Introduction to Fracture Mechanics pdf

Introduction to Fracture Mechanics 1 Introduction to Fracture Mechanics From Suresh: Fatigue of Materials Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com INTRODUCTION Importance of Fracture Mechanics :  All real materials contain defects: understand the influence of these defects on the strength of the material. Defect - tolerant design philosophy. 2 the material. Defect - tolerant design philosophy.  Relevance for Fatigue: understand the initiation and growth of fatigue cracks. We will use two approaches, an energy-based approach and a more rigorous mechanics approach. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Key Idea : Griffith (1921) postulated that for unit crack extension to occur under the influence of the applied stress, the decrease in potential energy of Griffith Fracture Theory Introduction 3 applied stress, the decrease in potential energy of the system, by virtue of the displacement of the outer boundaries and the change in the stored elastic energy, must equal the increase in surface energy due to crack extension. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Consider the center-cracked plate shown below. The in-plane dimensions of the plate are large compared to the crack length. 4 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Using the results of Inglis (1913) Griffith found that the net change in potential energy of the plate caused by the introduction of the crack is: . ' 22 E Ba W P   ' E E  5 = E Plane stress The surface energy of the crack system is 2 1 ' v E E   sS aBW  4  Plane strain where γ S is the free surface energy per unit surface area. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com The total system energy is then given by .4 ' 22 SSP aB E Ba WWU    6 Griffith noted that the critical condition for the onset of crack growth is: ,02 ' 2  S S P E a dA dW dA dW dA dU   Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com where A=2aB is the crack area and dA denotes an incremental increase in the crack area. Thus the stress required to initiate fracture is: . '2 a E S f     7 a  As the second derivative, d 2 U/da 2 is negative, the above equilibrium condition gives rise to unstable crack propagation. This applies for brittle materials; it must be modified for ductile materials such as metals. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Orowan (1952) extended Griffith’s brittle fracture concept to metals by simply adding a term representing plastic energy dissipation. The resultant expression for fracture initiation is , )('2E p s     8 , a p s f    where is the plastic work per unit area of surface created. Generally is much larger than p  . s  p  Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Energy Release Rate Crack Driving Force Consider an elastic plate with an edge crack of length as shown below: , a 9 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com The total mechanical potential energy of a cracked elastic body is given by the general expression FP wW    where is the stored elastic strain energy and is  F w 10 the work done by the external forces.  F Irwin (1956) proposed an approach for the characterization of the driving force for fracture in cracked bodies, which is conceptually equivalent to that of the Griffith model. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com [...]... and a stress σ2 is applied to the second plate If (1) ( 2) we choose σ1, σ1, σ2 and σ2 so that K I  K I then the fields at the crack tip are identical in both cases This is the principle of similitude, which is very important in fracture mechanics as it allows results from laboratory scale tests to be applied to large scale fracture problems 33 Stress Intensity Factors Simpo PDF Merge and Split Unregistered... the energy release rate to be evaluated 12 Modes of Fracture Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com The three basic modes of separation of the crack surfaces (modes of fracture) are depicted below: Combinations of modes (mixed-mode loading) are also possible 13 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Modes of Fracture Definitions Mode... Factors, D P Rooke 32 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Plane Crack Problem Similitude For a crack of length 2a1 in an infinite plate, subjected to an applied stress σ1 the stress intensity factor is known to be K I   1 a1 Consider two large plates, one with a center crack of length 2a1, the other with a center crack of length 2a2 A stress σ1 is applied to. .. normal to the plane of the crack The displacements are symmetric with respect to the x – z and x – y planes Mode II (in-plane sliding mode): The crack faces are mutually sheared in a direction normal to the crack front The displacements are symmetric with respect to the x – y plane and anti-symmetric with respect to the x – z plane 14 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com... of Fracture Mode III (tearing or anti-plane shear mode): The crack faces are sheared parallel to the crack front The displacements are antisymmetric with respect to the x – y and x – z planes The crack face displacements in modes II and III find an analogy to the motion of edge dislocations and screw dislocations, respectively 15 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com... preceding analysis considered fracture from an energy standpoint We now carry out a linear elastic stress analysis of the cracked body, which will allow us to formulate critical conditions for the growth of flaws more precisely An analysis of this type falls within the field of Linear Elastic Fracture Mechanics (LEFM) 16 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com We consider a... release rate, i.e., ς → ςc What are some typical values for KC? 34 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Material Glass Al2O3 Si3 N4 Polymers Al alloys Steels Kc (MPa m ) 1  34  48  0.5  2  10  100  30  300 35 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Fracture Mechanics #2: Role of Crack Tip Plasticity 1 ... Unregistered Version - http://www.simpopdf.com Plane Crack Problem How do we apply this analysis to the failure of actual materials? It has been found experimentally that when the stress intensity factor K (which depends on the geometry and loading) attains a critical value KC (a material property) the crack begins to grow, i.e., the critical condition for the onset of fracture is K → Kc The condition can... r 2r KI 2 r 31 Plane Crack Problem Stress Intensity Factors Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com The stress intensity factors for Modes I, II and III are defined as follows:  2r   0  ,  lim 2r   0  ,  lim 2r   0  K I  lim r 0 K II K III r 0 r 0 yy xy yz The stress intensity factor K depends on loading and geometry Many different geometries... here 24 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com   2  2 r    2   1r A1 cos   B1 cos  2     r   1 rx     r  r      1r A1 sin     2 B1 sin   2   Apply the boundary conditions:  A1  B1  cos   0, A1    2B1 sin   0 25 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 2Z  . Introduction to Fracture Mechanics 1 Introduction to Fracture Mechanics From Suresh: Fatigue of Materials Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com INTRODUCTION Importance. rate to be evaluated. G Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Modes of Fracture The three basic modes of separation of the crack surfaces ( modes of fracture) . symmetric with respect to the x – y plane and anti-symmetric with respect to the x – z plane. Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Modes of Fracture Definitions Mode