Chapter 2 Linear Elastic Fracture Mechanics

Transcription

1 Chapter 2 Linear Elasti Frature Mehanis 2.1 Introdution Beginning with the fabriation of stone-age axes, instint and experiene about the strength of various materials (as well as appearane, ost, availability and even divine properties) served as the basis for the design of many engineering strutures. The industrial revolution of the 19th entury led engineers to use iron and steel in plae of traditional materials like stone and wood. Unlike stone, iron and steel had the advantage of being strong in tension, whih meant that engineering strutures ould be made lighter and at less ost than was previously possible. In the years leading up to World War 2, engineers usually ensured that the maximum stress within a struture, as alulated using simple beam theory, was limited to a ertain perentage of the tensile strength of the material. Tensile strength for different materials ould be onveniently measured in the laboratory and the results for a variety of materials were made available in standard referene books. Unfortunately, strutural design on this basis resulted in many failures beause the effet of stress-raising orners and holes on the strength of a partiular struture was not appreiated by engineers. These failures led to the emergene of the field of frature mehanis. Frature mehanis attempts to haraterize a material s resistane to frature its toughness. 2.2 Stress Conentrations Progress toward a quantitative definition of toughness began with the work of Inglis 1 in Inglis showed that the loal stresses around a orner or hole in a stressed plate ould be many times higher than the average applied stress. The presene of sharp orners, nothes, or raks serves to onentrate the applied stress at these points. Inglis showed, using elastiity theory, that the degree of stress magnifiation at the edge of the hole in a stressed plate depended on the radius of urvature of the hole. The smaller the radius of urvature, the greater the stress onentration. Inglis found that the stress onentration fator,, for an elliptial hole is equal to:

2 32 Linear Elasti Frature Mehanis σ a σ yy 3σ a x σ a σ a Fig Stress onentration around a hole in a uniformly stressed plate. The ontours for yy shown here were generated using the finite-element method. The stress at the edge of the hole is 3 times the applied uniform stress. 1 2 (2.2a) where is the hole radius and is the radius of urvature of the tip of the hole. For a very narrow elliptial hole, the stress onentration fator may be very muh greater than one. For a irular hole, Eq. 2.2a gives = 3 (as shown in Fig ). It should be noted that the stress onentration fator does not depend on the absolute size or length of the hole but only on the ratio of the size to the radius of urvature. 2.3 Energy Balane Criterion In , A. A. Griffith of the Royal Airraft Establishment in England beame interested in the effet of srathes and surfae finish on the strength of mahine parts subjeted to alternating loads. Although Inglis s theory showed that the stress inrease at the tip of a rak or flaw depended only on the geometrial shape of the rak and not its absolute size, this seemed ontrary to the wellknown fat that larger raks are propagated more easily than smaller ones. This

3 2.3 Energy Balane Criterion 33 anomaly led Griffith to a theoretial analysis of frature based on the point of view of minimum potential energy. Griffith proposed that the redution in strain energy due to the formation of a rak must be equal to or greater than the inrease in surfae energy required by the new rak faes. Aording to Griffith, there are two onditions neessary for rak growth: i. The bonds at the rak tip must be stressed to the point of failure. The stress at the rak tip is a funtion of the stress onentration fator, whih depends on the ratio of its radius of urvature to its length. ii. For an inrement of rak extension, the amount of strain energy released must be greater than or equal to that required for the surfae energy of the two new rak faes. The seond ondition may be expressed mathematially as: du s d du (2.3a) d where U s is the strain energy, U is the surfae energy, and d is the rak length inrement. Equation 2.3a says that for a rak to extend, the rate of strain energy release per unit of rak extension must be at least equal to the rate of surfae energy requirement. Griffith used Inglis s stress field alulations for a very narrow elliptial rak to show that the strain energy released by introduing a double-ended rak of length 2 in an infinite plate of unit width under a uniformly applied stress a is [2]: 2 2 a U s Joules (per meter width) (2.3b) E We an obtain a semiquantitative appreiation of Eq. 2.3b by onsidering the strain energy released over an area of a irle of diameter 2, as shown in Fig The strain energy is U = (½ 2 /E)( 2 ). The atual strain energy omputed by rigorous means is exatly twie this value as indiated by Eq. 2.3b. As mentioned in Chapter 1, for ases of plane strain, where the thikness of the speimen is signifiant, E should be replaed by E/(1 2 ). In this hapter, we omit the (1 2 ) fator for brevity, although it should be noted that in most pratial appliations it should be inluded. The total surfae energy for two surfaes of unit width and length 2 is: U 4 Joules (per meter width) (2.3) The fator 4 in Eq. 2.3 arises beause of there being two rak surfaes of length 2. is the frature surfae energy of the solid. This is usually larger than the surfae free energy sine the proess of frature involves atoms loated a small distane into the solid away from the surfae. The frature surfae energy may additionally involve energy dissipative mehanisms suh as miroraking, phase transformations, and plasti deformation.

