for the triangular and tetrahedron elements of higher sophistication.

Stress Concentrations from Single-Filament Failures in Composite Materials

1969 ◽  
Vol 39 (7) ◽  
pp. 618-626 ◽  
Author(s):  
Peter Van Dyke ◽  
John M. Hedgepeth
Keyword(s):  
Matrix Material ◽  
Three Dimensional ◽  
Plastic Behavior ◽  
Two Dimensional ◽  
Failure Model ◽  
Stress Concentrations ◽  
Bond Failure ◽  
Single Filament ◽  
Concentration Problem ◽  
The Matrix

The solution of the two-dimensional, elastic, multiple-filament-failure stress concentration problem led to the treatment of three-dimensional, elastic failure models and a two-dimensional, plastic failure model where an ideally plastic behavior of the matrix material adjacent to a broken filament was assumed. Another plastic behavior is proposed wherein the bond between the broken filament and the adjacent matrix material fails completely after reaching a prescribed stress level. This failure formulation is applied to five- and seven-element-width models as well as to the infinite element case. Both the bond failure and matrix yield models are then extended to the three-dimensional cases with both square and hexagonal element configurations.

On the Correspondence between Two- and Three-Dimensional Elastic Solutions of Crack Problems

2016 ◽  
Vol 713 ◽  
pp. 18-21 ◽  
Author(s):  
Andrei G. Kotousov ◽  
Zhuang He ◽  
Aditya Khanna
Keyword(s):  
Scale Effects ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Governing Equations ◽  
Singular Stress ◽  
Linear Elastic ◽  
Crack Problems ◽  
Elastic Fracture ◽  
Stress States

The classical two-dimensional solutions of the theory of elasticity provide a framework of Linear Elastic Fracture Mechanics. However, these solutions, in fact, are approximations despite that the corresponding governing equations of the plane theories of elasticity are solved exactly. This paper aims to elucidate the main differences between the approximate (two-dimensional) and exact (three-dimensional) elastic solutions of crack problems. The latter demonstrates many interesting features, which cannot be analysed within the plane theories of elasticity. These features include the presence of scale effects of deterministic nature, the existence of new singular stress states and fracture modes. Furthermore, the deformation and stress fields near the tip of the crack is essentially three-dimensional and do not follow plane stress or plane strain simplifications. Moreover, in certain situations the two-dimensional solutions can provide misleading results; and several characteristic examples are outlined in this paper.

Tensile Fracture Loci for Brittle Materials Containing Spherical Voids

1986 ◽  
Vol 108 (3) ◽  
pp. 222-229 ◽  
Author(s):  
M. C. Shaw ◽  
J. P. Avery
Keyword(s):  
Three Dimensional ◽  
Brittle Materials ◽  
Dimensional Case ◽  
Tensile Fracture ◽  
Detailed Result ◽  
Three Dimension ◽  
Two Dimensional ◽  
Stress Criterion ◽  
Hydrostatic Tension ◽  
Spherical Voids

When very brittle materials are subjected to a complex state of stress they fail by maximum intensified tensile stress criterion first introduced by Griffith [1]. Nominal applied stresses are intensified by defects present in all real materials. It appears that defects controlling the strength of brittle materials are of two types—open ones characterized by circular voids found in sintered materials such as tungsten carbide and thin, essentially closed ones found in brittle polyphase rock such as granite. This paper is concerned with the extension of a very simple two dimensional theory for circular voids [3] to the three dimensional case involving spherical voids. While the fracture locus for the two dimensional case represents a conservative approximation sufficient for most engineering applications, the three dimension solution is necessary to give detailed result for cases involving near hydrostatic tension or compression.

Three-dimensional, elastic stress distribution in end-milled, keyed connections

1983 ◽  
Vol 18 (2) ◽  
pp. 143-149 ◽  
Author(s):  
H Fessler ◽  
M Eissa
Keyword(s):  
Elastic Stress ◽  
Three Dimensional ◽  
Empirical Equations ◽  
Two Dimensional ◽  
Stress Concentrations ◽  
Axial Distribution ◽  
Simple Method ◽  
Torque Transmission ◽  
Applied Torque ◽  
Stress Models

Three- and two-dimensional, photoelastic, frozen-stress models of Standard metric and inch keyed connections have been loaded in torsion. Results from models with three different key lengths are presented here and related to the axial distribution of torque transmission. Empirical equations for the elastic stress concentrations in the prismatic part of key and keyway at the positions of contact between key and shaft have been derived for any likely width, thickness, and length of key, keyway fillet size, and applied torque. A simple method of eliminating stress concentrations in the keyway end is described.

