Continuum approach to crack equilibriun continuum view of matter, retains the macroscopic, or thermodynamic view of crack propagation. It embraces two major needs ( For the routine analysis of a wide range of crack-loading geometries the Griffith concept needs to be placed within a more general theoretical ramework. The requirement is for functional quantities that characterise he driving force for fracture. Of these quantities, mechanical-energy- release rate G and stress-intensity factor K, with certain properties superposability, stand pre-eminent in present-day formalisms ii) A methodology for dealing with the complexity of stability conditions that define the nature of crack equilibria is required We have seen in chapter I how equilibrium cracks can be energetically stable as well as unstable. Many important crack systems pass through a sequence of different equilibrium states in their propagation to ultimate failure.A complete description of stability includes consideration of path, in addition to energetics, of fracture The basic principles underlying the above two elements of fracture mechanics have received insufficient attention from the materials com- munity. Accordingly, in the present chapter we shall outline these principles, the strict thermodynamical context of linearity and reversibility laid down by Griffith. In so doing we shall bypass detailed consideration of the nonlinear, dissipative terms that inevitably come into play when dealin with fundamental crack-tip separation processes in 'real materials. The means for incorporating such material-specific terms into appropriate fracture resistance parameters(analogous to Griffith's surface energy) will be discussed in chapter 3 2.1 Continuum approach to crack equilibrium: crack system thermodynamic cycle Let us begin by restating Griffith's thermodynamic concept of crack equilibrium in broader terms. Reconsider the plane-crack system of fig 1. 4. The solid is an isotropic linear elastic continuum, loaded arbitrarily at its outer boundary, and the crack is formed from an infinitesimally narrow For a specified crack length the problem reduces to a formal exercise
Continuum approach to crack equilibrium 17 continuum view of matter, retains the macroscopic, or thermodynamic, view of crack propagation. It embraces two major needs: (i) For the routine analysis of a wide range of crack-loading geometries the Griffith concept needs to be placed within a more general theoretical framework. The requirement is for functional quantities that characterise the driving force for fracture. Of these quantities, mechanical-energyrelease rate G and stress-intensity factor K, with certain properties of linear superposability, stand pre-eminent in present-day formalisms. (ii) A methodology for dealing with the complexity of stability conditions that define the nature of crack equilibria is required. We have seen in chapter 1 how equilibrium cracks can be energetically stable as well as unstable. Many important crack systems pass through a sequence of different equilibrium states in their propagation to ultimate failure. A complete description of stability includes consideration of path, in addition to energetics, of fracture. The basic principles underlying the above two elements of fracture mechanics have received insufficient attention from the materials community. Accordingly, in the present chapter we shall outline these principles, in the strict thermodynamical context of linearity and reversibility laid down by Griffith. In so doing we shall bypass detailed consideration of the nonlinear, dissipative terms that inevitably come into play when dealing with fundamental crack-tip separation processes in 'real materials'. The means for incorporating such material-specific terms into appropriate fracture resistance parameters (analogous to Griffith's surface energy) will be discussed in chapter 3. 2.1 Continuum approach to crack equilibrium: crack system as thermodynamic cycle Let us begin by restating Griffith's thermodynamic concept of crack equilibrium in broader terms. Reconsider the plane-crack system of fig. 1.4. The solid is an isotropic linear elastic continuum, loaded arbitrarily at its outer boundary, and the crack is formed from an infinitesimally narrow slit. For a specified crack length the problem reduces to a formal exercise
18 Continuum aspects I linear crack-tip field 卌 Fig.2.1. Reversible crack cycle,(a)→(b)→(c)→(b)-(a). Mechanical energy released in crack formation is determined by prior separation plane in elasticity theory, in which solutions may be found for the stress and strain(or displacement) fields in the loaded solid The question then arises as to how these fields, particularly in the vicinity of the crack tip, determine the energetics of crack propagation. Here the Inglis analysis(sect. 1.1) provides some foresight: the intensity of the field is largely determined by the outer boundary conditions (applied loader figuration), the distribution by the inner boundary conditions(stress-free crac Our approach is to treat the energetics of crack pr tion in terms of an operational, hypothetical opening and closing cycle. There are two ways in which such a cycle may be conceived. One is to consider the formation of the entire crack from the initially intact body(as done effectively by Griffith and Obreimoff in sects. 1. 3 and 1. 4). the other way is to consider an incremental extension of an existing crack. It will be implicit in our constructions,consistent with the griffith thesis, that the processes which determine the mechanical and surface energies operate independently of ach other while this may appear to be a trivial point, we will find cause in later chapters to question the decoupling of energy terms The first kind of opening and closing cycle, although not explicitly part the Irwin scheme, deserves attention if only because of its insistence that the mechanical energy term UM in(1. 5)is determined uniquely by the stresses in the loaded solid prior to cracking. At first sight this insistence may seem untenable, for it certainly can be argued that the progress of a crack must be determined by the highly modified stress state at the instant of extension. But the correspondence between crack energetics and prior stresses can be unequivocally demonstrated by the sequence in fig. 2. 1. We start with the crack-free state(a), for which it is presumed the elastic field is known. Suppose now that we make an infinitesimally narrow cut along the ultimate crack plane, and impose tractions equal and opposite to the
