Showing papers by "J. N. Reddy published in 2015"

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

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

An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics

Abstract: 1. General Introduction and Mathematical Preliminaries 2. Elements of Nonlinear Continuum Mechanics 3. The Finite Element Method: A Review 4. One-Dimensional Problems Involving a Single Variable 5. Nonlinear Bending of Straight Beams 6. Two-Dimensional Problems Involving a Single Variable 7. Nonlinear Bending of Elastic Plates 8. Nonlinear Bending of Elastic Shells 9. Finite Element Formulations of Solid Continua 10. Weak-Form Finite Element Models of Flows of Viscous Incompressible Fluids 11. Least-Squares Finite Element Models of Flows of Viscous Incompressible Fluids Appendix 1: Solution Procedures for Linear Equations Appendix 2: Solution Procedures for Nonlinear Equations

A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: theory and finite element analysis

Abstract: Finite element analysis of functionally graded plates based on a general third-order shear deformation plate theory with a modified couple stress effect and the von Karman nonlinearity is carried out to bring out the effects of couple stress, geometric nonlinearity and power-law variation of the material composition through the plate thickness on the bending deflections of plates. The theory requires no shear correction factors. The principle of virtual displacements is utilized to develop a nonlinear finite element model. The finite element model requires C 1 continuity of all dependent variables. The microstructural effects are captured using a length scale parameter via the modified couple stress theory. The variation of two-constituent material is assumed through the thickness direction according to a power-law distribution. Numerical results are presented for static bending problems of rectangular plates with various boundary conditions to bring out the parametric effects of the power-law index and length scale parameter on the load–deflection characteristics of plates with various boundary conditions.

Size-dependent vibration of a microplate under the action of a moving load based on the modified couple stress theory

Abstract: This study deals with forced vibration analysis of a microplate subjected to a moving load. The formulation is developed based on the modified couple stress theory in conjunction with Kirchhoff–Love plate theory. The equations of motion of the problem are derived using Lagrange’s equations. In order to obtain the response of the microplate, the trial function for the dynamic deflection is expressed in the polynomial form. The equations of motion are solved by using the implicit time integration Newmark-β method, and then displacements, velocities and accelerations of the microplate at the considered point and time are determined. Five different sets of boundary condition are considered. For this purpose, boundary conditions are satisfied by adding some auxiliary functions to the trial functions. A parametric study is conducted to study the effects of the material length scale parameter, plate aspect ratio, boundary conditions and the moving load velocity on the dynamic response of the microplate. Also, in order to validate the present formulation and solution method, some comparisons with those available in the literature are performed. Good agreement is found. The results show that the dynamic deflections are significantly affected by the scale parameter and the load velocity.

Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation

Abstract: We consider in this paper the free and forced vibration response of simply-supported functionally graded (FG) nanobeams resting on a non-linear elastic foundation. The two-constituent Functionally Graded Material (FGM) is assumed to follow a power-law distribution through the beam thickness. Eringen׳s non-local elasticity model with material length scales is used in conjunction with the Euler–Bernoulli beam theory with von Karman geometric non-linearity that accounts for moderate rotations. Non-linear natural frequencies of non-local FG nanobeams are obtained using He׳s Variational Iteration Method (VIM) and the direct and discretized Method of Multiple Scales (MMS), while the primary resonance analysis of an externally forced non-local FG nanobeam is performed only using the MMS. The effects of the non-local parameter, power-law index, and the parameters of the non-linear elastic foundation on the non-linear frequency-response are investigated.

