CRACK ARREST MODELS FOR INTERNAL AND EXTERNAL CRACKS WEAKENING A PLATE

Abstract

The problems investigated in this thesis are based on Dugdale model. The model was proposed to arrest opening of a quasi-static crack in Mode I type deformation. An infinite plate weakened by a single crack when subjected to tension at infinity, in a direction perpendicular to the rims of the crack, opens faces of the crack in Mode I type deformation. Consequently a plastic zone develops ahead each tip of the crack. The rims of developed plastic zones are subjected to closing normal uniform constant stress distribution. This arrests the plastic zone opening consequently the crack is stopped from further growing. Chapterwise description is given below. The work presented in the thesis is divided into eight chapters. Chapter One is written with the idea of presenting an overview of the subject. CHAPTER ONE introduces the basic concepts of the fracture mechanics relevant to the thesis. An account of developments, application and scope of the subject is presented. A brief survey of the literature and its development is presented in overview section of the chapter. The main emphasis is based on three types of crack problems, namely, (a) collinear hairline straight cracks weakening a plate (b) internal and external hairline collinear straight cracks and (c) co-circumferential circular arc cracks weakening a plate. The mathematical model obtained for the problems investigated in this thesis reduces to the problem of linear relationship and their solution. Chapter Two is written with the idea of making the thesis self-sufficient to a reader. CHAPTER TWO recapitulates stresses and displacements in terms of complex potentials. Formulation and solution of Hilbert problem for straight crack problem and circular arc crack problem are discussed in details. Concept of stress intensity factors and plastic zone are also introduced. The crack problems investigated are divided into three groups. In Chapter Three and Four Dugdale model is modified for the case of two unequal, collinear, hairline straight cracks weakening an infinite plate. CHAPTER THREE: A modified Dugdale model is proposed for two hairline, collinear, unequal straight cracks weakening an infinite elastic perfectly-plastic plate. Each rim of developed plastic zones is subjected to normal cohesive stress distribution PYY = t2aye and P. = 0, where t is any point on any rim of the plastic zone and aye is the yield point stress of the plate. Problem is solved using the complex variable technique discussed in Chapter 2. A qualitative analysis is carried out to find the behavior of load required to close the plastic zones versus affecting parameters viz. crack length, plastic zone length and inter crack distance. CHAPTER FOUR: The problem of two unequal, collinear, hairline cracks weakening a plate is discussed in this chapter. Tension applied at infinity causes coalescing of each plastic zone developed at the two adjacent tips of two cracks, The other rims of two plastic zones, each developed at the remaining tip of the two cracks, and the coalesced plastic zone are subjected to stress distribution PYr = t20Ye and Pxy = 0. Problem is solved using Muskhelishvili [62] technique. CHAPTER FIVE: Dugdale model solution is obtained for a plate weakened by two external and an internal collinear hairline straight cracks. Tension applied.at infinity opens the faces of the cracks in mode I type deformation. Consequently plastic zone is formed ahead of each tip, located at a finite distance for tip of two external cracks, and at both the tips of the internal crack. Solution of the problem is obtained from the solution of an Auxiliary problem. This Auxiliary problem is appropriately derived from the original problem and is solved using complex variable technique. Solution of original problem is then obtained from the solution of Auxiliary problem under certain constraints. A case study is carried out. The last category of crack problems considered is of co-circumferential arc cracks weakening an elastic perfectly-plastic plate and are discussed in Chapter Six, Seven and Eight. CHAFFER SIX: A Dugdale model solution is obtained for two unequal, hairline, co-circumferential cracks weakening an elastic perfectly-plastic plate. The faces of the cracks open forming a plastic zone ahead each tip of the two cracks. The tension applied at infinity is increased to the limit so that each plastic zone developed at the two adjacent tips of the two cracks gets coalesced. The problem is solved using principle of superimposition for stress intensity factors at the end tips of plastic zones for two component problems. An illustrative example is considered for qualitative studies. CHAFFER SEVEN gives the solution for modified Dugdale model for two unequal, hairline, co-circumferential circular arc cracks weakening a plate. Plastic zone developed ahead of each tip of cracks is closed by the stress distribution Prr = aye sin 0 and Pro = 0 . Problem is solved by solving two component problems appropriately derived from original problem and contributing towards the stress singularity at the each tip of plastic zone. Each problem is solved using complex variables. Solution of the original problem is then obtained using superimposition principle. CHAPTER EIGHT presents a crack arrest model for a plate weakened by two unequal circular arc cracks with coalesced plastic zones when rims of the developed plastic zones are subjected to normal cohesive stress distribution Prr = aye cos() and Pro = 0. Problem is solved using Dugdale hypothesis that the stresses remain finite at every point of the body.