18 Continuum aspects I: linear crack-tip field (a) Fig. 2.1. Reversible crack cycle, (a) -> (b) -> (c) -> (b) -> (a). Mechanical energy released in crack formation is determined by prior stresses on separation plane. in elasticity theory, in which solutions may be found for the stress and strain (or displacement) fields in the loaded solid. The question then arises as to how these fields, particularly in the vicinity of the crack tip, determine the energetics of crack propagation. Here the Inglis analysis (sect. 1.1) provides some foresight: the intensity of the field is largely determined by the outer boundary conditions (applied loading configuration), the distribution by the inner boundary conditions (stress-free crack walls). Our approach is to treat the energetics of crack propagation in terms of an operational, hypothetical opening and closing cycle. There are two ways in which such a cycle may be conceived. One is to consider the formation of the entire crack from the initially intact body (as done effectively by Griffith and Obreimoff in sects. 1.3 and 1.4). The other way is to consider an incremental extension of an existing crack. It will be implicit in our constructions, consistent with the Griffith thesis, that the processes which determine the mechanical and surface energies operate independently of each other. While this may appear to be a trivial point, we will find cause in later chapters to question the decoupling of energy terms. The first kind of opening and closing cycle, although not explicitly part of the Irwin scheme, deserves attention if only because of its insistence that the mechanical energy term UM in (1.5) is determined uniquely by the stresses in the loaded solid prior to cracking. At first sight this insistence may seem untenable, for it certainly can be argued that the progress of a crack must be determined by the highly modified stress state at the instant of extension. But the correspondence between crack energetics and prior stresses can be unequivocally demonstrated by the sequence in fig. 2.1. We start with the crack-free state (a), for which it is presumed the elastic field is known. Suppose now that we make an infinitesimally narrow cut along the ultimate crack plane, and impose tractions equal and opposite to the
ck equilibriu prior stresses there to maintain the system in equilibrium. This operation takes us to state(b), and the only energy involved thus far is the amount Us supplied by the cutting process in creating new fracture surfaces. We now relax the imposed tractions to zero(slowly, to avoid kinetic energy terms), applying constraints at the crack ends to prevent further extension. The resulting configuration is the equilibrium crack(c), and the mechanical nergy released in achieving this state is precisely UM. At this point the process is reversed. The tractions are re-applied, starting from zero and increasing linearly until the crack is closed again over its whole area. Since the elastic system is conservative the final stress state must be identical to the prior stress state (b). Thus the mechanical energy decrease with crack formation may, within the limits of Hooke's law, be expressed as an integral over the crack area of prior stresses multiplied by crack-wall displacements. Since the displacements are themselves related linearly hrough the elasticity equations to the crack-surface tractions, the prior stress distribution must uniquely determine the crack energetics. The final stage of the cycle merely involves a healing operation to recover the surface energy, and the removal of the imposed tractions to restore state(a) It is worth emphasising once more the implications of the above resu the entire propagation history of a crack is predestined by the existing stress state before fracture has even begun. Thus in many cases all that is needed to describe the fracture behaviour of an apparently complex system is a standard stress analysis of the system in its uncracked state. This principle will prove useful when we consider specific crack systems in sect. 2.5 The second kind of cycle, that involving the extension and closure over a small increment of slit-crack area, makes use of detailed linear elasticity solutions for the field at the tip of an existing crack. The presence of the crack assuredly complicates the elasticity analysis, but there is a certain universality in the near-tip solutions(foreshadowed in our allusion above to the Inglis analysis) that makes this an especially attractive route. It is the element of rsality that is the key to the innate power of Irwin's fracture mechanics We shall return later (sect. 2. 4) to this second application of the reversibility argument to incorporate the Griffith concept into our generalised description. At this point we turn our attention to specific details of fracture mechanics terminology
Continuum approach to crack equilibrium 19 prior stresses there to maintain the system in equilibrium. This operation takes us to state (b), and the only energy involved thus far is the amount Us supplied by the cutting process in creating new fracture surfaces. We now relax the imposed tractions to zero (slowly, to avoid kinetic energy terms), applying constraints at the crack ends to prevent further extension. The resulting configuration is the equilibrium crack (c), and the mechanical energy released in achieving this state is precisely UM. At this point the process is reversed. The tractions are re-applied, starting from zero and increasing linearly until the crack is closed again over its whole area. Since the elastic system is conservative the final stress state must be identical to the prior stress state (b). Thus the mechanical energy decrease associated with crack formation may, within the limits of Hooke's law, be expressed as an integral over the crack area of prior stresses multiplied by crack-wall displacements. Since the displacements are themselves related linearly through the elasticity equations to the crack-surface tractions, the prior stress distribution must uniquely determine the crack energetics. The final stage