Surface and non-local effects for non-linear analysis of Timoshenko beams

Abstract: In this paper, we present a non-local non-linear finite element formulation for the Timoshenko beam theory. The proposed formulation also takes into consideration the surface stress effects. Eringen׳s non-local differential model has been used to rewrite the non-local stress resultants in terms of non-local displacements. Geometric non-linearities are taken into account by using the Green–Lagrange strain tensor. A C0 beam element with three degrees of freedom has been developed. Numerical solutions are obtained by performing a non-linear analysis for bending and free vibration cases. Simply supported and clamped boundary conditions have been considered in the numerical examples. A parametric study has been performed to understand the effect of non-local parameter and surface stresses on deflection and vibration characteristics of the beam. The solutions are compared with the analytical solutions available in the literature. It has been shown that non-local effect does not exist in the nano-cantilever beam (Euler–Bernoulli beam) subjected to concentrated load at the end. However, there is a significant effect of non-local parameter on deflections for other load cases such as uniformly distributed load and sinusoidally distributed load (Cheng et al. (2015) [10]). In this work it has been shown that for a cantilever beam with concentrated load at free end, there is definitely a dependency on non-local parameter when Timoshenko beam theory is used. Also the effect of local and non-local boundary conditions has been demonstrated in this example. The example has also been worked out for other loading cases such as uniformly distributed force and sinusoidally varying force. The effect of the local or non-local boundary conditions on the end deflection in all these cases has also been brought out.

A peridynamic theory for linear elastic shells

Abstract: A state-based peridynamic formulation for linear elastic shells is presented. The emphasis is on introducing, possibly for the first time, a general surface based peridynamic model to represent the deformation characteristics of structures that have one physical dimension much smaller than the other two. A new notion of curved bonds is exploited to cater for force transfer between the peridynamic particles describing the shell. Starting with the three dimensional force and deformation states, appropriate surface based force, moment and several deformation states are arrived at. Upon application on the curved bonds, such states beget the necessary force and deformation vectors governing the motion of the shell. Correctness of our proposal on the peridynamic shell theory is numerically assessed against static deformation of spherical and cylindrical shells and flat plates.

Variational approach to dynamic analysis of third-order shear deformable plates within gradient elasticity

Abstract: A variational approach based on Hamilton’s principle is used to develop the governing equations for the dynamic analysis of plates using the Reddy third-order shear deformable plate theory with strain gradient and velocity gradient. The plate is made of homogeneous and isotropic elastic material. The stain energy, kinetic energy, and the external work are generalized to capture the gradient elasticity (i.e., size effect) in plates modeled using the third-order shear deformation theory. In this framework, both strain and velocity gradients are included in the strain energy and kinetic energy expressions, respectively. The equations of motion are derived, along with the consistent boundary equations. Finally, the resulting third-order shear deformation (strain and velocity) gradient plate theory is specialized to the first-order and classical strain gradient plate theories.

Active Structural Acoustic Control of a Softcore Sandwich Panel Using Multiple Piezoelectric Actuators and Reddy's Higher Order Theory

Abstract: The purpose of the present work is to theoretically investigate the active control of radiated sound power from a simply supported soft-core sandwich panel with a line moment excitation. Since noise transmission in the low frequency region through a soft-core sandwich panel mainly occurs due to flexural and dilatational modes, therefore, the focus of this study is to control these modes and achieve sound attenuation in a large frequency band. Two control methods, volume velocity and weighted sum of spatial gradients (WSSG) are used to drive three piezoelectric actuators (PZTs) attached on the exterior side of the bottom face plate. The governing equation of the sandwich panel with the PZTs is derived using the Hamilton’s principle considering Reddy’s third order shear deformation theory. Numerical studies indicate that while the line moment is at the mid vertical line, WSSG is able to attenuate the radiated sound power irrespective of core loss factor whereas volume velocity could not. However, both the c...

A Micropolar Cohesive Damage Model for Delamination of Composites

Abstract: A micropolar cohesive damage model for delamination of composites is proposed. The main idea is to embed micropolarity, which brings an additional layer of kinematics through the micro-rotation degrees of freedom within a continuum model to account for the micro-structural effects during delamination. The resulting cohesive model, describing the modified traction separation law, includes micro-rotational jumps in addition to displacement jumps across the interface. The incorporation of micro-rotation requires the model to be supplemented with physically relevant material length scale parameters, whose effects during delamination of modes I and II are brought forth using numerical simulations appropriately supported by experimental evidences.

