Elastodynamic Local Fields for a Crack Running in an Orthotopic Medium

1991 ◽  
Vol 58 (4) ◽  
pp. 982-987 ◽  
Author(s):  
A. Piva ◽  
E. Radi
Keyword(s):  
Dynamic Stress ◽  
Equations Of Motion ◽  
Local Fields ◽  
Stress Fields ◽  
Orthotropic Materials ◽  
Displacement Fields ◽  
First Order ◽  
Stress And Displacement Fields ◽  
Order Elliptic System ◽  
Separated Variables

The dynamic stress and displacement fields in the neighborhood of the tip of a crack propagating in an orthotropic medium are obtained. The approach deals with the methods of linear algebra to transform the equations of motion into a first-order elliptic system whose solution is sought under the assumption that the local displacement field may be represented under a scheme of separated variables. The analytical approach has enabled the distinction between two kinds of orthotropic materials for which explicit espressions of the near-tip stress fields are obtained. Some results are presented graphically also in order to compare them with the numerical solution given in a quoted reference.

Stress and Displacement Fields at a Transient Crack Tip Propagating in Functionally Graded Materials

2007 ◽  
Vol 345-346 ◽  
pp. 481-484
Author(s):  
Kwang Ho Lee ◽  
Gap Su Ban
Keyword(s):  
Functionally Graded Materials ◽  
Crack Tip ◽  
Functionally Graded ◽  
Stress Fields ◽  
Displacement Fields ◽  
Graded Materials ◽  
Transient Motion ◽  
Stress And Displacement Fields ◽  
Displacement Potentials ◽  
Nonhomogeneous Materials

Stress and displacement fields for a transient crack tip propagating along gradient in functionally graded materials (FGMs) with an exponential variation of shear modulus and density under a constant Poisson's ratio are developed. The equations of transient motion in nonhomogeneous materials are developed using displacement potentials and the solution to the displacement fields and the stress fields for a transient crack propagating at nonuniform speed though an asymptotic analysis.

The Stress Fields Caused by a Circular Cylindrical Inclusion

1992 ◽  
Vol 59 (2S) ◽  
pp. S107-S114 ◽  
Author(s):  
Hisao Hasegawa ◽  
Ven-Gen Lee ◽  
Toshio Mura
Keyword(s):  
Exact Solutions ◽  
Strain Energy ◽  
Elastic Solid ◽  
Cylindrical Inclusion ◽  
Stress Fields ◽  
Axisymmetric Stress ◽  
Displacement Fields ◽  
Elastic Fields ◽  
Stress And Displacement Fields ◽  
Interior Points

Exact solutions are presented in closed forms for the axisymmetric stress and displacement fields caused by a solid or hollow circular cylindrical inclusion (with uniform axial eigenstrain prescribed) in an infinite elastic solid. The same expressions are obtained for the elastic fields for interior and exterior points of the inclusion. Although Eshelby’s solutions for ellipsoidal inclusions are uniform in the interior points, the present solutions do not show the uniformity. When the length of inclusion becomes infinite, the present solutions agree with Eshelby ’s results. The strain energy is also shown. The method of Green’s function is used.

Stress and Displacement Fields for Propagating the Crack Along the Interface of Dissimilar Orthotropic Materials Under Dynamic Mode I and II Load

1999 ◽  
Vol 67 (1) ◽  
pp. 223-228 ◽  
Author(s):  
K. H. Lee
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Complex Function ◽  
Orthotropic Materials ◽  
Dynamic Mode ◽  
Mode I ◽  
Intensity Factors ◽  
Displacement Fields ◽  
Crack Propagation Velocity ◽  
Stress And Displacement Fields

General stress and displacement fields are derived as a crack steadily propagates along the interface of dissimilar orthotropic materials under a dynamic mode I and II load. They are obtained from the complex function formulation of steady plane motion problems for an orthotropic material and the complex eigenexpansion function. After the relationship between stress intensity factors and stress components for a propagating crack is defined, the stress, displacement components, and energy release rate with stress intensity factors are derived. The results are useful for both dissimilar isotropic and orthotropic and isotropic-orthotropic bimaterials, and homogeneous isotropic and orthotropic materials under subsonic crack propagation velocity. [S0021-8936(00)00601-2]

scholarly journals NONCONFORMING MIXED ELEMENTS FOR ELASTICITY

2003 ◽  
Vol 13 (03) ◽  
pp. 295-307 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
RAGNAR WINTHER
Keyword(s):  
Mixed Method ◽  
Variational Formulation ◽  
Degrees Of Freedom ◽  
Mixed Finite Elements ◽  
Plane Elasticity ◽  
Shape Functions ◽  
Displacement Fields ◽  
First Order ◽  
Stress And Displacement Fields ◽  
Vertex Degrees

We construct first-order, stable, nonconforming mixed finite elements for plane elasticity and analyze their convergence. The mixed method is based on the Hellinger–Reissner variational formulation in which the stress and displacement fields are the primary unknowns. The stress elements use polynomial shape functions but do not involve vertex degrees of freedom.

