Higher Order Singularities and Their Energetics in Elastic-Plastic Fracture

2000 ◽  
Author(s):  
Seyoung Im ◽  
Insu Jeon
Keyword(s):  
Conservation Laws ◽  
Interaction Energy ◽  
Higher Order ◽  
Configurational Forces ◽  
J Integral ◽  
Numerical Examples ◽  
Elastic Plastic ◽  
M Integral

Abstract The higher order singularities[1] are systematically examined, and discussed are their complementarity relation with the nonsingular eigenfunctions and their relations to the configurational forces like J-integral and M-integral. By use of the so-called two state conservation laws[2] or interaction energy, originally proposed by Eshelby[3] and later treated by Chen and Shield[4], the intensities of the higher order singularities are calculated, and their roles in elastic-plastic fracture are investigated. Numerical examples are presented for illustration.

Two-State Conservation Integrals and Stress Singularities in Generic Wedges

1999 ◽  
Author(s):  
Seyoung Im ◽  
Yongwoo Lee
Keyword(s):  
Conservation Laws ◽  
Interaction Energy ◽  
Series Expansion ◽  
Three Dimensional ◽  
J Integral ◽  
Stress Singularities ◽  
Material Interface ◽  
M Integral ◽  
Free Edges

Abstract The eigenvalues of Williams’ series expansion for generalized wedge problems, which include cracks, re-entrant corners, free edges, and cracks meeting with material interface, etc. are examined from the viewpoint of conservation laws like J-integral and M-integral. By use of the so-called two-state conservation laws or interaction energy, originally proposed by Eshelby and later treated by Chen and Shield, discussed is that the complementary pairs of eigenvalues exist in the J-integral sense and/or in the M-integral sense when these integrals are conserved. Similar results are shown to hold for the eigenvalues of three dimensional wedges.

Formulation of Three-Dimensional J-Integral for Finite Strain Elastic-Plastic Fracture Problems Under Any Load Histories (Monotonic and Cyclic Loads)

2018 ◽  
Author(s):  
Koichiro Arai ◽  
Hiroshi Okada ◽  
Yasunori Yusa
Keyword(s):  
Crack Front ◽  
Integral Domain ◽  
Crack Problem ◽  
Three Dimensional ◽  
Finite Domain ◽  
J Integral ◽  
Proportional Loading ◽  
Numerical Examples ◽  
Elastic Plastic ◽  
Tetrahedral Element

In this paper, a redefined three-dimensional J-integral for the quadratic tetrahedral element along with some numerical examples are presented. It is known that the J-integral represents the energy release rate and can be computed on an arbitrary path or a domain of integration. This feature is called the “path-independent property”. It requires the assumption of proportional loading in the case of elastic-plastic material. Because of this assumption, when the J-integral is applied to a problem under a non-proportional loading condition, the computed J-integral value depends on the integral path or the integral domain. To overcome this problem, the authors have proposed a formula to evaluate the energy dissipation inside a finite domain in the vicinity of the crack front, which is an extension of the conventional three-dimensional J-integral. The energy dissipating into a small but finite domain in the vicinity of crack front includes the energy released due to the opening of new crack faces and the deformation energy in the process zone. The numerical evaluation is carried out by the domain integral. In this formula, it is not necessary to evaluate any non-integrable terms at the crack front. Furthermore, the proposed formula has the feature of integral domain independence for any material models without any assumptions in their deformation histories. Therefore, it is possible to evaluate the J-integral for problems with non-proportional loads by using the proposed method. In this paper, computational method using the quadratic tetrahedral element for the redefined J-integral under non-proportional loading is presented first. Some results of three-dimensional semi-circular surface crack problem are presented as numerical examples.

scholarly journals A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

2018 ◽  
Vol 40 (1) ◽  
pp. 405-421 ◽  
Author(s):  
N Chatterjee ◽  
U S Fjordholm
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Numerical Examples ◽  
Continuity Equations ◽  
Multiple Dimensions ◽  
Nonlocal Conservation Laws ◽  
Volume Method ◽  
Singular Data

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

scholarly journals Hyperbolic three-string vertex

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Atakan Hilmi Fırat
Keyword(s):  
Boundary Value Problem ◽  
Conservation Laws ◽  
Riemann Surfaces ◽  
Higher Order ◽  
Local Coordinates ◽  
Pants Decomposition ◽  
Fuchsian Equation ◽  
String Amplitudes ◽  
Hyperbolic Riemann Surfaces ◽  
Liouville’S Equation

Abstract We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

Elastic-Plastic Analysis of an Elbow With Axial Part-Through Internal Crack at Crown Under In-Plane Bending

2008 ◽  
Author(s):  
K. M. Prabhakaran ◽  
S. R. Bhate ◽  
V. Bhasin ◽  
A. K. Ghosh
Keyword(s):  
Finite Element ◽  
Crack Depth ◽  
Bending Moment ◽  
Internal Crack ◽  
J Integral ◽  
Finite Element Analyses ◽  
Integrity Assessment ◽  
Elastic Plastic ◽  
Axial Part ◽  
Plane Bending

Piping elbows under bending moment are vulnerable to cracking at crown. The structural integrity assessment requires evaluation of J-integral. The J-integral values for elbows with axial part-through internal crack at crown under in-plane bending moment are limited in open literature. This paper presents the J-integral results of a thick and thin, 90-degree, long radius elbow subjected to in-plane opening bending moment based on number of finite element analyses covering different crack configurations. The non-linear elastic-plastic finite element analyses were performed using WARP3D software. Both geometrical and material nonlinearity were considered in the study. The geometry considered were for Rm/t = 5, and 12 with ratio of crack depth to wall thickness, a/t = 0.15, 0.25, 0.5 and 0.75 and ratio of crack length to crack depth, 2c/a = 6, 8, 10 and 12.

Prediction of Brittle Fracture Under Generalised Elastic Plastic Loading

2008 ◽  
Author(s):  
S. J. Lewis ◽  
C. E. Truman ◽  
D. J. Smith
Keyword(s):  
Fracture Mechanics ◽  
Brittle Fracture ◽  
Ferritic Steel ◽  
Failure Prediction ◽  
J Integral ◽  
Local Approach ◽  
Elastic Plastic ◽  
Failure Data ◽  
Stress States ◽  
Plastic Loading

This article describes an investigation into the ability of a number of different fracture mechanics approaches to predict failure by brittle fracture under general elastic/plastic loading. Data obtained from C(T) specimens of A508 ferritic steel subjected to warm pre-stressing and side punching were chosen as such prior loadings produce considerably non-proportionality in the resulting stress states. In addition, failure data from a number of round notched bar specimens of A508 steel were considered for failure with and without prior loading. Failure prediction, based on calibration to specimens in the as received state, was undertaken using two methods based on the J integral and two based on local approach methodologies.

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