Active attenuation of sound transmission through a soft-core sandwich panel into an acoustic enclosure using volume velocity cancellation

Abstract: In this paper, active control of harmonic sound transmitted through a soft-core sandwich panel into a rectangular enclosure is studied. As it has already been shown for the low frequency region, the noise transmission through a soft-core sandwich panel mainly occurs due to the flexural and the dilatational modes. Therefore, in this study, volume velocity cancellation control strategy is used to control these modes, and achieve sound attenuation in a broad frequency range. Point force and uniformly distributed force actuators are used as the secondary actuator to cancel the volume velocity of the bottom faceplate, which opens to the cavity, of the sandwich panel. Cancelling the net volume velocity of the bottom faceplate is compared not only in terms of the reduction in sound transmission through the sandwich panel into cavity but also in terms of the faceplate velocities. Also, the effectiveness of the volume velocity cancellation strategy has been studied for different values of isotropic loss factor of ...

Rate Constitutive Theories of Order Zero in Lagrangian Description for Thermoelastic Solids

Abstract: This article presents constitutive theories for the stress tensor and the heat vector for homogeneous, isotropic thermoelastic solids in Lagrangian description for finite deformation. The deforming solid is assumed to be in thermodynamic equilibrium during the evolution. Since conservation of mass, balance of momenta, and balance of energy are independent of the constitution of the matter, the second law of thermodynamics must form the basis for deriving the constitutive theories. We introduce the concept of rate constitutive theory and show that for thermoelastic solids the constitutive theories are, in fact, rate theories of order zero. These theories for stress tensor consider material derivative of order zero of the conjugate strain tensor as one of the argument tensors of the stress tensor established as a dependent variable in the constitutive theory. Generalization of this concept leading to higher order rate theories in Lagrangian description are considered in followup works [1, 2]. The conditions...

Introduction to Shape Memory Alloys

Abstract: Classical materials like metals and alloys have played a significant role as structural materials for many centuries. Engineers have designed components and selected alloys by employing the classical engineering approach of understanding the macroscopic properties of the material and selecting the appropriate one to match the desired functionality based on the application. With advancements in material science and with increasing space and logistical limitations, scientists have been constantly developing high performing materials for various applications.

Free Vibration Analysis of Elastic Plates in Contact with an Inviscid Fluid Medium

Abstract: In the present study, a finite element formulation is presented to investigate vibration response of elastic plates in contact with a fluid medium. The fluid is assumed to be incompressible and inviscid, and the impermeability condition of the plate is taken into account. The classical plate theory (CPT), first-order shear deformation plate theory (FSDT), and Reddy third-order shear deformation plate theory (RSDT) are considered for the kinematic description of the solid medium and the simplified Navier–Stokes equations are used as the governing equations for the fluid medium. For each plate theory, a coupled set of finite element equations is derived. The effect of the fluid pressure is considered as an added mass and its effect on natural frequencies and mode shapes is investigated through several numerical simulations by varying the boundary conditions.

Ordered rate constitutive theories for thermoviscoelastic solids without memory in Lagrangian description using Gibbs potential

