Initial Thermal Stresses Research Articles

Free Vibration Analysis of FGM Plates under Initial Thermal Stresses

Functionally graded materials (FGMs) are a new kind of composite materials which have a smooth variation of material properties along one or more directions. At each interface, the material is chosen according to specific applications and environment loadings. This paper presents some solutions to study the free vibration of FGM plates made of ceramic and metal. The formulation used is based on Reddys higher order shear deformation plate theory. Material properties are taken to be temperature-dependent, and vary continuously through the thickness direction according to a power law distribution (P-FGM). The plate is assumed to be initially stressed by temperature rise through the thickness. The energy functional of the system is obtained by using energy principles. Free vibration frequencies are then obtained by using a set of characteristic orthogonal polynomials and by applying Ritz method.

A 3D Discrete Layer-Differential Quadrature Free Vibration of Multi-Layered FG Annular Plates in Thermal Environment

A three-dimensional (3D) discrete layer approach coupled with the differential quadrature method (DQM) is developed for the free vibration analysis of the laminated functionally graded (FG) annular plates subjected to a thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses. After demonstrating the convergence behavior and accuracy of the method, two common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous core, are analyzed. The influences of different parameters on the solution are carried out.

OS0803 応力焦点化現象に及ぼす熱波の有限速度の影響

When an isotropic and homogeneous solid sphere and/or infinitely long solid cylinder are suddenly subjected to an instantaneous uniform heating, a stress wave occurs at the moment thermal impact is applied. The stress wave proceeds radially inward to the center of a sphere and/or cylinder. The wave may accumulate at the center and give rise to very large stress magnitudes, even though the initial thermal stress is relatively small. This phenomenon is called the stress-focusing effect. In this study, the stress focusing effect in a solid sphere and solid cylinder under instantaneous uniform heating at the free surface is studied on the basis of the generalized thermoelastic theories, that is, the Lord-Shulman's and the Green-Lindsay's theories. The combined governing equations of both theories are solved by the Laplace transform technique. The effects of the thermomechanical coupling and the relaxation time on the stress focusing phenomena as well as the singularity of stresses are discussed.

Stress and Strain Analysis of SRM Based on Monitored Shipboard Service Environment

Shipboard service environment of solid rocket motor (SRM) was monitored. Based on the monitored temperature and vibration data, the effects of temperature and low frequency vibration load on the stress and strain of SRM were analyzed by ABAQUS. It was shown by the results that the stress and strain of gain fluctuated near the initial thermal stress and strain with the low frequency vibration load. It was also found that the variations in stress and strain of the bonding interface were larger than that of the slot.

Dynamic response of functionally graded plates in thermal environment under moving load

As a first endeavor, the dynamic response of functionally graded (FG) plates in thermal environment under a moving load is investigated. The formulation is derived based on the first-order shear deformation theory (FSDT), which includes the effects of initial thermal stresses induced by the thermal environment and also the elastic foundation. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermal stresses are evaluated by means of the thermoelasticity theory. In order to develop a general solution procedure which can be employed for FG plates with general loading and boundary conditions, the finite element method (FEM) together with Newmark’s time integration scheme is adopted. The formulation and method of solution are validated by studying their convergence behavior and performing the comparison studies with existing results in the literature in the limit cases. Finally, the influences of temperature rise, material graded index, moving load velocity, elastic foundation parameters and boundary conditions on the dynamic behavior of FG plates in thermal environment and subjected to moving load is presented.

Study on the Interface Parameters of Composite Materials with Consideration of the Effect of Thermal Stresses

How to design the interface is a fundamental issue for the composite materials. Although a little work was concerned with the mechanical design of the interface, the effect of the initial thermal stresses has not been taken into account in those studies. While the thermal stresses due to the manufacturing process and the difference in coefficient of thermal expansion between matrix and reinforcing phase, do play an important role in the properties of the composites. In terms of a concept of the neutral inclusions that do not disturb the prescribed uniform stress field in the surrounding elastic body, the interface parameters are studied with consideration of the effect of an initial thermal stress field in this paper, indicating that the effect of thermal stresses may be eliminated by the design of interface parameters.

