{'fraction_non_alphanumeric': 0.011387900355871887, 'fraction_numerical': 0.0, 'mean_word_length': 4.882845188284519, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Thermal Stresses in an Infinite Elastic Pipe Weakened by Two Cylindrical Cracks

Axially symmetric thermal stresses in an elastic pipe weakened by two cylindrical cracks are provided. The surfaces of the cracks are assumed to be thermally insulated. The outer surface of the pipe is heated to maintain a constant temperature T d , and the inner surface of the pipe is cooled to maintain a constant temperature T b . As a first step, the boundary conditions related to the temperature field are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the cracks, the temperature difference at the crack surfaces is expanded into a series of functions that diminish to zero outside the cracks. The unknown coefficients in the series are determined by the Schmidt method so as to satisfy the thermal insulation inside the cracks. Next, the boundary conditions related to the stress field are reduced to dual integral equations. To solve the equations, the differences in the displacements at the crack surfaces are again expanded in a series of functions that diminish to zero outside the cracks. The Schmidt method is also used to solve the unknown coefficients in the series so as to satisfy the stress-free conditions inside the cracks. The stress intensity factors are defined and calculated numerically for several configurations of the pipe.