Rigid-Plastic Deformation of a Ring-Stiffened Shell Under Blast Loading

1997 ◽  
Vol 119 (4) ◽  
pp. 467-474 ◽  
Author(s):  
M. S. Hoo Fatt
Keyword(s):  
Plastic Deformation ◽  
Cylindrical Shell ◽  
Structural Model ◽  
Dynamic Equilibrium ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Body Motion ◽  
Fine Tuning ◽  
Rigid Plastic ◽  
Stiffened Cylindrical Shell

An approximate solution for the plastic deformation of a ring-stiffened cylindrical shell in response to a nonaxisymmetric, exponentially decaying pressure load, is presented. The analogy between the ring-stiffened cylindrical shell and a rigid-plastic string-on-foundation with discrete plastic resisting elements is used to find closed-form solutions for the transient and final deformations of the shell. Dynamic equilibrium of the central bay of the shell and the adjacent ring-stiffeners results in an inhomogeneous wave equation with inhomogeneous boundary conditions for the string. The initial-boundary value problem is solved by the method of eigenfunction expansion and a suitable orthogonality condition. The zeroth mode for the string describes rigid-body motion of the bay due to the remaining inertia of adjacent stiffeners. Permanent deformations are obtained using a plastic unloading criterion whereby the velocity and strain rate for each eigenmode vanish simultaneously. In the example problem, higher eigenmodes decay and vanish rapidly and final shell deformations are primarily governed by lower eigenmodes. The structural model gives qualitatively correct transient deflections and would be amenable to fine-tuning with numerical analysis and experimental evidence.

Shock-Wave Damage of Ring-Stiffened Cylindrical Shells

1994 ◽  
Vol 38 (03) ◽  
pp. 245-252
Author(s):  
Michelle S. Hoo Fatt
Keyword(s):  
Shock Wave ◽  
Cylindrical Shell ◽  
Structural Model ◽  
Dynamic Equilibrium ◽  
Galerkin Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Impulsive Loading ◽  
Stiffened Cylindrical Shell ◽  
Experimental Profile

A solution methodology for the nonlinear plastic response of the central bay of a ring-stiffened cylindrical shell subject to shock-wave loading is presented. The solution is based on a simple structural model that uses an analogy between a cylindrical shell and a string-on foundation in which ring stiffeners are modeled as lumped masses and springs. By requiring dynamic equilibrium within the central bay of the shell, one may reduce the problem to solving an inhomogeneous wave equation for which the motion of the ring stiffener is introduced into one of the boundary conditions of the string. The initial-boundary-value problem is solved by using a modified Galerkin approximation. The mode shape used to describe the local or bay deformation in the Galerkin approximation is determined from the experimental profile of an actual damaged shell. A Galerkin approximation not only yields a simple solution for the transient deformations of the shell, but it also has an advantage over an exact solution in that it can be easily extended to shells subject to asymmetric pressure loading with arbitrary time variation. The Galerkin solution is shown to approach two extreme cases of dynamic loading for the exponentially decaying pressure load: impulsive loading and static loading. A final deformed profile of the shell is obtained by using the concept of plastic unloading waves. The solution for the transient deflection is a stepping stone to the evaluation of strains and is therefore important in establishing a failure criterion for the shell. The analytical results presented herein may therefore be instrumental in establishing design criteria for prevention of failure of the ring-stiffened shell.

scholarly journals Plastic Deformation and Rupture of Ring-Stiffened Cylinders under Localized Pressure Pulse Loading

1994 ◽  
Vol 1 (3) ◽  
pp. 289-301 ◽  
Author(s):  
Michelle S. Hoo Fatt
Keyword(s):  
Plastic Deformation ◽  
Cylindrical Shell ◽  
Pressure Pulse ◽  
Fracture Initiation ◽  
Pulse Loading ◽  
Rigid Plastic ◽  
Stiffened Shell ◽  
Closed Form Solutions ◽  
Stiffened Cylindrical Shell ◽  
Lumped Masses

An analytical solution for the dynamic plastic deformation of a ring-stiffened cylindrical shell subject to high intensity pressure pulse loading is presented. By using an analogy between a cylindrical shell that undergoes large plastic deformation and a rigid-plastic string resting on a rigid-plastic foundation, one derives closed-form solutions for the transient and final deflection profiles and fracture initiation of the shell. Discrete masses' and springs are used to describe the ring stiffeners in the stiffened shell. The problem of finding the transient deflection profile of the central bay is reduced to solving an inhomogeneous wave equation with inhomogeneous boundary conditions using the method of eigenfunction expansion. The overall deflection profile consists of both global (stiffener) and local (bay) components. This division of the shell deflection profile reveals a complex interplay between the motions of the stiffener and the bay. Furthermore, a parametric study on a ring-stiffened shell damaged by a succession of underwater explosions shows that the string-on-foundation model with ring stiffeners described by lumped masses and springs is a promising method of analyzing the structure.

scholarly journals The Refined Model of Viscoelastic-Plastic Deformation of Reinforced Cylindrical Shells

