Blunting of a Plane Strain Crack Tip Into a Shape With Vertices

1977 ◽  
Vol 99 (4) ◽  
pp. 290-297 ◽  
Author(s):  
Robert M. McMeeking
Keyword(s):  
Stress Distribution ◽  
Plane Strain ◽  
Crack Tip ◽  
The Body ◽  
Rigid Plastic ◽  
Hardening Material ◽  
Crack Tips ◽  
Stress And Deformation ◽  
Material Points ◽  
Growth Of Voids

When monotonically increasing tensile opening loads are applied to a cracked, plane strain, elastic-plastic body, the crack tip will blunt until fracture occurs. At least within the rigid-plastic model for nonhardening material, the shape of the blunted tip is not unique. The blunted tip shape may have two or more sharp corners, or be smoothly curved. When the shape involves corners, the opening is predominantly accommodated by shearing of the material at the corners. This shearing transports material from the interior of the body onto the crack surface. In contrast, the smoothly blunted crack tip involves no such transfer of material points from the interior. However, the smoothly blunted crack, which was originally sharp, involves infinite strains on the crack tip surface. The crack with corners on the tip has large but finite strains on the crack tip surface. The stress and deformation field in front of a crack with two corners and with three corners on the tip, as calculated using the slip line method, is presented for the nonhardening, fully plastic, deeply cracked, double edge-notched thick panel. As in the case of the smoothly blunted crack tip, the elevated stress between the crack tips cannot be maintained very close to the crack tip, due to a lack of constraint. The stress distribution in the case of the crack tip with vertices on it differs from that of the smoothly blunted crack tip case. In particular, immediately in front of the crack tip with three corners, the stress is higher than that immediately in front of the smoothly blunted crack tip. An approximation for a power law hardening material indicates that the maximum stresses near the blunted crack tip is much the same for a crack with vertices on the tip as for a smoothly blunted crack tip. The details of the stress distribution, though, will depend on the mechanism by which the crack blunts. These results for stress and strain and some calculations of the growth of voids near the crack tips indicate the same fracture process could lead to different fracture toughnesses, depending on the type of mechanism by which the crack blunts.

Stress Distribution Near Internal Crack Tips for Longitudinal Shear Problems

1965 ◽  
Vol 32 (1) ◽  
pp. 51-58 ◽  
Author(s):  
G. C. Sih
Keyword(s):  
Stress Distribution ◽  
Stress Singularity ◽  
Crack Tip ◽  
Longitudinal Shear ◽  
Complex Stress ◽  
Internal Crack ◽  
Vertex Angle ◽  
Crack Tips ◽  
Bending Problems ◽  
Half Power

A method is developed for finding the stress distribution in a cracked body under longitudinal shear and applied to solve a number of problems. Stress solutions are obtained in closed form and discussed in connection with the Griffith-Irwin theory of fracture. The results indicate that current fracture-mechanics theories may be applied directly to longitudinal shear problems. More specifically, the character of the stress distribution near the vertex of a sector cylinder in shear is examined. The inverse half-power law of the stress singularity at a crack tip may be verified by taking a vertex angle of 2π. In addition, crack-tip, stress-intensity factors are defined and evaluated from a complex stress function in a manner similar to those previously given for extension and plate-bending problems. Results of such studies clarified the behavior of branched cracks and other crack systems of interest.

scholarly journals Plastic flow of isotropic rigid body at uniform deformation

2021 ◽  
Vol 66 (2) ◽  
pp. 186-193
Author(s):  
O. M. Dyakonov
Keyword(s):  
Plastic Flow ◽  
Shape Change ◽  
Plane Deformation ◽  
Wire Drawing ◽  
The Body ◽  
Shear Stresses ◽  
Rigid Plastic ◽  
Rectangular Profile ◽  
Deformation Force ◽  
Material Points

The mathematical analysis of plastic flow processes under uniform plane, axisymmetric and volumetric deformation is carried out. The analysis is based on the external shape change of the body, which determines the movement of material points. It is shown that the plastic flow of an isotropic rigid-plastic body under plane deformation obeys the hyperbolic law, and for axisymmetric and volumetric deformations – the inverse square law. Spatial-geometric expressions of these laws made it possible to reveal and explain in a new way the physical essence of plastic shear. It is proved that the stressed state of a body under uniform tension-compression deformation is complex and cannot be defined as “linear”. The normal stress, which coincides with the direction of the resulting deformation force, is not the main one, since in the areas perpendicular to this direction, the shear stresses are not equal to zero. Examples of solving technological problems are given: extrusion of cylindrical billets and wire drawing, rolling of a wide strip of rectangular profile. It is shown that the problems of determining the stress-strain state of isotropic rigid-plastic bodies along the known trajectories of movement of material points are statically definable.

scholarly journals Stress and deformation analysis of gob-side pre-backfill driving procedure of longwall mining: a case study