4 34 Linear Elasti Frature Mehanis Thus, taking the derivative with respet to in Eq. 2.3b and 2.3, this gives us the strain energy release rate (J/m per unit width) and the surfae energy reation rate (J/m per unit width). The ritial ondition for rak growth is: 2 a E 2 (2.3d) The left-hand side of Eq. 2.3d is the rate of strain energy release per rak tip and applies to a double-ended rak in an infinite solid loaded with a uniformly applied tensile stress. Equation 2.3d shows that strain energy release rate per inrement of rak length is a linear funtion of rak length and that the required rate of surfae energy per inrement of rak length is a onstant. Equation 2.3d is the Griffith energy balane riterion for rak growth, and the relationships between surfae energy, strain energy, and rak length are shown in Fig A rak will not extend until the strain energy release rate beomes equal to the surfae energy requirement. Beyond this point, more energy beomes available by the released strain energy than is required by the newly reated rak surfaes whih leads to unstable rak growth and frature of the speimen. σ a Strain energy released here (approximately) Two new surfaes σ a Fig The geometry of a straight, double-ended rak of unit width and total length 2 under a uniformly applied stress a Stress onentration exists at the rak tip. Strain energy is released over an approximately irular area of radius. Growth of rak reates new surfaes.

5 2.3 Energy Balane Criterion 35 Energy (U) Equilibrium (unstable) du g d U g = 4g Crak length du s d U γ +U s ps 2 U s = a 2 E Fig Energy versus rak length showing strain energy released and surfae energy required as rak length inreases for a uniformly applied stress as shown in Fig Craks with length below will not extend spontaneously. Maximum in the total rak energy denotes an unstable equilibrium ondition. The equilibrium ondition shown in Fig is unstable, and frature of the speimen will our at the equilibrium ondition. The presene of instability is given by the seond derivative of Eq. 2.3b. For d 2 U s /d 2 < 0, the equilibrium ondition is unstable. For d 2 U s /d 2 > 0, the equilibrium ondition is stable. Figure shows a onfiguration for whih the equilibrium ondition is stable. In this ase, rak growth ours at the equilibrium ondition, but the rak only extends into the material at the same rate as the wedge. The energy balane riterion indiates whether rak growth is possible, but whether it will atually our depends on the state of stress at the rak tip. A rak will not extend until the bonds at the rak tip are loaded to their tensile strength, even if there is suffiient strain energy stored to permit rak growth. For example, if the rak tip is blunted or rounded, then the rak may not extend beause of an insuffiient stress onentration. The energy balane riterion is a neessary, but not a suffiient ondition for frature. Frature only ours when the stress at the rak tip is suffiient to break the bonds there. It is ustomary to assume the presene of an infinitely sharp rak tip to approximate the worst-ase ondition. This does not mean, however, that all solids fail upon the immediate appliation of a load. In pratie, stress singularities that arise due to an infinitely sharp rak tip are avoided by plasti deformation of the material. However, if suh an infinitely sharp rak tip ould be obtained, then the rak would not extend unless there was suffiient energy for it to do so.

6 36 Linear Elasti Frature Mehanis (a) P h d (b) Energy (U) Equilbrium (Stable) U g = 4g Ed 3 h U 2 s = 8 3 Crak length Fig (a) Example of stable equilibrium (Obreimoff s experiment). (b) Energy versus rak length showing stable equilibrium as indiated by the minimum in the total rak energy. For a given stress, there is a minimum rak length that is not selfpropagating and is therefore safe. A rak will not extend if its length is less than the ritial rak length, whih, for a given uniform stress, is: 2 E (2.3e) 2 a In the analyses above, Eq. 2.3b impliitly assumes that the material is linearly elasti and in Eq. 2.3d is the frature surfae energy, whih is usually greater than the intrinsi surfae energy due to energy dissipative mehanisms in the viinity of the rak tip. The disussion above refers to a derease in strain potential energy with inreasing rak length. This type of loading would our in a fixed-grips apparatus, where the load is applied, and the apparatus lamped into position. It an be shown that exatly the same arguments apply for a dead-weight loading, where the frature surfae energy orresponds to a derease in potential energy of the loading system. The term mehanial energy release rate, may be more appropriate than strain energy release rate but the latter term is more ommonly used.

