Residual thermal stresses in the pull-out specimen: A finite element calculation

The residual thermal stress field in the pull-out specimen is calculated in the case of a high properties thermoset system (carbon-bismaleimide). The calculation is performed within the framework of the linear theory of elasticity by means of a finite element method. The specimen is modelled as a three-phase composite (holder-fibre-matrix). The meniscus which forms at the fibre entry is taken into account in order to provide a realistic stress concentration. The latter is far higher than the matrix strength. Evidence that fibre debonding propagates from the fibre end during cooling is then produced.Nomenclature T thermal load - L _e embedded length - r _f fibre radius - _c curvature radius of the meniscus (fibre entry) - r _c radial dimension of the finite element mesh - E _m,E _h matrix and holder moduli - E _A,E _T fibre axial and transverse moduli - _m, _h matrix and holder thermal expansion coefficients - _A, _T fibre axial and transverse thermal expansion coefficients - _rr, , _zz, _rz non-zero components of the residual stress field - _rr ⁱ , ^im , _zz ^im , _rz ⁱ stresses at the interface in the matrix (r=r _f + ) - _rr ⁱ , ^if , _zz ^if , _rz ⁱ stresses at the interface in the fibre (r=r _f–) - _p1 maximum principal stress - _zz ^f mean axial stress over the fibre section - _rupt ^m matrix strength - u _r ,u _z non-zero components of the displacement field     

Compressive strength of coated rigid-rod polymer fibres

One limitation to the use of high-strength/high-modulus rigid-rod polymer fibres like poly-(p-phenylene benzobisthiazole) (PBZT) and poly-(p-phenylene benzobisoxazole) (PBZO) in composite structures is their low compressive strength. Various theories have been developed to predict compressive strength of rigid-rod fibres. In this study the critical buckling stress for rigid-rod fibres with stiff external coatings has been theoretically modelled assuming that the failure mode in compression is the microbuckling of the fibrils in shear. Our model predicts that significant improvement in fibre compressive strength will occur only when relatively thick coatings, with thickness to diameter (t/D) ratios in excess of > 0.05, are used. Experimentally measured compressive strength of aluminium coated PBZT fibres shows values in good agreement to the theory at t/D ratios of 0.006 and below. Factors related to the selection of suitable coating materials and problems associated with establishing coating performance are identified.Nomenclature P axial compressive load - P _f axial compressive load on the fibre - P _c axial compressive load on the coating - P _cr ⁱ critical buckling load in the ith case - _cr critical buckling stress - _co compressive strength of the uncoated fibre - _c compressive strength of the coated fibre - v(x) lateral deflection of a buckled fibril or coating - V _m amplitude of the lateral deflection in the mth mode - m number of half-sine waves in the deflection mode - x coordinate distance along axial direction - y coordinate distance along radial direction - coordinate distance along circumferential direction - l length of the buckling unit - N number of fibrils in the fibre - D fibre diameter - d fibril diameter - t coating thickness - I _f moment of inertia of the fibril - A _f cross-sectional area of the fibril - E _f tensile modulus of the fibre - E _c tensile modulus of the coating material - E tensile modulus of the coated fibre - G torsional shear modulus of the fibre - v_c Poisson's ratio of the coating material - _f density of the fibre - _c density of the coating material - density of the coated fibre - U _f strain-energy change in the fibre - U _c strain-energy change in the coating - T _f external work done on the fibre - T _c external work done on the coating - d/D - t/D     

Use of adjoint functions in investigations of heat conduction and transfer processes

