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Thermoelastic damping in an auxeticrectangular plate with thermalrelaxation—free vibrations


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Smart Mater. Struct. 22 (2013) 000000 (6pp) UNCORRECTED PROOF

(Ed: Janet Thomas) Ascii/Word/SMS/ sms456486/SPE Printed 7/6/2013 Spelling US Issue no Total pages First page Last page File name Date req Artnum Cover date

Thermoelastic damping in an auxetic

rectangular plate with thermal

relaxation—free vibrations

Bogdan T Maruszewski, Andrzej Drzewiecki and Roman Starosta

Institute of Applied Mechanics, Poznan University of Technology, ulica Jana Pawla II 24, 60-965 Poznan, Poland



Received 16 December 2012, in final form 25 May 2013 Published

Online atstacks.iop.org/SMS/22/000000


This work describes an extended thermodynamical model to represent coupled thermomechanical interactions in continuum media having negative Poisson’s ratio. In particular, the Zener thermoelastic damping effect is considered for a plate with auxetic characteristics undergoing free vibrations. The extended thermodynamical model is characterized by a thermal relaxation time to avoid the propagation of thermal waves at infinite velocity. The thermal relaxation time used in this work is not Zener’s characteristic time constant. Strong dependence of the thermoelastic damping is observed for auxetic configurations, various plate thicknesses and ambient temperatures.

(Some figures may appear in colour only in the online journal) Q.1


1. Introduction

The definite geometry of a body can even be an origin of Q.3

certain unusual thermoelastic phenomena, which do not occur in infinite media (cf [1–4]). One of these phenomena is the so-called thermoelastic damping, which has nothing to do with the eventual viscous features of the body. It is observed in a pure elastic medium. The phenomenon of energy dissipation (not observed in a pure elastic body) in the case, for instance, of a thermoelastic plate during its vibrations, comes from an additional heat flux normal to the boundaries of the plate. The origin of this flux is the alternate compression and extension of the upper and lower fibers of the body. In this way, in the case of the plate the problem is 2D (plate)–3D (additional dimension resulting from its thickness). Zener [5] was the first to point out that one of the mechanisms of thermoelastic damping is based on the stress heterogeneities giving rise to fluctuations of temperature. That idea was developed by Alblas [6, 7] and then by Maruszewski [8], Bishop and Kinra [9], Milligan and Kinra [10], Kinra and Milligan [11] and Bishop and Kinra [12]. However, all the above considerations have been based on the classical irreversible thermodynamics from which the space–time

distribution (evolution) of the thermal field is described by the parabolic (transport, diffusion) heat equation. Such evolution is observed in the room temperature range and results, among others, from Fourier’s constitutive law for heat flux. Solutions of that equation suggest that the thermal signals propagate with infinite velocity. However, at low temperatures and for high local gradients the continual thermal field in a material characterizes the relaxation character, which allows the propagation of thermal signals with finite velocities. In order to model this character Vernotte, Cattaneo and others [14–16] generalized Fourier’s law. They introduced a time dealing with the thermal relaxation, calling it the thermal relaxation time. Discussion on the physical meaning of that time has been presented in [17]. To be more close to reality, which means that all physical signals should propagate with finite velocities, Ignaczak and Ostoja-Starzewski [13] (see also references therein) have proposed considerations dealing with the thermoelastic damping based on the extended thermodynamical model (cf [14–20]). Resulting from that model and the Vernotte–Cattaneo constitutive law, the heat conduction equation is now of the hyperbolic (wave) form. A detailed and broad discussion of the latter problems has been made by Ignaczak and Ostoja-Starzewski [13].



The phenomenon of the above described thermal relaxation is crucial in microscience, nanoscience and engineering (cf [21–24]).

In recent years materials with so-called negative properties have been of great interest, being accurately investigated for their very interesting and unexpected physical features and behavior. Such properties can, among others, be characterized by negative Poisson’s ratio, negative compressibility, negative stiffness, negative heat expansion coefficient, and the like (see [25–31]). The behavior of these materials strongly influences many mechanical structures, like laminar ones, composites, woven structures and other multiphase ones. From among the materials with the properties listed above we focus our attention in this paper only on those with negative Poisson’s ratio, called the auxetics.

This paper deals with the free bending vibrations of a thermoelastic rectangular plate in which the thermal field is also described by one relaxation time. This particular analysis has been made for both classic and auxetic materials.

