A Concentrated Force Problem of Plane Strain or Plane Stress

1947 ◽  
Vol 14 (3) ◽  
pp. A246
Author(s):  
A. E. Green
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Plane Stress ◽  
Complex Variable ◽  
Elliptical Hole ◽  
Minor Axis ◽  
Concentrated Force ◽  
Large Plate ◽  
Force Problem ◽  
Variable Analysis

Abstract The problem in plane strain or plane stress of a large plate containing an elliptical hole, which is loaded by line forces at the ends of the minor axis of the ellipse, is solved in closed form by using complex variable analysis.

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.

Elastic Equilibrium of a Plate With a Reinforced Elliptical Hole

1960 ◽  
Vol 27 (2) ◽  
pp. 283-288 ◽  
Author(s):  
Eugene Levin
Keyword(s):  
Circular Hole ◽  
Thin Plate ◽  
Plane Stress ◽  
Complex Variable ◽  
Elliptical Hole ◽  
Elastic Equilibrium ◽  
Stress Problem ◽  
Elliptical Ring ◽  
Plane Stress Problem ◽  
Digital Computers

An infinite thin plate with an elliptical hole reinforced by a confocal elliptical ring is subjected to loads in the plane. A solution to the generalized plane-stress problem is obtained using the complex variable techniques of Muskhelishvili. The result is presented in a form well suited to evaluation by digital computers. Specialization to a circular hole with a negligibly thin reinforcement is shown to be in agreement with results obtained by other authors.

Analysis of a Semi-Infinite Strip With Constant Stress Flanges Under Concentrated Loads

1966 ◽  
Vol 33 (3) ◽  
pp. 571-574 ◽  
Author(s):  
H. R. Meck
Keyword(s):  
Stress Distribution ◽  
Exact Solutions ◽  
Closed Form ◽  
Complex Variable ◽  
Constant Stress ◽  
Infinite Strip ◽  
Concentrated Loads ◽  
Simple Complex ◽  
Variable Analysis

An analysis is presented for a semi-infinite strip reinforced by flanges and subjected to concentrated in-plane loads at the ends of the flanges. The taper which results in constant flange stress is determined, and the stress distribution in the sheet is also found. A simple complex variable analysis is used which leads to exact solutions in closed form.

Green’s Functions for Holes/Cracks in Laminates With Stretching-Bending Coupling

2005 ◽  
Vol 72 (2) ◽  
pp. 282-289 ◽  
Author(s):  
Chyanbin Hwu
Keyword(s):  
Closed Form ◽  
Composite Laminates ◽  
Green's Functions ◽  
Green’S Functions ◽  
Continuation Method ◽  
Elliptical Hole ◽  
Arbitrary Point ◽  
Two Dimensional ◽  
Concentrated Force ◽  
Concentrated Forces

Consider an infinite composite laminate containing a traction-free elliptical hole subjected to concentrated forces and moments at an arbitrary point outside the hole. This problem for two-dimensional deformation has been solved analytically in the literature, while for the general unsymmetric composite laminates stretching and bending coupling may occur and due to the mathematical complexity the associated Green’s functions have never been found for complete loading cases. Recently, by employing Stroh-like formalism for coupled stretching-bending analysis, the Green’s functions for the infinite laminates (without holes) were obtained in closed-form. Based upon the nonhole Green’s functions, through the use of analytical continuation method the Green’s functions for holes are now obtained in explicit closed-form for complete loading cases and are valid for the full fields. The Green’s functions for cracks are then obtained by letting the minor axis of ellipse be zero. By proper differentiation, the stress resultants and moments along the hole boundary and the stress intensity factors of cracks are also solved explicitly. Like the Green’s functions for the infinite laminates, only the solutions associated with the in-plane concentrated forces f^1,f^2 and out-of-plane concentrated moments m^1,m^2 have exactly the same form as those of the corresponding two-dimensional problems. For the cases under the concentrated force f^3 and torsion m^3, new types of solutions are obtained.

Improved Compliance Solutions for C(T), SE(B) and Clamped SE(T) Specimens Including Side-Grooves, Varying Thicknesses and 3-D Effects

2014 ◽  
Author(s):  
Gustavo Henrique B. Donato ◽  
Felipe Cavalheiro Moreira
Keyword(s):  
Crack Growth ◽  
Plane Strain ◽  
Plane Stress ◽  
Potential Drop ◽  
Crack Size ◽  
Growth Conditions ◽  
Size Estimation ◽  
Unloading Compliance ◽  
Integrity Assessments ◽  
Elastic Unloading

Fracture toughness and Fatigue Crack Growth (FCG) experimental data represent the basis for accurate designs and integrity assessments of components containing crack-like defects. Considering ductile and high toughness structural materials, crack growing curves (e.g. J-R curves) and FCG data (in terms of da/dN vs. ΔK or ΔJ) assumed paramount relevance since characterize, respectively, ductile fracture and cyclic crack growth conditions. In common, these two types of mechanical properties severely depend on real-time and precise crack size estimations during laboratory testing. Optical, electric potential drop or (most commonly) elastic unloading compliance (C) techniques can be employed. In the latter method, crack size estimation derives from C using a dimensionless parameter (μ) which incorporates specimen’s thickness (B), elasticity (E) and compliance itself. Plane stress and plane strain solutions for μ are available in several standards regarding C(T), SE(B) and M(T) specimens, among others. Current challenges include: i) real specimens are in neither plane stress nor plane strain - modulus vary between E (plane stress) and E/(1-ν2) (plane strain), revealing effects of thickness and 3-D configurations; ii) furthermore, side-grooves affect specimen’s stiffness, leading to an “effective thickness”. Previous results from current authors revealed deviations larger than 10% in crack size estimations following existing practices, especially for shallow cracks and side-grooved samples. In addition, compliance solutions for the emerging clamped SE(T) specimens are not yet standardized. As a step in this direction, this work investigates 3-D, thickness and side-groove effects on compliance solutions applicable to C(T), SE(B) and clamped SE(T) specimens. Refined 3-D elastic FE-models provide Load-CMOD evolutions. The analysis matrix includes crack depths between a/W=0.1 and a/W=0.7 and varying thicknesses (W/B = 4, W/B = 2 and W/B = 1). Side-grooves of 5%, 10% and 20% are also considered. The results include compliance solutions incorporating all aforementioned effects to provide accurate crack size estimation during laboratory fracture and FCG testing. All proposals revealed reduced deviations if compared to existing solutions.

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