An examination is made of the use of adjoint functions in heat conduction and convection theory. Formulas of perturbation theory are obtained for steady and unsteady cases, an interpretation of the physical meaning of adjoint temperature is given, and some applications of the theory are discussed.Notation (r,) thermal conductivity - t(r,) temperature - t *(r,) adjoint temperature - q_V(r,) density of heat release sources - p(r,) a parameter of adjoint equation - r generalized coordinate - time - (r_s, ) heat transfer coefficient - I linear functional of temperature - (r,;r₀,₀) and *(r,; r₀,₀) Green's function for t(r, ) and t *(r, ) - C(r,) volume specific heat - W(r, ) vector distribution of flow velocities - V, S volume and surface areas of body - R radius of HRE - r, radial and angular coordinates - F_in, F_out inlet and outlet flow areas of channel     

Internal material damping of polymer matrix composites under off-axis loading

The objective of this paper is to determine theoretically the material damping of short fibre-reinforced polymer matrix composites. The major damping mechanism in such composites is the viscoelastic behaviour of the polymer matrix. The analysis was carried out by developing a finite-element program which is capable of evaluating the stress and strain distribution of short fibre composites under axial loading (see Fig. 1a). Using the concept of balance of force we can express the modulusE _x along the loading direction as a function of the mechanical properties of the fibre and matrix materials, fibre aspect ratio,l/d, loading angle,, and fibre volume fraction,V _f. Then we apply the elastic-viscoelastic correspondence principle to replace all the mechanical properties of the composite, fibre and matrix materials such asE _x,E _f,E _m,G _m, by the corresponding complex moduli such asE _x +iE _x , andE _f +iE _f . After separation of the real and imaginary parts, we can expressE _' ^x/t' andE _x ^t" as functions of the fibre aspect ratio,l/d, loading angle,, stiffness ratio,E _f/E _m, fibre volume fraction,V _f, and damping properties of the fibre and matrix materials such as _f and _m. Numerical results of the composite storage modulus,E _x , loss modulus,E _x , and loss factor (damping), _C, are plotted as functions of parameters such asl/d,,V _f, and are discussed in terms of variations ofl/d,, andE _f/E _m, in detail. It is observed that for a given composite, there exist optimum values ofl/d and at whichE _x and _c are maximized. The results of this paper can be used to optimize the performance of composite structures.Nomenclature A _c,A _f,A _m cross-sectional area of composite, fibre and matrix, respectively - d fibre diameter - E _L longitudinal modulus of composite (along the fibre direction) (see Fig. 1a) - E _T transverse modulus of composite (see Fig. 1a) - E _x modulus of composite along thex-direction (see Fig. 1b) - E _f tensile modulus of fibre - E _m tensile modulus of matrix - G _m shear modulus of matrix - G _LT in-plane shear modulus of composite (see Fig. 1a) - l fibre length - m tip to tip distance between fibres - i (–1)^1/2 - R one-half of centre-to-centre fibre spacing - V _f fibre volume fraction - x distance along fibre from end of fibre - defined in Equation 22 - defined in Equation 3 - ^* defined in Equation 19 - _L extensional (longitudinal strain) of composite - _f, _m extensional (longitudinal strain) of fibre and matrix, respectively - _c, _f, _m extensional loss factor of composite, fibre and matrix respectively - G _m shear loss factor of matrix - angle between fibre and thex-direction - ¯ _c, ¯ _f, ¯ _m average longitudinal stress in composite, fibre and matrix, respectively - longitudinal stress in fibre - shear stress at fibre-matrix interface - defined in Equation 23     

