The Stress Field Due to a Line Load Acting on a Half Space of a Power-Law Hardening Material (A Comparison Between Plane Strain and Plane Stress)

1973 ◽  
Vol 40 (1) ◽  
pp. 288-290 ◽  
Author(s):  
C. Atkinson
Keyword(s):  
Exact Solution ◽  
Stress Field ◽  
Plane Strain ◽  
Half Space ◽  
Power Law ◽  
Elastic Material ◽  
Plane Stress ◽  
Strain Fields ◽  
Line Load ◽  
Hardening Material

The exact solution is given for a line load acting on a half space of a power-law elastic material under conditions of plane stress. This solution is compared with the corresponding solution under plane-strain conditions; see Aruliunian [1]. A marked difference is found between the plane-stress and plane-strain fields for different values of the hardening exponent.

Influence of Porosity on Plane Strain Tensile Crack-Tip Stress Fields in Elastic-Plastic Materials: Part I

1992 ◽  
Vol 59 (3) ◽  
pp. 559-567 ◽  
Author(s):  
W. J. Drugan ◽  
Y. Miao
Keyword(s):  
Stress Field ◽  
Closed Form ◽  
Normal Stress ◽  
Plane Strain ◽  
Plane Stress ◽  
Entire Range ◽  
Crack Tip ◽  
Constant Stress ◽  
Tensile Crack ◽  
Stress Discontinuity

We perform an analytical first study of the influence of a uniform porosity distribution, for the entire range of porosity level, on the stress field near a plane strain tensile crack tip in ductile material. Such uniform porosity distributions (approximately) arise in incompletely sintered or previously deformed (e.g., during processing) ductile metals and alloys. The elastic-plastic Gurson-Tvergaard constitutive formulation is employed. This model has a sound micromechanical basis, and has been shown to agree well with detailed numerical finite element solutions of, and with experiments on, voided materials. To facilitate closed-form analytical results to the extent possible, we treat nonhardening material with constant, uniform porosity. We show that the assumption of singular plastic strain in the limit as the crack tip is approached renders the governing equations statically determinate with two permissible types of near-tip angular sector: one with constant Cartesian components of stress (“constant stress”); and one with radial stress characteristics (“generalized centered fan”). The former admits an exact asymptotic closed-form stress field representation, and although we prove the latter does not, we derive a highly accurate closed-form approximate representation. We show that complete near-tip solutions can be constructed from these two sector types for the entire range of porosity. These solutions are comprised of three asymptotic sector configurations: (i) “generalized Prandtlfield”for low porosities (0 ≤ f ≤ .02979), similar to the plane strain Prandtl field of fully dense materials, with a fully continuous stress field but sector extents that vary with porosity; (ii) “plane-stress-like field” for intermediate porosities (.02979 < f < .12029), resembling the plane stress solution for fully dense materials, with a ray of radial normal stress discontinuity but sector extents that vary with porosity; (iii) two constant stress sectors for the remaining high porosity range, with a ray of radial normal stress discontinuity and fixed sector extents. Among several interesting features, the solutions show that increasing porosity causes significant modification of the angular variation of stress components, particularly for a range of angles ahead of the crack tip, while also causing a drastic reduction in maximum hydrostatic stress level.

Cavities vis-a`-vis Rigid Inclusions and Some Related General Results in Plane Elasticity

1989 ◽  
Vol 56 (4) ◽  
pp. 786-790 ◽  
Author(s):  
John Dundurs
Keyword(s):  
Stress Field ◽  
Plane Strain ◽  
Elastic Constants ◽  
Plane Stress ◽  
Poisson Ratio ◽  
Plane Elasticity ◽  
By Products

There is a strange feature of plane elasticity that seems to have gone unnoticed: The stresses in a body that contains rigid inclusions and is loaded by specified surface tractions depend on the Poisson ratio of the material. If the Poisson ratio in this stress field is set equal to +1 for plane strain, or +∞ for plane stress, the rigid inclusions become cavities for elastic constants within the physical range. The paper pursues this circumstance, and in doing so also produces several useful by-products that are connected with the stretching and curvature change of a boundary.

Total Deformation, Plane-Strain Contact Analysis of Macroscopically Homogeneous, Compositionally Graded Materials With Constant Power-Law Strain Hardening

1997 ◽  
Vol 64 (4) ◽  
pp. 853-860 ◽  
Author(s):  
A. E. Giannakopoulos
Keyword(s):  
Strain Hardening ◽  
Plane Strain ◽  
Power Law ◽  
Protective Coatings ◽  
Contact Analysis ◽  
Engineering Structures ◽  
Graded Materials ◽  
Line Load ◽  
Structural Applications ◽  
Graded Material

Plane-strain contact analysis is presented for compositionally graded materials with power-law strain hardening. The half-space, y≤0, is modeled as an incompressible, nonlinear elastic material. The effective stress, σe, and the effective total strain, εe, are related through a power-law model, σe=K0εeμ;0<μ≤min(1,(1+k)). The material property K0 changes with depth, |y|, as K0=A|y|k;A>0,0≤|k|<1. This material description attempts to capture some features of the plane-strain indentation of elastoplastic or steady-state creeping materials that show monotonically increasing or decreasing hardness with depth. The analysis starts with the solution for the normal line load (Flamant’s problem) and continues with the rigid, frictionless, flat-strip problem. Finally, the general solution of normal indentation of graded material by a convex, symmetric, rigid, and frictionless two-dimensional punch is given. Applications of the present results range from surface treatments of engineering structures, protective coatings for corrosion and fretting fatigue, settling of beam type foundations in the context of soil and rock mechanics, to bioengineering as well as structural applications such as contact of railroad tracks.

