3.1 Smart Materials
3.1.2 Magnetoelectroelastic Materials
In order to consider the effect of magnetic field in the constitutive equations, Eq. (3.1) must be modified to:
rij¼CijkleklekijEkdkijHkbij#nijm Di¼eijkejkþijEjþgijHjþci#þvim Bi¼dijkejkþgijEjþlijHjþsi#þtim
i;j;k;l¼1;2;3
ð Þ ð3:7Þ
in whichBiandHkare respectively magnetic induction and magneticfield.dkij,gkij, andlijare piezomagnetic, magnetoelectric, and magnetic permeability coefficients, respectively;siandtiare pyromagnetic, and hygromagnetic coefficients. Similar to piezoelectric materials, the symmetry of stress and strain tensors leads to the fol- lowing symmetrical properties:
dkij¼dkji; gij¼gji; lij¼lji ð3:8Þ These symmetric properties result in 18 piezomagnetic, 6 magnetoelectric and 6 magnetic permeability constants for the most general case of triclinic system. As an example, the constitutive equations (3.7) for orthotropic and radially polarized and magnetized materials in a cylindrical coordinate systemðr;h;zÞcan be written as:
rrr
rhh
rzz
rzh
rrz rrh
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;
¼
c33 c13 c23 0 0 0 c13 c11 c12 0 0 0 c23 c12 c22 0 0 0
0 0 0 2c66 0 0
0 0 0 0 2c44 0
0 0 0 0 0 2c55
2 66 66 66 66 4
3 77 77 77 77 5
err
ehh
ezz
ezh
erz erh
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;
e33 0 0 e31 0 0 e32 0 0
0 0 0
0 0 e24 0 e15 0 2
66 66 66 66 4
3 77 77 77 77 5
Er
Eh Ez 8>
<
>: 9>
=
>;
d33 0 0 d31 0 0 d32 0 0
0 0 0
0 0 d24 0 d15 0 2
66 66 66 66 4
3 77 77 77 77 5
Hr
Hh Hz
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>: 9>
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>; b1
b2
b3
0 0 0 8>
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;
#
n1
n2
n3
0 0 0 8>
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#
Dr Dh Dz
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=
>;¼
e33 e31 e32 0 0 0
0 0 0 0 0 2e15
0 0 0 0 2e24 0
2 64
3 75
err
ehh
ezz
ezh
erz
erh
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; þ
33 0 0
0 11 0
0 0 22
2 64
3 75
Er Eh Ez
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>;
þ
g33 0 0 0 g11 0 0 0 g22
2 64
3 75
Hr
Hh Hz
8>
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>: 9>
=
>;þ c1
c2
c3
8>
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>: 9>
=
>;#þ v1
v2
v3
8>
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>: 9>
=
>;m
Br Bh Bz 8>
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=
>;¼
d33 d31 d32 0 0 0
0 0 0 0 0 2d15
0 0 0 0 2d24 0
2 64
3 75
err
ehh
ezz
ezh
erz
erh
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; þ
g33 0 0 0 g11 0 0 0 g22
2 64
3 75
Er Eh Ez 8>
<
>: 9>
=
>;
þ
l33 0 0 0 l11 0
0 0 l22
2 64
3 75
Hr
Hh Hz
8>
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>: 9>
=
>;þ s1
s2
s3
8>
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>: 9>
=
>;#þ t1
t2
t3
8>
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>: 9>
=
>;m
ð3:9Þ
3.1.2.1 Potential Field Equations
The quasi-stationary magnetic field equations in the absence of free conducting electromagnetic current are expressed as:
70 3 Multiphysics of Smart Materials and Structures
Hi¼ u;i ð3:10Þ whereuis the scalar magnetic potential.
3.1.2.2 Conservation Equations
The conservation or divergence equations for a hygrothermomagnetoelectroelastic medium are provided in this section. The equation of motion is:
rij;jþfi¼qui;tt ð3:11Þ where fi, q, and t, respectively, stand for body force, density, and time. In magneto-hygrothermoelectroelastic analysis, an electrically conducting elastic solid subjected to an external magnetic field experiences the Lorentz force via the electromagnetic-elastic interaction which works as a body force in Eq. (3.11) as follows [20]:
~J¼ r ~h; r ~e¼ l~h;t; r:~h¼0; ~e¼ l~u;t~H
~h¼ r ~uH~
; ~f ¼l~J~H ð3:12Þ
in which,~J,~h,~e,~u, ~H, and~f are, respectively, the electric current density, per- turbation of magnetic field, perturbation of electric field, displacement, magnetic intensity, and the Lorenz force vectors; l represents the magnetic permeability.
