SOME CIRCULAR ARC CRACK PROBLEMS IN ELASTIC - PLASTIC MEDIA

Abstract

The problems investigated in the thesis are based on Dugdale model and modified Dugdale model solution for infinite matrix containing circular arc crack(s). The chapterwise description is given below: Chapter 1 serves as introduction to fracture mechanics concepts. Its development, application and scope are briefly discussed. An overview of developments and work done by scholars are reported in chronological order which itself explains developments of the subject. Complex variable formulation of two-dimensional elasticity forms the subject matter of Chapter 2. The chapter is written to make the thesis self-sufficient. Stress components, displacements, stress-intensity factors are written in terms of complex potentials. The Hilbert problem (problem of linear relationship) is formulated for crack(s) in a matrix and solutions for different cases are discussed. Modes of crack tip deformation are included to enable reader to understand Dugdale's `strip yield. model'. Important parameters like plastic zone, crack opening displacement are described. Chapter 3 deals with problem of a circular arc crack in infinite plate subjected to prescribed loads at infinity. Plastic zones develop ahead of crack tips on account of these loads. Uniform yield stress is then applied to these plastic zones Abstract. causing their closure. The Dugdale model solution is obtained superimposing stress intensity factors at crack tip of the corresponding component problems. Complex variable technique of Muskhelishvili is used to solve the component problems. Analytic expressions for plastic zone size and crack opening displacement (COD) at tip of the crack are derived. Load required to close prescribed plastic zone is computed numerically for some cases of interest Variation of load required for closure -is studied with respect to important parameters like crack length, crack radius. It is observed as plastic zone length is increased, larger load at infinity is required for closure. While if the crack length is increased, for fixed plastic zone size, required load ratio decreases.] COD variation against plastic zone increase shows that bigger the plastic zone the crack opens more. The COD at crack tip for these required load is also depicted graphically. Chapter 4 introduces generalization of the Dugadale model for case of an infinite plate containing a circular arc crack. The plate is subjected to opening mode 1 loads at infinity. Plastic enclaves thus developed are closed by variable load ayecose. Here aye is the yield stress of matrix, e is angular variation along plastic enclaves. Using the boundary conditions dual Hilbert problems are obtained, solutions of which enable to Abstract calculate size of plastic enclave and COD analytically. Variation of required load ratio and corresponding COD versus increase in plastic enclave are studied for different crack length and crack radius, and presented graphically. By increasing crack radius the load ratio00ye ,required load for closure decreases while COD increases. Problem of circular arc crack in an infinite media subjected to prescribed loads at infinity is discussed in Chapter 5. The plastic enclaves developed are subjected to variable load ayesine. The solution is obtained in closed form usingLsomplex variable technique. Non-linear relations are obtained to compute length of fracture process zone and COD at crack tip. Some illustrative numerical work is done to study behaviour of required load ratio for closure of fracture process zone, when its size is increased. Effect of variation of important parameters (like crack length, crack radius) on required load ratio is also reported graphically. COD at crack tip is plotted varying fracture process zone, crack angle, radius of crack. Dugdale model for two equal and symmetrically situated circular arc cracks in an infinite elastic-perfectly plastic plate is proposed in Chapter 6. The plate is subjected to known loads at infinity which develop plastic zone ahead of the tips of Abstract cracks. These plastic zones are then closed by applying uniform yield stress over them. Employing complex variable technique, the complex potentials are obtained. Expressions for determining plastic zone length and COD at the crack tip are then derived in closed analytic form. Variation of required load ratio for closure versus plastic zone is depicted graphically. Variation of load ratio is also studied with respect to parameters viz. crack length, inter crack distance etc. As expected, the required load ratio increases with the increase in plastic zone. If the inter crack distance is increased the required load ratio for closure reduces. The crack tip opens more if plastic zone size is increased,is reaffirmed in this case too. Behaviour of COD versus crack length, inter crack distance is also reported. Chapter 7 presents the solution of two equal symmetrically situated circular arc cracks in a matrix acted upon by remotely applied loads giving rise to opening mode I deformation at crack tips. Cohesive zones formed ahead of cracks tips are closed by prescribing a variable load uyecose. Based on Dugdale's `strip yield model' and complex variable technique, analytic closed form solutions are obtained. A qualitative numerical study is carried out for required load for closure of cohesive zone and COD at the crack tip with respect to influencing parameters like plastic Abstract zone, crack angle, inter crack distance. Results obtained are presented graphically. Problem of two equal and symmetric circular arc cracks in an elastic medium is treated in Chapter 8. loads are prescribed at infinity causing plastic zones to develop _ahead of cracks tips. The plastic zones are acted upon by variable load distribution ayesine effecting closure of plastic zone. Solution is obtained using superimposition principle and complex variable technique. Closed form relations are derived for finding plastic zone length and COD at crack tip. Numerical studies are carried for load ratio required for closure of the plastic zones with respect to various parameters viz. half crack angle, inter distance between _ two crack tips. Behaviour of COD at crack tips for these parameters of practical interest is shown graphically. Chapter 9 describes modified Dugdale model for remotely loaded elastic-perfectly plastic infinite plate containing (i) a single circular arc crack and (ii) two equal symmetrically situated circular arc cracks. The plastic zones developed ahead of crack(s) tips in both the cases are closed by cohesive variable load u.yeexp(ie). The analytic expressions for calculating plastic zone length and COD at crack tips are derived.