Melek Usal

The Effects of Shear on the Deflection of Simply Supported Composite Beam Loaded Linearly

In this study two deflection functions due to both the flexure and the shear of an orthotropic simply supported beam loaded linearly are obtained by means of the anisotropic elasticity theory. In order to see the effect of shear, the deflections are calculated for different fiber directions. Two different composite materials are used during the deflection analysis. The error level and the shear deflection for the thermoplastic composite beam are the smallest for 45 orientation angle, and that for the polymer-matrix composite beam are the smallest for 90 orientation angle. When the cross-sectional height to the beam-length ratio increases, the shear effect also increases.

Static and Dynamic Analysis of Simply Supported Beams

In this study, two deflection functions due to flexure and shear have been obtained for the global form of composite materials. Two different composite materials are selected for comparison of these deflection functions. These composites are: polymer matrix composite simply supported beam, reinforced by unidirectional fibers; and thermoplastic simply supported beam, reinforced by woven Cr-Ni steel fibers. In accordance with these different material properties, analytical and numerical solutions have been carried out. For 0, 30, 45, 60, and 90 fiber orientation angles, static and dynamic behavior of the two different composite materials are examined. Numerical solutions are given as graphical forms. In addition to modal analysis, two different composite materials have been realized. Natural frequencies and vibration modes are given as graphical forms. ANSYS and MATLAB software are used for numerical analysis of the different composite materials.

A Mathematical Model for the Electrothermomechanical Behavior of Dielectric Viscoelastic Materials

This study examines the behavior of a material under electro-mechanical loads arising of the external medium in a viscoelastic and dielectric material, provided that isotropy constraint is imposed on the material. The reaction of the object to external loads is expressed in elastic stress, dissipative stress and electrical polarization. Based on constitutive axioms and concepts pertaining to the symmetry group of the material, constitutive functionals have been obtained. To materially determine arguments of these functionals, findings of the theory of invariants have been used as a routine method. As a result constitutive equations pertaining to elastic stress, polarization field and dissipative stress have been found in material and spatial coordinates whereas symmetric and asymmetric stresses have been found in terms of displacement gradient and its derivative by using the expressions of elastic stress, polarization field and dissipative stress.

A Mathematical Model for Thermomechanical Behavior of Arbitrary Fiber Reinforced Viscoelastic Composites - II

This study examines the behavior of a composite material under mechanical loads arising of the external medium in a viscoelastic medium reinforced by two arbitrary independent and inextensible fiber families, provided that isotropy constraint is imposed on the matrix material. Despite the fact that the matrix material is isotropic the model in consideration bears the characteristic of directed media included in the transverse isotropy symmetry group solely due to its fiber distribution. The reaction of the object to external loads is expressed in elastic stress and dissipative stress. Based on constitutive axioms and concepts pertaining to the symmetry group of the material, constitutive functionals have been obtained. To materially determine arguments of these functionals, findings of the theory of invariants have been used as a routine method. As a result constitutive equations pertaining to elastic stress and dissipative stress have been found in material and spatial coordinates whereas symmetric and asymmetric stresses have been found in terms of displacement gradient and its derivative by using the expressions of elastic stress and dissipative stress. K e y W o r d s : Viscoelasticity, balance equations, isotropy, balance equations, constitutive equations.

On Magneto-Viscoelastic Behavior of Fiber-Reinforced Composite Materials Part I: Anisotropie Matrix Material

In this work behavior of an arbitrarily fiber-reinforced viscoelastic and magneto-sensitive material is investigated systematically in the frame of modern continuum mechanics when they are subjected to external loadings. The matrix material is supposed to be made of viscoelastic material with magnetic sensitivity involving an artificial anisotropy due to fiber reinforcing by arbitrary distribution. Magneto-viscoelastic response of the material will show up as a stress consisting of a reversible and irreversible parts along with a magnetization field. Magnetic field and elastic stress field is derived from a thermodynamic potential, with the dissipative stress arising as a result of viscous properties of the material. Response functions in regard to objectivity axiom III depend in general to Green deformation and rate tensors, magnetic field, fiber distribution and temperature distribution. Constitutive equations for the stress and magnetization with the use of relevant balance equations yields field equations for the formulation and solution of boundary value problems for the bodies made of those materials under considerations.

On Continuum Damage Modeling of Fiber Reinforced Viscoelastic Composites with Microcracks in terms of Invariants

A continuum damage model is developed for the linear viscoelastic behavior of composites with microcracks consisting of an isotropic matrix reinforced by two arbitrarily independent and inextensible fiber families. Despite the fact that the matrix material is isotropic, the model in consideration bears the characteristic of directed media included in the transverse isotropy symmetry group solely due to its fibers distributions and the existence of microcracks. Using the basic laws of continuum damage mechanics and equations belonging to kinematics and deformation geometries of fibers, the constitutive functions have been obtained. It has been detected as a result of the thermodynamic constraints that the stress potential function is dependent on two symmetric tensors and two vectors, whereas the dissipative stress function is dependent on four symmetric tensors and two vectors. To determine arguments of the constitutive functionals, findings relating to the theory of invariants have been used as a method because of the fact that isotropy constraint is imposed on the material. As a result the linear constitutive equations of elastic stress, dissipative stress, and strain energy density release rate have been written in terms of material coordinate description. Using these expressions, total stress has been found.

