Response of Nonlinearly Supported Spherical Boundaries to Shock Waves

1957 ◽  
Vol 24 (4) ◽  
pp. 501-505
Author(s):  
M. L. Baron
Keyword(s):  
Integral Equation ◽  
Integral Transform ◽  
Boundary Value ◽  
Acoustic Medium ◽  
Plane Shock ◽  
Linear Boundary ◽  
Influence Coefficients ◽  
Pressure Time ◽  
Elastically Supported ◽  
Integral Transform Technique

Abstract An integral transform technique is used to solve a boundary-value problem in which the partial differential equation is linear but the associated boundary condition is nonlinear. A spherical cavity in an infinite acoustic medium has an elastically supported boundary such that the pressure-displacement relation on the boundary is nonlinear. The response of the boundary to a plane shock wave which progresses across the cavity and envelops it is obtained by solving two auxiliary boundary-value problems with linear boundary conditions. Using influence coefficients obtained from these solutions, a nonlinear integral equation for the response of the actual boundary is obtained. The integral equation is solved numerically for a set of parameters, and curves for the pressure-time and displacement-time responses of the boundary are presented.

Displacements and Velocities Produced by the Diffraction of a Pressure Wave by a Cylindrical Cavity in an Elastic Medium

1962 ◽  
Vol 29 (2) ◽  
pp. 385-395 ◽  
Author(s):  
M. L. Baron ◽  
R. Parnes
Keyword(s):  
Cylindrical Cavity ◽  
Integral Transform ◽  
Plane Shock ◽  
Displacement Fields ◽  
Influence Coefficients ◽  
Body Component ◽  
Cavity Boundary ◽  
Component Motion ◽  
The Mean ◽  
Integral Transform Technique

An infinitely long cylindrical cavity in an infinite elastic homogeneous and isotropic medium is enveloped by a plane shock wave whose front is parallel to the axis of the cavity. The displacement and velocity fields produced by the diffraction of the wave by the cavity are determined by means of an integral transform technique. Expressions for the radial and tangential components of the displacements and velocities are derived, and numerical results are presented for these quantities at the cavity boundary. Results for the mean (rigid-body component) motion of the cavity boundary are also presented. The problem is considered for pressure waves with a step distribution in time. The results may be used as influence coefficients to determine, by means of Duhamel integrals, the velocity and displacement fields produced by waves with time-varying pressure.

An Integral Equation Approach to the Semi-Infinite Strip Problem

1973 ◽  
Vol 40 (4) ◽  
pp. 948-954 ◽  
Author(s):  
G. D. Gupta
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Laminate Composite ◽  
Integral Transform ◽  
Stress Singularity ◽  
Infinite Strip ◽  
Integral Equation Approach ◽  
Fixed End ◽  
Integral Transform Technique

A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. Integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. Stress singularity at the strip corner is obtained from the singular integral equation which is then solved numerically. Stresses along the rigid end are determined and the effect of the material properties on the stress-intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.

Analytical Solution of Forced Vibration of a Column Subjected to Conservative and Nonconservative Forces

2001 ◽  
Author(s):  
Zhixiang Xu ◽  
Kunisato Seto ◽  
Hideyuki Tamura
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problem ◽  
Integral Transform ◽  
Boundary Value ◽  
Suspension Bridge ◽  
Follower Force ◽  
Analysis Process ◽  
Finite Integral ◽  
Excitation Force ◽  
Integral Transform Technique

Abstract This paper presents analytical results of forced transverse vibration of a column with a mass attached at free-end subjected to a tangential follower force and a transverse distributed excitation force, that is a simplified model of some structures in civil and mechanical engineering, e.g., a column of a suspension bridge, a launched rocket in the atmosphere. Because the tangential follower force is nonconservative, it is very difficult to get the analytical solution of the problem by usually-used analysis methods with which the adjoint boundary value problem can not be directly obtained. However, by applying the finite integral transform technique, we directly obtained the adjoint boundary value problem in the analysis process, and successfully obtained the analytical solution of the column’s vibration excited by the transverse distributed force.