A Two-Property Yield, Failure (Fracture) Criterion for Homogeneous, Isotropic Materials

2004 ◽  
Vol 126 (1) ◽  
pp. 45-52 ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Experimental Data ◽  
Failure Criterion ◽  
Fracture Criterion ◽  
Three Dimensional ◽  
Fracture Type ◽  
Elastic Materials ◽  
Dimensional Theory ◽  
Stress States ◽  
Ductile Behavior ◽  
Two Parameter

A critical review is given of the various historical attempts to formulate a general, three-dimensional theory of failure for broad classes of homogeneous, isotropic elastic materials. Following that, a recently developed two-parameter yield/failure criterion is compared with the historical efforts and it is further interpreted and extended. Specifically, the yield/failure criterion is combined with a fracture restriction that places limits on certain tensile stress states, without involving any additional parameters. An evaluation is conducted using available experimental data obtained from a variety of materials types. The two materials parameters are given a primary designation as yield type properties over a specified range of ductile behavior, and as failure or fracture type properties over the complementary brittle range.

Nonlinear Through-Thickness Behavior of a Toroidal Shell Using 2-D Finite Elements

2001 ◽  
Author(s):  
James M. Greer ◽  
Anthony N. Palazotto
Keyword(s):  
Finite Elements ◽  
Transverse Shear ◽  
Strain Energy ◽  
Shell Thickness ◽  
Three Dimensional ◽  
Toroidal Shell ◽  
Two Dimensional ◽  
Qualitative And Quantitative ◽  
Pipe Elbow ◽  
State Of Stress

Abstract The use of two-dimensional shell finite elements is explored for finding the three-dimensional state of stress in a toroidal shell. The torus under study represents a 90-degree pipe elbow with a pressure load on a portion of its surface. Layer-wise polynomials are used to represent the transverse shear and normal stretch deformations in the shell. These functions are chosen such that displacements and stresses (but not strains) are continuous at the ply interfaces. Both isotropic and composite (cross-ply) versions of the shell are investigated, and the thicknesses of each are varied to see the effect on through-thickness behavior. Significant qualitative and quantitative differences in these behaviors are observed, particularly in the important direct through-thickness (peeling) stress. The contribution of the transverse deformations to strain energy is investigated and, in most of the shells studied, the thickness stretch component is found to be a greater contributor to strain energy than the transverse shear, though the transverse shear contribution is seen to vary more dramatically with changes in shell thickness.

Some Applications of Photoelasticity in Turbine-Generator Design

1939 ◽  
Vol 6 (4) ◽  
pp. A151-A155
Author(s):  
M. Hetényi
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Stress Concentrations ◽  
Turbine Generator ◽  
Transverse Grooves ◽  
Generator Design

Abstract This paper deals with three particular problems in turbine-generator design, namely (a) stress concentrations in transverse grooves of two-pole generator rotors, (b) centrifugal stresses in rotors, and (c) stress concentrations in T-heads (bolts and blade fastenings). The connection between these problems is only their mutual sphere of application and the fact that in each case the solution has been obtained by photoelasticity. The problems have been investigated as two-dimensional ones, the first and third being analyzed by the standard photoelastic technique, while in the second problem the fundamentally three-dimensional method of “freezing” the stresses in bakelite samples was applied.

Notched Fatigue Behavior under Multiaxial Stress States

2014 ◽  
Vol 891-892 ◽  
pp. 185-190
Author(s):  
Nicholas R. Gates ◽  
Ali Fatemi ◽  
Darrell F. Socie ◽  
Nam Phan
Keyword(s):  
Crack Growth ◽  
Fatigue Behavior ◽  
Damage Analysis ◽  
Stress Concentrations ◽  
Axial Torsion ◽  
State Of Stress ◽  
Notch Geometry ◽  
Fatigue Damage Analysis ◽  
Evolution Stage ◽  
Stress States

Most engineering components and structures contain stress concentrations, such as notches. The state of stress at such concentrations is typically multiaxial due to the notch geometry, and/or multiaxiality of the loading. Significant portions of the fatigue life of notched members are usually spent in crack initiation (crack formation and microscopic growth) and macroscopic crack growth. Synergistic complexity of combined stress and stress concentration has been evaluated in a limited number of studies. Available experimental evidence suggests the current life estimation and fatigue damage analysis techniques commonly used may not be capable of accurate predictions for such complex and yet highly practical conditions. This paper investigates notched fatigue behavior under multiaxial loads using aluminum alloys. Many effects involved in such loading conditions are included. These include the effects of stress state (axial, torsion, combined axial-torsion), geometry condition (smooth versus notched), and damage evolution stage (nucleation and micro-crack growth versus long crack growth).

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