Abstract: The paper presents rate constitutive theories for finite deformation of homogeneous, isotropic, compressible, and incompressible thermoviscoelastic solids without memory in Lagrangian description derived using the second law of thermodynamics expressed in terms of Gibbs potential Ψ. To ensure thermodynamic equilibrium during evolution, the rate constitutive theories must be derived using entropy inequality [as other three conservation and balance laws are do not provide a mechanism for deriving constitutive theories for the deforming matter (Surana in Advanced mechanics of continuua. in preparation, 2014)]. The two forms of the entropy inequality in Ψ derived using conjugate pairs \({\mathbf{\sigma}^*}\), \({[\dot{J}]}\) : first Piola–Kirchhoff stress tensor and material derivative of the Jacobian of deformation and \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) ; second Piola–Kirchhoff stress tensor and material derivative of Green’s strain tensor are precisely equivalent as the conjugate pairs \({\mathbf{\sigma}^*}\), \({[\dot{J}]}\) and \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) are transformable from each other. In the present work, we use \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) as conjugate pair. Two possible choices of dependent variables in the constitutive theories: Ψ, η, \({\mathbf{\sigma}^{[0]}}\), \({\mathbf{q}}\) and Ψ, η, \({\mathbf{\varepsilon}_{[0]}}\), \({\mathbf{q}}\) (in which η is entropy density and \({\mathbf{q}}\) is heat vector) are explored based on conservation and balance laws. It is shown that the choice of Ψ, η, \({\mathbf{\varepsilon}_{[0]}}\), \({\mathbf{q}}\) is essential when the entropy inequality is expressed in terms of Ψ. The arguments of these dependent variables are decided based on desired physics. Viscoelastic behavior requires considerations of at least \({\mathbf{\varepsilon}_{[0]}}\) and \({\dot{\mathbf{\varepsilon}}_{[0]}}\) (or \({\mathbf{\varepsilon}_{[1]}}\)) in the constitutive theories. We generalize and consider strain rates \({\mathbf{\varepsilon}_{[i]}}\); i = 0, 1, …, n−1 as arguments of the dependent variables in the derivations of the ordered rate theories of up to orders n. At the onset, \({\mathbf{\sigma}^{[0]}}\), \({\mathbf{\varepsilon}_{[i]}}\) ; i = 0, 1, …, n−1, θ and \({\mathbf{g}}\) are considered as arguments of Ψ, η, \({\mathbf{\varepsilon}_{[n]}}\) and \({\mathbf{q}}\). When \({\dot{\Psi}}\) is substituted in the entropy inequality, the resulting conditions eliminate η as a dependent variable, reduce arguments of some of the dependent variables in the constitutive theory etc. but do not provide a mechanism to derive constitutive theories for \({\mathbf{\varepsilon}_{[i]}}\) and \({\mathbf{q}}\). The stress tensor \({\mathbf{\sigma}^{[0]}}\) is decomposed into equilibrium stress \({{}_e \mathbf{\sigma}^{[0]}}\) and deviatoric stress \({{}_d \mathbf{\sigma}^{[0]}}\). Upon substituting this in the entropy inequality, we finally arrive at the inequality that must be satisfied by \({{}_e \mathbf{\sigma}^{[0]}}\), \({{}_d \mathbf{\sigma}^{[0]}}\) and \({\mathbf{q}}\). Derivations of the constitutive theory for \({{}_e \mathbf{\sigma}^{[0]}}\) follow directly from \({{}_e \mathbf{\sigma}^{(0)}}\), equilibrium Cauchy stress tensor, and the constitutive theory for \({\mathbf{\varepsilon}_{[n]}}\) is derived using the theory of generators and invariants. Constitutive theories for the heat vector \({\mathbf{q}}\) of up to orders n that are consistent (in terms of the argument tensors) with the constitutive theories for \({\mathbf{\varepsilon}_{[n]}}\) are also derived. Many simplified forms of the rate theories of orders n are presented. Material coefficients are derived by considering Taylor series expansions of the coefficients in the linear combinations representing \({\mathbf{\varepsilon}_{[n]}}\) and \({\mathbf{q}}\) using the combined generators of the argument tensors about a known configuration \({\underline{\Omega}}\) in the combined invariants of the argument tensors and temperature. It is shown that the rate constitutive theories of order one (n = 1) when further simplified results in constitutive theories that resemble currently used theories but are in fact different. The solid materials characterized by these theories have mechanisms of elasticity and dissipation but have no memory, i.e., no relaxation behavior or rheology. Fourier heat conduction law is shown to be an over-simplified case of the rate theory of order one for \({\mathbf{q}}\). The paper establishes when there is equivalence between the constitutive theories derived here using Ψ and those presented in Surana et al. (Acta Mech 224(11):2785—2816, 2013), that are derived using Helmholtz free energy density Φ.

