Showing papers by "J. N. Reddy published in 2016"

Theory and Analysis of Elastic Plates and Shells

Abstract: Because plates and shells are common structural elements in aerospace, automotive, and civil engineering structures, engineers must understand the behavior of such structures through the study of theory and analysis. Compiling this information into a single volume, Theory and Analysis of Elastic Plates and Shells, Second Edition presents a complete

Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory

Abstract: Finite element models of microstructure-dependent geometrically nonlinear theories for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a two-constituent material, the von Karman nonlinearity, and the strain gradient effects are developed for the classical and first-order plate theories. The strain gradient effects are included through the modified couple stress theory that contains a single material length scale parameter which can capture the size effect in a functionally graded material plate. The developed finite element models are used to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on the bending response of functionally graded circular plates with different boundary conditions.

An Equivalent Layer-Wise Approach for the Free Vibration Analysis of Thick and Thin Laminated and Sandwich Shells

Abstract: The main purpose of the paper is to present an innovative higher-order structural theory to accurately evaluate the natural frequencies of laminated composite shells. A new kinematic model is developed starting from the theoretical framework given by a unified formulation. The kinematic expansion is taken as a free parameter, and the three-dimensional displacement field is described by using alternatively the Legendre or Lagrange polynomials, following the key points of the most typical Layer-wise (LW) approaches. The structure is considered as a unique body and all the geometric and mechanical properties are evaluated on the shell middle surface, following the idea of the well-known Equivalent Single Layer (ESL) models. For this purpose, the name Equivalent Layer-Wise (ELW) is introduced to define the present approach. The governing equations are solved numerically by means of the Generalized Differential Quadrature (GDQ) method and the solutions are compared with the results available in the literature or obtained through a commercial finite element program. Due to the generality of the current method, several boundary conditions and various mechanical and geometric configurations are considered. Finally, it should be underlined that different doubly-curved surfaces may be considered following the mathematical framework given by the differential geometry.

Eringen’s Stress Gradient Model for Bending of Nonlocal Beams

Abstract: This paper is concerned with the bending response of nonlocal elastic beams under transverse loads, where the nonlocal elastic model of Eringen, also called the stress gradient model, is used. This model is known to exhibit some paradoxical responses when applied to beams with certain types of boundary conditions. In particular, for clamped-free boundary condition, this nonlocal model is not able to predict scale effects in the presence of concentrated loads, or it leads to an apparent stiffening effect for distributed loads in contrast to other boundary conditions for which softening effect is observed. In the literature, these paradoxes have been resolved by changing the kernel of the nonlocal model or by modifying the standard boundary conditions. In this paper, the paradox is solved from the nonlocal differential model itself via some related discontinuous nonlocal kinematics. It is shown that the kinematics related to the nonlocal constitutive law lead to the use of moment or shear discontinuities. With such a nonlocal differential model coupled with the nonlocal discontinuity requirements, the beam effectively shows a softening response irrespective of the boundary conditions studied, including the clamped-free boundary conditions, and thereby resolves the paradox. The model is also compared to lattice-based solutions where an excellent agreement between the present nonlocal model and the lattice one is obtained. Finally, the stress gradient model is shown to be cast in a stress-based variational framework, which coincides with a Timoshenko-type model where the shear effect is shown to play the nonlocal role.

GraFEA: a graph-based finite element approach for the study of damage and fracture in brittle materials

Abstract: In this paper the conventional finite element method for linear elastic response is reformulated in such a way that makes it favorable for the study of damage and fracture in brittle materials This modified finite element framework is based on the idea presented by Reddy and Srinivasa (Finite Elem Anal Des 104:35–40, 2015), where it was shown that for discretized hyperelastic materials, the magnitude of the nodal forces (in the discretized form) can be written in terms of the axial strains along the edges of the elements and that the equilibrium equations at each node can be written in terms of the forces along the edges alone Using this concept and by exploiting the fact that FEM discretization leads to an undirected cyclic graph with nodes and edges whose connectivity is related to the elements of the FEM, one can reformulate the displacement-based finite element framework with constant strain triangular elements to represent the continuum as a nonlocal network The network representation of the continuum is “nonlocal” in the sense that the force along any given edge doesn’t only depend on the strain along that edge, but on a collective behavior of the strains along the edges neighboring the edge of interest This method is named as GraFEA (for graph-based finite element analysis) Damage is introduced using a nonlocal damage criterion originating from the idea of the weakest links statistics proposed by Lin, Evans, and Ritchie (J Mech Phys Solids 34(5):477–497, 1986) This idea, which was very successful in studying cleavage fracture of mild steel at very low temperatures, can be used to impose a damage criterion to the nolocal network GraFEA has the major advantage that one can impose an edge-based failure criterion using the weakest link thoery directly on the discretized body, and potentially simulating crack initiation, crack growth, and branching without the need for extra enrichment functions (as with other methods) The simplicity of the method and the fact that it is based on conventional finite element method makes it suitable for integration into commercial softwares The governing equations for this approach are derived and applied to two simple crack growth simulations (as a proof of concept) in two-dimensional regions with a hole