7 2.4 Linear Elasti Frature Mehanis Stress intensity fator 2.4 Linear Elasti Frature Mehanis 37 During the Seond World War, George R. Irwin 3 beame interested in the frature of steel armor plating during penetration by ammunition. His experimental work at the U.S. Naval Researh Laboratory in Washington, D.C. led, in , to a theoretial formulation of frature that ontinues to find wide appliation. Irwin showed that the stress field (r,) in the viinity of an infinitely sharp rak tip ould be desribed mathematially by: K 1 3 os 1 sin sin yy (2.4.1a) 2r The first term on the right hand side of Eq a desribes the magnitude of the stress whereas the terms involving desribe its distribution. K 1 is defined as * : K1 ay (2.4.1b) The oordinate system for Eqs a and 2.4.1b is shown in Fig In Eqn b, a is the externally applied stress, Y is a geometry fator, and is the rak half-length. K 1 is alled the stress intensity fator. There is an important reason for the stress intensity fator to be defined in this way. For a partiular rak system, and Y are onstants so the stress intensity fator tells us that the magnitude of the stress at position (r,) depends only on the external stress applied and the square root of the rak length. For example, doubling the externally applied stress a will double the magnitude of the stress in the viinity of the rak tip at oordinates (r,) for a given rak size. Inreasing the rak length by four times will double the stress at (r,) for the same value of applied stress. The stress intensity fator K 1, whih inludes both applied stress and rak length, is a ombined sale fator, whih haraterizes the magnitude of the stress at some oordinates (r,) near the rak tip. The shape of the stress distribution around the rak tip is exatly the same for raks of all lengths. Equation 2.4.1a shows that, for all sizes of raks, the stresses at the rak tip are infinite. Despite this, the Griffith energy balane riterion must be satisfied for suh a rak to extend in the presene of an applied stress a. The stress intensity fator K 1 thus provides a numerial value, whih quantifies the magnitude of the effet of the stress singularity at the rak tip. We shall see later that there is a ritial value for K 1 for different materials whih orresponds to the energy balane riterion being met. In this way, this ritial value of K 1 haraterizes the frature strength of different materials. * Some authors prefer to define K 1 without 1/2 in Eq b. In this ase, 1/2 does not appear in Eq a.

8 38 Linear Elasti Frature Mehanis In Eq b, Y is a funtion whose value depends on the geometry of the speimen, and a is the applied stress. For a straight double-ended rak in an infinite solid, Y = 1. For a small single-ended surfae rak (i.e., a semi-infinite solid), Y = ,6. This 12% orretion arises due to the additional release in strain potential energy (ompared with a ompletely embedded rak) aused by the presene of the free surfae as indiated by the shaded portion in Fig This orretion has a diminished effet as the rak extends deeper into the material. For embedded penny-shaped raks, Y = 2/. For half-penny-shaped surfae flaws in a semi-infinite solid, the appropriate value is Y = Values of Y for ommon rak geometries and loading onditions an be found in standard engineering texts. σ a Additional strain energy released σ yy σ ys σ a x plasti zone σ a Fig Semi-infinite plate under a uniformly applied stress with single-ended surfae rak of half-length. Dark shaded area indiates additional release in strain energy due to the presene of the surfae ompared to a fully embedded rak in an infinite solid. A further orretion an be made for the effet of a free surfae in front of the rak (i.e., the surfae to whih the rak is approahing). This orretion fator is very lose to 1 for raks with a length less than one-tenth the width of the speimen.

