tailieunhanh - Introduction to Elasticity Part 14

Tham khảo tài liệu 'introduction to elasticity part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | agree with the Inglis solution and it turns out that for plane stress loading p n. The total strain energy U released is then the strain energy per unit volume times the volume in both triangular regions ơ 2E na2 Here the dimension normal to the x-y plane is taken to be unity so U is the strain energy released per unit thickness of specimen. This strain energy is liberated by crack growth. But in forming the crack bonds must be broken and the requisite bond energy is in effect absorbed by the material. The surface energy S associated with a crack of length a and unit depth is S 2ya where Y is the surface energy . Joules meter2 and the factor 2 is needed since two free surfaces have been formed. As shown in Fig. 2 the total energy associated with the crack is then the sum of the positive enery absorbed to create the new surfaces plus the j i tree strain energy liberated by allowing the regions near the crack flanks to become unloaded. Figure 2 The fracture energy balance. As the crack grows longer a increases the quadratic dependence of strain energy on a eventually dominates the surface energy and beyond a critical crack length ac the system can lower its energy by letting the crack grow still longer. Up to the point where a ac the crack will grow only if the stress in increased. Beyond that point crack growth is spontaneous and catastrophic. The value of the critical crack length can be found by setting the derivative of the total energy S U to zero d S U da 2y f na 0 E Since fast fracture is imminent when this condition is satisfied we write the stress as ơf. Solving 2Ey ơf y na Griffith s original work dealt with very brittle materials specifically glass rods. When the material exhibits more ductility consideration of the surface energy alone fails to provide an 3 accurate model for fracture. This deficiency was later remedied at least in part independently by Irwin4 and Orowan5. They suggested that in a ductile material a good deal - in fact the vast .