Analysis of anisotropic gradient elastic shear deformable plates

Abstract: In this paper, Reddy’s third-order shear deformable plate theory is employed for the analysis of centrosymmetric anisotropic plate structures within strain gradient elasticity. The general three-dimensional anisotropic gradient theory is reduced to a two-dimensional formulation for the analysis of thick plate structures. The third-order shear deformation theory (TSDT) takes into account quadratic variation of the transverse shear strains of the plate and does not require shear correction factors. In order to investigate the case of small strains but moderate rotations, the von Karman strains are considered. The TSDT is also simplified to anisotropic Kirchhoff plate theory within gradient elasticity. To study specific material properties in more detail, the (Kirchhoff and TSDT) gradient plate theory of general anisotropy is simplified to the more practical case of orthotropic plates. It is observed that the gradient theory provides the capability to capture the size effects in anisotropic plate structures. As case studies, the bending and buckling behaviors of the simply supported orthotropic (Kirchhoff and TSDT) plates are studied. Variationally consistent boundary conditions are also discussed. Finally, analytical solutions are presented for the bending and buckling of simply supported orthotropic Kirchhoff plates. The effects of internal length scales on deflections and buckling loads are presented.

Buckling of Functionally-Graded Beams with Partially Delaminated Piezoelectric Layers

Abstract: The critical buckling load of a functionally-graded simply-supported beam with partially delaminated piezoelectric layers is discussed. The governing equations of motion are derived using two different, i.e. Euler–Bernoulli and Timoshenko beam theories, and the buckling load was evaluated from the exact solution to the corresponding eigenvalue equation. The equations were simplified to some extent by shifting the coordinate origin such that there is zero bending-extension coupling. Effects induced by the delaminated length, asymmetry, piezoelectric thickness, voltage, and the functionally graded materials (FGMs) volume fraction are evaluated. The validity of results and the invoked assumptions were successfully verified with existing results and finite element calculations. There is some difference between the analytical buckling load and that calculated with the finite element method (FEM), proven small in amount but predictable in terms of the piezoelectric thickness. Further, a general formula is derived to evaluate the dimensionless critical buckling load with the parameters obtained from regression analysis of the solution that can be utilized for all cases where the materials are not far different in their mechanical properties. The results can be utilized as benchmark design tool.

Optimal design of laminated piezocomposite energy harvesting devices considering stress constraints

Abstract: Summary Energy harvesting devices are smart structures capable of converting the mechanical energy (generally, in the form of vibrations) that would be wasted otherwise in the environment into usable electrical energy. Laminated piezoelectric plate and shell structures have been largely used in the design of these devices because of their large generation areas. The design of energy harvesting devices is complex, and they can be efficiently designed by using topology optimization methods (TOM). In this work, the design of laminated piezocomposite energy harvesting devices has been studied using TOM. The energy harvesting performance is improved by maximizing the effective electric power generated by the piezoelectric material, measured at a coupled electric resistor, when subjected to a harmonic excitation. However, harmonic vibrations generate mechanical stress distribution that, depending on the frequency and the amplitude of vibration, may lead to piezoceramic failure. This study advocates using a global stress constraint, which accounts for different failure criteria for different types of materials (isotropic, piezoelectric, and orthotropic). Thus, the electric power is maximized by optimally distributing piezoelectric material, by choosing its polarization sign, and by properly choosing the fiber angles of composite materials to satisfy the global stress constraint. In the TOM formulation, the Piezoelectric Material with Penalization and Polarization material model is applied to distribute piezoelectric material and to choose its polarization sign, and the Discrete Material Optimization method is applied to optimize the composite fiber orientation. The finite element method is adopted to model the structure with a piezoelectric multilayered shell element. Numerical examples are presented to illustrate the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.

Fluctuation relation based continuum model for thermoviscoplasticity in metals

Abstract: A continuum plasticity model for metals is presented from considerations of non-equilibrium thermodynamics. Of specific interest is the application of a fluctuation relation that subsumes the second law of thermodynamics en route to deriving the evolution equations for the internal state variables. The modelling itself is accomplished in a two-temperature framework that appears naturally by considering the thermodynamic system to be composed of two weakly interacting subsystems, viz. a kinetic vibrational subsystem corresponding to the atomic lattice vibrations and a configurational subsystem of the slower degrees of freedom describing the motion of defects in a plastically deforming metal. An apparently physical nature of the present model derives upon considering the dislocation density, which characterizes the configurational subsystem, as a state variable. Unlike the usual constitutive modelling aided by the second law of thermodynamics that merely provides a guideline to select the admissible (though possibly non-unique) processes, the present formalism strictly determines the process or the evolution equations for the thermodynamic states while including the effect of fluctuations. The continuum model accommodates finite deformation and describes plastic deformation in a yield-free setup. The theory here is essentially limited to face-centered cubic metals modelled with a single dislocation density as the internal variable. Limited numerical simulations are presented with validation against relevant experimental data. (C) 2016 Elsevier Ltd. All rights reserved.