2. Basic equations

The equations governing thermoelastic processes with the relaxation of the thermal field have the following form [18] (the index tensor notation for both the vector components and space differentiation has been used; i, j = 1, 2, 3; the superimposed dot denotes the partial time derivative): ui,jj+ 1 1 − 2νuj,ij− m µθ,i= ρ µu¨i, (2.1) θ,jj−  τ ∂ ∂t+1  ρcv k ˙ θ +m kT0u˙k,k  =0, (2.2) m =(3λ + 2µ)αT =2µ 1 +ν 1 − 2ναT, θ = T − T0, θ T0 1.

In equations (2.1) and (2.2)θ represents a small change in temperature occurring during the vibration process. ui

denotes the elastic displacement vector, T is the absolute temperature,ν is Poisson’s ratio, ρ is the density, λ, µ denote Lam´e’s constants,αT is the heat expansion coefficient, T0 is

the reference temperature (room, surrounding, etc), k is the heat conductivity coefficient and cv is the specific heat for

constant volume. The mass forces and heat sources have been neglected. The boundary conditions for (2.1) read

bµ(ui,j+uj,i) + λ(uk,k−mθ)δijcnj=pi, (2.3)

where piis the traction, ni is the normal unit vector positive

outwards andδijis the Kronecker symbol. Then, the boundary

condition for equation (2.2) in the general form is as follows [13]: ∂θ ∂n+η  τ ∂ ∂t+1  θ = f , (2.4)

whereη is the surface heat exchange coefficient and f denotes an arbitrary function.

Figure 1. Geometry of the problem.

The thermal relaxation time τ is treated as a free parameter, because it is defined by the Vernotte–Cattaneo law (cf [14–16]). It is a fundamental parameter in the extended thermodynamical model of interactions dealing with so-called ‘second sound’ effects [18–21] and it is measured only at low temperatures (cf [13]). Please note that the relaxation time τ is not the characteristic time in Zener’s model of thermoelastic damping dependent on the thickness of the plate. The comparison of these times and description of the difference between their physical meanings is very well presented by Ignaczak and Ostoja-Starzewski [13].

Let us now adjust equations (2.1) and (2.2) to the dynamics of a rectangular plate (see figure1).

In this way equations (2.1) take the form (cf [2,7]) D0w,ααββ+ρh ¨w + 1 1 −νT MT,αα=p, (2.5) θ,ii−  τ ∂ ∂t +1 ρc v k ˙ θ +m kT0e˙  =0, (2.6) whereα, β = 1, 2, i = 1, 2, 3, D0= ETh 3 12(1−νT2), m = ETαT 1−2νT and the moment MT, due to the temperature distribution, is given

by MT =αTET Z h2 −h 2 θ(x1, x2, x3)x3dx3. (2.7)

The general form of the dilatation e for the thermoelastic plate is as follows:

e = 1 − 2ν

1 −ν (u,1+ν,2−x3w,αα) + αT 1 +ν

1 −νθ. (2.8) uand v denote displacements corresponding to the elongation of the middle surface and w is the deflection of the plate. However, in the following we confine ourselves to only a simplified form of e, i.e. (cf [7])

e = −1 − 2ν

1 −ν x3w,αα, (2.9)

because in the case of small plate bending the states described by u and v are approximately independent of the state Q.4 described by w (we still consider only pure bending of the plate). In such a situation one frequently assumes that the


middle surface area of the plate becomes constant during a small bending process (see [1–4]).

Moreover, the influence of the thermal contribution in (2.8) on the dilatation e in the case of small bending and temperature changes coming only from extension and compression of the plate fibers during vibrations can also be neglected as it is very small [7]. Note that the parameters ET

andνT in (2.5) and (2.6) are isothermal.

Poisson’s ratioν in (2.8) and (2.9) has an effective value dependent on the vibrational mode (see [7,32]),

(1 − ν2 T)(1 − 2νs) = (1 − ν)(1 − 2νs) − (νs−νT)(1 − 2ν) ×(1 − ν)(1 + νT)ω 2[1 −τ22ω2 x)] τ22 x−ω2)2+ω2 , (2.10)

whereνsis the adiabatic Poisson’s ratio (cf [1,2]),

ω2 x = Hk τ%cv, H = π 2 a2 + π2 b2; (2.11)

ωx denotes a new frequency connected with the relaxation

character of the thermal field in the description within the extended thermodynamical model of interactions. However, ν does not differ much from νT, so we assume thatν = νT in

the following (see [7]).

