Analytical Solution for Size-Dependent Elastic Field of a Nanoscale Circular Inhomogeneity

2006 ◽  
Vol 74 (3) ◽  
pp. 568-574 ◽  
Author(s):  
L. Tian ◽  
R. K. N. D. Rajapakse
Keyword(s):  
Analytical Solution ◽  
Elastic Field ◽  
Infinite Matrix ◽  
Surface Residual Stress ◽  
Residual Surface Stress ◽  
Stress Effects ◽  
Size Dependent ◽  
Closed Form Analytical Solution ◽  
Remote Loading ◽  
Interface Elasticity

Two-dimensional elastic field of a nanoscale circular hole/inhomogeneity in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. A closed-form analytical solution is obtained by using the complex potential function method of Muskhelishvili. Selected numerical results are presented to investigate the size dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. Stress state is found to depend on the radius of the inhomogeneity/hole, surface elastic constants, surface residual stress, and magnitude of far-field loading.

scholarly journals Nanoparticles and the influence of interface elasticity

2008 ◽  
Vol 35 (1-3) ◽  
pp. 267-286 ◽  
Author(s):  
Mi Changwen ◽  
Demitris Kouris
Keyword(s):  
Material Properties ◽  
Axial Symmetry ◽  
Outer Boundary ◽  
Elastic Field ◽  
Interface Stress ◽  
Axially Symmetric ◽  
Displacement Potential ◽  
Surface And Interface ◽  
Stress Effects ◽  
Interface Elasticity

In this manuscript, we discuss the influence of surface and interface stress on the elastic field of a nanoparticle, embedded in a finite spherical substrate. We consider an axially symmetric traction field acting along the outer boundary of the substrate and a non-shear uniform eigenstrain field inside the particle. As a result of axial symmetry, two Papkovitch-Neuber displacement potential functions are sufficient to represent the elastic solution. The surface and interface stress effects are fully represented utilizing Gurtin and Murdoch's theory of surface and interface elasticity. These effects modify the traction-continuity boundary conditions associated with the classical continuum elasticity theory. A complete methodology is presented resulting in the solution of the elastostatic Navier's equations. In contrast to the classical solution, the modified version introduces additional dependencies on the size of the nanoparticles as well as the surface and interface material properties.

Surface Stress Effect on the Pull-In Instability of Hydrostatically and Electrostatically Actuated Rectangular Nanoplates With Various Edge Supports

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
M. Faghih Shojaei ◽  
V. Mohammadi ◽  
M. A. Darabi
Keyword(s):  
Boundary Conditions ◽  
Governing Equation ◽  
Surface Stress ◽  
Plate Thickness ◽  
Stress Effect ◽  
Residual Surface Stress ◽  
Electrostatically Actuated ◽  
Stress Effects ◽  
Size Dependent ◽  
Surface Stress Effect

This paper is aimed to investigate the size-dependent pull-in behavior of hydrostatically and electrostatically actuated rectangular nanoplates including surface stress effects based on a modified continuum model. To this end, based on the Gurtin–Murdoch theory and Hamilton's principle, the governing equation and corresponding boundary conditions of an actuated nanoplate are derived; the step-by-step linearization scheme and the differential quadrature (GDQ) method are used to discretize the governing equation and associated boundary conditions. The effects of the thickness of the nanoplate, surface elastic modulus and residual surface stress on the pull-in instability of the nanoplate are investigated. Plates made of two different materials including aluminum (Al) and silicon (Si) are selected to explain the variation of the pull-in voltage and pressure with respect to plate thickness.

