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that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic 6 griffith s theory provides excellent agreement with experimental data for brittle materials such as glass for ductile materials such as steel although the relation σ f a c displaystyle sigma _ f sqrt a c still holds the surface energy γ predicted by griffith s theory is usually unrealistically high a group working under g r irwin 7 at the u s naval research laboratory nrl during world war ii realized that plasticity must play a significant role in the fracture of ductile materials in ductile materials and even in materials that appear to be brittle 8 a plastic zone develops at the tip of the crack as the applied load increases the plastic zone increases in size until the crack grows and the elastically strained material behind the crack tip unloads the plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat hence a dissipative term has to be added to the energy balance relation devised by griffith for brittle materials in physical terms additional energy is needed for crack growth in ductile materials as compared to brittle materials irwin s strategy was to partition the energy into two parts the stored elastic strain energy which is released as a crack grows this is the thermodynamic driving force for fracture the dissipated energy which includes plastic dissipation and the surface energy and any other dissipative forces that may be at work the dissipated energy provides the thermodynamic resistance to fracture then the total energy is g 2 γ g p displaystyle g 2 gamma g_ p where γ displaystyle gamma is the surface energy and g p displaystyle g_ p is the plastic dissipation and dissipation from other sources per unit area of crack growth the modified version of griffith s energy criterion can then be written as σ f a e g π displaystyle sigma _ f sqrt a sqrt cfrac e g pi for brittle materials such as glass the surface energy term dominates and g 2 γ 2 j m 2 displaystyle g approx 2 gamma 2 text j m 2 for ductile materials such as steel the plastic dissipation term dominates and g g p 1000 j m 2 displaystyle g approx g_ p 1000 text j m 2 for polymers close to the glass transition temperature we have intermediate values of g displaystyle g between 2 and 1000 j m 2 displaystyle text j m 2 stress intensity factor edit main article stress intensity factor another significant achievement of irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid 7 this asymptotic expression for the stress field in mode i loading is related to the stress intensity factor k i displaystyle k_ i following 9 σ i j k i 2 π r f i j θ displaystyle sigma _ ij left cfrac k_ i sqrt 2 pi r right f_ ij theta where σ i j displaystyle sigma _ ij are the cauchy stresses r displaystyle r is the distance from the crack tip θ displaystyle theta is the angle with respect to the plane of the crack and f i j displaystyle f_ ij are functions that depend on the crack geometry and loading conditions irwin called the quantity k displaystyle k the stress intensity factor since the quantity f i j displaystyle f_ ij is dimensionless the stress intensity factor can be expressed in units of mpa m displaystyle text mpa sqrt text m stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface weakness energy both of these terms are simply related to the energy terms that griffith used k i σ π a displaystyle k_ i sigma sqrt pi a and k c e g c for plane stress e g c 1 ν 2 for plane strain displaystyle k_ c begin cases sqrt eg_ c text for plane stress sqrt cfrac eg_ c 1 nu 2 text for plane strain end cases where k i displaystyle k_ i is the mode i displaystyle i stress intensity k c displaystyle k_ c the fracture toughness and ν displaystyle nu is poisson s ratio fracture occurs when k i k c displaystyle k_ i geq k_ c for the special case of plane strain deformation k c displaystyle k_ c becomes k i c displaystyle k_ ic and is considered a material property the subscript i displaystyle i arises because of the different ways of loading a material to enable a crack to propagate it refers to so called mode i displaystyle i loading as opposed to mode i i displaystyle ii or i i i displaystyle iii the expression for k i displaystyle k_ i will be different for geometries other than the center cracked infinite plate as discussed in the article on the stress intensity factor consequently it is necessary to introduce a dimensionless correction factor y displaystyle y in order to characterize the geometry this correction factor also often referred to as the geometric shape factor is given by empirically determined series and accounts for the type and geometry of the crack or notch we thus have k i y σ π a displaystyle k_ i y sigma sqrt pi a where y displaystyle y is a function of the crack length and width of sheet given for a sheet of finite width w displaystyle w containing a through thickness crack of length 2 a displaystyle 2a by y a w sec π a w displaystyle y left frac a w right sqrt sec left frac pi a w right strain energy release edit main article strain energy release rate irwin was the first to observe that if the size of the plastic zone around a crack is small compared to the size of the crack the energy required to grow the crack will not be critically dependent on the state of stress the plastic zone at the crack tip 6 in other words a purely elastic solution may be used to calculate the amount of energy available for fracture the energy release rate for crack growth or strain energy release rate may then be calculated as the change in elastic strain energy per unit area of crack growth i e g u a p u a u displaystyle g left cfrac partial u partial a right _ p left cfrac partial u partial a right _ u where u is the elastic energy of the system and a is the crack length either the load p or the displacement u are constant while evaluating the above expressions irwin showed that for a mode i crack opening mode the strain energy release rate and the stress intensity factor are related by g g i k i 2 e plane stress 1 ν 2 k i 2 e plane strain displaystyle g g_ i begin cases cfrac k_ i 2 e text plane stress cfrac 1 nu 2 k_ i 2 e text plane strain end cases where e is the young s modulus ν is poisson s ratio and k i is the stress intensity factor in mode i irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode i mode ii sliding mode and mode iii tearing mode stress intensity factors for the most general loading conditions next irwin adopted the additional assumption that the size and shape of the energy dissipation zone remains approximately constant during brittle fracture this assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material this new material property was given the name fracture toughness and designated g ic today it is the critical stress intensity factor k ic found in the plane strain condition which is accepted as the defining property in linear elastic fracture mechanics crack tip plastic zone edit in theory the stress at the crack tip where the radius is nearly zero would tend to infinity this would be considered a stress singularity which is not possible in real world applications for this reason in numerical studies in the field of fracture