Showing papers by "J. N. Reddy published in 2011"

Microstructure-dependent couple stress theories of functionally graded beams

Abstract: A microstructure-dependent nonlinear Euler–Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Karman geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler–Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.

A non-classical Mindlin plate model based on a modified couple stress theory

Abstract: A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton’s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.

A nonlocal curved beam model based on a modified couple stress theory

Abstract: A nonlocal Timoshenko curved beam model is developed using a modified couple stress theory and Hamilton's principle. The model contains a material length scale parameter that can capture the size effect, unlike the classical Timoshenko beam theory. Both bending and axial deformations are considered, and the Poisson effect is incorporated in the model. The newly developed nonlocal model recovers the classical model when the material length scale parameter and Poisson's ratio are both taken to be zero and the straight beam model when the radius of curvature is set to infinity. In addition, the nonlocal Bernoulli–Euler curved beam model can be realized when the normal cross-section assumption is restated. To illustrate the new model, the static bending and free vibration problems of a simply supported curved beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko curved beam model. Also, the differences in both the deflection and rotation predicted by the current and classical Timoshenko model are very large when the beam thickness is small, but they diminish with the increase of the beam height. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the nonlocal model is higher than that by the classical model, and the difference between them is significantly large only for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.

Buckling analysis of isotropic and laminated plates by radial basis functions according to a higher-order shear deformation theory

Abstract: The third-order shear deformation theory of Reddy and collocation with radial basis functions is used to predict the buckling loads of elastic plates. The theory accounts for parabolic distribution of the transverse strains through the thickness of the plate. It is shown that the collocation method with radial basis functions produces highly accurate critical buckling loads and modes.

A General Nonlinear Third-Order Theory of Functionally Graded Plates

Abstract: A general third-order plate theory that accounts for geometric nonlinearity and twoconstituent material variation through the plate thickness (i.e., functionally graded plates) is presented using the dynamic version of the principle of virtual displacements. The formulation is based on power-law variation of the material through the thickness and the von Karman nonlinear strains. The governing equations of motion derived herein for a general third-order theory with geometric nonlinearity and material gradation through the thickness are specialized to the existing classical and shear deformation plate theories in the literature. The theoretical developments presented herein can be used to develop finite element models and determine the effect of the geometric nonlinearity and material grading through the thickness on the bending, vibration, and buckling and postbuckling response of elastic plates.

Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity

Abstract: The nonlinear structural–acoustic problem considered in this study is the large amplitude free vibration of a rectangular elastic plate backed by a cavity. Very few classical solutions for this nonlinear structural–acoustic problem have been developed, although there are many for nonlinear plate or linear structural–acoustic problems. Thus, the main contributions of this study paper include (1) a concise multi-acoustic single structural modal formulation that is derived from two coupled partial differential equations representing the nonlinear structural free vibration and the acoustic pressure induced and (2) the approximate elliptical integral solution that is obtained by solving one residual equation only, and well agrees with that obtained from a harmonic balance finite element analysis. It is found that the natural frequency convergences with the increase in the numbers of acoustic modes and harmonic terms, and the effects of vibration amplitude, air cavity depth, and aspect ratio on the nonlinear natural frequency are also examined.

Transient analysis of composite and sandwich plates by radial basis functions

Abstract: This article presents a study of the linear transient response of composite plates using radial basis functions and collocation method. We use the Kansa method and radial basis functions in a pseud...

Nonlocal continuum crystal plasticity with internal residual stresses

Abstract: We derive a three-dimensional constitutive theory accounting for length-scale dependent internal residual stresses in crystalline materials that develop due to a non-homogeneous spatial distribution of the excess dislocation (edge and screw) density. The second-order internal stress tensor is derived using the Beltrami stress function tensor φ that is related to the Nye dislocation density tensor. The formulation is derived explicitly in a three-dimensional continuum setting for elastically isotropic materials. The internal stresses appear as additional resolved shear stresses in the crystallographic visco-plastic constitutive law for individual slip systems. Using this formulation, we investigate two boundary value problems involving single crystals under symmetric double slip. In the first problem, the response of a geometrically imperfect specimen subjected to monotonic and cyclic loading is investigated. The internal stresses affect the overall strengthening and hardening under monotonic loading, which is mediated by the severity of initial imperfections. Such imperfections are common in miniaturized specimens in the form of tapered surfaces, fillets, fabrication induced damage, etc., which may produce strong gradients in an otherwise nominally homogeneous loading condition. Under cyclic loading the asymmetry in the tensile and compressive strengths due to this internal stress is also strongly influenced by the degree of imperfection. In the second example, we consider simple shear of a single crystalline lamella from a layered specimen. The lamella exhibits strengthening with decreasing thickness and increasing lattice incompatibility with shearing direction. However, as the thickness to internal length-scale ratio becomes small the strengthening saturates due to the saturation of the internal stress. Finally, we present the extension of this approach for crystalline materials exhibiting elastic anisotropy, which essentially depends on the appropriate Green function within φ.

Alternative least‐squares finite element models of Navier‐Stokes equations for power‐law fluids

Abstract: Purpose – Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.Design/methodology/approach – For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is use...

