The Wiener-Hopf Equation in the Nevanlinna and Smirnov Algebras and Ultra-Distributions

1988 ◽  
pp. 83-96
Author(s):  
V. S. Vladimirov
Keyword(s):  
Hopf Equation

Mutual influence of the faces contact area and the pre-fracture zone near the tip of the interfacial crack

2020 ◽  
Vol 13 (3) ◽  
pp. 143-161
Author(s):  
M.V. Dudyk
Keyword(s):  
Contact Area ◽  
Fracture Zone ◽  
Mutual Influence ◽  
Interfacial Crack ◽  
Normal Displacement ◽  
Crack Model ◽  
Hopf Equation ◽  
Resistant Material ◽  
Order Of Magnitude ◽  
Crack Faces

BACKGROUND: Under plane strain conditions, a crack model was developed on a plane interface between two different materials, which assumes the existence near its tip of the faces contact area and a narrow lateral pre-fracture zone in a less crack-resistant material of the composite compound. The pre-fracture zone is modeled by the line of normal displacement rupture, on which the normal stress is equal to the tensile strength of the material. Assuming that the dimensions of the pre-fracture zone and the contact zone have the same order of magnitude and are significantly smaller than the crack length, the problem is reduced to the vector Wiener–Hopf equation. METHODS: An approximate method for solving the vector Wiener–Hopf equation was developed, which was used to obtain the equations for determining the sizes of the pre-fracture zone and the contact faces area. The pre-fracture zone orientation was determined from the condition of the potential energy maximum accumulated in the zone. Numerical calculations of the indicated parameters and analysis of their dependences on the configuration and module of external load are executed. RESULTS: A significant mutual influence of the pre-fracture zone and crack faces contact on their sizes and orientation of the zone was revealed.

On the scattering of sound by two semi-infinite parallel staggered plates - I. Explicit matrix Wiener-Hopf factorization

1988 ◽  
Vol 420 (1858) ◽  
pp. 131-156 ◽  
Keyword(s):  
Asymptotic Expression ◽  
Series Representation ◽  
Auxiliary Function ◽  
Sound Waves ◽  
Parallel Plates ◽  
Hopf Equation ◽  
Product Decomposition ◽  
Matrix Kernel ◽  
Wave Fields ◽  
The Matrix

This paper discusses the two-dimensional scattering of sound waves by two semi-infinite rigid parallel plates. The plates are staggered, so that a line in the plane of the motion passing through both edges is not in general perpendicular to the plane of either plate. The problem is formulated as a matrix Wiener-Hopf functional equation, which exhibits the difficulty of a kernel containing exponentially growing elements. We show how this difficulty may be overcome by constructing an explicit product decomposition of the matrix kernel with both factors having algebraic behaviour at infinity. This factorization is written in terms of a single entire auxiliary function that has a simple infinite series representation. The Wiener-Hopf equation is solved for arbitrary incident wave fields and we derive an asymptotic expression for the field scattered to infinity; the latter includes the possibility of propagating modes in the region between the plates. In part II of this work we will evaluate our solution numerically and obtain some analytical estimates in a number of physically interesting limits.

scholarly journals Alternating Strain Regimes for Failure Propagation in Flexural Systems

2019 ◽  
Vol 72 (3) ◽  
pp. 305-339 ◽  
Author(s):  
M Garau ◽  
M J Nieves ◽  
I S Jones
Keyword(s):  
Steady State ◽  
Transient Analysis ◽  
Fracture Process ◽  
Mechanical Load ◽  
Dynamic Features ◽  
Hopf Equation ◽  
Beam Structure ◽  
Numerical Studies ◽  
Failure Propagation ◽  
Discrete Mass

Summary We consider both analytical and numerical studies of a steady-state fracture process inside a discrete mass-beam structure, composed of periodically placed masses connected by Euler–Bernoulli beams. A fault inside the structure is assumed to propagate with a constant speed and this occurs as a result of the action of a remote sinusoidal, mechanical load. The established regime of fracture corresponds to the case of an alternating generalised strain regime. The model is reduced to a Wiener–Hopf equation and its solution is presented. We determine the minimum feeding wave energy required for the steady-state fracture process to occur. In addition, we identify the dynamic features of the structure during the steady-state fracture regime. A transient analysis of this problem is also presented, where the existence of steady-state fracture regimes, revealed by the analytical model, are verified and the associated transient features of this process are discussed.

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