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 influence of crack size on the ductile-brittle transition

1988 ◽  
Vol 415 (1848) ◽  
pp. 197-226 ◽  
Keyword(s):  
Boundary Layer ◽  
Fracture Toughness ◽  
Transition Temperature ◽  
Plane Strain ◽  
Crack Size ◽  
Specimen Geometry ◽  
External Loading ◽  
Stress And Strain ◽  
Strain Fields ◽  
Penny Shaped Crack

Simple criteria for brittle and ductile crack extension are applied to the stress and strain fields adjacent to the tip of a crack. They are applied at a specified distance from the crack tip, which should be related to the material’s microstructure. The basic approach is to examine each criterion and find which is satisfied first, as the external loading is increased; the predicted fracture is classified either brittle or ductile accordingly. The stress and strain fields depend upon temperature, principally through the variation of flow stress σ 0 with temperature and, to avoid excessive computation, a constitutive relation is constructed which allows stresses and strains both to be scaled in terms of σ 0 , so that major computations need to be done only at a reference temperature, for a range of applied loads. For any given crack configuration, the result of the calculation is a theoretical prediction of fracture toughness as a function of temperature. At low temperatures, the fracture toughness is low and rises rapidly with temperature, corresponding to satisfaction of the criterion for brittle failure. Above a transition temperature, T T , the ductile criterion is satisfied first, and the toughness variation thereafter falls slowly as temperature increases, corresponding to failure ‘on the upper shelf’. Both the absolute level of the toughness at a given temperature and the transition temperature T T are sensitive to crack size as well as specimen geometry. Although this is self-evident for cracks of microstructural dimensions, the striking feature of this work is the prediction that substantial sensitivity to size and geometry may well be displayed for cracks as large as 1 cm in materials of significance for major engineering structures. Generally, toughness increases and transition temperature decreases as crack size decreases, but these beneficial effects can be nullified by stress triaxiality. Detailed calculations are performed for a buried crack and an edge crack under conditions of plane strain and for a penny-shaped crack loaded axisymmetrically. The plane strain calculations are supplemented by ‘boundary layer’ calculations, in which the effect of specimen geometry appears through a single parameter. The close agreement of the ‘boundary layer’ calculations with the full specimen calculations offers the prospect of a simple characterization of specimen geometry and loading, without the need for geometry-specific computations. The calculations that are reported are, of course, based upon a particular model, chosen in part for com­putational convenience. Thus, their status is that they display possible trends which may be considered to merit further investigation, both theoretical and experimental.

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.

scholarly journals The Penny-Shaped Crack and the Plane Strain Crack in an Infinite Body of Power-Law Material

1981 ◽  
Vol 48 (4) ◽  
pp. 830-840 ◽  
Author(s):  
M. Y. He ◽  
J. W. Hutchinson
Keyword(s):  
Plane Strain ◽  
Power Law ◽  
Deformation Theory ◽  
Small Strain ◽  
Stress And Strain ◽  
Infinite Body ◽  
Penny Shaped Crack ◽  
Shaped Crack ◽  
Remote Stress ◽  
Plane Strain Crack

A study is carried out of the problem of a penny-shaped crack in an infinite body of power-law material subject to general remote axisymmetric stressing conditions. The plane strain version of the problem is also examined. The material is incompressible and is characterized by small strain deformation theory with a pure power relation between stress and strain. The solutions presented also apply to power-law creeping materials and to a class of strain-rate sensitive hardening materials. Both numerical and analytical procedures are employed to obtain the main results. A perturbation solution obtained by expanding about the trivial state in which the stress is everywhere parallel to the crack leads to simple formulas which are highly accurate even when the remote stress is perpendicular to the crack.