Some expectation on the mechanism of cross-flow instability in a swept wing flow

Summary Three-dimensional boundary layer transition on axisymmetric rotating bodies is the subject of a comprehensive experimental study. Based on this study, hypotheses are made on the mechanism of cross-flow instability for swept wing flow. These new results are combined with past explanations to provide a rough sketch for the entire flow field over the swept wing. From this new viewpoint there appears the mechanism of traveling waves, being induced by a stationary disturbance. Some uncertainties appearing in recent papers concerning this flow field are discussed. Among these uncertainties for which an explanation is provided, is the discrepancy of frequencies between the hot wire signal and the visualized flow pattern.Nomenclature x direction along a potential flow stream line - y direction normal to a potential flow stream line - z direction normal to bothx andy directions - U mean velocity inx-direction - V mean velocity iny-direction - x direction along a disturbance - y direction normal tox direction - u, v, w fluctuating velocity components inx, y, z directions - U velocity inx-direction with wall fixed coordinate - U _e velocity of outer edge of boundary-layer - U uniform flow velocity normal to leading edge - V uniform flow velocity parallel to leading edge - Q upstream velocity - N rotation speed of an axisymmetric body - P arbitrary point on a disk surface - r radius to a pointP - R ₀ radius of a disk or a cylinder - U _p phase velocity of ring like vortices - T position where wall streaks appear in the case of oil flow visualization - Re _c,t critical and transitional Reynolds numbers - angle of the spiral disturbance - boundary-layer thickness - angular velocity - sweep angle of a body - wave length of disturbance - kinematic viscosity of a fluid With 11 Figures     

The tensile strength and ductility of continuous fibre composites

The plastic instability approach has been applied to the tensile behaviour of a continuous fibre composite. It is shown that the combination of two components with different strengths and degrees of work-hardening produces a new material with a new degree of work-hardening, which may be determined by the present analysis. Expressions for the elongation at rupture and the strength of a composite have been obtained and the results of the calculation are compared with some experimental data.List of symbols V _f volume fraction of fibres in composite - , , true strain of fibre, matrix and composite - s true stress - , , nominal stress on fibre, matrix and composite - _*, _*, _* critical stress of fibre, matrix and composite (ultimate tensile strength) - _*, _* critical strain of separate fibre and matrix - _* critical strain of composite - Q external load - A cross-sectional area - A ₀ initial value of area     

Relative diffusion resistance in drying wheat

An apparatus is described for examining various methods of convective drying.Notation tan=N_I drying rate in the first period - tan =(dW^c/d)_II drying rate in the second period - drying time - W _e ^c equilibrium water content - W^c water content of grain on dry mass - N^*=(1/N_I)(dW^c/d) dimensionless drying rate - T_sur surface temperature - T_a ambient temperature - T_w wet-bulb temperature - A,, experimental coefficients Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 31, No. 5, pp. 839–843, November, 1976.     

Aerodynamic detuning for control of supersonic rotor forced response

A mathematical model is developed to analyze the aerodynamically forced response characteristics of a detuned rotor operating in a supersonic inlet flow field with a supersonic axial component. Alternate blade aerodynamic detuning is considered, with the aerodynamic detuning accomplished by alternating the circumferential spacing of adjacent blades. The unsteady aerodynamics are determined by developing an influence coefficient technique which is appropriate for both aerodynamically tuned and detuned rotor configurations. Rotor forced response is then analyzed by combining this unsteady aerodynamic model with a single-degree-of-freedom structural model. The effects of this detuning on the forced response characteristics of supersonic axial flow rotors is then demonstrated by applying this model to baseline twelve bladed rotors.List of symbols c airfoil chord - [CM] ⁿ gust influence coefficient of airfoil, n - [CM] ⁿ motion-induced influence coefficient of airfoil, n - I mass moment of inertia - K linear spring constant - k reduced frequency _c/U - M dimensionless unsteady aerodynamic moment - M inlet Mach number - r dimensionless radius of gyration - S airfoil spacing - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - w complex amplitude of gust - x ₀ elastic axis location measured from leading edge - amplitude of airfoil oscillations - complex oscillatory displacement - interblade phase angle - _d detuned interblade phase angle - level of aerodynamic detuning - mass ratio - oscillatory frequency - ₀ reference frequency     

Determination of the thermophysical characteristics of alternatively freezing and thawing dispersed media by the method of solving inverse problems of heat conduction

