The grifith concept look at events within the boundaries of a critically loaded solid. How, for example, are the applied stresses transmitted to the inner regions where fracture actually takes place? What is the nature of the fracture mechanism itself? The answers to such questions were to hold the key to an understanding of all fracture phenomena The breakthrough came in 1920 with a classic paper by AA. Griffith Griffith considered an isolated crack in a solid subjected to an applied stress and formulated a criterion for its extension from the fundamental energy theorems of classical mechanics and thermodynamics. The prin- ciples laid down in that pioneering work, and the implications drawn from those principles, effectively foreshadowed the entire field of present day fracture mechanics. In our introductory chapter we critically analyse the contributions of Griffith and some of his contemporaries. This serves to introduce the reader to many of the basic concepts of fracture theory, and thus to set the scene for the remainder of the book 1.1 Stress concentrators An important precursor to the griffith study was the stress by Inglis(1913)of an ellipti nalysis showed that the local stresses about a sharp notch or corner could rise to a level several times that of the applied stress. It thus be- came apparent that even submicroscopic flaws might be potential sources of weakness in solids. More importantly, the Inglis equations provided he first real insight into the mechanics of fracture; the limiting case of an infinitesimally narrow ellipse might be considered to represent a Let us summarise briefly the essential results of the Inglis analysis. We consider in fig. l I a plate containing an elliptical cavity of semi-axes b, subjected to a uniform applied tension oa along the Y-axis. The objective is to examine the modifying effect of the hole on the distribution of stress in the solid. If it is assumed that Hooke's law holds everywhere in the plate that the boundary of the hole is stress-free, and that b and c are small in comparison with the plate dimensions, the problem reduces to a relatively straightforward exercise in linear elasticity theory. Although the math ematical treatment becomes somewhat unwieldy, involving as it does the use of elliptical coordinates, some basic results of striking simplicity emerge from the analysis
2 The Griffith concept look at events within the boundaries of a critically loaded solid. How, for example, are the applied stresses transmitted to the inner regions where fracture actually takes place? What is the nature of the fracture mechanism itself? The answers to such questions were to hold the key to an understanding of all fracture phenomena. The breakthrough came in 1920 with a classic paper by A. A. Griffith. Griffith considered an isolated crack in a solid subjected to an applied stress, and formulated a criterion for its extension from the fundamental energy theorems of classical mechanics and thermodynamics. The principles laid down in that pioneering work, and the implications drawn from those principles, effectively foreshadowed the entire field of presentday fracture mechanics. In our introductory chapter we critically analyse the contributions of Griffith and some of his contemporaries. This serves to introduce the reader to many of the basic concepts of fracture theory, and thus to set the scene for the remainder of the book. 1.1 Stress concentrators An important precursor to the Griffith study was the stress analysis by Inglis (1913) of an elliptical cavity in a uniformly stressed plate. His analysis showed that the local stresses about a sharp notch or corner could rise to a level several times that of the applied stress. It thus became apparent that even submicroscopic flaws might be potential sources of weakness in solids. More importantly, the Inglis equations provided the first real insight into the mechanics of fracture; the limiting case of an infinitesimally narrow ellipse might be considered to represent a crack. Let us summarise briefly the essential results of the Inglis analysis. We consider in fig. 1.1 a plate containing an elliptical cavity of semi-axes b, c, subjected to a uniform applied tension aA along the Y-axis. The objective is to examine the modifying effect of the hole on the distribution of stress in the solid. If it is assumed that Hooke's law holds everywhere in the plate, that the boundary of the hole is stress-free, and that b and c are small in comparison with the plate dimensions, the problem reduces to a relatively straightforward exercise in linear elasticity theory. Although the mathematical treatment becomes somewhat unwieldy, involving as it does the use of elliptical coordinates, some basic results of striking simplicity emerge from the analysis