9 2.4 Linear Elasti Frature Mehanis 39 Equation 2.4.1a arises from Westergaard s solution 7 for the Airy stress funtion, whih fulfills the equilibrium equations of stresses subjet to the boundary onditions assoiated with a sharp rak, = 0, in an infinite, biaxially loaded plate. Equation 2.4.1a applies only to the material in the viinity of the rak tip. A ursory examination of Eq a shows that yy approahes zero for large values of r rather than the applied stress a. To obtain values for stresses further from the rak tip, additional terms in the series solution must be inluded. However, near the rak tip, the loalized stresses are usually very muh greater than the applied uniform stress that may exist elsewhere, and the error is thus negligible. The subsript 1 in K 1 is assoiated with tensile loading, as shown in Fig Stress intensity fators exist for other types of loading, as also shown in this figure, but our interest enters mainly on type 1 loading the most ommon type that leads to brittle failure. (a) (b) () Fig Three modes of frature. (a) Mode I, (b) Mode II, and () Mode III. Type I is the most ommon. The figures on the right indiate displaements of atoms on a plane normal to the rak near the rak tip.

10 40 Linear Elasti Frature Mehanis An important property of the stress intensity fators is that they are additive for the same type of loading. This means that the stress intensity fator for a ompliated system of loads may be derived from the addition of the stress intensity fators determined for eah load onsidered individually. It shall be later shown how the additive property of K 1 permits the stress field in the viinity of a rak an be alulated on the basis of the stress field that existed in the solid prior to the introdution of the rak. The power of Eq b annot be overestimated. It provides information about events at the rak tip in terms of easily measured marosopi variables. It implies that the magnitude and distribution of stress in the viinity of the rak tip an be onsidered separately and that a riterion for failure need only be onerned with the magnitude or intensity of stress at the rak tip. Although the stress at an infinitely sharp rak tip may be infinite due to the singularity that ours there, the stress intensity fator is a measure of the strength of the singularity Crak tip plasti zone Equation 2.4.1a implies that at r = 0 (i.e., at the rak tip) yy approahes infinity. However, in pratie, the stress at the rak tip is limited to at least the yield strength of the material, and hene linear elastiity annot be assumed within a ertain distane of the rak tip (see Fig ). This nonlinear region is sometimes alled the rak tip plasti zone 8. Outside the plasti zone, displaements under the externally applied stress mostly follow Hooke s law, and the equations of linear elastiity apply. The elasti material outside the plasti zone transmits stress to the material inside the zone, where nonlinear events our that may prelude the stress field from being determined exatly. Equation 2.4.1a shows that the stress is proportional to 1/r 1/2. The strain energy release rate is not influened muh by events within the plasti zone if the plasti zone is relatively small. It an be shown that an approximate size of the plasti zone is given by: r p 2 K1 2 2 ys (2.4.2a) where ys is the yield strength (or yield stress) of the material. The onept of a plasti zone in the viinity of the rak tip is one favored by many engineers and materials sientists and has useful impliations for frature in metals. However, the existene of a rak tip plasti zone in brittle solids appears to be objetionable on physial grounds. The stress singularity predited by Eq a may be avoided in brittle solids by nonlinear, but elasti, deformations. In Chapter 1, we saw how linear elastiity applies between two atoms for small displaements around the equilibrium position. At the rak tip, the displaements are not small on an atomi sale, and nonlinear behavior is to be

11 2.4 Linear Elasti Frature Mehanis 41 expeted. In brittle solids, strain energy is absorbed by the nonlinear strething of atomi bonds, not plasti events, suh as disloation movements, that may be expeted in a dutile metal. Hene, brittle materials do not fall to piees under the appliation of even the smallest of loads even though an infinitely large stress appears to exist at the tip of any surfae flaws or raks within it. The energy balane riterion must be satisfied for suh flaws to extend Crak resistane The assumption that all the strain energy is available for surfae energy of new rak faes does not apply to dutile solids where other energy dissipative mehanisms exist. For example, in rystalline solids, onsiderable energy is onsumed in the movement of disloations in the rystal lattie and this may happen at applied stresses well below the ultimate strength of the material. Disloation movement in a dutile material is an indiation of yield or plasti deformation, or plasti flow. Irwin and Orowan 9 modified Griffith s equation to take into aount the nonreversible energy mehanisms assoiated with the plasti zone by simply inluding this term in the original Griffith equation: du du du s p (2.4.3a) d d d The right-hand side of Eq a is given the symbol R and is alled the rak resistane. At the point where the Griffith riterion is met, the rak resistane indiates the minimum amount of energy required for rak extension in J/m 2 (i.e., J/m per unit rak width). This energy is alled the work of frature (units J/m 2 ) whih is a measure of toughness. Dutile materials are tougher than brittle materials beause they an absorb energy in the plasti zone, as what we might all plasti strain energy, whih is no longer available for surfae (i.e., rak) reation. By ontrast, brittle materials an only dissipate stored elasti strain energy by surfae area reation. The work of frature is diffiult to measure experimentally K 1C, the ritial value of K 1 The stress intensity fator K 1 is a sale fator whih haraterizes the magnitude of the stress at some oordinates (r,) near the rak tip. If eah of two raks in two different speimens are loaded so that K 1 is the same in eah speimen, then the magnitude of the stresses in the viinity of eah rak is preisely the same. Now, if the applied stresses are inreased, keeping the same value of K 1 in eah speimen, then eventually the energy balane riterion will be satisfied and the rak in eah will extend. The stresses at the rak tip are exatly the same at this point although unknown (theoretially infinite for a