Three-dimensional free vibration of functionally graded truncated conical shells subjected to thermal environment

A three-dimensional (3D) free vibration analysis of the functionally graded (FG) truncated conical shells subjected to thermal environment is presented. The material properties are assumed to be temperature-dependent and graded in the radius direction, which can vary according to a simple power law distribution. The initial thermal stresses are obtained accurately by solving the thermoelastic equilibrium equations and by considering the two-dimensional axisymmetric temperature distribution in the shell. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to solve the thermal and thermo-mechanical governing equations. For this purpose, a mapping technique is employed to transform the cross section of the shell into the computational domain of DQM. The convergence behavior of the method is numerically demonstrated and comparison studies with the available solutions in the literature are performed. The effects of temperature dependence of material properties, geometrical parameters, material graded index, thermal and mechanical boundary conditions on the frequency parameters of the FG truncated conical shells are carried out.

Thermal effect on axisymmetric bending of functionally graded circular and annular plates using DQM

This paper presents the effects of thermal environment and temperature-dependence of the material properties on axisymmetric bending of functionally graded (FG) circular and annular plates. The material properties are assumed to be temperature-dependent and graded in the thickness direction. In order to accurately evaluate the effect of thermal environment, the initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. Governing equations and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the virtual work principle based on the elasticity theory. The differential quadrature method (DQM) as an efficient and robust numerical tool is used to obtain the initial thermal stresses and response of the plate. Comparison studies with some available results for FG plates are performed. The influences of temperature rise, temperature-dependence of material properties, material graded index and different geometrical parameters are carried out.

An Analytical Method for Free Vibration Analysis of Composite Beams Subjected to Initial Thermal Stresses

The purpose of this paper is to present exact solutions for the free vibration of symmetrically laminated composite beams. The present analysis includes the first shear deformation theory and the rotary inertia. The analytical solutions take into account the thermal effect on the free vibration characteristics of the composite beams. In particular, the aim of this work is to derive the exact closed-form characteristic equations for common boundary conditions. The different parameters that could affect the natural frequencies are included as factors (aspect ratio, thermal load-to-shear coefficient, ply orientation) to better perform dynamic analysis to have a good understanding of dynamic behavior of composite beams. In order to derive the governing set of equations of motion, the Hamilton’s principle is used. The system of ordinary differential equations of the laminated beams is then solved and the natural frequencies’ equations are obtained analytically for different boundary conditions. Numerical results are presented to show the influence of temperature rise, aspect ratio, boundary conditions and ply orientation on the natural frequencies of composite beams.

Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment

Free vibration analysis of functionally graded (FG) thin-to-moderately thick annular plates subjected to thermal environment and supported on two-parameter elastic foundation is investigated. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle based on the first order shear deformation theory (FSDT). The initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. Differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with the available solutions in the literature for isotropic and FG circular and annular plates. The effects of the temperature rise, elastic foundation coefficients, the material graded index and different geometrical parameters on the frequency parameters of the FG annular plates are investigated. The new results can be used as benchmark solutions for future researches.

Free vibration of functionally graded arbitrary straight-sided quadrilateral plates in thermal environment

As a first endeavor, the free vibration of functionally graded (FG) arbitrary straight-sided quadrilateral plates under thermal environment and based on the first order shear deformation theory (FSDT) is presented. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The initial thermal stresses are evaluated by solving the thermo-elastic equilibrium equations. The solution procedure is based on transformation of the governing equations from physical domain to computational domain and then the discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The accuracy of the present method is demonstrated by studying the free vibration of isotropic and FG plates with various shapes and comparing the solutions obtained against existing results in literature. Then, the effects of thickness-to-length ratio, volume fraction index, temperature rise, geometrical shape and the boundary conditions on the frequency parameters of the plate are studied.