For the model of interactions taken in the paper we assume that the changes of temperature come only from vibrations of the plate and there is no thermal influence from the surroundings.

3. Free vibrations of the thermoelastic plate

The object of our considerations is a rectangular plate of thickness h simply supported on all edges. For the equations (2.5) and (2.6), putting pi=0, the boundary conditions are as

follows [3]: w(0, x2) = w(a, x2) = w(x1, 0) = w(x1, b) = 0, (3.1) w,11+ 1 D0(1 − νT) MT =0 at x1=0, a, (3.2) w,22+ 1 D0(1 − νT) MT =0 at x2=0, b, (3.3) θ = 0 at x1=0, a, (3.4) θ = 0 at x2=0, b, (3.5) ∂θ ∂x3 +η  τ ∂ ∂t +1  θ = 0 at x3= ± h 2. (3.6) The conditions (3.4)–(3.6) indicate that the temperature at lateral surfaces has been determined. At the remaining boundaries the temperature varies because of alternate extension and compression of the upper and lower fibers of the plate during vibrations. At these boundaries free heat exchange has been assumed.

Since we are interested in the description of the thermoelastic damping of a rectangular thermoelastic plate having Poisson’s ratio ν = νT ∈ (−1; 0.5) during free

bending vibrations, solutions of equations (2.5) and (2.6) with conditions (3.1)–(3.6) are looked for in the forms

w = ∞ X m=1 ∞ X n=1 w00mnsin mπ a x1  sinnπ b x2  eiωt, (3.7) θ = ∞ X m=1 ∞ X n=1 θ00mn(x3) sin mπ a x1  sinnπ b x2  eiωt. (3.8) For the sake of simplicity we only take the first terms of expansions (3.7) and (3.8) into consideration and we denote in the following w0011=w00andθ0011=θ00.

Substituting the first terms of (3.7) and (3.8) into equations (2.5) and (2.6) we obtain

(ω2 0−ω2)w00−β Z h2 −h 2 θ00(x3)x3dx3=0, (3.9) ∂2θ 00 ∂x2 3 − [H +(iω − τω2)γ ]θ00 = (iω − τω2)δx3w00, (3.10) where ω2 0= D0 ρhH2, β = αTETH (1 − νT)ρh , γ = ρcv k , δ = ETαTT0H k 1 − 2ν 1 − 2νT. Denoting B = Z h2 −h 2 θ00(x3)x3dx3, (3.11)

the solution (3.9) becomes

w00= βB

ω2 0−ω2

. (3.12)

Substituting now (3.12) into (3.10) yields d2θ00 dx23 −ε 2θ 00=Cx3, (3.13) where ε2=H +(iω − τω2)γ, C = βδB(iω − τω2) ω2 0−ω2 . (3.14) Denotingθ = ¯C ¯θ00, equation (3.13) takes the form


dx23 −ε2θ¯

00 =x3. (3.15)

Assuming homogeneous boundary conditions at h/2 and −h/2, the solution of (3.15) reads

¯ θ00=

−e12(h−2x3)εh +e12(h+2x3)εh +2x



Figure 2. First self-frequency versus Poisson’s ratio for different plate thicknesses. Hence, B = ¯C ¯D, where ¯ D = −h(12 − h 2ε26hε(cothhε 2)) 12ε4 , (3.17) ¯ C = βδB(iω − τω 2) ω2 0−ω2 , (3.18)

and the first self-frequency of the plate should satisfy the following equation:

ω2 0−ω

2βδ(iω − τω2) ¯D = 0. (3.19)

4. Numerical results

The main aim of the simulations presented in this section is to check how changes of one of the mechanical coefficients (Poisson’s ratio) influences the dynamics and thermoelastic damping of materials. As the reference classical medium we have chosen a steel-like material with ν = 0.3. Then, changing the Poisson’s ratioν within the range −1 < ν < 0.5 we assume that each of its values (except for ν = 0.3) is related to another material. We are still interested in testing the influence of a single elastic coefficient variation on the behavior of thermoelastic (practically hypothetical) materials. Just variation of the Poisson’s ratio gives a possibility to design and produce materials that are very important in new

technologies, namely materials called auxetics. Moreover, since our description has extended character which is based on the thermal relaxation, we are dealing with processes running in reasonably low temperatures where the ‘second sound’

effects are observed [14–16]. Q.5

Let us consider now a material with the following properties (a thin plate):

ET =1011N m−2, αT =3 × 10−6K−1,

ρ = 7860 kg m−3, k =58 J s−1m−1K−1,

cv =460 J kg−1K−1, τ = 10−10s,

h =0.005 m, a =1 m, b =1 m, T0=100 K.

