Plane Thermal Stress Analysis of an Orthotropic Cylinder Subjected to an Arbitrary, Transient, Asymmetric Temperature Distribution

A closed-form, analytical solution is presented for the transient, plane thermal stress analysis of a linearly elastic, homogeneously orthotropic hollow cylinder subjected to an arbitrary temperature distribution. The thermoelastic solution, obtained by a stress function approach, can be used as the basis for the corresponding thermoviscoelastic solution for thermorheologically simple viscoelastic materials by invoking the viscoelastic Correspondence Principle. This solution can also be directly extended to the class of weakly inhomogeneously orthotropic cylinders using perturbation methods. The transient asymmetric temperature field is characterized by Fourier-Bessel eigenfunction expansions. The analytically derived stress function satisfies a linear, fourth-order inhomogeneous partial differential equation and the Cesaro integral conditions, which assure the existence of a single-valued displacement field. The corresponding thermal stresses are then computed by the stress-stress function relations. A key feature of the analytical solution is that the hoop, radial, and shear stresses, due to the transient arbitrary temperature distribution, are expressed explicitly in terms of the scalar temperature field. A polymer composite example is presented to validate the current method and to qualitatively illustrate the distribution of thermal stresses due to an asymmetric temperature distribution. Numerical results are presented for the thermally driven hoop, radial and (interlaminar) shear stresses in a hollow, hoop-wound glass/epoxy cylinder. This analysis demonstrates that potentially debilitating interlaminar shear stresses can develop in laminated composites when subjected to an even modest transient asymmetric temperature distribution. Their magnitudes depend on the severity of the spatial and temporal thermal gradients in the circumferential direction. While still relatively low compared to the hoop stress, the shear stress may cause thermal failure due to the typically low interlaminar shear strengths of laminated composite materials.