Showing papers by "J. N. Reddy published in 2012"

Bending relationships between the modified couple stress-based functionally graded Timoshenko beams and homogeneous Bernoulli–Euler beams

Abstract: The Bernoulli–Euler and Timoshenko beam theories are reformulated using a modified couple stress theory and through-thickness power-law variation of a two-constituent material [functionally graded material (FGM)]. The model contains a material length scale parameter that can capture the size effect in a FGM. The equations are then used to develop algebraic relationships for the deflections, slopes, stress resultants of the Timoshenko beam theory (TBT) for microstructure-dependent FGM beams in terms of the same quantities of the conventional Bernoulli–Euler beam theory (BET). The relationships allow determination of the solutions of the TBT for microstructure-dependent FGM beams whenever solutions based on the BET are available. Examples of the use of the relationships are presented using straight beams with simply supported and clamped boundary conditions.

Free vibration and buckling analysis of beams with a modified couple-stress theory

Abstract: A model based on a modified couple stress theory for the free vibration and buckling analyses of beams is presented. The model also incorporates the Poisson's effect and allows the analysis of Timoshenko beams with any arbitrary end boundary condition. The natural frequencies and buckling loads are computed using the Ritz method. Parametric studies show that, while the natural frequencies and the buckling loads increase monotonically with the increase of the material length scale, they present a minimum in certain values of the Poisson's ratio. A study relating the classical elasticity and the couple stress strain energies is also presented. By establishing this relation, explicit formulas to obtain the natural frequencies and buckling loads, in which the couple stress and Poisson's effects are accounted for, in terms of the buckling loads of the classical elasticity are found. These formulas, which are valid when the shear strain and stress are zero, allow an expedite computation of natural frequencies and buckling loads of beams with couple stress and Poisson's effect.

Design of laminated piezocomposite shell transducers with arbitrary fiber orientation using topology optimization approach

Abstract: SUMMARY Sensor and actuator based on laminated piezocomposite shells have shown increasing demand in the field of smart structures. The distribution of piezoelectric material within material layers affects the performance of these structures; therefore, its amount, shape, size, placement, and polarization should be simultaneously considered in an optimization problem. In addition, previous works suggest the concept of laminated piezocomposite structure that includes fiber-reinforced composite layer can increase the performance of these piezoelectric transducers; however, the design optimization of these devices has not been fully explored yet. Thus, this work aims the development of a methodology using topology optimization techniques for static design of laminated piezocomposite shell structures by considering the optimization of piezoelectric material and polarization distributions together with the optimization of the fiber angle of the composite orthotropic layers, which is free to assume different values along the same composite layer. The finite element model is based on the laminated piezoelectric shell theory, using the degenerate three-dimensional solid approach and first-order shell theory kinematics that accounts for the transverse shear deformation and rotary inertia effects. The topology optimization formulation is implemented by combining the piezoelectric material with penalization and polarization model and the discrete material optimization, where the design variables describe the amount of piezoelectric material and polarization sign at each finite element, with the fiber angles, respectively. Three different objective functions are formulated for the design of actuators, sensors, and energy harvesters. Results of laminated piezocomposite shell transducers are presented to illustrate the method. Copyright © 2012 John Wiley & Sons, Ltd.

Vibration of Timoshenko beams using non-classical elasticity theories

Abstract: This paper presents a comparison among classical elasticity, nonlocal elasticity, and modified couple stress theories for free vibration analysis of Timoshenko beams. A study of the influence of rotary inertia and nonlocal parameters on fundamental and higher natural frequencies is carried out. The nonlocal natural frequencies are found to be lower than the classical ones, while the natural frequencies estimated by the modified couple stress theory are higher. The modified couple stress theory results depend on the beam cross-sectional size while those of the nonlocal theory do not. Convergence of both non-classical theories to the classical theory is observed as the beam global dimension increases.

Rate constitutive theory for ordered thermoelastic solids

Abstract: When the mathematical models for the deforming solids are constructed using the Eulerian description, the material particle displacements and hence the strain measures are not known. In such cases the constitutive theory must utilize convected time derivatives of the strain measures. The entropy inequality provides a mechanism for determining constitutive equations for the equilibrium stress with the additional requirement that the work expanded due to the deviatoric part of the Cauchy stress tensor be positive, but provides no mechanism for establishing the constitutive theory for it. In the development of the constitutive theory in the Eulerian description for thermoelastic solids, one must consider a coordinate system in the current configuration in which the deformed material lines can be identified. Thus the covariant, contravariant and Jaumann convected coordinate systems are natural choices for the development of the constitutive theory. The compatible conjugate pairs of convected time derivatives of the stress and strain measures in these bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of the constitutive theory for thermoelastic solids. This framework has a foundation based on the basic principles and axioms of continuum mechanics but the resulting constitutive theory must satisfy the conditions resulting from the entropy inequality to ensure thermodynamic equilibrium of the deforming matter. This paper presents development of rate constitutive theories for compressible as well as incompressible, homogeneous, isotropic solids. The density, temperature, and temperature gradient in the current configuration and the convected time derivatives of the strain tensor up to any desired order in the chosen basis are considered as the argument tensors of the first convected time derivative of the deviatoric Cauchy stress tensor and heat vector. The thermoelastic solids described by these constitutive theories are termed ordered thermoelastic solids due to the fact that the constitutive theories for the deviatoric Cauchy stress tensor and heat vector are dependent on the convected time derivatives of the strain tensor up to any desired order, the highest order defining the order of the solid.