An Automated Procedure for Determining Asymptotic Elastic Stress Fields at Singular Points

2006 ◽  
Vol 41 (4) ◽  
pp. 287-295 ◽  
Author(s):  
Donghee Lee ◽  
J. R Barber
Keyword(s):  
Elastic Stress ◽  
Analytical Tool ◽  
Singular Points ◽  
Stress Fields ◽  
Local Geometry ◽  
Dominant Eigenvalue ◽  
Displacement Fields ◽  
Singular Stress ◽  
Stress And Displacement Fields ◽  
Contour Plots

The paper describes an analytical tool in MATLAB for determining the nature of the stress and displacement fields near a fairly general singular point in linear elasticity. The user is prompted to input the local geometry of the system, the material properties, and the boundary conditions (and interface conditions in the case of composite bodies or problems involving contact between two or more bodies). The tool then computes the dominant eigenvalue and provides as output the equations defining the singular stress and displacement fields and contour plots of these fields. No knowledge of the asymptotic analysis procedure is required of the user. The tool is tested against previously published results where available and proves to be robust and accurate. It is potentially useful for the development of special finite elements for singular points or for characterizing failure at such points. It can be downloaded from the website http://www-personal.umich.edu/∼jbarber/asymptotics/intro.html

Elastodynamic Near-Tip Stress and Displacement Fields for Rapidly Propagating Cracks in Orthotropic Materials

1975 ◽  
Vol 42 (1) ◽  
pp. 183-189 ◽  
Author(s):  
J. D. Achenbach ◽  
Z. P. Bazˇant
Keyword(s):  
Crack Propagation ◽  
Numerical Procedure ◽  
Orthotropic Materials ◽  
Time Varying ◽  
Two Dimensional ◽  
Displacement Fields ◽  
Stress And Displacement Fields ◽  
Curved Paths ◽  
Isotropic And Orthotropic Materials ◽  
Maximum Stresses

The near-tip angular variations of elastodynamic stress and displacement fields are investigated for rapid transient crack propagation in isotropic and orthotropic materials. The two-dimensional near-tip displacement fields are assumed in the general form rp T(t, c) K(θ, c), where c is a time-varying velocity of crack propagation, and it is shown that p = 0.5. For isotropic materials, K(θ, c) is determined explicitly by analytical considerations. A numerical procedure is employed to determine K(θ, c) for orthotropic materials. The tendency of the maximum stresses to move out of the plane of crack propagation as the speed of crack propagation increases is more pronounced for orthotropic materials, for the case that the crack propagates in the direction of the larger elastic modulus. The angular variations of the near-tip fields are the same for steady-state and transient crack propagation, and for propagation along straight and curved paths, provided that the direction of crack propagation and the speed of the crack tip vary continuously.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Photogrammetric Process to Monitor Stress Fields Inside Structural Systems

Sensors ◽  
2021 ◽  
Vol 21 (12) ◽  
pp. 4023
Author(s):  
Leonardo M. Honório ◽  
Milena F. Pinto ◽  
Maicon J. Hillesheim ◽  
Francisco C. de Araújo ◽  
Alexandre B. Santos ◽  
...  
Keyword(s):  
Boundary Element ◽  
Real Time ◽  
Stress Fields ◽  
Structural Systems ◽  
Unknown Boundary ◽  
Stress Distributions ◽  
Displacement Fields ◽  
Time Stress ◽  
Beam Surface ◽  
Measured Displacement

This research employs displacement fields photogrammetrically captured on the surface of a solid or structure to estimate real-time stress distributions it undergoes during a given loading period. The displacement fields are determined based on a series of images taken from the solid surface while it experiences deformation. Image displacements are used to estimate the deformations in the plane of the beam surface, and Poisson’s Method is subsequently applied to reconstruct these surfaces, at a given time, by extracting triangular meshes from the corresponding points clouds. With the aid of the measured displacement fields, the Boundary Element Method (BEM) is considered to evaluate stress values throughout the solid. Herein, the unknown boundary forces must be additionally calculated. As the photogrammetrically reconstructed deformed surfaces may be defined by several million points, the boundary displacement values of boundary-element models having a convenient number of nodes are determined based on an optimized displacement surface that best fits the real measured data. The results showed the effectiveness and potential application of the proposed methodology in several tasks to determine real-time stress distributions in structures.

Analysis of Linear Nonconservative Vibrations

1995 ◽  
Vol 62 (3) ◽  
pp. 685-691 ◽  
Author(s):  
F. Ma ◽  
T. K. Caughey
Keyword(s):  
Modal Analysis ◽  
Equations Of Motion ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Equivalence Transformations ◽  
State Space Approach ◽  
Nonconservative System ◽  
First Order ◽  
Physical Insight ◽  
Space Approach

The coefficients of a linear nonconservative system are arbitrary matrices lacking the usual properties of symmetry and definiteness. Classical modal analysis is extended in this paper so as to apply to systems with nonsymmetric coefficients. The extension utilizes equivalence transformations and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, the generalized modal analysis can offer substantial reduction in computational effort and ample physical insight.

scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

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