On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves
01 Sep 1983-Journal of Applied Physics (American Institute of Physics)-Vol. 54, Iss: 9, pp 4703-4710
TL;DR: In this article, the integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels.
Abstract: Integropartial differential equations of the linear theory of nonlocal elasticity are reduced to singular partial differential equations for a special class of physically admissible kernels. Solutions are obtained for the screw dislocation and surface waves. Experimental observations and atomic lattice dynamics appear to support the theoretical results very nicely.
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01 Jan 2002TL;DR: Memory-dependent nonlocal nonlocal Electromagnetic Elastic Solids as mentioned in this paper have been shown to be memory-dependent on nonlocal elasticity and nonlocal linear elasticity, as well as nonlocal Linear Elasticity and Nonlocal Fluid Dynamics.
Abstract: 1. Motion and Deformation.- 2. Stress.- 3. Constitutive Axioms.- 4. Nonlocal Electromagnetic Theory.- 5. Constitutive Equations of Memory-Dependent Nonlocal Electromagnetic Elastic Solids.- 6. Nonlocal Linear Elasticity.- 7. Nonlocal Fluid Dynamics.- 8. Nonlocal Linear Electromagnetic Theory.- 9. Memory-Dependent Nonlocal Thermoelastic Solids.- 10. Memory-Dependent Nonlocal Fluids.- 11. Memory-Dependent Nonlocal Electromagnetic Elastic Solids.- 12. Memory-Dependent Nonlocal Electromagnetic Thermofluids.- 13. Nonlocal Microcontinua.- 14. Memory-Dependent Nonlocal Micropolar Electromagnetic Elastic Solids.- 15. Nonlocal Continuum Theory of Liquid Crystals.
1,967 citations
TL;DR: An overview, comparison and critical review of the different approaches to topology optimization, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.
Abstract: Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.
1,816 citations
TL;DR: In this article, the Euler-Bernoulli, Timoshenko, Reddy, and Levinson beam theories are reformulated using the nonlocal differential constitutive relations of Eringen.
1,519 citations
TL;DR: In this paper, a nonlocal elasticity theory is employed to develop a non-local Benoulli/Euler beam model and some representative problems are solved to illustrate the magnitude of predicted nonlocal effects.
1,171 citations
TL;DR: In this paper, a higher-order non-local strain gradient elasticity model is proposed, which is based on the nonlocal effects of the strain field and first gradient strain field.
Abstract: In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed It is based on the nonlocal effects of the strain field and first gradient strain field This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects
1,085 citations
References
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01 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.
7,426 citations
TL;DR: In this article, the dispersion relations for one dimensional plane waves were obtained by fitting the nonlocal material moduli to exactly the acoustical branch of elastic waves within one Brillouin zone in periodic one dimensional lattices.
1,101 citations
TL;DR: In this paper, the authors explore the properties of a semi-infinite non-local dielectric to assess the effect of spatial dispersion on the reflectivity of the material and on properties of surface polaritons.
Abstract: In this paper, we explore the properties of a model of a semi-infinite nonlocal dielectric to assess the effect of spatial dispersion on the reflectivity of the material and on the properties of surface polaritons. For the model, the nonlocal form of Maxwell's equations may be solved exactly. The additional boundary conditions follow from Maxwell's equations, and it is not necessary to introduce microscopic considerations to complete the theory. We exhibit closed expressions for the reflectivity of the material, for the case where the electric field is parallel to the plane of incidence, and for the case where it is perpendicular to the plane of incidence. At non-normal incidence, when the electric field vector is parallel to the plane of incidence, structure which owes its origin to spatial-dispersion effects appears in the reflectivity. We show that in the presence of spatial dispersion, the surface polaritons acquire a finite lifetime even in the case where the dielectric is lossless; i.e., in the presence of spatial dispersion the surface polaritons become virtual surface waves. In the quasistatic limit, we obtain an analytic expression for the dependence of the real and imaginary part of the surface-polariton frequency on wave vector in the long-wavelength limit. We then present the theory of frustrated internal reflection of radiation from a prism and crystal configuration similar to that employed in several recent experiments. In a final section, we present the results of some numerical calculations of the reflectivity of the crystal, and the width and position of the dip observed in the frustrated-internal-reflection method, for parameters characteristic of the fundamental exciton line in ZnSe.
168 citations
TL;DR: In this paper, field equations of nonlocal elasticity are solved to determine the state of stress in the neighborhood of a line crack in an elastic plate subject to a uniform shear at the surface of the crack tip.
Abstract: Field equations of nonlocal elasticity are solved to determine the state of stress in the neighborhood of a line crack in an elastic plate subject to a uniform shear at the surface of the crack tip. A fracture criterion based on the maximum shear stress gives the critical value of the applied shear for which the crack becomes unstable. Cohesive stress necessary to break the atomic bonds is calculated for brittle materials.
148 citations
01 Jan 1976
113 citations