The Singularity Strength at the Apex of a Wedge Undergoing Finite Deformations

1990 ◽  
Vol 57 (3) ◽  
pp. 577-580 ◽  
Author(s):  
J. M. Duva
Keyword(s):  
Boundary Conditions ◽  
Plane Strain ◽  
Finite Deformation ◽  
Finite Deformations ◽  
The Other ◽  
Nonlinear Material ◽  
Singular Behavior ◽  
Arbitrary Size

Herein we establish general formulae for characterizing the singular behavior at the apex of a wedge of nonlinear material of arbitrary size undergoing plane-strain finite deformation. Two sets of boundary conditions are considered: (a) both wedge flanks are clamped, and (b) one flank is clamped and the other free. The mode of deformation is obtained in one simple case for illustration.

Examples of nonlinear homogenization involving degenerate energies. I. Plane strain

2005 ◽  
Vol 461 (2063) ◽  
pp. 3681-3703 ◽  
Author(s):  
Isaac V Chenchiah ◽  
Kaushik Bhattacharya
Keyword(s):  
Shape Memory Alloys ◽  
Shape Memory ◽  
Plane Strain ◽  
Stress And Strain ◽  
Strain Fields ◽  
Perfectly Plastic ◽  
Plastic Materials ◽  
Nonlinear Homogenization

The study of polycrystals of shape-memory alloys and rigid-perfectly plastic materials gives rise to problems of nonlinear homogenization involving degenerate energies. This paper presents a characterization of the stress and strain fields in a class of problems in plane strain, and uses it to study examples including checkerboards and hexagonal microstructures. Consequences for shape-memory alloys and rigid-perfectly plastic materials are discussed.

Mathematical Model of the Effective Properties of a Fiber Reinforced Composite with a Linearly Graded Transition Zone

2010 ◽  
Vol 38 (4) ◽  
pp. 286-307
Author(s):  
Carey F. Childers
Keyword(s):  
Mathematical Model ◽  
Transition Zone ◽  
Effective Properties ◽  
Local Stress ◽  
Stress And Strain ◽  
Fiber Reinforced ◽  
Strain Fields ◽  
Shear Moduli ◽  
Fiber Reinforced Composite ◽  
Reinforced Composite

Abstract Tires are fabricated using single ply fiber reinforced composite materials, which consist of a set of aligned stiff fibers of steel material embedded in a softer matrix of rubber material. The main goal is to develop a mathematical model to determine the local stress and strain fields for this isotropic fiber and matrix separated by a linearly graded transition zone. This model will then yield expressions for the internal stress and strain fields surrounding a single fiber. The fields will be obtained when radial, axial, and shear loads are applied. The composite is then homogenized to determine its effective mechanical properties—elastic moduli, Poisson ratios, and shear moduli. The model allows for analysis of how composites interact in order to design composites which gain full advantage of their properties.

The stress and strain fields in the neighbourhood of a notch in polyethylene

Polymer ◽  
1989 ◽  
Vol 30 (8) ◽  
pp. 1456-1461 ◽  
Author(s):  
Xue-qin Wang ◽  
Norman Brown
Keyword(s):  
Stress And Strain ◽  
Strain Fields

Endochronic Analysis of Cyclic Elastoplastic Strain Fields in a Notched Plate

1983 ◽  
Vol 50 (4a) ◽  
pp. 789-794 ◽  
Author(s):  
K. C. Valanis ◽  
J. Fan
Keyword(s):  
Residual Stress ◽  
Numerical Scheme ◽  
External Loading ◽  
Stress And Strain ◽  
Elastoplastic Strain ◽  
Residual Stress Field ◽  
Progressive Change ◽  
Strain Fields ◽  
Cyclic Tension ◽  
Cyclic Elastoplastic Strain

In this paper we present an analytical cum-numerical scheme, based on endochronic plasticity and the finite element formalism. The scheme is used to calculate the stress and elastoplastic strain fields in a plate loaded cyclically in its own plane along its outer edges and bearing two symmetrically disposed edge notches. One most important result that stands out is that while the external loading conditions are symmetric and periodic, the histories of stress and strain at the notch tip are neither symmetric nor periodic in character. In cyclic tension ratcheting phenomena at the tip of the notches prevail and a progressive change of the residual stress field at the notch line is shown to occur.

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