Heat Conduction Basic Research Part 11 ppt

Energy Transfer in Pyroelectric Material 11which is the rate of kinetic energy density.. Substitution the constitutive equation Equation 72into the above equation yields T0γ ij ε˙ij+T0ξ

Energy Transfer in Pyroelectric Material 11

which is the rate of kinetic energy density

II Multiplyingϕ by the time derivative of Equation (3), integrating the resulting expression

over volume Ω and using the identity equation D˙k ϕ,k = D˙k,k ϕ+D˙k ϕ ,k and GaussianTheorem, we have

where n kis the unit outward normal of dS

Substitution the constitutive equation Equation (7)2into the above equation yields

T0γ ij ε˙ij+T0ξ i ˙E i+ρC ˙θ = − q i,i

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Energy Transfer in Pyroelectric Material

Applying the operator L on both sides of this equation and using Equation (6) yields

κ ij θ ,ij − L(T0γ ij ε˙ij+T0ξ i ˙E i+ρC ˙θ) =0 (22)Multiplying Equation (22) byθ and apply volume integral on this expression, we obtain

which is the energy balance law for pyroelectric medium with generalized Fourier conductionlaw for arbitrary time dependent wave field

As the general energy balance states:

Ω

which is the law governing the energy transformation The physical significance of Equation

(26) is that the rate of heat or dissipation energy Q equals to the reduction of the rate of

entire energy ˙W within the volume plus the reduction of this energy flux outward the surface

Poyting-Umov vector) and its direction indicates the direction of energy flow at that point,the length being numerically equal to the amount of energy passing in unit time through unit

area perpendicular to P.

In this chapter, important conclusions can be made from Equation (25): the energy density W

in the the pyroelectric medium:

Energy Transfer in Pyroelectric Material 13

The physical meaning of E i ξ i θ can be seen from constitutive equation in Equation (7)3, from

which E i ξ i is found to contribute entropy Therefore the result E i ξ i θ, by its multiplication with

temperature disturbanceθ, is the dissipation due to the pyroelectric effect Therefore Q the

rate of energy dissipation per unit volume is represented by

3.1 Energy balance law for the real-valued inhomogeneous harmonic wave

In previous section, we derived the energy balance equation for the pyroelectric medium anddefined the total energy, dissipation energy and energy flux vector explicitly Keeping in mindthat the real part is indeed the physical part of any quantity, and considering Equation (10),

we can define the corresponding fundamental field functions as

ϕ= 1

2

Ψexp(ix i k i)exp(iωt) +Ψ∗exp(− ix i k ∗ i)exp(− iωt)

which are the real-valued inhomogeneous harmonic waves assumed on the basis of the pair

of complex vector fields for Equation (8)

The velocity of plane of constant phase is defined by

and the maximum attenuation isA, where indicates the norm(or length) of a vector.The quantities of the rate of energy density, the dissipation energy and the energy flux vectorcan be expressed by inserting Equation (30) into Equations (27), (28) and (29)

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Energy Transfer in Pyroelectric Material

The mechanical potential energy density W e

is time harmonic with frequency 2ω The first term, expressed as W e afterwards, represents

quantity over one period The notation Re stands for the real part and Im the imaginary part

Similarly, the kinetic energy density K takes the form

K= 1

2ρω2U i U i ∗exp(− 2x s A s) −1

2ρω2Re[U i U iexp(i2x s k s)exp(i2ωt)] (33)

The electric energy density W E

W E= 1

2λ ikRe[k i k ∗ kΨΨ∗exp(− 2x s A s)] −1

2λ ikRe[k i k kΨΨexp(i2x s k s)exp(i2ωt)] (34)

The heat energy density W θ

k i k jΘΘexp(i2x s k s)exp(i2ωt) (36)

Q (τ)because of the relaxation

U i k jΘexp(i2x s k s)exp(i2ωt)+(37)

+ξ iIm(k ∗ iΨ∗Θ)exp(− 2x s A s) + ξ i Im[k ∗ iΨ∗Θ∗exp(i2x s k s)exp(i2ωt)] +

−ΘΘ∗exp(− 2x s A s) −Re[ΘΘexpi(2x i k i)exp(i2ωt )]}

At last, Q (ξ)attributed by the pyroelectric effect

Q (ξ)=2ξ i Re(k i ωΨΘ ∗)exp(− 2x s A s) +2ξ i Re[(k i ωΨΘ)exp(2ix s k s)exp(i2ωt)] (38)

The energy flux vector P i consists of three different parts: P i (u)is generated in the elastic field;

P i (ϕ) in the electric field; P i (θ)in the thermal field, which are expressed as

P j (u) = − σ ji ˙u j

= − ωc jikl {Re(U i ∗ U k k l)exp(− 2x s A s) +Re[U i U k k lexp(i2x s k s)exp(i2ωt )]} + (39)

