The Analytical Solutions For Simply Supported Beam With A Fracture Process Zone Under Uniform Load

In the process of crack propagation of concrete materials,there is a cohesive zone(known as the fracture process zone)at the crack tip.Due to the complexity of the crack tip process zone,the linear elastic fracture mechanics is no longer suitable for concrete materials.The analytical solution of the stress function at the crack tip of the zone model is difficult or even impossible to solve.Based on "Duan and Nakagawa's Model",this paper takes the simply supported beam with fracture process zone under uniform load as the research object.The function superposition method and boundary configuration method are used to study the full-field analytical solution of the fracture process zone.The main research contents and conclusions are as follows:(1)For solving the problem of uniformly distributed load of simply supported beam with an edge-crack and the fracture process zone,the stress functions of four kinds of basic problems are chosen.Based on the ‘Duan and Nakagawa's model,a group of the analytic solutions is yielded for a simply supported beam with the fracture process zone by means of mathematical analysis and selection.The obtained tensile strain softening curves and natural vibration frequencies of the simply supported beam are compared with the different cohesive distribution with water pressure type,constant type or the assumed weight function being linear in fracture process zone.The tensile strain softening curve of the water pressure distribution is similar to the one under the linear weight function.The frequency of non-cracked simply supported beam is the biggest;the cohesion constant type is followed;and the water pressure type of cohesive and the one under the linear weight function are the lowest.(2)The tensile strain softening curve and self-vibration characteristics of simply supported beams with no-cracked simply supported beams and fractures in the process of water pressure,constant distribution or primary function is analyzed and compared.It is found that the softening curve has a similar trend of water pressure distribution and one weight function pull-down strain softening curve;The crack-free simply supported beam has the highest natural vibration frequency,the cohesive force is the second,and the water pressure type and the first weight function type are the lowest.(3)Based on Williams stress function,a group of the analytic solutions and the corresponding stress intensity factor is yielded for a simply supported beam with no-crack body stress boundary conditions by means of boundary collocation method.Based on the ‘Duan and Nakagawa's model,a group of the analytic solutions of Cohesion Zone model is yielded by weighting the first item(singular item),Eliminating the singularity of the crack tip stress infinitely.Through the comparison of normal stress distribution of typical section under different solution methods,By comparing the normal stress distributions of typical sections under different solutions,the rationality and accuracy of solutions in cohesive zone model are demonstrated.Such a mechanical model can macroscopically reflect the fracture characteristics of concrete and other materials.(4)The spring unit is used to simulate the fracture process area,and the Ansys software is used to simulate the simply supported beam problem with edge-crack under uniform load.In the typical section,the numerical simulation solution,the boundary configuration method and the function superposition method have a good agreement on the stress distribution..