Effect of Finite Velocity of Thermal Wave on Stress Focusing Phenomena

The analysis of a long cylindrical rod subjected to a sudden rise in temperature uniformly over its cross section has been studied by Ho [1]. Due to the instantaneous heating, the stress waves reflected from the cylindrical surface of the rod may accumulate at the center and give rise to very high stresses, even though the initial thermal stress is relatively small. This phenomenon is called the stress focusing effect. Hata has studied this effect for several cases of cylindrical rod [2-4]. The stress focusing effect for spheres has been studied by Mann-Nachbar [5] and Hata [6, 7]. However, these series of papers have been studied in the context of classical uncoupled theory of dynamic thermoelasticity. The theory of dynamic thermoelasticity which takes into account the coupling effects between temperature and strain fields involves the infinite thermal wave speed. The theory of generalized thermoelasticity has been developed in an attempt to eliminate the physical paradox of the infinite velocity of thermal propagation. At present, there are two theories of the generalized thermoelasticity: the first is proposed by Lord and Shulman [8], the second is proposed by Green and Lindsay [9]. Recently, other theories have presented (see Ignaczak and Hetnarski [10]). Furukawa et al. used the fundamental equations of generalized thermoelasticity introduced by Noda et al. [11], which include the Lord-Shulman theory and Green-Lindsay theory, and analyzed the one- and two-dimensional problems for plate, for example. In this paper, we treat an isotropic and homogeneous solid sphere. We use the fundamental equations of generalized thermoelasticity which include two theories. The effects of the thermo-mechanical coupling and the relaxation times on the stress focusing phenomena are examined.

Three-dimensional free vibration of thick functionally graded annular plates in thermal environment

The free vibration analysis of functionally graded (FG) thick annular plates subjected to thermal environment is studied based on the 3D elasticity theory. The material properties are assumed to be temperature dependent and graded in the thickness direction. Considering the thermal environment effects and using Hamilton's principle, the equations of motion are derived. The effects of the initial thermal stresses are considered accurately by obtaining them from the 3D thermoelastic equilibrium equations. The differential quadrature method (DQM) as an efficient and accurate numerical tool is used to solve both the thermoelastic equilibrium and free vibration equations. Very fast rate of convergence of the method is demonstrated. Also, the formulation is validated by comparing the results with those obtained based on the first-order shear deformation theory and also with those available in the literature for the limit cases, i.e. annular plates without thermal effects. The effects of temperature rise, material and geometrical parameters on the natural frequencies are investigated. The new results can be used as benchmark solutions for future researches.

Out-of-plane free vibration of functionally graded circular curved beams in thermal environment

Abstract A first known formulation for the out-of-plane free vibration analysis of functionally graded (FG) circular curved beams in thermal environment is presented. The formulation is based on the first order shear deformation theory (FSDT), which includes the effects of shear deformation and rotary inertia due to both torsional and flexural vibrations. The material properties are assumed to be temperature dependent and graded in the direction normal to the plane of the beam curvature. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle. Differential quadrature method (DQM), as an efficient and accurate numerical method, is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results, in the limit cases, with the available solutions in the literature for isotropic circular curved beams. In addition, for FG circular curved beams with soft simply supported edges, the results are compared with the obtained exact solutions. Then, the effects of temperature rise, boundary conditions, material and geometrical parameters on the natural frequencies are investigated.

Three-dimensional thermal buckling analysis of piezoelectric antisymmetric angle-ply laminates using finite layer method

Finite layer method is the most efficient numerical method for 3D analysis of simply supported rectangular plates. Using this method, the 3D analysis is transformed into one dimensional analysis by virtue of the orthogonal properties of trigonometric interpolation functions. In the present study, the finite layer method is extended to the thermal buckling analysis of piezoelectric antisymmetric angle-ply laminates, which may be combined with some symmetrical cross-plies. Full coupling between the thermal, electrical and mechanical fields is taken into consideration. Pre-buckling state is assumed to be steady, and initial thermal stresses are computed accordingly. The geometrical stiffness matrix is then formed, and the critical temperature and buckling mode are obtained. Numerical examples are presented to verify the proposed method. The critical temperature is determined for both the adiabatic and isothermal buckling processes. The thermal buckling behaviours of some piezoelectric laminates and the effects of the thermo-electro-mechanical coupling are investigated.

Evolution of internal strain during plastic deformation in magnesium matrix composites

Synchrotron radiation diffraction during in situ tensile tests has been used to evaluate the internal elastic strains within the grains of magnesium alloy, AZ31, unreinforced and reinforced with 5 and 10% volume of SiC particles. Composites present initial thermal residual stresses, which are positive (tensile) in the matrix and negative (compressive) in the reinforcing particles. Internal elastic strains evolve in a similar behaviour in the unreinforced AZ31 and in both composites. However, the accumulated elastic strains are reduced in the case of the composite because a part of the applied load is borne by the ceramic particles.

