The Generalized Lame´ Problem—Part I: Coupled Poromechanical Solutions

2004 ◽  
Vol 71 (2) ◽  
pp. 168-179 ◽  
Author(s):  
Mazen Y. Kanj ◽  
Younane N. Abousleiman
Keyword(s):  
Plane Strain ◽  
Hollow Cylinder ◽  
Closed Form Solution ◽  
Confining Pressure ◽  
Special Problem ◽  
Form Solution ◽  
Geomechanical Properties ◽  
Lamé Problem ◽  
Loading Modes ◽  
Generalized Plane Strain

Cylindrical geometries are known to present special problem simulation capabilities in engineering design. (For example, solid and hollow cylinder tests are routinely studied in soil and rock mechanics to gain insights into the geomechanical properties and to assess the stability of boreholes and cylindrical openings in the project design geomedia.) This paper identifies a unified and universal solution to all the three recognized right-cylindrical problem objects in poromechanics. A closed-form solution to the problem of the finite, homogeneous, isotropic, fully saturated, thick-walled hollow cylinder subjected to various loading modes is readily presented and described. The assumed loading modes encompass arbitrary temporal functions of uniformly distributed inner/outer pore pressure, inner/outer confining pressure, inner/outer deviatoric stress, and end axial compaction or extension. The time-dependent response derivations are outlined within the frameworks of the Biot’s theory of linear poroelasticity and facilitated by the governing generalized plane-strain (GPS) principle. The (as presented) solution is shown to converge asymptotically to those of the two essential problem setups in geomechanics: the finite solid cylinder and the borehole core in an infinite medium. As such, a complete/explicit solution to a generalized statement of the Lame´ problem is presented. The solution utilizes a fairly simple loading decomposition scheme which leads to two basic problem forms: a generalized poroelastic axisymmetric problem and a generalized, plane-strain, poroelastic deviatoric problem.

On the Application of Stress Triaxiality Formula for Plane Strain Fracture Testing

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
Yuanli Bai ◽  
Xiaoqing Teng ◽  
Tomasz Wierzbicki
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Fracture Strain ◽  
Closed Form Solution ◽  
Stress Triaxiality ◽  
Form Solution ◽  
Fracture Locus ◽  
Strain Fracture ◽  
Lode Angle ◽  
Angle Parameter

Theoretical and experimental studies have shown that stress triaxiality is the key parameter controlling the magnitude of the fracture strain. Smooth and notched round bar specimens are mostly often used to quantify the effect of stress triaxiality on ductile fracture strain. There is a mounting evidence (Bai and Wierzbicki, 2008, “A New Model of Metal Plasticity and Fracture With Pressure and Lode Dependence,” Int. J. Plast., 24(6), pp. 1071–1096) that, in addition to the stress triaxiality, the normalized third deviatoric stress invariant (equivalent to the Lode angle parameter) should also be included in characterization of ductile fracture. The calibration using round notched bars covers only a small range of possible stress states. Plane strain fracture tests provide additional important data. Following Bridgman’s stress analysis inside the necking of a plane strain specimen, a closed-form solution is derived for the stress triaxiality inside the notch of a flat-grooved plane strain specimen. The newly derived formula is verified by finite element simulations. The range of stress triaxiality in round notched bars and flat-grooved specimens is similar, but the values of the Lode angle parameter are different. These two groups of tests are therefore very useful in constructing a general 3D fracture locus. The results of experiments and numerical simulations on 1045 and DH36 steels have proved the applicability of the closed-form solution and have demonstrated the effect of the Lode angle parameter on the fracture locus.

The Generalized Lame´ Problem—Part II: Applications in Poromechanics

2004 ◽  
Vol 71 (2) ◽  
pp. 180-189 ◽  
Author(s):  
Younane N. Abousleiman ◽  
Mazen Y. Kanj
Keyword(s):  
Plane Strain ◽  
Laboratory Testing ◽  
Infinite Medium ◽  
Time Dependent ◽  
Solid Cylinder ◽  
Testing Procedures ◽  
Lamé Problem ◽  
Essential Problem ◽  
Generalized Plane Strain ◽  
Borehole Core

The cylinder is the geometry most widely used in laboratory testing procedures for rocks and other geomaterials. This paper applies a unified and universal Lame´ solution to all the three recognized right-cylindrical problems in poromechanics. As such, the solution of the hollow-cylinder features itself converging asymptotically to the exact values predicted by the solutions of the two other essential problem setups in geomechanics; namely, the finite solid cylinder case and the borehole core in an infinite medium. The time-dependent response derivations were “scripted” within the frameworks of the Biot’s theory of linear poroelasticity and facilitated by the governing generalized plane-strain (GPS) principle.

Plane-Strain Bending With Isotropic Strain Hardening at Large Strains

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Sergei Alexandrov ◽  
Yeong-Maw Hwang
Keyword(s):  
Strain Hardening ◽  
Plane Strain ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Form Solution ◽  
Isotropic Hardening ◽  
Large Strains ◽  
Rigid Plastic ◽  
Elastic Plastic ◽  
Hardening Law

Finite deformation elastic-plastic analysis of plane-strain pure bending of a strain hardening sheet is presented. The general closed-form solution is proposed for an arbitrary isotropic hardening law assuming that the material is incompressible. Explicit relations are given for most popular conventional laws. The stage of unloading is included in the analysis to investigate the distribution of residual stresses and springback. The paper emphasizes the method of solution and the general qualitative features of elastic-plastic solutions rather than the study of the bending process for a specific material. In particular, it is shown that rigid-plastic solutions can be used to predict the bending moment at sufficiently large strains.

scholarly journals The Part-Through Surface Crack in an Elastic Plate

1972 ◽  
Vol 39 (1) ◽  
pp. 185-194 ◽  
Author(s):  
J. R. Rice ◽  
N. Levy
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Plane Strain ◽  
Surface Crack ◽  
Unit Length ◽  
Mathematical Formulation ◽  
Closed Form Solution ◽  
Form Solution ◽  
Bending Loads

An elastic analysis is presented for the tensile stretching and bending of a plate containing a surface crack penetrating part-through the thickness, Fig. 1. The treatment is approximate, in that the two-dimensional generalized plane stress and Kirchhoff-Poisson plate bending theories are employed, with the part-through cracked section represented as a continuous line spring. The spring has both stretching and bending resistance, its compliance coefficients being chosen to match those of an edge cracked strip in plane strain. The mathematical formulation reduces finally to two-coupled integral equations for the thickness averaged force and moment per unit length along the cracked section. These are solved numerically for the case of a semi-elliptical part-through crack, with results compared to a simple but approximate closed-form solution. Extensive results are given for the stress intensity factor at the midpoint of the part-through crack for both remote tensile and bending loads on the plate. These results indicate that the stress-intensity factor is substantially lower, in general, than for a similarly loaded strip in plane strain with a crack of the same depth.

Sign in / Sign up

Export Citation Format

Share Document