mechanics it is often appropriate to represent cracks as round tipped notches with a geometry dependent region of stress concentration replacing the crack tip singularity 9 in actuality the stress concentration at the tip of a crack within real materials has been found to have a finite value but larger than the nominal stress applied to the specimen nevertheless there must be some sort of mechanism or property of the material that prevents such a crack from propagating spontaneously the assumption is the plastic deformation at the crack tip effectively blunts the crack tip this deformation depends primarily on the applied stress in the applicable direction in most cases this is the y direction of a regular cartesian coordinate system the crack length and the geometry of the specimen 10 to estimate how this plastic deformation zone extended from the crack tip irwin equated the yield strength of the material to the far field stresses of the y direction along the crack x direction and solved for the effective radius from this relationship and assuming that the crack is loaded to the critical stress intensity factor irwin developed the following expression for the idealized radius of the zone of plastic deformation at the crack tip r p k c 2 2 π σ y 2 displaystyle r_ p frac k_ c 2 2 pi sigma _ y 2 models of ideal materials have shown that this zone of plasticity is centered at the crack tip 11 this equation gives the approximate ideal radius of the plastic zone deformation beyond the crack tip which is useful to many structural scientists because it gives a good estimate of how the material behaves when subjected to stress in the above equation the parameters of the stress intensity factor and indicator of material toughness k c displaystyle k_ c and the yield stress σ y displaystyle sigma _ y are of importance because they illustrate many things about the material and its properties as well as about the plastic zone size for example if k c displaystyle k_ c is high then it can be deduced that the material is tough and if σ y displaystyle sigma _ y is low one knows that the material is more ductile the ratio of these two parameters is important to the radius of the plastic zone for instance if σ y displaystyle sigma _ y is small then the squared ratio of k c displaystyle k_ c to σ y displaystyle sigma _ y is large which results in a larger plastic radius this implies that the material can plastically deform and therefore is tough 10 this estimate of the size of the plastic zone beyond the crack tip can then be used to more accurately analyze how a material will behave in the presence of a crack the same process as described above for a single event loading also applies and to cyclic loading if a crack is present in a specimen that undergoes cyclic loading the specimen will plastically deform at the crack tip and delay the crack growth in the event of an overload or excursion this model changes slightly to accommodate the sudden increase in stress from that which the material previously experienced at a sufficiently high load overload the crack grows out of the plastic zone that contained it and leaves behind the pocket of the original plastic deformation now assuming that the overload stress is not sufficiently high as to completely fracture the specimen the crack will undergo further plastic deformation around the new crack tip enlarging the zone of residual plastic stresses this process further toughens and prolongs the life of the material because the new plastic zone is larger than what it would be under the usual stress conditions this allows the material to undergo more cycles of loading this idea can be illustrated further by the graph of aluminum with a center crack undergoing overloading events 12 limitations edit the s s schenectady split apart by brittle fracture while in harbor 1943 but a problem arose for the nrl researchers because naval materials e g ship plate steel are not perfectly elastic but undergo significant plastic deformation at the tip of a crack one basic assumption in irwin s linear elastic fracture mechanics is small scale yielding the condition that the size of the plastic zone is small compared to the crack length however this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture which has led to a number of catastrophic failures linear elastic fracture mechanics is of limited practical use for structural steels and fracture toughness testing can be expensive elastic plastic fracture mechanics edit vertical stabilizer which separated from american airlines flight 587 leading to a fatal crash most engineering materials show some nonlinear elastic and inelastic behavior under operating conditions that involve large loads citation needed in such materials the assumptions of linear elastic fracture mechanics may not hold that is the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size the size and shape of the plastic zone may change as the applied load is increased and also as the crack length increases therefore a more general theory of crack growth is needed for elastic plastic materials that can account for the local conditions for initial crack growth which include the nucleation growth and coalescence of voids decohesion at a crack tip a global energy balance criterion for further crack growth and unstable fracture ctod edit main article crack tip opening displacement historically the first parameter for the determination of fracture toughness in the elasto plastic region was the crack tip opening displacement ctod or opening at the apex of the crack indicated this parameter was determined by wells during the studies of structural steels which due to the high toughness could not be characterized with the linear elastic fracture mechanics model he noted that before the fracture happened the walls of the crack were leaving clarification needed and that the crack tip after fracture ranged from acute to rounded off due to plastic deformation in addition the rounding of the crack tip was more pronounced in steels with superior toughness there are a number of alternative definitions of ctod in the two most common definitions ctod is the displacement at the original crack tip and the 90 degree intercept the latter definition was suggested by rice and is commonly used to infer ctod in finite element models of such note that these two definitions are equivalent if the crack tip blunts in a semicircle most laboratory measurements of ctod have been made on edge cracked specimens loaded in three point bending early experiments used a flat paddle shaped gage that was inserted into the crack as the crack opened the paddle gage rotated and an electronic signal was sent to an x y plotter this method was inaccurate however because it was difficult to reach the crack tip with the paddle gage today the displacement v at the crack mouth is measured and the ctod is inferred by assuming the specimen halves are rigid and rotate about a hinge point the crack tip r curve edit main article crack growth resistance curve an early attempt in the direction of elastic plastic fracture mechanics was irwin s crack extension resistance curve crack growth resistance curve or r curve this curve acknowledges the fact...