Crack growth resistance curve

In materials modeled by linear elastic fracture mechanics (LEFM), crack extension occurs when the applied energy release rate G {displaystyle G} exceeds G R {displaystyle G_{R}} , where G R {displaystyle G_{R}} is the material's resistance to crack extension. In materials modeled by linear elastic fracture mechanics (LEFM), crack extension occurs when the applied energy release rate G {displaystyle G} exceeds G R {displaystyle G_{R}} , where G R {displaystyle G_{R}} is the material's resistance to crack extension. Conceptually G {displaystyle G} can be thought of as the energetic gain associated with an additional infinitesimal increment of crack extension, while G R {displaystyle G_{R}} can be thought of as the energetic penalty of an additional infinitesimal increment of crack extension. At any moment in time, if G ≥ G R {displaystyle Ggeq G_{R}} then crack extension is energetically favorable. A complication to this process is that in some materials, G R {displaystyle G_{R}} is not a constant value during the crack extension process. A plot of crack growth resistance G R {displaystyle G_{R}} versus crack extension Δ a {displaystyle Delta a} is called a crack growth resistance curve, or R-curve. A plot of energy release rate G {displaystyle G} versus crack extension Δ a {displaystyle Delta a} for a particular loading configuration is called the driving force curve. The nature of the applied driving force curve relative to the material's R-curve determines the stability of a given crack. The usage of R-curves in fracture analysis is a more complex, but more comprehensive failure criteria compared to the common failure criteria that fracture occurs when G ≥ G c {displaystyle Ggeq G_{c}} where G c {displaystyle G_{c}} is simply a constant value called the critical energy release rate. An R-curve based failure analysis takes into account the notion that a material's resistance to fracture is not necessarily constant during crack growth. R-curves can alternatively be discussed in terms of stress intensity factors ( K ) {displaystyle (K)} rather than energy release rates ( G ) {displaystyle (G)} , where the R-curves can be expressed as the fracture toughness ( K I c {displaystyle K_{Ic}} , sometimes referred to as K R {displaystyle K_{R}} ) as a function of crack length a {displaystyle a} . The simplest case of a material's crack resistance curve would be materials which exhibit a 'flat R-curve' ( G R {displaystyle G_{R}} is constant with respect to Δ a {displaystyle Delta a} ). In materials with flat R-curves, as a crack propagates, the resistance to further crack propagation remains constant and thus, the common failure criteria of G ≥ G c {displaystyle Ggeq G_{c}} is largely valid. In these materials, if G {displaystyle G} increases as a function of Δ a {displaystyle Delta a} (which is the case in many loading configurations and crack geometries), then as soon as the applied G {displaystyle G} exceeds G c {displaystyle G_{c}} the crack will unstably grow to failure without ever halting. Physically, the independence of G R {displaystyle G_{R}} from Δ a {displaystyle Delta a} is indicative that in these materials the phenomena which are energetically costly during crack propagation do not evolve during crack propagation. This tends to be an accurate model for perfectly brittle materials such as ceramics, in which the principal energetic cost of fracture is the development of new free surfaces on the crack faces. The character of the energetic cost of the creation of new surfaces remains largely unchanged regardless of how long the crack has propagated from its initial length. Another category of R-curve that is common in real materials is a 'rising R-curve' ( G R {displaystyle G_{R}} increases as Δ a {displaystyle Delta a} increases). In materials with rising R-curves, as a crack propagates, the resistance to further crack propagation increases, and it requires a higher and higher applied G {displaystyle G} in order to achieve each subsequent increment of crack extension δ a {displaystyle delta a} . As such, it can be technically challenging in these materials in practice to define a single value to quantify resistance to fracture (i.e. G c {displaystyle G_{c}} or K I c {displaystyle K_{Ic}} ) as the resistance to fracture rises continuously as any given crack propagates. Materials with rising R-curves can also more easily exhibit stable crack growth than materials with flat R-curves, even if G {displaystyle G} strictly increases as a function of a {displaystyle a} . If at some moment in time a crack exists with initial length a 0 {displaystyle a_{0}} and an applied energy release rate which is infinitesimally exceeding the R-curve at this crack length [ G ( a 0 ) = G R ( a 0 ) + δ G ] {displaystyle } then this material would immediately fail if it exhibited flat R-curve behavior. If instead it exhibits rising R-curve behavior, then the crack has an added criteria for crack growth that the instantaneous slope of the driving force curve must be greater than the instantaneous slope of the crack resistance curve ( δ G ( a 0 ) δ a ≥ δ G R ( a 0 ) δ a ) {displaystyle {Biggl (}{frac {delta G(a_{0})}{delta a}}geq {frac {delta G_{R}(a_{0})}{delta a}}{Biggr )}} or else it is energetically unfavorable to grow the crack further. If G ( a 0 ) {displaystyle G(a_{0})} is infinitesimally greater than G R ( a 0 ) {displaystyle G_{R}(a_{0})} but δ G ( a 0 ) δ a ≤ δ G R ( a 0 ) δ a {displaystyle {frac {delta G(a_{0})}{delta a}}leq {frac {delta G_{R}(a_{0})}{delta a}}} then the crack will grow by an infinitesimally small increment δ a {displaystyle delta a} such that G ( a 0 + δ a ) = G R ( a 0 + δ a ) {displaystyle G(a_{0}+delta a)=G_{R}(a_{0}+delta a)} and then crack growth will arrest. If the applied crack driving force G ( a ) {displaystyle G(a)} was gradually increased over time (through increasing the applied force for example) then this would lead to stable crack growth in this material as long as the instantaneous slope of the driving force curve continued to be less than the slope of the crack resistance curve.