Nonlinear Waves in Solid Continua with Finite Deformation

Abstract: This work considers initiation of nonlinear waves, their propagation, reflection, and their interactions in thermoelastic solids and thermoviscoelastic solids with and without memory. The conservation and balance laws constituting the mathematical models as well as the constitutive theories are derived for finite deformation and finite strain using second Piola-Kirchoff stress tensor and Green’s strain tensor and their material derivatives [1]. Fourier heat conduction law with constant conductivity is used as the constitutive theory for heat vector. Numerical studies are performed using space-time variationally consistent finite element formulations derived using space-time residual functionals and the non-linear equations resulting from the first variation of the residual functional are solved using Newton’s Linear Method with line search. Space-time local approximations are considered in higher order scalar product spaces that permit desired order of global differentiability in space and time. Computed results for non-linear wave propagation, reflection, and interaction are compared with linear wave propagation to demonstrate significant differences between the two, the importance of the nonlinear wave propagation over linear wave propagation as well as to illustrate the meritorious features of the mathematical models and the space-time variationally consistent space-time finite element process with time marching in obtaining the numerical solutions of the evolutions.

Graphical Description of Temperature Controlled Actuation of SMA Wires

Abstract: In this chapter, we will demonstrate how the SMA wires and springs are actuated and the way a given stroke is achieved by means of temperature changes.

Case Studies in the Preliminary Design of SMA Actuators

Abstract: In spite of the rather complex and hysteretic behavior of SMA components (wires and springs), and the complex changes that occur in their microstructures, the preliminary design of SMA wires and springs turns out to be reasonably simple provided we utilize their shape memory characteristics in specific ways. Designing SMA actuators requires defining actuation intervals (temperature profile), load levels and/or displacement outputs over its intended designed life. The alloy composition dictates the transformation temperatures and the component’s operating range. SMA components could be functional under load controlled or displacement controlled or simultaneous load and displacement controlled setups.

Factors Influencing Design of SMA Actuators

Abstract: Given the complex nonlinear nature of SMA responses, understanding their coupled thermomechanical responses of different components has been of significant interest for both researchers and application developers.

Fatigue of SMAs

Abstract: With growing applications of SMA components in different engineering applications, the issue of material performance over its designed life is of great concern to researchers lately. Fatigue studies in SMAs is still an unsolved puzzle and we wish to highlight some important items that affect the designers and some open questions in SMA fatigue areas.

Computational Methods for Engineering Science

Abstract: This special issue attempts to cover the recent advances on various aspects of theories, analyses, and applications of computational methods in engineering science, reflecting the state of the art in computational methods and their frameworks, applications, networking technologies, and new and advanced engineering applications in emerging technologies such as the bioscience and biotechnology, nanoscience and nanotechnology, numerical modeling, simulation and analysis, and material sciences. The topics covered by the articles including in this special issue encompass computationalmechanics, ocean and offshore engineering, computational fluid dynamics, computational mathematics and statistics, computational physics, computational material sciences, multiscale modeling, disaster simulation and analysis, element-free/meshless/mesh-free methods and dimension-reduction methods, geometric and material nonlinear analyses, damage, fracture and fatigue, contact mechanics and friction, smart structures and health monitoring, structural optimization, nanomechanics, biomechanics, inverse and coupling problems, and reliability theory and application. We hope that this special issue will be cited for recent advances in these research areas.

Basic SMA Component Geometries and Responses

Abstract: SMA components must allow for a large surface area compared to their volume in order for them to be able to be cooled rapidly and repeated use at reasonable actuation frequencies. Thus SMAs are commonly used in the form of wires/rods, springs, tubes or beams under different loading conditions (tension, torsion or bending) for exploiting their unique characteristics in many practical applications. All these geometries are governed by their high surface to volume ratios.

Manufacturing and Post Treatment of SMA Components

Abstract: We would like to briefly review the different manufacturing techniques and post-treatment techniques employed in the SMA community. This is by no means an exhaustive discussion but a broad overview of some important items. As designers, it is important to having an understanding of these details especially as many of these techniques/treatments discussed here influence the performance of the component immensely.