A Mathematical Model for the Magnetoelastic Behavior of Anisotropic Magnetic Sensitive Materials Based on Continuum Theory

Abstract Magnetoelastic behavior of a magnetic sensitive material has been analyzed theoretically in the present paper. The theory is formulated in the context of continuum electromagnetics. The solid medium is supposed to be made of elastic material with magnetic sensitivity and to be nonlinear, homogeneous, compressible, isothermal, has anisotropy. Magneto-elastic response of the material will show up as a stress and a magnetization field. From the formulation belonging to the constitutive equations, it has been observed that the stress and the magnetization have been derived from a scalar-valued thermodynamic potential defined in calculations. As a result of thermodynamic constraints, it has been determined that the free energy function is dependent on Green deformation tensor, magnetic field, and temperature distribution. The free energy function has been represented by a power series expansion and the type and number of terms taken into consideration in this series expansion has determined the non-linearity of the medium. Constitutive equations of symmetric stress, magnetization field and asymmetric stress have been obtained in both material and spatial coordinates. The quasi-linear constitutive equations which on material coordinates have been obtained by expressions (63)–(65). The quasi-linear constitutive equations have been given in expressions (70)–(72) on spatial coordinates. Finally, the quasi-linear constitutive equations of the symmetric stress and magnetization field are substituted in the relevant balance equations to obtain the field equations. The field equations containing the unknowns uk and ϕ coordinates have been obtained by expressions (75) and (76). Solution of these field equations under initial and boundary conditions forms the mathematical structure of specified a boundary value problem.

A Constitutive Formulation for the Linear Thermoelastic Behavior of Arbitrary Fiber-Reinforced Composites

The linear thermoelastic behavior of a composite material reinforced by two independent and inextensible fiber families has been analyzed theoretically. The composite material is assumed to be anisotropic, compressible, dependent on temperature gradient, and showing linear elastic behavior. Basic principles and axioms of modern continuum mechanics and equations belonging to kinematics and deformation geometries of fibers have provided guidance and have been determining in the process of this study. The matrix material is supposed to be made of elastic material involving an artificial anisotropy due to fibers reinforcing by arbitrary distributions. As a result of thermodynamic constraints, it has been determined that the free energy function is dependent on a symmetric tensor and two vectors whereas the heat flux vector function is dependent on a symmetric tensor and three vectors. The free energy and heat flux vector functions have been represented by a power series expansion, and the type and the number of terms taken into consideration in this series expansion have determined the linearity of the medium. The linear constitutive equations of the stress and heat flux vector are substituted in the Cauchy equation of motion and in the equation of conservation of energy to obtain the field equations.

Determination of constitutive equations dependent on temperature changes for dielectric materials having micro-voids

Abstract A damage constitutive model based on continuum damage mechanics (CDM) is proposed to investigate the electro-thermomechanical behavior of a thermoelastic dielectric structure in the present paper. The solid medium is assumed to be dielectric, incompressible, homogeneous, dependent on temperature gradient, having micro-voids, and showing linear elastic behavior. The matrix material made of elastic material involving an artificial anisotropy due to the existence of micro-voids has been assumed as an isotropic medium. Damage is incorporated by two symmetric, second-rank, tensor-valued, internal state variables which represent the total areas of “active” and “passive” voids contained within a representative volume element. Using fundamental concepts of continuum electrodynamics, CDMs and irreversible thermodynamics, the constitutive functionals have been obtained. It has been detected as a result of the thermodynamic constraints that stress potential function depends on two symmetric tensors and a vector, whereas the heat flux vector function depends on two symmetric tensors and two vectors. Since the matrix material has been assumed as an isotropic medium, the constitutive equations based on the constitutive functionals, which are stress potential and heat flux vector, have been obtained using the theory of invariants. Finally, the constitutive equations belonging to symmetric stress, polarization field, asymmetric stress, heat flux vector, and strain energy density release rate have been written in terms of material coordinate description.

Comparison of Analytical and Numerical Solutions for Simply Supported Composite Beams under Uniformly Distributed Loads

The deflections due to bending are considered in the majority of beam applications. However, we must also examine a secondary deflection due to shear for short, deep beams. In this study, total deflections at the mid point of the simply supported composite beams subjected to uniformly distributed forces were compared analytically and numerically for beams with different geometrical parameters and composite with different orientation angles. The numerical study was performed using ANSYS software. It is shown that the magnitude of total deflection depends on the length and height of the beam. The analytical and numerical solutions are in good agreement with each other.