8.1 Thermal Expansion 281 to be unable to change its length by the amount dictated by its CTE(Eq.8.2). As a result,each component experiences a thermal stress.The component that expands more than the amount indicated by its CTE experiences thermal stress that is tensile,whereas the component that expands less than the amount indicated by its CTE experiences thermal stress that is compressive.The thermal stress is equal to the thermal force divided by the cross-sectional area.In the absence of an applied force, F1+F2=0, (8.9) where F and Fare the thermal forces in component I(all of the strips of com- ponent I together)and component 2(all of the strips of component 2 together), respectively.Hence, F1=-F2. (8.10) Since force is the product of stress and cross-sectional area,Eq.8.10 can be written U1VIA U2V2A, (8.11) where U and U2 are the stresses in components 1 and 2,respectively,and A is the area of the overall composite.Dividing by A gives U1=U22. (8.12) Using Eq.8.2,the strain in component 1 is given by (ac-)AT and the strain in component 2 is given by(ac-a2)AT.Since stress is the product of the strain and the modulus,Eq.7.12 becomes (ae-a1)△TM1h=-(ae-a2)△TM22, (8.13) where M and M2 are the elastic moduli of components 1 and 2,respectively. Dividing Eq.8.13 by AT gives (ac-a1)M1v1 =-(ac-a2)M2v2. (8.14) Rearrangement gives ac=(a1M1V1+c2M2V2)/(M11+M2V2). (8.15) Equation 8.15 is the rule of mixtures expression for the CTE of a composite in the parallel configuration. In the case that M M2,Eq.8.15 becomes xc=(1M+c22)/(v1+v2)=a1+a22, (8.16) since 1+V2=1. (8.17)8.1 Thermal Expansion 281 to be unable to change its length by the amount dictated by its CTE (Eq. 8.2). As a result, each component experiences a thermal stress. The component that expands more than the amount indicated by its CTE experiences thermal stress that is tensile, whereas the component that expands less than the amount indicated by its CTE experiences thermal stress that is compressive. The thermal stress is equal to the thermal force divided by the cross-sectional area. In the absence of an applied force, F1 + F2 = 0 , (8.9) where F1 and F2are the thermal forces in component 1 (all of the strips of com￾ponent 1 together) and component 2 (all of the strips of component 2 together), respectively. Hence, F1 = −F2 . (8.10) Since force is the product of stress and cross-sectional area, Eq. 8.10 can be written as U1v1A = U2v2A , (8.11) where U1 and U2 are the stresses in components 1 and 2, respectively, and A is the area of the overall composite. Dividing by A gives U1v1 = U2v2 . (8.12) Using Eq. 8.2, the strain in component 1 is given by (αc − α1)ΔT and the strain in component 2 is given by (αc − α2)ΔT. Since stress is the product of the strain and the modulus, Eq. 7.12 becomes (αc − α1)ΔTM1v1 = −(αc − α2)ΔTM2v2 , (8.13) where M1 and M2 are the elastic moduli of components 1 and 2, respectively. Dividing Eq. 8.13 by ΔT gives (αc − α1)M1v1 = −(αc − α2)M2v2 . (8.14) Rearrangement gives αc = (α1M1v1 + α2M2v2)/(M1v1 + M2v2) . (8.15) Equation 8.15 is the rule of mixtures expression for the CTE of a composite in the parallel configuration. In the case that M1 = M2, Eq. 8.15 becomes αc = (α1v1 + α2v2)/(v1 + v2) = α1v1 + α2v2 , (8.16) since v1 + v2 = 1 . (8.17)