The article explains an algorithm for determining the thermophysical characteristics of dispersed media with phase transitions based on the method of solving inverse problems of heat conduction.Notation r space coordinate - time - T temperature of the specimen - T₀ initial temperature - c_i, c_w, c_sk specific heat of ice, water, and of the organic-mineral skeleton, respectively - c_f, c_m, _f, _m specific heat and thermal conductivity in the frozen and melted zones, respectively - c effective heat capacity - thermal conductivity - p density - ₀, _sb bound and strongly bound moisture, respectively - (T) amount of nonfrozen water - R radius of the cylinder - q() heat flux - I functional - u₁(), U₂() measured temperatures of the specimen at the points r = 0 and r = R, respectively, at the instant - ₁, ₂ degree of confidence of the supplementary information - final instant of time - a, b, k, _s positive constants - L specific heat of melting - N number of grid nodes over space - n number of grid nodes over time - h grid step over space - grid step over time - solution of the conjugate system - s number of iteration Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 292–297, August, 1980.     

Carbon fibre-reinforced cement

This review describes fabrication processes for aligned fibre and random fibre carbonreinforced cement and links important process parameters with composite theory. The way in which the material fits into the general framework of crack constraint and matrix cracking theories is discussed. A broad survey is made of the mechanical properties, durability and dimensional stability of a variety of carbon-reinforced cement composites, and economic constraints on potential applications are considered.List of symbols b breadth of three-point bend specimen - d depth of three-point bend specimen - E _c composite Young's modulus - E _f fibre Young's modulus - E _m matrix Young's modulus - l fibre length - l _c fibre critical transfer length - l _s specimen span in three-point bend test - m Weibull modulus - r fibre radius - P applied load - V _f fibre volume fraction - V _m matrix volume fraction - x length of fibre needed to transfer load _mu V _m - x _d crack spacing in a composite with short, aligned fibres - _fu fibre ultimate strain - _mu matrix ultimate strain - _fu fibre ultimate strength - _mu matrix ultimate strength - _cu composite ultimate strength - _MOR modulus of rupture - _T tensile strength - interlaminar shear strength - _i interfacial shear strength - _m matrix work of fracture - _F work of fracture     

Microstructure and mechanical properties of metastable lath martensite wires in the Fe-Ni-Cr-Al-C system produced by melt spinning in rotating water

Metastable lath martensite ( _L ) phase wires with high strengths have been produced in the Fe-Ni-Cr-Al-C alloy system by melt spinning in rotating water. These wires have a circular cross section and a white lustre and the wire diameter is in the range of 100 to 140m. The width and length of each lath in the _L phase are as small as about 0.3 and 2m, respectively. The _y, _f and _p are about 900 and 1650 MPa and 2.0% for the _L wires. The subsequent annealing causes an increase in _p as well as _y and _f and the attained values are about 1000 and 1700 MPa and 3.0% for Fe-10Ni-10Cr-6.5 Al-1.0C wire annealed at 773 K for 1 h owing to the precipitation strengthening of a very fine unidentified carbide and to a high density of dislocations and lath boundaries in the _L phase. Further annealing causes a significant decrease in _p through decomposition of _L to+M₇C₃+M₂₃C₆. Therefore, the high strength combined with relatively good ductility for the _L wires is interpreted as due to the suppression of the phase transformation of _L to a mixed structure of+M₇C₃+M₂₃C₆ by melt quenching.     

The preparation of alumina fibre by sol-gel processing

Attempts have been made to prepare alumina fibre from the colloidal sol and polymerized alkoxides. The aluminium chloride or aluminium nitrate systems were found to be potential methods for producing continuous alumina fibre: the aluminium nitrate system had a better sintering behaviour than the aluminium chloride system. The aluminium isopropoxide system, however, was unsuitable for preparing alumina fibre but was suitable for the preparation of monoliths, membranes, powders, and multicomponent ceramics. The thermal changes of these precursors were studied by transmission electron microscopy, Fourier-transform infrared spectroscopy and X-ray diffraction. The results demonstrated the different routes of phase transformation as the temperature increases. The aluminium chloride system exhibits two routes for phase transformation: (a) boehmite -Al₂O₃, and (b) gibbsite -Al₂O₃.     