Analysis of Thermal Interfacial Resistance Between Nanofins and Various Coolants

Abstract: Molecular dynamic simulations are performed to study the thermal interfacial resistance between a (5,5) carbon nanotube (CNT) and different matrix materials (water, ethyl alcohol, and 1-hexene). After the matrix-CNT ensemble is equilibrated to a base temperature of 300K, the temperature of the nanotube is raised to 750K by scaling the velocities of the carbon atoms. The system is then allowed to relax under constant energy. The exponential decay of temperature is used to calculate the thermal interfacial resistance. The interfacial resistance for water, ethyl alcohol and 1-hexene are found to be 2.13 × 10−8, 4.74 × 10−8, and 7.29 × 10−8 m2K/W, respectively, from the analysis.

Impact performance of Al/SiC functionally graded circular plates

Abstract: This experimental study addresses the effect of ceramic reinforcement on the impact performance of functionally graded (FG) circular plates. The FG circular plates were produced from two different ceramic (SiC or Al 2 O 3 ) and metal (Al) combinations using powder stacking-hot pressing technique. The Al/SiC FG circular plates exhibited better impact resistance than Al/Al 2 O 3 FG circular plates. The effect of ceramic particle size was also investigated on the impact performance of specimens with different composition variations. A ceramic particle size of 50 μm improved the impact performance. In addition, the material composition and impactor velocity also affected the impact performance. Thus, the contact force increases as the composition changes from metal-rich to ceramic-rich. The peak contact force increases with increasing the velocity of impactor.

Nonconventional finite element models for nonlinear analysis of beams

Abstract: In this study, mixed finite element models of beam bending are developed to include the membrane forces and shear forces in addition to the bending moments and displacements. Mixed finite element models were developed based on the weighted residual statements of governing equations. The Euler–Bernoulli beam theory (EBT) and the Timoshenko beam theory (TBT) are used. The effectiveness of the new mixed models is evaluated in light of other mixed models to show the advantages. Each newly developed model is examined and compared with other models to verify its performance under various boundary conditions. In the linear analysis, solutions are compared with available analytical solutions and solutions of existing models. In the nonlinear case, direct and Newton–Raphson methods are used to solve the nonlinear equations. The converged solutions are compared with available solutions of the displacement models. Post-processed data of the mixed model developed herein shows better accuracy than the conventional dis...

Theoretical Formulations for Finite Element Models of Functionally Graded Beams with Piezoelectric Layers

Abstract: In this paper an overview of functionally graded materials and constitutive relations of electro elasticity for three-dimensional deformable solids is presented, and governing equations of the Bernoulli–Euler and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material and piezoelectric layers are developed using the principle of virtual displacements. The formulation is based on a power-law variation of the material in the core with piezoelectric layers at the top and bottom. Virtual work statements of the two theories are also developed and their finite element models are presented. The theoretical formulations and finite element models presented herein can be used in the analysis of piezolaminated and adaptive structures such as beams and plates. © 2011 IAU, Arak Branch. All rights reserved.

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

Abstract: The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method ( GM ), Galerkin method with weak form ( / GM WF ), Petrov-Galerkin method ( PGM ), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspondence is established between these integral forms and the elements of the calculus of variations: 1) to determine which methods of approximation yield unconditionally stable (variationally consistent integral forms, VC ) computational processes for which types of operators and, 2) to establish which integral forms do not yield unconditionally stable computations (variationally inconsistent integral forms, VIC ). It is shown that variationally consistent time integral forms in hpk framework yield computational processes for ODEs in time that are unconditionally stable, provide a mechanism of higher order global differentiability approximations as well as higher degree local approximations in time, provide control over approximation error when used as a time marching process and can indeed yield time accurate solutions of the evolution. Numerical studies are presented using standard model problems from the literature and the results are compared with Wilson’s  method as well as Newmark method to demonstrate highly meritorious features of the proposed methodology.

Thermomechanical response of functionally graded structures

Abstract: Functionally gradient materials (FGM) are a class of composites that have a gradual variation of material properties from one surface to another. These novel materials were proposed as thermal barrier materials for applications in space planes, space structures, nuclear reactors, turbine rotors, flywheels, and gears, to name only a few. As conceived and manufactured today, these materials are isotropic and nonhomogeneous Twoconstituent FGMs are usually made of a mixture of ceramic and metals for use in thermal environments. The ceramic constituent of the material provides the high temperature resistance due to its low thermal conductivity. The ductile metal constituent, on the other hand, prevents fracture due to high temperature gradient ina very short period of time. Typical situations where thermal shock occurs are during reentry of space vehicles, where the temperature changes from 273 o C to about 1,100 o C in a few minutes, and the advanced gas turbine, wherein a severe temperature transient of a change in temperature of 1,500 o C occurs over a time period of 15 s. The present lecture is an overview of the recent developments in the numerical modeling of functionally graded structures [1-5]. The lecture will present detailed discussion of the influence of geometric nonlinearity and temperature-dependent material properties on the response of functionally graded structures. Acknowledgement. The research reported herein was carried out under a research projects from the NSF, Grant CMMI-1030836 and MURI09 project from the AFOSR under grant FA9550-09-1-0686.The support is gratefully acknowledged. References