Maxwell’s electromagnetic equations or equations of charge and current conser- vation are written as [21]:
Di;i¼qe; Bi;i¼0 ð3:13Þ in whichqeis the charge density.
Furthermore, the classical energy conservation equation is [22]:
qi;iþqcm#;tR
¼0 ð3:14Þ whereqi,cm, andRare heatflux component, specific heat at constant volume, and internal heat source per unit mass, respectively. However, Biot [23] introduced the effects of elastic term in the energy equation to obtain more accurate results for thermoelastic analysis. The energy equation (3.14) was modified for the classical, coupled thermoelasticity as follows:
qi;iþqS;tT0R
¼0 ð3:15Þ whereSdenotes the entropy per unit mass and is defined as:
qS¼bijeijþqcm
T0 # ð3:16Þ
Through Eqs. (3.15) and (3.16), the energy equation is coupled with the strain rate. Considering the advent of smart materials with coupled multiphysical inter- actions, the classical coupled thermoelasticity equations could be modified to also consider the coupling effects of electric, magnetic, and hygroscopic fields on the energy equations [24]. Accordingly, Eq. (3.16) could be written in the following form for classical, coupled hygrothermomagnetoelectroelasticity:
qS¼bijeijþciEiþsiHiþqcm T0
#þdtm ð3:17Þ
wheredtis the specific heat-moisture coefficient. On the other hand, the conservation law for the mass of moisture in the absence of a moisture source is given by [25]:
pi;iþm;t¼0 ð3:18Þ in which, pi represents the moisture flux component that is the rate of moisture transfer per unit area.
3.1.2.3 Fourier Heat Conduction and Fickian Moisture Diffusion
The following Fourier heat conduction theory which relates the heatfluxqi to the temperature gradient is the most widely used theory in the literature:
qi¼ kTij#;j ð3:19Þ Furthermore, the diffusion of moisture in a solid is basically the same as that of temperature. As a result, the Fickian moisture diffusion equation for moisturefluxpi
can be defined similar to Fourier heat conduction equation as follows:
pi¼ fHijm;j ð3:20Þ In the above equations, kijT and fHij are the thermal conductivity and moisture diffusivity coefficients, respectively. Substituting Eq. (3.19) into (3.14) and Eq. (3.20) into (3.18) lead to a diffusion-like equations with parabolic-type gov- erning differential equations for temperature and moisture concentration. To con- sider the possible effect of other physical fields on the heat and mass flux, Eqs. (3.19) and (3.20) could be modified as [3,18]:
qi¼kMijklekl;jkEijkEk;jkijkBHk;jkTij#;jkijHm;j
pi¼fMijklekl;jfEijkEk;jfBijkHk;jfTij#;jfHijm;j ð3:21Þ
72 3 Multiphysics of Smart Materials and Structures
where kijklM, kijkE, kBijk, kijH, fMijkl, fEijk, fBijk, fTijði;j;k;l¼1;2;3Þ are, respectively, strain-thermal conductivity, electric-thermal conductivity, magnetic-thermal con- ductivity, moisture-thermal conductivity (Dufour effect), strain-moisture diffusivity, electric-moisture diffusivity, magnetic-moisture diffusivity, heat-moisture diffusiv- ity (Soret effect) coefficients. These coefficients represent the degree of thermal and mechanical, thermal and electrical, thermal and magnetic, thermal and hygroscopic, hygroscopic and mechanical, hygroscopic and electrical, hygroscopic and magnetic, and hygroscopic and thermalfield interactions.
The conventional heat conduction and moisture diffusion theories based on the classical Fourier and Fickian laws lead to an infinite speed of thermal and moisture wave propagation due to the parabolic-type heat and mass transport equations.
Fourier and Fickian laws assume instantaneous hygrothermal responses and a quasi-equilibrium thermodynamic condition. The classical diffusion theories have been widely used in heat and mass transfer problems; however, the heat and mass transmission is observed to be a non-equilibrium phenomenon, and they propagate with a finite speed for applications involving very low temperature, high temper- ature gradients, short-pulse heating, laser drying, laser melting and welding, rapid solidification, very high frequencies of heat and mass flux densities, and micro temporal and spatial scales [26]. Consequently, different non-Fourier and non-Fickian heat and mass transfer theories have been developed to remove these drawbacks.