The thermal expansion of carbon-fibre reinforced plastics

The thermal expansion characteristics of a series of carbon-fibre fabric reinforced plastic laminates over the approximate temperature range 90 K to 440 K have been determined. The reinforcements included Morganite Type II fibres in a plain weave and a two-by-two twill weave and Courtaulds Grafil E/XAS fibres in a two-by-two twill weave, a five-shaft satin weave and in an unwoven unidirectional disposition. The results show that the ratio of fibre tow densities in the principal fibre directions, the crimp in the reinforcing fibres and the laminate stacking sequence all influence the magnitudes and temperature dependences of the linear thermal expansion coefficients, as well as the detailed manner in which the dimensions respond to changes of temperature. Volume shrinkage effects resulting from temperature cycling are also reported. The linear thermal expansion coefficients of Courtaulds Grafil E/XAS carbon fibres in directions parallel, ^f , and perpendicular, _¶ ^f , to the fibre axis have been estimated as ^f =–2.6× 10^–7 K^–1 and ^f =2.6×10^–5 K^–1.     

Stress distributions along a short fibre in fibre reinforced plastics

This paper develops an analysis for predicting the normal stress and interfacial shearing stress distribution along a single reinforcing fibre of a randomly oriented chopped-fibre composite, such as sheet moulding compound (SMC), from a knowledge of the constituent properties and the length-to-diameter ratio of the fibres. The analysis is useful in analysing the tensile strength of SMC, and as a guide to increasing the tensile strength by altering the elastic characteristics. The model is based on a generalized shear-lag analysis. Numerical values of the normal stress and interfacial shearing stress are presented as functions of various parameters. It is observed that the maximum normal stress occurs at the middle of the fibre and the maximum shear stress occurs at the end. The analysis is restricted to loading which does not result in buckling of the fibre; i.e., axial loads on the fibre can be at most only slightly compressive.List of symbols a _f Ratio of the fibre length to diameter (aspect ratio, l _f/d _f) - E _a Young's modulus of the composite (defined in Equation 21) - E _f Young's modulus of the fibre material - E _m Young's modulus of the matrix material - G _f Shear modulus of the fibre material - G _m Shear modulus of the matrix material - l Half the length of the matrix sheath which surrounds the fibre - l _f Half of the length of the fibre - Q Defined in Equation 14. - R Ratio of the length of the fibre to the matrix in a representative volume element; a parameter 0R[(1/V _f–1) ] - r _a Radius of the composite body (we assume r _ar _m, r _f) - r _f Radius of the fibre - r _m Radius of the matrix sheath which surrounds the fibre - u _a Displacement of the composite along the fibre direction - u _f Displacement of the fibre along the fibre direction - V _f Fibre volume fraction - (XYZ) Co-ordinate system with Z-axis parallel to the direction of the applied load (Fig. 1a) - (xyz) Co-ordinate system which is rotated by about the X-axis (Fig. 1a) - (¯x¯y¯z) Co-ordinate system which is rotated by about the z-axis (Fig. 1b) - Fibre orientation angle measured from the Z-axis - _m Engineering shear strain in the matrix - Defined in Equation 8 - Polar angle measured from the x–z plane - Defined in Equation 9 - Applied normal stress - _a Normal stress in the composite along the fibre axis - _f Normal stress in the fibre along the fibre axis - _m Normal stress in the matrix along the fibre axis - Shear stress on the fibre—matrix interface     

Numerical method for calculating the filling of a mold and heat transfer of a melt under the action of centrifugal forces