Stress concentra Applied stress, oa Fig.1.1. Plate containing elliptical cavity, semi-axes b, c, subjected to Beginning with the equation of the ellipse, x2/c2+y2/b2=1 one may readily show the radius of curvature to have a minimum value P=b/c,(b< c) at C. It is at c that the greatest concentration of stress occurs 4(1+2c/b =aA[+2(c/p)12 For the interesting case b< c this equation reduces to a/o4=2c/b=2(c/p)12 The ratio in(1. 4)is an elastic stress-concentration factor. It is immediately evident that this factor can take on values much larger than unity narrow holes. We note that the stress concentration depends on the of the hole rather than the size
Stress concentrators Applied stress, <rA I I I I I Fig. 1.1. Plate containing elliptical cavity, semi-axes b, c, subjected to uniform applied tension aA. C denotes 'notch tip'. Beginning with the equation of the ellipse, x*/c2 +y2 /b*=l, (1.1) one may readily show the radius of curvature to have a minimum value p = b2 /c, (b<c) (1.2) at C. It is at C that the greatest concentration of stress occurs: (1.3) (1.4) For the interesting case b <^ c this equation reduces to aJaA ~ 2c/b = 2(c/p)1/2 The ratio in (1.4) is an elastic stress-concentration factor. It is immediately evident that this factor can take on values much larger than unity for narrow holes. We note that the stress concentration depends on the shape of the hole rather than the size
he Griffith concept Fig. 1. 2. Stress concentration at elliptical cavity, c= 3b. Note that concentrated stress field is localised within A c from tip, highest gradients within≈p The variation of the local stresses along the X-axis is also of interest. Fig 1. 2 illustrates the particular case c= 36. The stress o,w drops from its maximum value ac 7o at C and approaches aa asymptotically at large x, while a rises to a sharp peak within a small distance from the stress-free rface and subsequently drops toward zero with the same tendency as o, The example of fig. 1. 2 reflects the general result that significant perturbations to the applied stress field occur only within a distance c from the boundary of the hole, with the greatest gradients confined to a highly localised region of dimension x p surrounding the position of maximum concentration Inglis went on to consider a number of stress-raising configurations, and concluded that the only geometrical feature that had a marked influence on the concentrating power was the highly curved region where the stresses were actually focussed. Thus(1. 4)could be used to estimate the stress- concentration factors of such systems as the surface notch and surface step in fig. 1.3, with p interpreted as a characteristic radius of curvature and c as a characteristic notch length. a tool was now available for appraising the potential weakening effect of a wide range of structural irregularities. including, presumably, a real crack
The Griffith concept Fig. 1.2. Stress concentration at elliptical cavity, c = 3b. Note that concentrated stress field is localised within « c from tip, highest gradients within « /?. The variation of the local stresses along the X-axis is also of interest. Fig. 1.2 illustrates the particular case c = 3b. The stress ayy drops from its maximum value ac = loK at C and approaches aA asymptotically at large x, while GXX rises to a sharp peak within a small distance from the stress-free surface and subsequently drops toward zero with the same tendency as ayy. The example of fig. 1.2 reflects the general result that significant perturbations to the applied stress field occur only within a distance « c from the boundary of the hole, with the greatest gradients confined to a highly localised region of dimension « p surrounding the position of maximum concentration. Inglis went on to consider a number of stress-raising configurations, and concluded that the only geometrical feature that had a marked influence on the concentrating power was the highly curved region where the stresses were actually focussed. Thus (1.4) could be used to estimate the stressconcentration factors of such systems as the surface notch and surface step in fig. 1.3, with p interpreted as a characteristic radius of curvature and c as a characteristic notch length. A tool was now available for appraising the potential weakening effect of a wide range of structural irregularities, including, presumably, a real crack
Griffith energy-balance concept Fig. 1.3. Stress concentration half-systems: surface cavity and surface step of characteristic length c and notch radius p Despite this step forward the fundamental nature of the fracture mechanism remained obscure. If the Inglis analysis were indeed to be applicable to a crack system, then why in practice did large cracks tend to propagate more easily than small ones? Did not such behaviour violate the size-independence property of the stress-concentration factor? What is the ohysical significance of the radius of curvature at the tip of a real crack? These were some of the obstacles which stood between the Inglis approach and a fundamental criterion for fracture 1. 2 Griffith energy-balance concept: equilibrium fracture Griffith's idea was to model a static crack as a reversible thermodynamic system. The important elements of the system are defined in fig. 1.4:an elastic body B containing a plane-crack surface S of length c is subjected to loads applied at the outer boundary A. Griffith simply sought the configuration that minimised the total free energy of the system the crack would then be in a state of equilibrium, and thus on the verge of extension The first step in the treatment is to write down an expression for the total energy U of the system. To do this we consider the individual energy term that are subject to change as the crack is allowed to undergo virtual
