Solution Procedure

3.4 Effect of Hygrothermal Excitation on One-Dimensional Smart

3.4.1 Solution Procedure

The solution procedure for hollow and solid MEE cylinders is given in this section similar to those reported in [40] for magnetoelectroelastic cylinders. Substituting Eqs. (3.103) and (3.25) into Eq. (3.24) results in the following coupled ordinary differential equations in terms of moisture concentration, temperature change, magnetic potential, electrical potential, and displacement:

c₃₃r²u_;rrþc₃₃ru_;rc₁₁uþe₃₃r²u_;rrþðe₃₃e₃₁Þru_;r

þd33r²/_;rrþðd33d31Þr/_;rb1r²#_;r

ðb1b3Þr#n1r²m_;rðn1n3Þrmþqdr³x²¼0 ð3:104aÞ e33r²u_;rrþðe31þe33Þru_;r 233r²u_;rr 233r/_;r

g₃₃r²u_;rrg₃₃ru_;rþc1r²#_;rþc1r#þv1r²m_;rþv1rm¼0 ð3:104bÞ d33r²u_;rrþðd31þd33Þru_;rg33r²/_;rrg33r/_;rl33r²u_;rr

l33ru_;rþs1r²#_;rþs1r#þt1r²m_;rþt1rm¼0 ð3:104cÞ The non-dimensional parameters are introduced in Eq. (3.28) and new electric potential, magnetic potential, temperature change, and moisture concentration change as:

U¼e₃₃

c33/; W¼d₃₃

c33u; H¼ b1

c33#; M¼ n1

c33

m ð3:105Þ Using Eqs. (3.28) and (3.105), Eq. (3.104) can be rewritten in the following form:

r²u_;rrþru_;rauþr²U;rrþð1bÞrU;rþr²W;rrþð1mÞrW;r

r²H;rð1gÞrHr²M_;rð11ÞrMþXr³

a²¼0 ð3:106aÞ r²u_;rrþð1þmÞru_;rfr²U_;rrfrU_;rkr²W_;rrkrW_;r

þYr²H_;rþYrHþWr²M_;rþWrM¼0 ð3:106bÞ r²u_;rrþð1þbÞru_;rcr²U;rrcrU;rfr²W;rrfrW;r

þXr²H_;rþXrHþVr²M_;rþVrM¼0 ð3:106cÞ in which,

X¼qdx²a² c33

ð3:107Þ

The following set of second-order coupled ordinary differential equations with constant coefficients is obtained by using the non-dimensional radial coordinate q¼^r_aand then changing variableq withsbyq¼e^s as follows:

€

uauþU€ bU_ þW€ mW_ ¼ae^sH_ þð1gÞae^sH

þae^sM_ þð11Þae^sMXae^3s ð3:108aÞ

€

uþbu_cU€fW€ ¼ Xae^sH_ Xae^sHVae^sM_ Vae^sM ð3:108bÞ

€

uþm_ufU€kW€ ¼ Yae^sH_ Yae^sHWae^sM_ Wae^sM ð3:108cÞ in which, the overdot stands for differentiation with respect to s. For uncoupled hygrothermomagnetoelectroelastic problems, temperature and moisture concentra- tion distributions are obtained separately by solving heat conduction and moisture diffusion equations. The axisymmetric and steady state Fourier heat conduction and Fickian moisture diffusion equations for an infinitely long hollow cylinder are written as [17]:

1

rrk^Th;r

;r¼0 ðHeat conduction equationÞ ð3:109aÞ 1

rrk^Cm_;r

;r¼0 ðMoisture diffusion equationÞ ð3:109bÞ where, k^T and k^C are thermal conductivity and moisture diffusivity coefficients, respectively. Solving Eq. (3.109) and using the non-dimensional radial coordinate as well as the new parameters defined in Eq. (3.105) result in the following tem- perature and moisture concentration distribution along the radial direction for a homogenous cylinder:

102 3 Multiphysics of Smart Materials and Structures

H¼C1lnð Þ þq C2 ð3:110aÞ M¼C3lnð Þ þq C4 ð3:110bÞ whereC1,C2,C3, andC4 are integration constants which are determined by satis- fying the hygrothermal boundary conditions. The non-dimensional temperature change on the inner and outer surfaces of the hollow cylinder are, respectively, assumed to be Ha and Hb. Similarly, non-dimensional moisture concentration change are assigned to beMaandMbon the inner and outer surfaces. The integration constants according to the hygrothermal boundary conditions can be obtained as:

C1¼HbHa

lnð Þi ; C2¼Ha; C3 ¼MbMa

lnð Þi ; C4¼Ma ð3:111Þ in which,i¼^b_ais the aspect ratio of the hollow cylinder. It should be mentioned that for solid cylinders as well as hollow cylinders with uniform temperature and moisture concentration rise, we haveC1 ¼C3¼0. For more general hygrothermal boundary conditions, one may refer to the Robin-type boundary conditions con- sidered in [17,32]. It is worth noting that temperature and moisture concentration have similar effects on magnetoelectroelastic responses in uncoupled hygrother- momagnetoelectroelasticity according to Eqs. (3.108) through (3.111). The solution of Eq. (3.108) can be found analytically by successive decoupling method [40].

Eliminating Ubetween equations in (3.108) leads to the following two ordinary differential equations aboutu andW:

1þc

ð Þu^vb²þac _

uþðcfÞW^vþðbfmcÞW€

¼ðcXÞae^sH€þðcð2gÞ þXðb2ÞÞae^sH_ þðcð1gÞ þXðb1ÞÞae^sH þðcVÞae^sM€ þðcð21Þ þVðb2ÞÞae^sM_

þðcð11Þ þVðb1ÞÞae^sM3aXce^3s

ð3:112aÞ cf

ð Þ€uþðmcbfÞu_þf²kcW€ ¼ðXfYcÞae^sH_

þðXfYcÞae^sHðVfWcÞae^sM_ þðVfWcÞae^sM ð3:112bÞ By eliminatingWbetween Eqs. (3.112) and considering the temperature change and moisture concentration change distributions according to Eq. (3.110), the fol- lowing ordinary differential equation with constant coefficients for radial dis- placementuis achieved:

a₂u^vþa₁u_ ¼ðb₁C₂þb₂C₁þb₄C₄þb₅C₃Þe^sþðb₁C₁þb₄c₃Þse^sþb₅e^3s ð3:113Þ

where,

a₁¼ðmcbfÞ²

f²kc b²þac

; a₂¼1þcðcfÞ² f²kc b1¼a cð1gÞ þXðb1Þ ðXfYcÞðbfmcþcfÞ

f²kc

b2¼a cð2gÞ þXðb2Þ ðXfYcÞðbfmcþ2c2fÞ f²kc

b₃¼a cð11Þ þVðb1Þ ðVfWcÞðbfmcþcfÞ f²kc

b4¼a cð21Þ þVðb2Þ ðVfWcÞðbfmcþ2c2fÞ f²kc

b₅¼ 3acX

ð3:114Þ

The solution of Eq. (3.113) in terms of variableqcan be expressed as:

u¼AþCq^mþDq^mþðK₁lnð Þ þq K₂ÞqþK₃q³ ð3:115Þ in which,

m¼i ffiffiffiffiffi a1

a2

r

m2R and i¼ ffiffiffiffiffiffiffi 1

p

ð3:116Þ

andA,C, andDare integration constants. Employing Eqs. (3.113) and (3.115), the following expression forWcan be obtained:

W¼FþElnð Þ þq Cc1q^mþDc2q^mþðc3lnð Þ þq c4ÞqþK3c5q³ ð3:117Þ where,Eand Fare new integration constants and:

c1¼ 1

mkcf²ðmðcfÞ þmcbfÞ;c2 ¼ 1

mkcf²ðmðcfÞ mcþbfÞ c3¼ 1

kcf²ðK1ðcð1þmÞ fð1þbÞÞ aC1ðXfYcÞ aC3ðVfWcÞÞ c4¼ 1

kcf²ðK2ðcfÞ þðK2K1ÞðmcbfÞ þa Cð 1C2ÞðXfYcÞ þa Cð ₃C₄ÞðVfWcÞÞ

c5¼ 1

3kcf²ð3ðcfÞ þmcbfÞ

ð3:118Þ

104 3 Multiphysics of Smart Materials and Structures

Using Eq. (3.108b) and considering Eqs. (3.115) and (3.117), U can be expressed by:

U¼HþGlnð Þ þq Cl1q^mþDl2q^mþðl3lnð Þ þq l4ÞqþK3l5q³ ð3:119Þ in which,Gand Hare integration constants and:

l1 ¼ 1

mcðmþbmfc1Þ;l2¼ 1

mcðmbmfc2Þ l3 ¼1

cðK1ð1þbÞ fc3þaXC1þaVC3Þ l₄ ¼1

cðK₂þðK₂K₁Þbc₄fþaX Cð ₂C₁Þ þaV Cð ₄C₃ÞÞ l5 ¼ 1

3cð3þb3fc5Þ

ð3:120Þ

For convenience, the following non-dimensional stresses, displacement, electric potential, and magnetic potential are used:

Rrr¼rrr

c₃₃;Rhh¼rhh

c₃₃;Rzz¼rzz

c₃₃;U¼u

a;U1 ¼U

a;W1¼W

a ð3:121Þ Then, we reach:

Rrr¼q¹

a ðdAþGþEÞ þq^m1

a ðmð1þl1þc1Þ þdÞC þq^m1

a ðmð1þl2þc2Þ þdÞDþlnð Þq

a ðK1ð1þdÞ þl3þc3a Cð 1þC3ÞÞ þ1

aðK1þð1þdÞK2þl3þl4þc3þc4a Cð 2þC4ÞÞ þq²

a ð3 1ð þl5þc5Þ þdÞK3

ð3:122aÞ Rhh¼q¹

a ðdAþbGþmEÞ þ q^m1

a ðmðdþbl1þmc1Þ þdÞC þq^m1

a ðmðdþbl2þmc2Þ þdÞDþlnð Þq

a ðK1ðdþaÞ þbl3þmc3aðgC1þ1C3ÞÞ þ 1

aðdK₁þðdþaÞK₂þbðl₃þl₄Þ þmðc₃þc₄Þ aðgC₂1C₄ÞÞ þ q²

a ð3ðdþbl5þmc5Þ þaÞK3

ð3:122bÞ

Rzz¼q¹

a ðdAþbGþmEÞ þq^m1

a ðmðdþbl₁þmc₁Þ þdÞC þq^m1

a ðmðdþbl2þmc2Þ þdÞDþlnð Þq

a ðK1ðdþdÞ þbl3þmc3aðgC1þ1C3ÞÞ þ1

aðdK₁þðdþdÞK₂þbðl₃þl₄Þ þmðc₃þc₄Þ aðgC₂þ1C₄ÞÞ þq²

að3ðdþbl5þmc5Þ þdÞK3

ð3:122cÞ W1¼F

a þE

alnð Þ þq C

ac1q^mþD

aC2q^mþðc3lnð Þ þq c4Þq a þK3c5

a q³ ð3:122dÞ U1¼H

a þG

alnð Þ þq C

al1q^mþD

al2q^mþðl3lnð Þ þq l4Þq a þK3l5

a q³ ð3:122eÞ U¼A

a þC aq^mþD

aq^mþ K₁

a lnð Þ þq K₂ a

qþK₃

a q³ ð3:122fÞ We have seven integration constants in the aforementioned equations; however, there exist only six boundary conditions in the magnetoelectroelastic medium.

Therefore, one complimentary equation is needed which is obtained by substituting Eqs. (3.115), (3.117), and (3.119) into Eq. (3.108):

aAþmEþbG¼0 ð3:123Þ The integration constants are obtained in the following subsections for hollow and solid cylinders. Although the solution procedure for this hygrothermomagne- toelectroelastic analysis under steady-state condition is the same as [40], the work is a pioneer in such emerging multiphysical analysis. The analytical solutions given in Eq. (3.122) could be employed for the design of MEE structures as well as a benchmark solution for verification of the other analytical and numerical results which will be used later for the multiphysical problem.

To consider the effect of temperature and moisture dependency of elastic coef- ficients on the magnetoelectroelastic response, the elastic coefficients are expressed in the following form [41,42]:

Cij¼Cij0ð1þa#þbMÞ ð3:124Þ in which,C_ij0 is an elastic coefficient at stress-free temperature and moisture con- centration;aandbare empirical material constants for temperature and moisture dependency. In the current work, the temperature and moisture dependency is only considered for uniform temperature and moisture concentration rise to avoid dealing with non-linear–problems [43].

106 3 Multiphysics of Smart Materials and Structures