of the cycle merely involves a healing operation to recover the surface energy, and the removal of the imposed tractions to restore state (a). It is worth emphasising once more the implications of the above result: the entire propagation history of a crack is predestined by the existing stress state before fracture has even begun. Thus in many cases all that is needed to describe the fracture behaviour of an apparently complex system is a standard stress analysis of the system in its uncracked state. This principle will prove useful when we consider specific crack systems in sect. 2.5. The second kind of cycle, that involving the extension and closure over a small increment of slit-crack area, makes use of detailed linear elasticity solutions for the field at the tip of an existing crack. The presence of the crack assuredly complicates the elasticity analysis, but there is a certain universality in the near-tip solutions (foreshadowed in our allusion above to the Inglis analysis) that makes this an especially attractive route. It is the element of universality that is the key to the innate power of Irwin's fracture mechanics. We shall return later (sect. 2.4) to this second application of the reversibility argument to incorporate the Griffith concept into our generalised description. At this point we turn our attention to specific details of fracture mechanics terminology
Continuum aspects 1: linear crack-tip field 2.2 Mechanical-energy-release rate G Consider now the elemental crack system of fig. 2. 2. The body contains a slit of length c, the walls of which are traction-free Consider the lower end to be rigidly fixed, the upper end to be loaded with a tensile point force P If workless constraints are imposed at the ends of the slit to pre extension the specimen will behave as an equilibrium elastic spring accordance with hookes law where uo is the load-point displacement and 2=2(c) is the elastic compliance. The strain energy in the system is equal to the work of elastic loading P=是P2=11 (22) Now suppose, with the body maintained in a loaded configuration, we release our end constraints on the slit and allow incremental extensions through de. We should expect the compliance to increase. To show this formally we differentiate(2.1), thus duo= 2dP+dAP so that for duo>0, dP < o(general loading conditions for dc >0)we have d>0 always. At the same time we should expect the composite mechanical energy term UM=Ur+UA to decrease (sect. 1. 2). It is convenient to consider two extreme loading configurations Constant force ('dead-weight'loading). The applied force remains constant as the crack extends At P=const the change in potential energy of the loading system, i.e. the negative of the work associated with the load point displacement, is determinable from(2. 3)as dU - Pd and the corresponding change in elastic strain energy from(2.2)and (2.3) dUg=是P2d (2.4b)
20 Continuum aspects I: linear crack-tip field 2.2 Mechanical-energy-release rate, G Consider now the elemental crack system of fig. 2.2. The body contains a slit of length c, the walls of which are traction-free. Consider the lower end to be rigidly fixed, the upper end to be loaded with a tensile point force P. If workless constraints are imposed at the ends of the slit to prevent extension the specimen will behave as an equilibrium elastic spring in accordance with Hooke's law uQ = XP (2.1) where u0 is the load-point displacement and X = X(c) is the elastic compliance. The strain energy in the system is equal to the work of elastic loading; UK = ^°P(u0)du0 = \PuQ = \P*l = \u\IL (2.2) Now suppose, with the body maintained in a loaded configuration, we release our end constraints on the slit and allow incremental extensions through dc. We should expect the compliance to increase. To show this formally we differentiate (2.1), thus; so that for du0 ^ 0, dP ^ 0 (general loading conditions for dc > 0) we have dl ^ 0 always. At the same time we should expect the composite mechanical energy term UM= U^+UA to decrease (sect. 1.2). It is convenient to consider two extreme loading configurations: (i) Constant force ('dead-weight' loading). The applied force remains constant as the crack extends. At P = const the change in potential energy of the loading system, i.e. the negative of the work associated with the loadpoint displacement, is determinable from (2.3) as dUA = -Pdu0 = -P2 dA, (2.4a) and the corresponding change in elastic strain energy from (2.2) and (2.3) as d£/E = |P2 d/L (2.4b)
Mechanical-energy-release rate, G Fig. 2.2. Simple specimen for defin Applied point load P displaces through u, during crack formation c, increasing system compliance. The total mechanical energy change dum=dUe+dua is therefore dUM=-lPdλ (2.5) ()Constant displacement (fixed-grips'loading). The applied loading system suffers zero displacement as the crack extends. At u,=const the energy changes are dU=0 dUE=-l(2/13)d=-lP2 26b) again using(2. 2) to compute the strain energy term. This gives du=-iP2d2 e see that (2.5)and(2.7) are identical that is, the released during incremental crack extension is independent of loading configuration. We leave it to the reader to prove this result for the more complex case in which neither P nor uo are held constant We have considered only one particular specimen configuration here, that of loading at a point, but a more rigorous analysis shows our
Mechanical-energy-release rate, G Pi 21 Fig. 2.2. Simple specimen for defining mechanical-energy-release rate. Applied point load P displaces through u0 during crack formation c, increasing system compliance. The total mechanical energy change dUM = d£/E + d£/A is therefore (2.5) (ii) Constant displacement ('fixed-grips' loading). The applied loading system suffers zero displacement as the crack extends. At u0 = const the energy changes are dUA = again using (2.2) to compute the strain energy term. This gives dUM= -\P2 dL (2.6a) (2.6b) (2.7) We see that (2.5) and (2.7) are identical: that is, the mechanical energy released during incremental crack extension is independent of loading configuration. We leave it to the reader to prove this result for the more complex case in which neither P nor u0 are held constant. We have considered only one particular specimen configuration here, that of loading at a point, but a more rigorous analysis shows our