Diffraction of a Pressure Wave by a Cylindrical Cavity in an Elastic Medium

1961 ◽  
Vol 28 (3) ◽  
pp. 347-354 ◽  
Author(s):  
M. L. Baron ◽  
A. T. Matthews
Keyword(s):  
Shock Wave ◽  
Stress Field ◽  
Cylindrical Cavity ◽  
Hoop Stress ◽  
Integral Transform ◽  
Time History ◽  
Plane Shock ◽  
Influence Coefficients ◽  
History Of ◽  
Stress Concentration Factors

An infinitely long cylindrical cavity in an infinite elastic homogeneous and isotropic medium is enveloped by a plane shock wave whose front is parallel to the axis of the cavity. An integral transform technique is used to determine the stress field produced in the medium by the diffraction of the incoming shock wave by the cavity. Expressions for the radial stress σrr, the hoop stress σθθ, and the shear stress σrθ are derived as inversion integrals, and numerical results are presented for the time-history of the hoop stress σθθ at the boundary of the cavity. The amplifications of the hoop-stress concentration factors due to the dynamic loading are noted. The problem is considered for pressure waves with a step distribution in time. These results may be used as influence coefficients to determine, by means of Duhamel integrals, the stress field produced by waves with time-varying pressures.

scholarly journals On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

2021 ◽  
Vol 273 ◽  
pp. 04002
Author(s):  
Vyacheslav Berkovich ◽  
Viсtor Poltinnikov
Keyword(s):  
Integral Equation ◽  
Wave Field ◽  
Boundary Integral Equation ◽  
Integral Transform ◽  
Boundary Integral ◽  
Geometric Characteristics ◽  
Dynamic Elasticity ◽  
Elasticity Problems ◽  
Steady Oscillations ◽  
Integral Transform Technique

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.

Dynamic Response of Circular Rigid-Plastic Plates Resting on Fluid

1976 ◽  
Vol 43 (1) ◽  
pp. 102-106 ◽  
Author(s):  
D. Krajcinovic
Keyword(s):  
Integral Equation ◽  
Dynamic Response ◽  
Integral Transform ◽  
Specific Problem ◽  
Integral Equation Method ◽  
Dual Integral Equation ◽  
Inviscid Fluid ◽  
Rigid Plastic ◽  
Interaction Problem ◽  
Integral Transform Technique

Paper concerns a specific problem in the study of dynamic response of rigid-plastic plates in contact with incompressible and inviscid fluid. The dynamic load is assumed to be of high intensity and short duration. Employing integral transform technique in conjunction with the dual integral equation method this interaction problem is reduced to the problem of plate deforming in vacuum for which the solution is well known.

scholarly journals A Class of Linear Boundary Value Problem for $k$-regular Functions in Clifford Analysis

2016 ◽  
Vol 8 (3) ◽  
pp. 57
Author(s):  
Yan Zhang
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Clifford Analysis ◽  
Boundary Value ◽  
Regular Function ◽  
Integral Equation Method ◽  
Linear Boundary ◽  
Regular Functions ◽  
Linear Boundary Value Problem

In this paper, we introduce the linear boundary value problem for $k$-regular function, and give an unique solution for this problem by integral equation method and fixed-point theorem.

Axisymmetric Singular Stresses in a Thick-Walled Spherical Shell With an Internal Edge Crack

1983 ◽  
Vol 50 (1) ◽  
pp. 37-42 ◽  
Author(s):  
A. Atsumi ◽  
Y. Shindo
Keyword(s):  
Integral Equation ◽  
Spherical Shell ◽  
Edge Crack ◽  
Integral Transform ◽  
Crack Opening ◽  
Cauchy Kernel ◽  
Opening Displacement ◽  
Inner Surface ◽  
Dominant Part ◽  
Integral Transform Technique

The paper considers the elastostatic axisymmetric problem for a thick-walled spherical shell containing a circumferential edge crack on the inner surface. The ring-shaped edge crack and the inner surface of the spherical shell are subjected to internal pressure. Using an integral transform technique we obtain a singular integral equation of the first kind which has a generalized Cauchy kernel as the dominant part. The integral equation is solved numerically, and the influence of the geometrical configuration on the stress-intensity factor and the crack-opening displacement is shown graphically in detail.

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