Mechanics of Solids and Structures

Abstract: Introduction Modeling of Engineering Systems Review of Statics Concepts of Stress and Strain Influence of Material Properties Principles of Mechanics of Solids Use of Numerical Methods and Computers Statically Determinate Systems Pin-Jointed Structures Uniformly Loaded Thin Shells Flexible Cables Relationships between Stress and Strain Hydrostatic Stress and Volumetric Strain Elastic Stress-Strain Equations Other Stress-Strain Relationships Deformations of Statically Determinate Systems Statically Indeterminate Systems Pin-Jointed Structures Other Statically Indeterminate Systems Bending of Beams: Moments, Forces, and Stresses Some Practical Examples of Beams Shear Forces and Bending Moments in Beams Stresses Due To Bending Combined Bending and Axial Loads Bending of Beams-Deflections Relationship between Curvature and Bending Moment Deflection of Statically Determinate Beams Deflection of Statically Indeterminate Beams Computer Method for Beam Deflections Torsion Torsion of Shafts Statically Determinate Torsion Problems Statically Indeterminate Torsion Problems Combined Bending and Torsion Instability and the Buckling of Struts and Columns Stable, Neutral, and Unstable Equilibrium Buckling of Pin-Ended Struts Struts and Columns with Other End Conditions Transformations of Stress and Strain Transformation of Stress Transformation of Strain Computer Method for Stresses and Strains at a Point Yield and Fracture Criteria Equilibrium and Compatibility Equations: Beams and Thick-Walled Cylinders Stress Equilibrium Equations Strain Compatibility Equations Application to Beam Bending Application to Thick-Walled Cylinders and Disks Energy Methods of Structural Mechanics Concepts of Work and Energy Strain Energy and Complementary Strain Energy Virtual Work and Complementary Virtual Work Variational Operator and Fundamental Lemma The Principle of Virtual Displacements and its Special Cases The Principle of Virtual Forces and its Special Cases Appendices Appendix A: Properties of Materials Appendix B: Moments of Area Appendix D: Deflections and Slopes for Some Common Cases of the Bending of Uniform Beams Index

A SPECTRAL/hp NONLINEAR FINITE ELEMENT ANALYSIS OF HIGHER-ORDER BEAM THEORY WITH VISCOELASTICITY

Abstract: In this paper, a finite element model for efficient nonlinear analysis of the mechanical response of viscoelastic beams is presented. The principle of virtual work is utilized in conjunction with the third-order beam theory to develop displacement-based, weak-form Galerkin finite element model for both quasi-static and fully-transient analysis. The displacement field is assumed such that the third-order beam theory admits C0 Lagrange interpolation of all dependent variables and the constitutive equation can be that of an isotropic material. Also, higher-order interpolation functions of spectral/hp type are employed to efficiently eliminate numerical locking. The mechanical properties are considered to be linear viscoelastic while the beam may undergo von Karman nonlinear geometric deformations. The constitutive equations are modeled using Prony exponential series with general n-parameter Kelvin chain as its mechanical analogy for quasi-static cases and a simple two-element Maxwell model for dynamic cases. The fully discretized finite element equations are obtained by approximating the convolution integrals from the viscous part of the constitutive relations using a trapezoidal rule. A two-point recurrence scheme is developed that uses the approximation of relaxation moduli with Prony series. This necessitates the data storage for only the last time step and not for the entire deformation history.

Biomechanics of breast tumor: effect of collagen and tissue density

Abstract: Various clinical studies have established the influence of breast tissue density on the onset and formation of breast tumor. In this paper we develop a mathematical correlation of the change in density with the change in collagen content and the corresponding change in the bio-mechanical characteristics of the breast tissue. The developed computational model is used in understanding the mechanical response of the breast tissue using multiscale mathematical homogenization techniques. The mathematical modeling is carried out using hyperelastic material formulations for the macro-scale finite element simulations and small strain material models for the lower order homogenization.