Griffith energy-balance concept m m tttit i nTTTT TTTT Fig. 1.3. Stress concentration half-systems: surface cavity and surface step of characteristic length c and notch radius p. Despite this step forward the fundamental nature of the fracture mechanism remained obscure. If the Inglis analysis were indeed to be applicable to a crack system, then why in practice did large cracks tend to propagate more easily than small ones? Did not such behaviour violate the size-independence property of the stress-concentration factor? What is the physical significance of the radius of curvature at the tip of a real crack? These were some of the obstacles which stood between the Inglis approach and a fundamental criterion for fracture. 1.2 Griffith energy-balance concept: equilibrium fracture Griffith's idea was to model a static crack as a reversible thermodynamic system. The important elements of the system are defined in fig. 1.4: an elastic body B containing a plane-crack surface S of length c is subjected to loads applied at the outer boundary A. Griffith simply sought the configuration that minimised the total free energy of the system; the crack would then be in a state of equilibrium, and thus on the verge of extension. The first step in the treatment is to write down an expression for the total energy U of the system. To do this we consider the individual energy terms that are subject to change as the crack is allowed to undergo virtual
he griffith System boundary Fig. 1. 4. Static plane-crack system, showing incremental extension of length c through dc: B, elas loading extension. Generally, the system energy associated with crack formation may be partitioned into mechanical or surface terms. The mechanical energy itself consists of two terms, UM= UR+UA: UR is the strain potential energy stored in the elastic medium Ua is the potential energy of the outer applied loading system, expressible as the negative of the work associated with any displacements of the loading points. The term Us is the free energy expended in creating the new crack surfaces. We may therefore U+U Thermodynamic equilibrium is then attained by balancing the mech anical and surface energy terms over a virtual crack extension dc(fig. 1. 4) It is not difficult to see that the mechanical energy will generally decrease as the crack extends(dUM/de<0). For if the restraining tractions across the incremental crack boundary dc were suddenly to relax, the crack walls would in the general case, accelerate outward and ultimately come to rest In figuration of lower energy On the other hand, the surface energy term will generally increase with crack extension, since cohesive forces of molecular attraction across dc must be overcome during the creation of the new fracture surfaces(dUs/dc>0). Thus the first term in (1.5)favours crack extension, while the second opposes it. This is the
The Griffith concept System boundary Fig. 1.4. Static plane-crack system, showing incremental extension of crack length c through dc: B, elastic body; S, crack surface; A, applied loading. extension. Generally, the system energy associated with crack formation may be partitioned into mechanical or surface terms. The mechanical energy itself consists of two terms, UM= UE+UA: UE is the strain potential energy stored in the elastic medium; UA is the potential energy of the outer applied loading system, expressible as the negative of the work associated with any displacements of the loading points. The term C/s is the free energy expended in creating the new crack surfaces. We may therefore write u=uM+us. (1.5) Thermodynamic equilibrium is then attained by balancing the mechanical and surface energy terms over a virtual crack extension dc (fig. 1.4). It is not difficult to see that the mechanical energy will generally decrease as the crack extends (dUM/dc < 0). For if the restraining tractions across the incremental crack boundary dc were suddenly to relax, the crack walls would, in the general case, accelerate outward and ultimately come to rest in a new configuration of lower energy. On the other hand, the surface energy term will generally increase with crack extension, since cohesive forces of molecular attraction across dc must be overcome during the creation of the new fracture surfaces (dUs/dc > 0). Thus the first term in (1.5) favours crack extension, while the second opposes it. This is the