ωe kji {−Re(k k U i ∗Ψ)exp(− 2x s A s) +Re[k k ΨU iexp(i2x s k s)exp(i2ωt )]} +

ωγ ji[[Im(U i ∗Θ)exp(− 2x s A s) − ωImU iΘexp(i2x s k s)exp(i2ωt)]]

Energy Transfer in Pyroelectric Material 15

In the electric field, P j (ϕ)

P j (ϕ)=ϕ ˙D j

= − ωe jmn {Re(U m k nΨ∗)exp[− 2x s A s] +Re[ΨU m k nexp[i(2x s k s)]expi2 ωt ]} + (40)

ωλ mj {Re(k mΨΨ∗)exp(− 2x s A s) +Re[k mΨΨexp(i2x s k s)exp(i2ωt )]} +

ωξ j {Im(Θ∗Ψ)exp(− 2x s A s) −Im[ΘΨexp[i(2x i k i)]exp(i2ωt )]}

In the thermal field, P j (θ)

density over one period, that is

which corresponds to the average local velocity of energy transport From an experimentalpoint of view, it is more interesting to define velocity from averaged quantities(Deschamps et al., 1997)

We can substitute the expressions in Equations (32)-(35) and (39)-(41) into (42), which yields

a lengthy formulation Comparing the expression of phase velocity in Equation (31) with theenergy velocity in Equation (42), it is obvious that they are different from each other in moduli

as well as directions

3.2 Results and discussion

According to previous studies, it is already known that there are waves of four modes, whichare quasilongitudinal, quasitransverse I, II and temperature In this section, we’d like to

discuss phase velocity vp, energy velocity vErelated to the four mode waves They are studied

as functions defined in propagation angleθ and attenuation angle γ After wave vector k is

determined, Equations (31) and (42) yield the phase velocity and energy velocity respectively.The material constants under study is transversely isotropic material, see Section 2.4

The variation of phase and energy velocity of quasilongitudinal wave is presented in Fig 6(a) which shows that the phase velocity does not vary with attenuation angleγ , while the

corresponding energy velocity can be influenced byγ With γ increasing, the energy velocity

turns small It is also noted that the phase velocity is a little bigger than the energy velocityfor quasilongitudinal wave mode

The case of temperature wave is shown in Fig 6 (b) Different from quasilongitudinal wave,the phase velocity and energy velocity of temperature wave are influenced by propagationangleθ and attenuation angle γ Both phase velocity and energy velocity decay with γ For

givenγ, the phase velocity is also bigger than energy velocity.

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Energy Transfer in Pyroelectric Material

(b)

Fig 6 Variations of velocity with propagation angleθ at γ=0 ◦, 30◦.

Energy Transfer in Pyroelectric Material 17

Plots of the computed velocities of quasitransverse wave I and II are given in Fig 7 The phase

small withγ increasing.

4 Conclusion

In this chapter, the energy process of the pyroelectric medium with generalized heatconduction theory is addressed in the framework of the inhomogeneous wave resultsoriginally The characters of inhomogeneous waves lie in that its propagation direction

determined by four parameters The range of attenuation angle should be confined in(-90◦,90◦) to make waves attenuate Further analysis demonstrates that, in anisotropic plane,the positive and negative attenuation angle have different influences on waves, while, in theisotropic plane, they are the same Based on the governing equations and state equations, thedynamic energy conservation law is derived The energy transfer, in an arbitrary instant, isdescribed explicitly by the energy conservation relation From this relation, it is found thatenergy density in pyroelectric medium consists of the electric energy density, the heat energydensity, the mechanical potential energy density, the kinetic energy density The heat loss ordissipation energy is equal to the reduction of the entire energy within a fixed volume plusthe reduction of this energy flux outward the surface bounding this volume The dissipationenergy in pyroelectric medium are attributed by the heat conduction, relaxation time andpyroelectric effect The energy flux is obtained and it can not be influenced by the relaxationtime The phase velocity and energy velocity of four wave modes in pyroelectric mediumare studied Results demonstrate that the attenuation angle almost doesn’t influence phasevelocity of quasilongitudinal, quasitransverse I, II wave modes, while plays large role onthe temperature wave The energy velocities of the four wave modes all decay with theattenuation angle

5 References

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11

Steady-State Heat Transfer and Thermo-Elastic Analysis of Inhomogeneous Semi-Infinite Solids