On multigrid methods for the solution of least-squares finite element models for viscous flows

Abstract: There is a vast literature on least-squares finite element models (LSFEM) applied to fluid dynamics problems. The hp version of the least-squares models is computationally expensive, which necessitates the usage of elegant methods for solving resulting systems of equations. Amongst some of the schemes used for solving large systems of equations is the element-by-element (EBE) solution technique, which has found widespread use in least-squares applications. However, the use of EBE techniques with Jacobi preconditioning leads to very little performance gains as compared to solving a non-preconditioned system. Because of such considerations, the hp version LSFEM solutions are computationally intensive. In this study, we propose to solve the LSFEM systems using the multigrid method, which offers superior convergence rates compared to the EBE–JCG. We demonstrate the superior convergence of the Multigrid solver compared to Jacobi preconditioning for the wall-driven cavity and backward facing step problems using...

Finite element model for nutrient distribution analysis of a hollow fiber membrane bioreactor.

Abstract: SUMMARY Hollow fiber membrane bioreactors (HFMB) are extensively used for the development of tissue substitutes for bones and cartilages. In an HFMB, the nutrient transport is dependent on the material properties of the porous scaffold and fiber membrane and also on the fluid flow through the hollow fiber. The difficulty in obtaining real-time data along with the presence of large number of variables in experimental studies have lead to increased application of computational models for the performance analysis of bioreactors. A major difficulty in the computational analysis of HFMB is the modeling of the interactions at the fluid and porous scaffold interfaces, which has often been neglected or incorporated using specific boundary conditions. In this study, a new FEM is developed to analyze the fluid flow in the fluid-porous region with the interface coupled directly into the FEM. Distribution of nutrients in the bioreactor is studied by coupling mass transport equations to the fluid-porous finite element framework. The new model is implemented to study the influence of permeability, cell density, and flow rate on the nutrient concentration distribution in the HFMB. The developed computational framework is an ideal tool to study fluid flow through porous-open channels and can also be used for the design of bioreactors for optimal tissue growth. Copyright © 2011 John Wiley & Sons, Ltd.

Computations of Evolutions for Isothermal Viscous and Viscoelastic Flows in Open Domains

Abstract: This paper presents numerical computations of evolutions described by the initial value problems (IVPs) in isothermal incompressible viscous and viscoelastic flows in open domains using a space-time finite element model in hpk framework with space-time variationally consistent (STVC) integral forms. The mathematical models for viscous and viscoelastic liquids include the continuity and momentum equations in terms of the total stress tensor and the velocity vector. The second law of thermodynamics, namely the Clausius-Duhem in-equality, forms the basis for the constitutive equations for all matters including the ones considered here. For the liquids considered here, it becomes necessary to decompose the total stress tensor into equilibrium stress tensor and the deviatoric stress tensor. Determination of equilibrium stress using the Clausius-Duhem inequality with incompressibility constraint yields mechanical pressure as a function of temperature in the case of thermofluids otherwise constant but a function...

Fluid-Solid Interaction of Incompressible Media Using h, p, k Mathematical and Computational Framework

Abstract: This paper considers multi-media interaction processes in which the interacting media are incompressible elastic solids and incompressible liquids such as Newtonian fluids, generalized Newtonian fluids, dilute polymeric liquids described by Maxwell, and Oldroyd-B models or dense polymeric liquids with Giesekus constitutive model. The mathematical models for both solids and liquids are developed using conservation laws in Eulerian description for isothermal conditions with velocities, pressure, and deviatoric stresses as dependent variables. The constitutive equations for the solids and the liquids provide closure to the governing differential equations resulting from the conservation laws. For Newtonian and generalized Newtonian fluids, the commonly used constitutive equations are well known in terms of first convected derivative of the strain tensor, stress tensor, and the transport properties of the fluids. For dilute and dense polymeric liquids that are viscous as well as elastic, the mathematical mode...

A critical evaluation of various nonlinear beam finite elements

Abstract: The characteristics of interdependent interpolation and mixed interpolation nonlinear beam finite elements are investigated in comparison with the equal-order interpolation element with uniform reduced integration. The stiffness matrix of the 3-noded and 4-noded equal order interpolation elements is identical to that of the 2-noded interdependent interpolation element if the internal nodal degrees-of-freedom are eliminated. The extension of the latter to include nonlinear kinematics by approximating the extensional displacement and the twist rotation with quadratic and cubic Lagrange polynomials yields unsatisfactory results. The 2-noded, 3-noded, and 4-noded mixed interpolation elements using one-, two-, and three-point quadrature rules, respectively, are shown to be equivalent to the corresponding uniform interpolation elements employing the same quadrature rules. The equivalence is established in the framework of nonlinear kinematics and anisotropic couplings.