In-plane free vibration of functionally graded circular arches with temperature-dependent properties under thermal environment

Analysis of in-plane free vibration of functionally graded (FG) thin-to-moderately thick deep circular arches in thermal environments is presented based on first-order shear deformation theory (FSDT). The material properties are assumed to be temperature-dependent and graded in the thickness direction. Hamilton's principle is employed to derive the equations of motion and the related boundary conditions including the effects of initial thermal stresses. The temperature is assumed to be uniform through the arch surface and varied through the thickness. The initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. The differential quadrature method (DQM) as an efficient numerical tool is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The numerical solutions are validated by comparing to the solutions of the limited cases for isotropic arches, as well as by examining the solutions convergence behavior. Parametric studies are also conducted to study the effects of the temperature rise, boundary conditions and the material graded index on the frequency of the FG arches. Also, the impact of geometrical parameters such as the thickness-to-mean radius ratio and the opening angle are examined.

Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties

Abstract In this paper, the in-plane free vibration analysis of functionally graded (FG) thick circular arches subjected to initial stresses due to thermal environment is studied. The formulations are based on the two-dimensional elasticity theory. The material properties are assumed to be temperature-dependent and graded in the thickness direction. Considering the thermal environment effects, the equations of motion are derived using the Hamilton’s principle. The initial thermal stresses are obtained by solving the two-dimensional thermoelastic equilibrium equations. Two types of temperature rise, uniform and variable through the thickness, is considered. Differential quadrature method (DQM) is adopted to solve the equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with those available in the literature for isotropic arches. The effects of temperature rise, material and geometrical parameters on the natural frequencies are investigated. The new results can be used as benchmark solutions for future researches.

Formation of mountains on Io: Variable volcanism and thermal stresses

Thermal stresses are potentially important drivers of Io's tectonics and mountain building. It has been hypothesized that sustained local or regional shut down of heat-pipe volcanism on Io could lead to deep crustal heating and large compressive stresses [McKinnon, W.B., Schenk, P.M., Dombard, A.J., 2001. Geology 29, 103–106]. Such large stresses would then be relieved by thrust faulting and uplifting of crustal blocks, producing mountains like those observed on Io. Here we analyze the tectonic consequences of the heat-pipe model in detail, considering both the initial thermal stress state of a basalt or peridotite crust created by heat-pipe volcanism, and relative roles of subsidence stresses (due to burial of preexisting layers) and thermal stresses arising from variable volcanism and changes in crustal (∼lithosphere) thickness. We limit the magnitude of the potential subsidence stresses in our study, because the magnitude of subsidence stresses can be quite large, if not dominant. Results indicate that for a fixed crustal thickness, the region of failure and faulting moves closer to the surface as eruption rate decreases and time increases. When the crust melts at its base as volcanism decreases (as might occur under steady state tidal heating), resulting in crustal thinning, the region of failure is brought even closer to the surface. Naturally, when compressive, subsidence stresses are included, the vertical extent of crust in brittle failure thickens to include most of the lithosphere. In contrast, increases in eruption rate cause the extent of the region in compressional failure to decrease and be driven very deep in the crust (in the absence of sufficient subsidence stress). Therefore, regions of declining volcanism are more likely to produce mountains, whereas regions of extensive or increasing volcanism are less likely to do so. This is consistent with the observation of a global anticorrelation between mountains and volcanic centers on Io. Finally, we find that the choice of crustal composition/rheology (dry basalt vs. dry peridotite) has little effect on our results implying that basalt, peridotite and komatiite are all similarly “stiff” in the Io environment.

O.491 Long-term outcome following resection for IF malignancies

Based on the modified strain gradient theory (MSGT) in conjunction with the Timoshenko beam theory, the vibrational behavior under thermal environment of variable section functionally graded (FG) microbeams carrying microparticles is studied. The eigenfrequency equations are derived using the Chebyshev–Ritz method. The effects of temperature dependence of material properties and the initial thermal stresses are considered. After validating the method, the influences of the width taper ratio, the position and values of the microparticle masses, temperature rise, length scale parameters and material gradient index on the free vibration behavior of microbeams are studied. In addition, comparisons between the results of the MSGT, the modified couple stress theory (MCST), and the classical theory (CT) are performed. It is found that in addition to the microparticle masses and locations, their influences on the natural frequencies depend on the vibration mode number and the edge restraints of microbeams. Also, it is shown that the temperature rise, length scale parameters and material gradient index have considerable effects on the natural frequencies of microbeams. Moreover, the results demonstrate that the MSGT and the CT provide the largest and the smallest natural frequencies, respectively.