Combined Loading of a Fully Plastic Ligament Ahead of an Edge-Crack

1986 ◽  
Vol 53 (2) ◽  
pp. 271-277 ◽  
Author(s):  
C. F. Shih ◽  
J. W. Hutchinson
Keyword(s):  
Plane Strain ◽  
Half Space ◽  
Power Law ◽  
Edge Crack ◽  
Deformation Theory ◽  
Approximate Solutions ◽  
Free Edge ◽  
Numerical Results ◽  
Combined Loading

Complete, accurate numerical results are given for the solution to the problem of a semi-infinite crack aligned perpendicularly to the free-edge of a semi-infinite half space in which the ligament is subject to arbitrary combinations of bending and tension or compression. The material is an incompressible, pure power-law deformation theory solid. Conditions of plane strain are assumed. Approximate solutions are proposed for predominantly bending loadings and also for predominantly stretching loadings.

Ductile Tearing Instability of a Center-Cracked Panel of an Elastic-Plastic Strain Hardening Material

1981 ◽  
Vol 103 (1) ◽  
pp. 46-54 ◽  
Author(s):  
Akram Zahoor ◽  
Paul C. Paris
Keyword(s):  
Plastic Strain ◽  
Strain Hardening ◽  
Plane Strain ◽  
Plane Stress ◽  
Hardening Material ◽  
Ductile Tearing ◽  
Elastic Plastic ◽  
Linear Elastic ◽  
Tearing Instability ◽  
Crack Stability

An analysis for crack instability in an elastic-plastic strain hardening material is presented which utilizes the J-integral and the tearing modulus parameter, T. A center-cracked panel of finite dimensions with Ramberg-Osgood material representation is analyzed for plane stress as well as plane strain. The analysis is applicable in the entire range of elastic-plastic loading from linear elastic to full yield. Crack instability is strongly influenced by the elastic compliance of the system, the conditions of plane stress or plane strain, and the hardening characteristics of the material. Numerical results indicate that if crack stability is ensured in a plane strain situation, then under the same circumstances a geometrically identical but plane stress panel will be stable.

The Singularity at the Apex of a Rigid Wedge Embedded in a Nonlinear Material

1988 ◽  
Vol 55 (2) ◽  
pp. 361-364 ◽  
Author(s):  
J. M. Duva
Keyword(s):  
Plane Strain ◽  
Power Law ◽  
Nonlinear Material ◽  
Stress And Strain ◽  
Singular Behavior ◽  
Strain Fields

The singular behavior of the stress and strain fields at the apex of a square rigid wedge embedded in a nonlinear material under plane-strain conditions is described. Both a power law and a bilinear law for the nonlinear material are considered.

The Buckling of an Elastic Layer Bonded to an Elastic Substrate in Plane Strain

1994 ◽  
Vol 61 (2) ◽  
pp. 231-235 ◽  
Author(s):  
T. W. Shield ◽  
K. S. Kim ◽  
R. T. Shield
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Plane Strain ◽  
Half Space ◽  
Elastic Layer ◽  
Beam Theory ◽  
Elastic Half Space ◽  
Theory Model ◽  
Beam Model ◽  
Elastic Substrate

The solution for buckling of a stiff elastic layer bonded to an elastic half-space under a transverse compressive plane strain is presented. The results are compared to an approximate solution that models the layer using beam theory. This comparison shows that the beam theory model is adequate until the buckling strain exceeds three percent, which occurs for modulus ratios less than 100. In these cases the beam theory predicts a larger buckling strain than the exact solution. In all cases the wavelength of the buckled shape is accurately predicted by the beam model. A buckling experiment is described and a discussion of buckling-induced delamination is given.

Transverse Stress for the Elastic-Plastic Plane Strain Problems

2011 ◽  
Vol 121-126 ◽  
pp. 550-553
Author(s):  
Chang Lu Tian ◽  
Shu Li Wang
Keyword(s):  
Plane Strain ◽  
Power Law ◽  
Transverse Stress ◽  
Hardening Material ◽  
Elastic Plastic

A relation to determine the transverse stress in terms of in-plane stresses for elastic-plastic plane strain problems in a power-law hardening material is presented. The results might prove useful in the elastic-plastic analyses of plane strain problems

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