Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur
Lecture 34 Thermal Stresses in Plates
Lecture Overview Introduction Mechanical and Thermal Strains Stiffness Matrix for a Lamina Thermal Forces and Moments
Introduction A change in a body s temperature causes its dimensions to grow or contract, therebygeneratingthermalstrains thermal strains. For a linear system, i.e. where thermal strain is directly proportional to change in temperature, the expression for thermal strain (ε T ) for isotropic material is: ε T = αδt where, α is coefficient of thermal expansion, and ΔT is rise in body s temperature. For an orthotropic unidirectional lamina, thermal strains in longitudinal and transverse directions are defined as: ε T L= L α L ΔT ε T T= α T ΔT where, α L and α T are coefficients of thermal expansion in longitudinal and transverse directions, respectively. Also, it should be noted here that there are no shear strains in the L T plane associated with thermal expansion. Hence, γ T LT = 0.
Introduction In earlier equations for ε T L and ε T L, the values of α L and α L, have been defined ed earlier e in Eqs. 17.8 and 17.9. These relations eato saebe are being reproduced here as well. α L = (E f V f α f + E m V m α m )/E L (Eq. 17.8) α T = (1+ν f )V f α f + (1+ν m )V m α m α L ν LT (Eq. 17.9) Equations 17.8and 17.9 9are valid for composites with fibers having isotropic properties. In case, fiber used in the composite are orthotropic (carbon and Kevlar), they have different axial and transverse properties. For composites made fromorthotropic orthotropic fibers and isotropic matrix, a different equation for α T, as proposed by Hasin should be used. This equation is given below. α T = (1+ν LTf α Lf /α Tf )V f α Tf +(1+ν m )V m α m (ν LTf V f α f + ν m V m )(Eα) L /E L (Eq. 34.1) where, E L = E Lf V f + E m V m (Eα) L = E Lf V f α Lf + E m V m α m
Introduction Looking at expressions for thermal strains with respect to material axes, it can be stated that α L and α T are identical to thermal strains corresponding to temperature increment of unity. Hence, these coefficients follow the same transformation law as that followed by the strain vector. Thus, coefficient of thermal expansions measured with respect to an arbitrarycoordinate system (x y) can bewritten as: Eq. 34.1 Here, [T] 1 is the inverse of [T] as defined in Eq. 10.7. Using these relations, thermal strains in arbitrary coordinate system can be written as: Eq. 34.2
Mechanical and Thermal Strains Thermal strains by themselves cannot generate a force or a moment, unless the body is not completely free to deform due to temperature. Thus, at the level of a whole laminate, there are no resultant forces and moments due to temperature alone. However, at the level of a lamina, the same may not be true. This is because a lamina by itself is not entirely free to bend ortwist or expand due to changes in temperature. These stresses in a lamina are attributable to strains which are in excess of thermal strains as defined in Eq. 34.2. These excessive strains are known as mechanical strains, and are denoted by a superscript M. Thus: Eq. 34.3
Mechanical and Thermal Strains In Eq. 34.3, we see that mechanical strain is the difference of total strain, and thermal strain. Further, thermal stresses at individual ply level may be calculated by multiplying mechanical strain vector with lamina stiffness matrix. Thus: Eq. 34.4 4 However, in Eq. 34.4, 4 mid plane strains andmid plane curvatures arenot known. Equation 34.44 may be integrated t over thickness (assuming temperature t is constant over thickness) as per Eq. 26.1 to yield relations for forceresultant vector. Similarly, Eq. 34.4 may be multiplied with z, and then integrated over thickness as per Eq. 26.2, 2 to yield relations for momentresultant vector.
Stiffness Matrices Thus, we get following relations for force and moment resultants. Eq. 34.5 Eq. 34.6
Thermal Forces and Moments Here, thermal force and moment vectors are defined as: (Eq. q 34.7 19.6) Fi ll th ilib i ti f th l t ti ll th Finally, the equilibrium equations for the plate are essentially the same as defined in Eqs. 27.6, 27.7 and 27.8. This is so, since resultant force and moment equations already include thermal effects on account of their revised definitions.
Thermal Forces and Moments Thermal stresses get induced in a laminate whenever the temperature of laminate differs from that of its free stress state. During the fabrication of composite plates, plies are stacked together at elevated temperatures. At these temperatures, matrix material permeates into different layers, and binds them because it gets cured. Later, the cured laminate is cooled to roomtemperature. As a result of this process, thermal stresses get induced in a laminate between different layers, because individual layers are no longer, after curing, free to contract during the cooling process. These stresses, which get induced in the laminate due to such cooling are known as residual or curing stresses. Residual stresses, if not properly managed, may lead to failure of laminate.
Rf References 1. Analysis and Performance of Fiber Composites, Agarwal, B.D.and Broutman, L. J., John Wiley & Sons. 2. Mechanics of Composite Materials, Jones, R. M., Mc Graw Hill. 3. Engineering Mechanics of Composite Materials, Daniel, I. M. and Ishai, O., Oxford University Press.