12 42 Linear Elasti Frature Mehanis perfetly elasti material but limited in pratie by inelasti deformations). The value of K 1 at the point of rak extension is alled the ritial value: K 1C. K 1C then defines the onset of rak extension. It does not neessarily indiate frature of the speimen this depends on the rak stability. It is usually regarded as a material property and an be used to haraterize toughness. In ontrast to the work of frature, its determination does not depend on exat knowledge of events within the plasti zone. Consistent and reproduible values of K 1C an only be obtained when speimens are tested in plane strain. In plane stress, the ritial value of K 1 for frature depends on the thikness of the plate. Hene, K 1C is often alled the plane strain frature toughness and has units MPam 1/2. Low values of K 1C mean that, for a given stress, a material an only withstand a small length of rak before a rak extends. The ondition K 1 = K 1C does not neessarily orrespond to frature, or failure, of the speimen. K 1C desribes the onset of rak extension. Whether this is a stable or unstable ondition depends upon the rak system. Catastrophi frature ours when the equilibrium ondition is unstable. For raks in brittle materials initiated by ontat stresses, the rak may be initially unstable and then beome stable due to the sharply diminishing stress field. For example, in Chapter 7, we find that the variation in strain energy release rate (diretly related to K 1 ), the quantity dg/d, is initially positive and then beomes negative as the rak beomes longer. In terms of stress intensity fator, the rak is stable when dk 1 /d < 0 and unstable when dk 1 /d > 0. The ondition K 1 = K 1C for the stable onfiguration means that the rak is on the point of extension but will not extend unless the applied stress is inreased. If this happens, a new stable equilibrium rak length will result. Under these onditions, eah inrement of rak extension is suffiient to aount for the attendant release in strain potential energy. For the unstable onfiguration, the rak will immediately extend rapidly throughout the speimen and lead to failure. Under these onditions, for eah inrement of rak extension there is insuffiient surfae energy to aount for the release in strain potential energy Equivalene of G and K Let G be defined as being equal to the strain energy release rate per rak tip and given by the left-hand side of Eq. 2.3d, that is, for a double-ended rak within an infinite solid, the rate of release in strain energy per rak tip is: 2 G (2.4.5a) E Thus, substituting Eq b into Eq a, we have: 2 K1 G (2.4.5b) E

13 2.5 Determining Stress Intensity Fators 43 When K 1 = K 1C, then G beomes the ritial value of the rate of release in strain energy for the material whih leads to rak extension and possibly frature of the speimen. The relationship between K 1 and G is signifiant beause it means that the K 1C ondition is a neessary and suffiient riterion for rak growth sine it embodies both the stress and energy balane riteria. The value of K 1C desribes the stresses (indiretly) at the rak tip as well as the strain energy release rate at the onset of rak extension. It should be remembered that various orretions to K, and hene G, are required for raks in bodies of finite dimensions. Whatever the orretion, the orrespondene between G and K is given in Eq b. A fator of sometimes appears in Eq b depending on the partiular definition of K 1 used. Consistent use of in all these formulae is essential, espeially when omparing equations from different soures. Again, we should reognize that Eq b applies to plane stress onditions. In pratie, a ondition of plane strain is more usual, in whih ase one must inlude the fator (1 2 ) in the numerator. 2.5 Determining Stress Intensity Fators Measuring stress intensity fators experimentally Diret appliation of Griffith s energy balane riterion is seldom pratial beause of diffiulties in determining work of frature. Furthermore, the Griffith riterion is a neessary but not suffiient ondition for rak growth. However, stress intensity fators are more easily determined and represent a neessary and suffiient ondition for rak growth, but in determining the stress intensity fator, Eq b annot be used diretly beause the shape fator Y is not generally known. As mentioned previously, Y = 2/ applies for an embedded penny shaped irular rak of radius in an infinite plate. Expressions suh as this for other types of raks and loading geometries are available in standard texts. To find the ritial value of K 1, it is neessary simply to apply an inreasing load P to a prepared speimen, whih has a rak of known length already introdued, and reord the load at whih the speimen fratures. Figure shows a beam speimen loaded so that the side in whih a rak has been introdued is plaed in tension. Equation allows the frature toughness to be alulated from the rak length and load P at whih frature of the speimen ours. Note that in pratie the length of the beam speimen is made approximately 4 times its height to avoid edge effets.

