446 MICROMECHANICS Since cr =0,Egs.(11.66)and (11.62)give the following correspondence be- tween the thermal and moisture expansion coefficients: a→月 a6→0 (11.67) dm→ m」 Vm Longitudinal moisture expansion coefficient.The longitudinal moisture ex- pansion coefficient is obtained by replacing in Eq.(11.54)ai by B1,and @m by Bm/vm,and by setting n=0 as follows: 成，-侧 (11.68) Transverse moisture expansion coefficient.The transverse moisture expan- sion coefficient is obtained by replacing in Eq.(11.60)ai by Bi,m by Bm/vm,and by setting a =0: B2 =8m+vrvr2(-B1)+vm(Bm-vmB1) (11.69) 11.1.9 Thermal Conductivity The heat conducted(per unit time)in the longitudinal and transverse directions is given by Fourier's law =AK)(-) =A(K2) (11.70) where A is the cross-sectional area,aT/ax is the temperature gradient,and K1, K2 are the longitudinal and transverse heat conduction coefficients,respectively. The longitudinal and transverse normal forces are related to the longitudinal and transverse normal strains by Hooke's law as follows: F=A(E)(e1)F=A(E)(e2). (11.71) Equations (11.70)and(11.71)show the analogy between Fourier's and Hooke's laws.This analogy is used to obtain the micromechanical expressions for the ther- mal conductivities.The heat-conducted q corresponds to the force F,the thermal conductivity K to the Young modulus E and the temperature gradient aT/ax to the strain e. q→F K→E (11.72) aT ax →e.446 MICROMECHANICS Since cf = 0, Eqs. (11.66) and (11.62) give the following correspondence be￾tween the thermal and moisture expansion coefficients: αi =⇒ βi αfi =⇒ 0 αm =⇒ βm vm . (11.67) Longitudinal moisture expansion coefficient. The longitudinal moisture ex￾pansion coefficient is obtained by replacing in Eq. (11.54) α1 by β1, and αm by βm/vm, and by setting αf1 = 0 as follows: β1 = Em E1 βm. (11.68) Transverse moisture expansion coefficient. The transverse moisture expan￾sion coefficient is obtained by replacing in Eq. (11.60) αi by βi , αm by βm/vm, and by setting αfi = 0: β2 = βm + vfνf12(−β1) + νm(βm − vmβ1). (11.69) 11.1.9 Thermal Conductivity The heat conducted (per unit time) in the longitudinal and transverse directions is given by Fourier’s law q1 = A(K1)  − ∂T ∂x1  q2 = A(K2)  − ∂T ∂x2  , (11.70) where A is the cross-sectional area, ∂T/∂x is the temperature gradient, and K1, K2 are the longitudinal and transverse heat conduction coefficients, respectively. The longitudinal and transverse normal forces are related to the longitudinal and transverse normal strains by Hooke’s law as follows: F1 = A(E1) (1) F2 = A(E2) (2). (11.71) Equations (11.70) and (11.71) show the analogy between Fourier’s and Hooke’s laws. This analogy is used to obtain the micromechanical expressions for the ther￾mal conductivities. The heat-conducted q corresponds to the force F, the thermal conductivity K to the Young modulus E and the temperature gradient ∂T/∂x to the strain . q =⇒ F K =⇒ E ∂T ∂x =⇒ . (11.72)