2021 ◽  
Author(s):  
Rui Wu ◽  
Penghui Zhang ◽  
Pinnaduwa H. S. W. Kulatilake ◽  
Hao Luo ◽  
Qingyuan He
Keyword(s):  
Rock Mass ◽  
Stress Distribution ◽  
Coal Seam ◽  
Abutment Pressure ◽  
Longwall Mining ◽  
Vertical Stress ◽  
Floor Level ◽  
Working Face ◽  
Floor Heave ◽  
Stress And Deformation

AbstractAt present, non-pillar entry protection in longwall mining is mainly achieved through either the gob-side entry retaining (GER) procedure or the gob-side entry driving (GED) procedure. The GER procedure leads to difficulties in maintaining the roadway in mining both the previous and current panels. A narrow coal pillar about 5–7 m must be left in the GED procedure; therefore, it causes permanent loss of some coal. The gob-side pre-backfill driving (GPD) procedure effectively removes the wasting of coal resources that exists in the GED procedure and finds an alternative way to handle the roadway maintenance problem that exists in the GER procedure. The FLAC3D software was used to numerically investigate the stress and deformation distributions and failure of the rock mass surrounding the previous and current panel roadways during each stage of the GPD procedure which requires "twice excavation and mining". The results show that the stress distribution is slightly asymmetric around the previous panel roadway after the “primary excavation”. The stronger and stiffer backfill compared to the coal turned out to be the main bearing body of the previous panel roadway during the "primary mining". The highest vertical stresses of 32.6 and 23.1 MPa, compared to the in-situ stress of 10.5 MPa, appeared in the backfill wall and coal seam, respectively. After the "primary mining", the peak vertical stress under the coal seam at the floor level was slightly higher (18.1 MPa) than that under the backfill (17.8 MPa). After the "secondary excavation", the peak vertical stress under the coal seam at the floor level was slightly lower (18.7 MPa) than that under the backfill (19.8 MPa); the maximum floor heave and maximum roof sag of the current panel roadway were 252.9 and 322.1 mm, respectively. During the "secondary mining", the stress distribution in the rock mass surrounding the current panel roadway was mainly affected by the superposition of the front abutment pressure from the current panel and the side abutment pressure from the previous panel. The floor heave of the current panel roadway reached a maximum of 321.8 mm at 5 m ahead of the working face; the roof sag increased to 828.4 mm at the working face. The peak abutment pressure appeared alternately in the backfill and the coal seam during the whole procedure of "twice excavation and mining" of the GPD procedure. The backfill provided strong bearing capacity during all stages of the GPD procedure and exhibited reliable support for the roadway. The results provide scientific insight for engineering practice of the GPD procedure.

scholarly journals Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 145
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Yeong-Maw Hwang
Keyword(s):  
Plane Strain ◽  
Yield Criterion ◽  
Large Strains ◽  
Rigid Plastic ◽  
Plastic Properties ◽  
Elastic Plastic ◽  
Starting Point ◽  
The Difference ◽  
Inside Radius ◽  
Elastic Unloading

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

scholarly journals An Unstructured Finite Volume Method for Incompressible Flows With Complex Immersed Boundaries

2009 ◽  
Author(s):  
Lin Sun ◽  
Sanjay R. Mathur ◽  
Jayathi Y. Murthy
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Incompressible Flows ◽  
Simple Algorithm ◽  
Flow Region ◽  
Solid Material ◽  
The Body ◽  
Immersed Boundary ◽  
Material Points ◽  
Volume Method

A numerical method is developed for solving the 3D, unsteady, incompressible flows with immersed moving solids of arbitrary geometrical complexity. A co-located (non-staggered) finite volume method is employed to solve the Navier-Stokes governing equations for flow region using arbitrary convex polyhedral meshes. The solid region is represented by a set of material points with known position and velocity. Faces in the flow region located in the immediate vicinity of the solid body are marked as immersed boundary (IB) faces. At every instant in time, the influence of the body on the flow is accounted for by reconstructing implicitly the velocity the IB faces from a stencil of fluid cells and solid material points. Specific numerical issues related to the non-staggered formulation are addressed, including the specification of face mass fluxes, and corrections to the continuity equation to ensure overall mass balance. Incorporation of this immersed boundary technique within the framework of the SIMPLE algorithm is described. Canonical test cases of laminar flow around stationary and moving spheres and cylinders are used to verify the implementation. Mesh convergence tests are carried out. The simulation results are shown to agree well with experiments for the case of micro-cantilevers vibrating in a viscous fluid.

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková
Keyword(s):  
Stress Distribution ◽  
Fracture Mechanics ◽  
Boundary Conditions ◽  
Displacement Field ◽  
Singular Term ◽  
Crack Tip ◽  
Higher Order ◽  
Precise Description ◽  
Accurate Knowledge ◽  
Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

Sign in / Sign up

Export Citation Format

Share Document