Plane Deformations of an Inhomogeneity–Matrix System Incorporating a Compressible Liquid Inhomogeneity and Complete Gurtin–Murdoch Interface Model

2018 ◽  
Vol 85 (12) ◽  
Author(s):  
Ming Dai ◽  
Min Li ◽  
Peter Schiavone
Keyword(s):  
Analytic Solution ◽  
Elastic Solid ◽  
Compressible Liquid ◽  
External Loading ◽  
Interface Model ◽  
Interface Tension ◽  
Soft Solids ◽  
Remote Loading ◽  
Interface Elasticity ◽  
Plane Deformations

We consider the plane deformations of an infinite elastic solid containing an arbitrarily shaped compressible liquid inhomogeneity in the presence of uniform remote in-plane loading. The effects of residual interface tension and interface elasticity are incorporated into the model of deformation via the complete Gurtin–Murdoch (G–M) interface model. The corresponding boundary value problem is reformulated and analyzed in the complex plane. A concise analytical solution describing the entire stress field in the surrounding solid is found in the particular case involving a circular inhomogeneity. Numerical examples are presented to illustrate the analytic solution when the uniform remote loading takes the form of a uniaxial compression. It is shown that using the simplified G–M interface model instead of the complete version may lead to significant errors in predicting the external loading-induced stress concentration in gel-like soft solids containing submicro- (or smaller) liquid inhomogeneities.

Dynamic Interaction of Two Independent SDOF Oscillators Supported by a Flexible Foundation Embedded in a Half-Space: Closed-Form Analytical Solution for Incident Plane SH-Waves

2021 ◽  
pp. 1-24
Author(s):  
Liguo Jin ◽  
Liting Du ◽  
Haiyan Wang
Keyword(s):  
Analytical Solution ◽  
Closed Form ◽  
Half Space ◽  
Strong Interaction ◽  
Rigid Foundation ◽  
Structural Stiffness ◽  
Sh Waves ◽  
Closed Form Analytical Solution ◽  
Flexible Foundation ◽  
Plane Sh Waves

This paper presents a closed-form analytical solution for the dynamic response of two independent SDOF oscillators standing on one flexible foundation embedded in an elastic half-space and excited by plane SH waves. The solution is obtained by the wave function expansion method and is verified by comparison with the results of the special cases of a rigid foundation and the published research result of a flexible foundation. The model is utilized to investigate how the foundation stiffness influences the system response. The results show that there will be a significant interaction between the two independent structures on one flexible foundation and the intensity of the interaction is mainly dependent on foundation stiffness and structural stiffness. For a system with more flexible foundation, strong interaction will exist between the two structures; larger structural stiffness will also lead to a strong interaction between the two structures. When the structural mass and the structural stiffness are all larger, the flexible foundation cannot be treated as a rigid foundation even if the foundation stiffness is many times larger than that of soil. This model may be useful to get insight into the effects of foundation flexibility on the interaction of two independent structures standing on one flexible foundation.

An Improved Analytical Model for Deflections of a Circular Multi-Layer Piezoelectric Actuator

2005 ◽  
Author(s):  
Mandar Deshpande ◽  
Laxman Saggere
Keyword(s):  
Analytical Solution ◽  
Closed Form ◽  
Piezoelectric Actuator ◽  
Plate Theory ◽  
Analytical Models ◽  
Sensors And Actuators ◽  
Closed Form Solutions ◽  
Design And Optimization ◽  
Closed Form Analytical Solution ◽  
Model Correlation

Models for simple closed-form analytical solutions for accurately predicting static deflections of circular thin-film piezoelectric microactuators are very useful in design and optimization of a variety of MEMS sensors and actuators utilizing piezoelectric actuators. While closed-form solutions treating actuators with simple geometries such as cantilevers and beams are available, simple analytical models treating circular bending-type actuators commonly used in MEMS applications are generally lacking. This paper presents a closed-form analytical solution for accurately estimating the deflections and the volume displacements of a circular multi-layer piezoelectric actuator under combined voltage and pressure loading. The model for the analytical solution presented in this paper, which is based on classical laminated plate theory, allows for inclusion of multiple layers and non-uniform diameters of various layers in the actuator including bonding and electrode layers, unlike other models previously reported in the literature. The analytical solution presented is validated experimentally as well as through a finite element solution and excellent experiment-model correlation within 1% variation is demonstrated. General guidelines for optimization of circular piezoelectric actuator are also discussed. The utility of the model for design optimization of a multi-layered piezoelectric actuator is demonstrated through a numerical example wherein the dimensions of a test actuator are optimized to improve the displaced volume by three-fold under combined voltage and resisting pressure loads.

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