This paper presents an explicit difference method for solving the conjugate problem of pouring molten metal into a casting mold and its solidification under the action of centrifugal forces with allowance for the free surface.Notation r, z transverse and longitudinal coordinates - u, v horizontal and vertical velocity components - V velocity vector - angular velocity of rotation - P pressure - P normalized pressure - T temperature - time - g free fall acceleration - coefficient of volumetric expansion - C heat capacity - thermal conductivity - density - L crystallization heat - v viscosity - Re=|V|h/v Reynolds grid number - h grid spacing - l mixing length in a turbulent flow - relaxation parameter - coefficient of convective heat transfer - coefficient defining the boundary conditions at the solid wall - D flow divergence - volumetric velocity of filling - emissivity - ₀ Stefan-Boltzmann constant - thickness of a layer Indices r, z, I, j numbers of grid nodes - n number of the integration step with respect to time - L, S temperatures of the liquidus and solidus, respectively - s temperature on the surface - med temperature of the medium - 0 initial state of the system - m metal - mol molten state of the metal - red reduced emissivity in the gap - rad radiant component of the heat transfer coefficient; g, gas-air gap - coat heat-insulating coating - fil filling Dneprodzerzhinsk Industrial Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 4, pp. 678–686, July–August, 1995.     

Thermoelastic stresses in ribbons and tubes grown from the melt by the Stepanov method

Ribbons and tubes grown from the melt by the Stepanov technique have a wide range of technical applications. Sapphire ribbons are used as substrates in microelectronics and sapphire tubes are used as gas-discharge balloons in laser engineering, fine chemical technology and high-vacuum equipment. Practice has shown that misorientation angles of small-angle boundaries in sapphire crystals should not exceed several degrees because an increase in the misorientation angles between blocks drastically lowers the strength and worsens the dielectric properties of these crystals. One of the main mechanisms of formation of the block structure of melt-grown crystals, including shaped sapphire crystals, is dislocation polygonization that begins when the dislocation density exceeds a certain critical value. In turn, dislocations are formed under deformations due to thermal stresses. Calculations of thermal fields in crystals and the corresponding thermoelastic stress fields can be used as an input to improve and optimize the growth process. The dependence of thermoelastic stresses in ribbons and tubes on the technological parameters has been calculated.Nomenclature ₁ Thermal diffusivity of the melt - ₂ Thermal diffusivity of the crystal - k ₁ Thermal conductivity of the melt - k ₂ Thermal conductivity of the crystal - V ₁ Velocity vector of the melt - V ₂ Velocity vector of the growing crystal - V ₀ Crystal pulling rate - H _f Latent heat of fusion - ₁ Density of the melt - ₂ Density of the crystal - in Interface normal vector - Crystal-melt interface normal vector - t Interface tangential vector - s Sided crystal-melt tangential vector - T _m Melting temperature - T _e Ambient temperature - T ₁ ⁰ Temperature at the bottom of the meniscus - T ₂ ⁰ Crystal temperature at the top of the meniscus - Normal vector at lateral surfaces of the crystal and meniscus - Stefan-Boltzmann constant - ₁ Emissivity of the meniscus lateral surface - ₂ Emissivity of the crystal lateral surface - g Acceleration due to gravity - _LG Melt-gas surface tension - a Die half dimension - ₀ Angle of growth - _t Thermal expansion coefficient - h ₁ Heat transfer coefficient of the melt - h ₂ Heat transfer coefficient of the crystal - C s Heat capacity - E Young's modulus - Poisson's coefficient - Melt kinematic viscosity - P Pressure in the melt     

The role of silicon in the formation of the (Al5Cu6Mg2) σ phase in Al-Cu-Mg alloys