Contribution of material properties of cellular components on the viscoelastic, stress-relaxation response of a cell during AFM indentation

Abstract: The close relationship between the mechanical properties of biological cells, namely, elasticity, viscosity, and the state of its disease condition has been widely investigated using atomic force microscopy (AFM). In this study, computational simulation of the AFM indentation is carried out using a finite element (FE) model of an adherent cell. A parametric evaluation of the material properties of the cellular components on the viscoelastic, stress-relaxation response during AFM indentation is performed. In addition, the loading rate, the size of the nucleus, and the geometry of the cell are varied. From the present study, it is found that when comparing the material properties derived from experimental force-deflection curves, the influence of loading rates should be accommodated. It also provides a framework that can quantify the variation of the mechanical property with various stages of malignancy of the cancer cell, a potential procedure for cancer diagnosis.

Transient analysis of Cosserat rod with inextensibility and unshearability constraints using the least-squares finite element model

Abstract: In this study, time-dependent fully discretized least-squares finite element model is developed for the transient response of Cosserat rod having inextensibility and unshearability constraints to simulate a surgical thread in space. Starting from the kinematics of the rod for large deformation, the linear and angular momentum equations along with constraint conditions for the sake of completeness are derived. Then, the α-family of time derivarive approximation is used to reduce the governing equations of motion to obtain a semi-discretized system of equations, which are then fully discretized using the least-squares approach to obtain the non-linear finite element equations. Newton׳s method is utilized to solve the non-linear finite element equations. Dynamic response due to impulse force and time-dependent follower force at the free end of the rod is presented as numerical examples.

Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory

Abstract: In recent papers, Surana et al presented internal polar non-classical Continuum theory in which velocity gradient tensor in its entirety was incorporated in the conservation and balance laws Thus, this theory incorporated symmetric part of the velocity gradient tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials In another recent paper Surana et al presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to orders n and Cauchy moment tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient tensor In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient tensor are assumed not to contribute to dissipation Secondly, the skew symmetric part of the velocity gradient tensor is used as rotation rates to determine rate of rotation gradient tensor This is an approximation to true convected time derivatives of the rotation gradient tensor The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient tensor; (2) first convected derivative of the symmetric part of the rotation gradient tensor as used by Surana et al is only approximate; (3) has inconsistent treatment of dissipation due to Cauchy moment tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress tensor in which convected time derivatives of up to order n are considered in the theory The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n are conjugate with the moment tensor Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress tensor, moment tensor and heat vector Stress tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress tensor and the moment tensor are further decomposed into equilibrium and deviatoric tensors It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress tensor, moment tensor and heat vector Density in the current configuration, convected time derivatives of the strain tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n, temperature gradient tensor and temperature are considered as argument tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al for internal polar non-classical incompressible thermofluids

A Higher Order Equilibrium Finite Element Method

Abstract: In this paper a mixed spectral element formulation is presented for planar, linear elasticity. The degrees of freedom for the stress are integrated traction components, i.e. surface force components. As a result the tractions between elements are continuous. The formulation is based on minimization of the complementary energy subject to the constraints that the stress field should satisfy equilibrium of forces and moments. The Lagrange multiplier which enforces equilibrium of forces is the displacement field and the Lagrange multiplier which enforces equilibrium of moments is the rotation. The formulation satisfies equilibrium of forces pointwise if the body forces are piecewise polynomial. Equilibrium of moments is weakly satisfied. Results of the method are given on orthogonal and curvilinear domains and an example with a point singularity is given.

Adaptive isogeometric analysis based on a combined r-h strategy

Abstract: In the present work, an r-h adaptive isogeometric analysis is proposed for plane elasticity problems. For performing the r-adaption, the control net is considered to be a network of springs with the individual spring stiffness values being proportional to the error estimated at the control points. While preserving the boundary control points, relocation of only the interior control points is made by adopting a successive relaxation approach to achieve the equilibrium of spring system. To suit the noninterpolatory nature of the isogeometric approximation, a new point-wise error estimate for the h-refinement is proposed. To evaluate the point-wise error, hierarchical B-spline functions in Sobolev spaces are considered. The proposed adaptive h-refinement strategy is based on using De-Casteljau’s algorithm for obtaining the new control points. The subsequent control meshes are thus obtained by using a recursive subdivision of reference control mesh. Such a strategy ensures that the control points lie ...