14 44 Linear Elasti Frature Mehanis P W S B Fig Single edge nothed beam (SENB) PS W W K 1 (2.5.1) BW W W W Consistent and reproduible results for frature toughness an only be obtained under onditions of plane strain. In plane stress, the values of K 1 at frature depend on the thikness of the speimen. For this reason, values of K 1C are measured in plane strain, hene the term plane strain frature toughness Calulating stress intensity fators from prior stresses Under some irumstanes, it is possible 10 to alulate the stress intensity fator for a given rak path using the stress field in the solid before the rak atually exists. The proedure makes use of the property of superposition of stress intensity fators. Consider an internal rak of length 2 within an infinite solid, loaded by a uniform externally applied stress a, as shown in Fig a. The presene of the rak intensifies the stress in the viinity of the rak tip, and the stress intensity fator K 1 is readily determined from Eq b. Now, imagine a series of surfae trations in the diretion opposite the stress and applied to the rak faes so as to lose the rak ompletely, as shown in Fig b. At this point, the stress distribution within the solid, uniform or otherwise, is preisely equal to what would have existed in the absene of the rak beause the rak is now ompletely losed. The stress intensity fator thus drops to zero, sine there is no longer a onentration of stress at the rak tip. Thus, in one ase, the presene of the rak auses the applied stress to be intensified in the viinity of the rak, and in the other, appliation of the surfae trations auses this intensifiation to be redued to zero.

15 2.5 Determining Stress Intensity Fators 45 (a) σ (b) σ () A F A F B F A A b σ σ Fig (a) Internal rak in a solid loaded with an external stress. (b) Crak losed by the appliation of a distribution of surfae trations F. () Internal rak loaded with surfae trations F A and F B. Consider now the situation illustrated in Fig Wells 11 determined the stress intensity fator K 1 at one of the rak tips A for a symmetri internal rak of total length 2 being loaded by fores F A applied on the rak faes at a distane b from the enter. The value for K 1 for this ondition is: K F b b A 1A (2.5.2a) Fores F B also ontribute to the stress field at A, and the stress intensity fator due to those fores is: K F b b B 1B (2.5.2b) Due to the additive nature of stress intensity fators, the total stress intensity fator at rak tip A shown in Fig due to fores F A and F B, where F A = F B = F, is : 2F K1 K1A K1B b 12 (2.5.2) It is important to note that the Green s weighting funtions here apply to a double-ended rak in an infinite solid. For example, Eq a applies to a fore F A applied to a double-ended symmetri rak and not F A applied to a single rak tip alone.