2020 ◽  
pp. 138-149
Author(s):  
A P Yankovskii
Keyword(s):  
Plastic Deformation ◽  
Cylindrical Shell ◽  
Boundary Value Problem ◽  
Elastoplastic Deformation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Residual Deflection ◽  
Initial Boundary Value ◽  
Reddy Theory ◽  
Reinforced Shell

The paper formulates the initial-boundary-value problem of the viscoelastic-plastic bending behavior of cylindrical circular shells cross-reinforced along equidistant surfaces. The instant elastoplastic deformation of the shell composition components is described by the governing equations of the theory of plastic flow with isotropic hardening. The viscoelastic deformation of these materials is described by the defining relations of the Maxwell - Boltzmann model of body. The geometric nonlinearity of the problem is taken into account in the Karman approximation. The used system of two-dimensional resolving equations and the corresponding initial and boundary conditions make it possible to determine displacements and stress-strain state (including residual one) in materials of the composition of flexible cylindrical shells with varying degrees of accuracy. In this case, the weak resistance of the considered composite structures to transverse shears is taken into account. In the first approximation, the equations are used, the initial and boundary conditions correspond to the relations of the widely used non-classical Reddy theory. A numerical solution of the initial-boundary-value problem posed is constructed using an explicit step-by-step "cross" scheme. The elastoplastic and viscoelastic-plastic dynamic deformation of a relatively thin long circular cylindrical shell is investigated. The structure is rationally reinforced in the circumferential direction and is loaded with an internal pressure of an explosive type. It has been demonstrated that under intense short-term loading even of a relatively thin cylindrical reinforced shell by internal pressure, the traditional Reddy theory does not guarantee that the maximum residual deflection and the intensity of residual deformations of the components of the composition are accurate to within 10% compared to calculations performed by the refined theory. The difference in the results of the corresponding calculations increases with an increase in the relative thickness of the composite shell. It was found that after plastic deformation of a long reinforced cylindrical shell in its residual state, not only appear zones of edge effects, but also a local zone of an intense deformation located in the vicinity of the central section of the shell. The length of the local central zone is comparable with the length of the zones of edge effects. It is shown that the amplitude of the transverse vibrations of the reinforced shell in the vicinity of the initial moment of time significantly (by an order of magnitude) exceeds the value of the maximum modulus of the residual deflection. Therefore, the calculations performed in the framework of the theory of elastoplastic deformation of composition materials do not allow a very approximate determination of the magnitude of the residual displacements and the magnitude of the residual deformed state of the components of the composition of the cylindrical shell.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

2019 ◽  
Vol 84 (5) ◽  
pp. 873-911 ◽  
Author(s):  
Marianna A Shubov ◽  
Laszlo P Kindrat
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Control Input ◽  
Vibrational Modes ◽  
Boundary Feedback ◽  
Feedback Matrix ◽  
Dissipative Boundary ◽  
Initial Boundary Value ◽  
The Right ◽  
Euler Bernoulli Beam

Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($\alpha ,\beta ,k_1,k_2$) linear boundary feedback law at the right end. The $2 \times 2$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$. The role of the control parameters is examined and the following results have been proven: (i) when $\beta \neq 0$, the set of vibrational modes is asymptotically close to the vertical line on the complex $\nu$-plane given by the equation $\Re \nu = \alpha + (1-k_1k_2)/\beta$; (ii) when $\beta = 0$ and the parameter $K = (1-k_1 k_2)/(k_1+k_2)$ is such that $\left |K\right |\neq 1$ then the following relations are valid: $\Re (\nu _n/n) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iii) when $\beta =0$, $|K| = 1$, and $\alpha = 0$, then the following relations are valid: $\Re (\nu _n/n^2) = O\left (1\right )$ and $\Im (\nu _n/n) = O\left (1\right )$ as $\left |n\right |\to \infty$; (iv) when $\beta =0$, $|K| = 1$, and $\alpha>0$, then the following relations are valid: $\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$ and $\Im (\nu _n/n^2) = O\left (1\right )$ as $\left |n\right |\to \infty$.

scholarly journals On the flow of MHD generalized maxwell fluid via porous rectangular duct

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 989-1002
Author(s):  
Aamir Farooq ◽  
Muhammad Kamran ◽  
Yasir Bashir ◽  
Hijaz Ahmad ◽  
Azeem Shahzad ◽  
...  
Keyword(s):  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rectangular Duct ◽  
Modified Darcy’S Law ◽  
Initial Boundary Value ◽  
Generalized Maxwell Fluid ◽  
Limiting Case ◽  
The Impact

Abstract The purpose of this proposed investigation is to study unsteady magneto hydrodynamic (MHD) mixed initial-boundary value problem for incompressible fractional Maxwell fluid model via oscillatory porous rectangular duct. Considering the modified Darcy’s law, the problem is simplified by using the method of the double finite Fourier sine and Laplace transforms. As a limiting case of the general solutions, the same results can be obtained for the classical Maxwell fluid. Also, the impact of magnetic parameter, porosity of medium, and the impact of various material parameters on the velocity profile and the corresponding tangential tensions are illuminated graphically. At the end, we will give the conclusion of the whole paper.

scholarly journals An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

2021 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Emilia Bazhlekova
Keyword(s):  
Boundary Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimates ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Generalized Eigenfunction ◽  
The One ◽  
Initial Boundary Value

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

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