The role of silicon in the precipitation of the phase (Al₅Cu₆Mg₂) has been investigated through comparative studies on Al-3.63Cu-1.67Mg (wt%) and Al-3.63Cu-1.67Mg-0.5Si alloys. Both alloys were extensively examined after solution treating at 525°C for 2.5 h followed by ageing at 265°C for times up to 650 h. Limited studies were also undertaken on both alloys after ageing at 200 and 305°C. Precipitation of was observed in Al-3.62%Cu-1.66%Mg-0.5%Si for all ageing conditions studied but was absent in Si-free Al-3.62%Cu-1.66%Mg. In addition, S and phases were observed in both alloys. The volume fraction of phase in the Si containing alloy was substantially reduced by a pre-age stretch followed by ageing for 24 h at 265°C with S being the dominant precipitate type. The volume fraction of phase in the Si containing alloy was lower after ageing 24 h at 200°C than after 24 h at 265 and 305°C. Peak hardness was higher for the Si free alloy on ageing at 200 and 265°C, but the Si free alloy softened more rapidly, reflecting the more rapid coarsening kinetics of S compared with .     

Viscoelastic-plastic behaviour with mean strain changes in polypropylene

The stress-strain curves and stress-relaxation curves of polypropylene are obtained by using a closed-loop, electrohydraulic, servo-controlled testing machine. Effects of mean strain changes on deformation behaviour are examined in a tension-compression mode under strain control at room temperature (18–23 °C). The hysteresis loops of three mean strains show a steady-state response from the stress-strain curves at a strain rate of 1 × 10^–3 s^–1 at a strain width of 5%, at a number of cycles of N=50 and at three mean strains (_m=0, + 1.0 and + 2.0%). The drop of stress at the mean strain of _m= -1.0% is larger in magnitude than that at _m=+1.0%; this is caused by the higher stress level at _m=- 1.0% as compared with the stress level at _m=+1.0%. From the results of stress amplitude and the stress drop behaviour, the magnitude of stress drop is hardly affected by the mean strain.     

A finite volume technique to simulate the flow of a viscoelastic fluid

Hydrodynamically developing flow of Oldroyd B fluid in the planar die entrance region has been investigated numerically using SIMPLER algorithm in a non-uniform staggered grid system. It has been shown that for constant values of the Reynolds number, the entrance length increases as the Weissenberg number increases. For small Reynolds number flows the center line velocity distribution exhibit overshoot near the inlet, which seems to be related to the occurrence of numerical breakdown at small values of the limiting Weissenberg number than those for large Reynolds number flows. The distributions of the first normal stress difference display clearly the development of the flow characteristics from extensional flow to shear flow.List of symbols D rate of strain tensor - L slit halfheight - P pressure, indeterminate part of the Cauchy stress tensor - R the Reynolds number - t time - U average velocity in the slit - u velocity vector - u,v velocity components - W the Weissenberg number based on the difference between stress relaxation time and retardation time - W ₁ the Weissenberg number based on stress relaxation time - x,y rectangular Cartesian coordinates - ratio of retardation time to stress relaxation time - zero-shear-rate viscosity, ₁ + ₂ - ₁ non-Newtonian contribution to - ₂ Newtonian contribution to - ₁ stress relaxation time - ₂ retardation time - density - (, , ) xx, yy and xy components of ₁, respectively - determinate part of the Cauchy stress tensor - ₁ non-Newtonian contribution to - ₂ Newtonian contribution to      

Transient thermal stress analysis of multilayered hollow cylinder

Summary This paper deals with the transient response of one-dimensional axisymmetric quasistatic coupled thermoelastic problems. Laplace transform and finite difference methods are used to analyze the problems. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal stress distribution in a transient state. Moreover, the computational procedures established in this article can solve the generalized thermoelasticity problem for a multilayered hollow cylinder with orthotropic material properties.Nomenclature Lame's constant - density - C _v specific heat - k _r ,k radial and circumferential thermal conductivity - _r , linear radial and circumferential thermal expansion coefficient - E _r ,E radial and circumferential Young's modulus - v _r Poisson's ratio - ₀ reference temperature - ,T dimensional and nondimensional temperature - r ^*,r dimensional and nondimensional radial coordinate - ,t dimensional and nondimensional time - _r ^* , _r dimensional and nondimensional radial stress - ^* , dimensional and nondimensional circumferential stress - U, u dimensional and nondimensional radial component of displacement