16 46 Linear Elasti Frature Mehanis Now, if the trations F are ontinuous along the length of the rak, then the fore per unit length may be assoiated with a stress applied (b) normal to the rak. The total stress intensity fator is given by integrating Eq with F replaed by df = (b)db ( b) K1 12 db b (2.5.2d) However, if the fores F are reversed in sign suh that they lose the rak ompletely, then the assoiated stress distribution (b) must be that whih existed prior to the introdution of the rak. The stress intensity fator, as alulated by Eq d, for ontinuous surfae trations applied so as to lose the rak, is preisely the same as that (exept for a reversal in sign) alulated for the rak using the marosopi stress a in the absene of suh trations. For example, for the uniform stress ase, where (b) = a, Eq d redues to Eq b. As long as the prior stress field within the solid is known, the stress intensity fator for any proposed rak path an be determined using Eq d. The strain energy release rate G an be alulated from Eq b. Of ourse, one annot always immediately determine whether a rak will follow any partiular path within the solid. It may be neessary to alulate strain energy release rates for a number of proposed paths to determine the maximum value for G. The rak extension that results in the maximum value for G is that whih an atual rak will follow. In brittle materials, raks usually initiate from surfae flaws. The strain energy release rate as alulated from the prior stress field (i.e., prior to there being any flaws) applies to the omplete growth of the subsequent rak. The onditions determining subsequent rak growth depend on the prior stress field. The strain energy release rate, G, an be used to desribe the rak growth for all flaws that exist in the prior stress field but an only be onsidered appliable for the subsequent growth of the flaw that atually first extends. Assuming there is a large number of raks or surfae flaws to onsider, the one that first extends is that giving the highest value for G (as alulated using the prior stress field) for an inrement of rak growth. Subsequent growth of that flaw depends upon the Griffith energy balane riterion (i.e., G 2) being met as alulated along the rak path still using the prior stress field, even though the atual stress field is now different due to the presene of the extending rak. To show this, one must make use of the standard integral: 1 1 x dx sin C a x a

17 2.5.3 Determining stress intensity fators using the finite-element method 2.5 Determining Stress Intensity Fators 47 Stress intensity fators may also be alulated using the finite-element method. The finite-element method is useful for determining the state of stress within a solid where the geometry and loading is suh that a simple analytial solution for the stress field is not available. The finite-element solution onsists of values for loal stresses and displaements at predetermined node oordinates. A value for the loal stress yy at a judiious hoie of oordinates (r,) an be used to determine the stress intensity fator K 1. For example, at = 0, Eq a beomes: 1 2 K1 yy 2r (2.5.3a) where yy is the magnitude of the loal stress at r. It should be noted that the stress at the node that orresponds to the loation of the rak tip (r = 0) annot be used beause of the stress singularity there. Stress intensity fators determined for points away from the rak tip, outside the plasti zone, or more orretly the nonlinear zone, may only be used. However, one annot use values that are too far away from the rak tip sine Eq a applies only for small values of r. At large r, yy as given by Eq a approahes zero, and not as is atually the ase, a. Values of K 1 determined from finite-element results and using Eq a should be the same no matter whih node is used for the alulation, subjet to the onditions regarding the hoie of r mentioned previously. However, it is not always easy to hoose whih value of r and the assoiated value of yy to use. In a finite-element model, the speimen geometry, density of nodes in the viinity of the rak tip, and the types of elements used are just some of the things that affet the auray of the resultant stress field. One method of estimation is to determine values for K 1 at different values of r along a line ahead of the rak tip at = 0. These values for K 1 are then fitted to a smooth urve and extrapolated to r = 0, as shown in Fig K 1 best estimate r r Fig Estimating K 1 from finite-element results. For elements near the rak tip, Eq a is valid and K 1 an be determined from the stresses at any of the nodes near the rak tip. In pratie, one needs to determine a range of K 1 for a fixed (e.g., = 0) for a range of r and extrapolate bak to r = 0.

18 48 Linear Elasti Frature Mehanis Referenes 1. C.E. Inglis, Stresses in a plate due to the presene of raks and sharp orners, Trans. Inst. Nav. Arhit. London 55, 1913, pp A.A. Griffith, Phenomena of rupture and flow in solids, Philos. Trans. R. So. London Ser. A221, 1920, pp G.R. Irwin, Frature dynamis, Trans. Am. So. Met. 40A, 1948, pp G.R. Irwin, Analysis of stresses and strains near the end of a rak traversing in a plate, J. Appl. Meh. 24, 1957, pp B.R. Lawn, Frature of Brittle Solids, 2nd Ed., Cambridge University Press, Cambridge, U.K., I.N. Sneddon, The distribution of stress in the neighbourhood of a rak in an elasti solid, Pro. R. So. London, Ser. A187, 1946, pp H.M. Westergaard, Bearing pressures and raks, Trans. Am. So. Meh. Eng. 61, 1939, pp. A49 A D.M. Marsh, Plasti flow and frature of glass, Pro. R. So. London, Ser. A282, 1964, pp E. Orowan, Energy riteria of frature, Weld. J. 34, 1955, pp F.C. Frank and B.R. Lawn, On the theory of hertzian frature, Pro. R. So. London, Ser. A229, 1967, pp A.A. Wells, Br. Weld. J. 12, 1965, p. 2.

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