Yuriy Tokovyy1 and Chien-Ching Ma2

1Pidstryhach Institute for Applied Problems of Mechanics and Mathematics

National Academy of Sciences of Ukraine, Lviv,

2Department of Mechanical Engineering, National Taiwan University

in the need to solve the governing equations in the differential form with variable coefficients which are not pre-given for arbitrary dependence of thermo-physical and thermo-elastic material properties on the coordinate Particularly, for functionally graded materials, whose properties vary continuously from one surface to another, it is impossible, except for few particular cases, to solve the mentioned problems analytically (Tanigawa, 1995) The analytical, semi-analytical, and numerical methods for solving the heat conduction and thermoelasticity problems in inhomogeneous solids attract considerable attention in recent years The overview

of the relevant literature is given in our publications (Tokovyy & Ma, 2008, 2008a, 2009, 2009a)

On the other hand, determination of temperature gradients, stresses and deformations is usually an intermediate step of a complex engineering investigation Therefore analytical methods are of particular importance representing the solutions in a most convenient form The great many of existing analytical methods are developed for particular cases of inhomogeneity (e.g., in the form of power or exponential functions of a coordinate, etc.) The methods applicable for wider ranges of material properties are oriented mostly on representation the inhomogeneous solid as a composite consisting of tailored homogeneous layers However, such approaches are inconvenient for applying to inhomogeneous materials with large gradients of inhomogeneity due to the weak convergence of the solution with increasing the number of layers

A general method for solution of the elasticity and thermoelasticity problems in terms of stresses has been developed by Prof Vihak (Vigak) and his followers in (Vihak, 1996; Vihak

et al., 1998, 2001, 2002; Vigak, 1999; Vigak & Rychagivskii, 2000, 2002) The method consists

in construction of analytical solutions to the problems of thermoelasticity based on direct integration of the original equilibrium and compatibility equations Originally the equilibrium equations are in terms of stresses, and they do not depend on the physical stress-strain relations, as well as on the material properties At the same time the general equilibrium relates all the stress-tensor components This enables one to express all the stresses in terms of the governing stresses The compatibility equations in terms of strain are then reduced to the governing equations for the governing stresses When these equations are solved, all the stress-tensor components can be found by means of the aforementioned

expressions In addition, the method enables the derivation of: a) fundamental integral

equilibrium and compatibility conditions for the imposed thermal and mechanical loadings

and the stresses and strains; b) one-to-one relations between the stress-tensor and

displacement-vector components Therefore, when the stress-tensor components are found, then the displacement-vector components are also found automatically Such relations have been derived for the case of one-dimensional problem for a thermoelastic hollow cylinder (Vigak, 1999a) and plane problems for elastic and thermoelastic semi-plane (Vihak & Rychahivskyy, 2001; Vigak, 2004; Rychahivskyy & Tokovyy, 2008)

Since application of this method rests upon the direct integration of the equilibrium equations, the proposed solution scheme offers ample opportunities for efficient analysis of inhomogeneous solids In contrast to homogeneous materials, the compatibility equations in terms of stresses are with variable coefficients This causes that the governing equations, obtained on the basis of the compatibility ones, appear as integral equations of Volterra type By following this solution strategy, the one-dimensional thermoelasticity problem for a radially-inhomogeneous cylinder has been analyzed (Vihak & Kalyniak, 1999; Kalynyak, 2000) In the same manner, the two-dimensional elasticity and thermoelasticity problems for inhomogeneous cylinders, strips, planes and semi-planes were solved in (Tokovyy & Rychahivskyy, 2005; Tokovyy & Ma, 2008, 2008a, 2009, 2009a) The same method has also been extended for three-dimensional problems (Tokovyy & Ma, 2010, 2010a) Application of this method for analysis of inhomogeneous solids exhibits several advantages First of all, this method is unified for various kinds of inhomogeneity and different shape of domain and it does not impose any restriction on the material properties Moreover, when applying the resolvent-kernel algorithm for solution of the governing Volterra-type integral equation, the solutions of corresponding elasticity and thermoelasticity problems for inhomogeneous solids appear in the form of explicit functional dependences on the thermal and mechanical loadings, which makes them to be rather usable for complex engineering analysis

Herein, we consider an application of the direct integration method for analysis of thermoelastic response of an inhomogeneous semi-plane within the framework of linear uncoupled thermoelasticity (Nowacki, 1962) The solution of this problem consists of two

stages: i) solution of the in-plane steady-state heat conduction problem under certain boundary conditions, and ii) solution of the plane thermal stress problem due to the above

determined temperature field and appropriate boundary conditions Solution of both problems is reduced to the governing Volterra integral equation By making use of the resolvent-kernel solution technique, the governing equation is solved and the solution of the original problem is presented in an explicit form Due to the later result, the one-to-one relationships are set up between the tractions and displacements on the boundary of the inhomogeneous semi-plane Using these relations and the solution in terms of stresses, we find solutions for the boundary value problems with displacements or mixed conditions

pyroelectricity, Acta Mechanica 214(3): 275–289.

piezoelectric materials, Journal of Sound and Vibration 330(6): 111 1? ?112 0.

Lord,...