Note thatθ00is in K and w00is in m.

From figure 2 it results that the first self-frequency achieves the highest values if the plate is in the auxetic state, reaching a minimum forν = 0 and being almost constant for the non-auxetic state.

Figure3indicates that the thermoelastic damping Im(ω), as the imaginary part of the frequency, is lowest in the auxetic state, increasing if the Poisson’s ratio also increases, not depending much on the plate thickness.

The distribution of the temperature amplitude along the plate thickness is shown for both the auxetic and non-auxetic material states in figure4.

We see (figure 4(b)) that if the temperature amplitude and bending are phase shifted by π/2, the temperature in a reasonably large central region of the plate is practically equal to zero because of the thermoelastic damping.

From figure 5 it results that the first self-frequency practically does not depend on the reference temperature (is independent of the surrounding temperature) because curves for different reference temperatures T0 from a reasonably

large interval cover each other. That is one of the results of the simulations. However, the thermoelastic damping is strongly dependent on the surrounding temperature (see figure6).

The results shown in figure 6 come from calculations made in the paper and indicate that for very low reference temperatures the dependence of the thermoelastic damping on the Poisson’s ratio is almost constant.

In contrast to the mechanical interactions Re(ω) (figure5), the thermoelastic damping strongly depends on the reference temperature.


Figure 4. Distribution of the temperature amplitude along the plate thickness: (a) in the same phase as the bending, (b) with the phases shifted byπ/2 with respect to each other (cf [9–12]).

Figure 5. First self-frequency versus Poisson’s ratio for three reference temperatures: T0=10, 100 and 273 K.

Figure 6. Thermoelastic damping versus Poisson’s ratio for different reference temperatures.

Then, the thermoelastic damping depends almost linearly on the reference temperature T0 for different values of the

Poisson’s ratio (see figure7).

5. Conclusions

The thermoelastic interactions in a plate of thickness h present the following situation: the mechanical field is described in two dimensions but the thermal one is distributed in three dimensions. Therefore, we were interested in the third

Figure 7. Thermoelastic damping versus reference temperature T0

for different Poisson’s ratios.

component of the heat flux along the thickness of the plate. That component is responsible for the energy dissipation occurring during pure mechanical dynamics of the plate during bending—it is the basis of the so-called thermoelastic damping. Research has been carried out for both normal and auxetic plate material. During free vibrations when the plate was alternately bending and its upper and lower fibers were also alternately extended and compressed we noticed considerable influence of the Poisson’s ratio on the first self-frequency and the thermoelastic damping. This means that the mechanical energy dissipation determined by the ‘extra’ component of the heat flux (the component normal to the middle surface) strongly depends on the generalized auxeticity (the property that characterizes materials with Poisson’s ratio in the interval −1 < ν < 0.5). However, the greatest changes of quantities shown in figures 2–7 are noticed for pure auxetics (ν ∈ (−1, 0)). With regard to the Q.6 influence of the thermal relaxation time, which is significant at ultrasonic frequencies, on the dispersion of vibrations and the thermoelastic damping, the analysis of the results obtained in this paper shows that it is negligible in both auxetic and normal states in the case of free vibrations of the plate. This fact results from relation (3.19) and figures 2, 3, 5–7, where a difference between the normal and auxetic states for various reference temperatures and values of h occurs only at frequencies much lower than ultrasonic ones.


The authors have not compared with results obtained by others in detail since the paper consists of a description of the thermoelastic damping taking into account both the auxetic properties of the elastic field and the relaxation properties of the temperature field. Moreover, no experimental validation has been carried out in this paper.


The paper has been supported by MNSzW 2363/B/T02/ 2010/39 grant.



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Figure 1. Geometry of the problem.
Figure 3. Thermoelastic damping versus Poisson’s ratio for different plate thicknesses.
Figure 4. Distribution of the temperature amplitude along the plate thickness: (a) in the same phase as the bending, (b) with the phases shifted by π/2 with respect to each other (cf [9–12]).


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