S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo Stess Intensity Facto We have modeled a body by using the linea elastic theoy We have modeled a cack in the body by a flat plane, and the font of the cack by a staight line Within this idealized model, the field aound the font of the cack is singula The singula field is clealy an atifact of the idealized model, but Iwin and othes made the singula field a centepiece of factue mechanics The mathematics of this singula field had been known long befoe Iwin enteed the field We will focus on the mathematics in this lectue, and will descibe Iwin s way of using the singula field in the following lectue In pevious lectues, we have descibed seminal ideas of Inglis http://imechanicaog/node/7457) and of Giffith http://imechanicaog/node/7470) We have also descibed the eintepetation of these ideas due to Iwin and Oowan http://imechanicaog/node/7507) Ou desciptions have centeed on two quantities: enegy elease ate and factue enegy This enegy-based appoach leads us into applications of factue mechanics http://imechanicaog/node/7531) Befoe showing you moe applications, I d like to tell you about a basic concept, possibly also due to Iwin, concened with the modes of factue Modes of factue Depending on the symmety of the field aound the tip of a cack, factue may be classified into thee modes Mode I: tensile mode, o opening mode Mode II: in-plane shea mode, o sliding mode Mode III: anti-plane shea mode, o teaing mode The modes descibe the local condition aound a point on the font Fo example, conside a penny-shaped cack in an infinite body The font of the cack is a cicle When a load pulls the body in the diection pependicula to the plane of the cack, evey point along the font of the cack is unde the mode I condition When a load sheas the body in the diection paallel to the plane of the cack, evey point along the font of the cack is unde a mixed mode II and mode III condition Only a few special points on the font ae unde eithe a pue mode II condition, o a pue mode III condition The concept of the modes appeals to ou intuition, but this concept is absent in the enegy-based appoach negy elease ate by itself does not diffeentiate the modes of factue negy elease ate chaacteizes the amplitude of the applied load, but not the mode of the applied load To descibe the mode of the load will equie us to talk about the field nea the tip of the cack negy elease ate fo a cack in a linea elastic mateial Conside a cack of length a in a sheet subject to applied stess appl The body is made of a homogeneous, isotopic, linealy elastic mateial, with Young s modulus and Poisson s atio ν Lineaity of the bounday-value poblem and /1/14 1
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo dimensional consideation dictate that the enegy elease ate should take the fom G β appl) a The dimensionless numbe β must be detemined by solving the bounday-value poblem Linea elastic field aound the tip of a cack Focus on the field nea one tip of the cack We model the tip of the cack as a point, and the two faces of the cack as staight lines Let, ) be the pola coodinates centeed at the tip of the cack The two faces of the cack coincide with the lines π and π A mateial paticle in the body is at a distance fom the tip of the cack The mateial paticle is said to be nea the tip of the cack if is much smalle than the length chaacteistic of the bounday-value poblem; fo example, << a, whee a is the length of the cack We want to detemine the fields nea the tip of the cack, including the stess ij, ), stain ij, ) and displacement u i, ) To this end, we egad the body as infinite, and the cack as semi-infinite The faces of the cack ae taction-fee, and the load is applied emotely fom the tip of the cack We epesent the applied load by a single paamete: the enegy elease ate G In doing so we change ou pespective somewhat In the past, we egad G as pat of the solution of a bounday-value poblem We now egad G as the applied load in a bounday-value poblem This mathematical model diffes fom a eal cack in a eal mateial in many aspects, but fo the time being we stick to the model itself body π cack π tip The nea-tip stess field is squae-oot singula I find a way to each the squae-oot singulaity without solving the bounday-value poblem; athe, we invoke two elementay consideations: the bounday-value poblem is linea, and the geomety of the poblem is scale-fee You can skip this section if you do not like this line of easoning The enegy elease ate G is the loading paamete of the bounday-value poblem As G inceases, the fields of stess, stain and displacement incease In the linea theoy of elasticity, the stess, stain, and displacement ae linealy /1/14
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo popotional to the applied stess, but G is quadatic in the applied stess Thus, the stess field scales with G as ij, ) G The infinite body and the semi-infinite cack povide no length scales that is, the geomety of the bounday-value poblem is scale-fee The atio G / has the dimension of length The thee quantities G, and fom a single dimensionless goup: G A combination of lineaity and dimensional consideation dictates that the field of stess nea the tip of the cack should take the fom ij G f ij ), ) ae dimensionless functions of, and possibly also depend on whee f ij Poisson s atio ν Thus, fom these elementay consideations, we find out that the stess field nea the tip of the cack is squae-oot singula in Simila consideations show that the fields of stain and displacement nea the tip of the cack take the following foms: ij, ) G g ij u i, ) G h i ) These angula functions can be detemined by solving the bounday-value poblem, as descibed below Govening equations of linea elasticity The state of a body is x, x x x, x x, and chaacteized by thee fields: displacement u i 1, 3), stain ij 1, 3 ) stess x, x x ) ij 1, 3 ), In the body, the thee fields satisfy the govening equations: Stain-displacement elations u u )/ Balance of foces 0 ij ij, j i, j j, i Stess-stain elations ij Cijpq pq At evey point on the suface of the body, in each diection, we pescibe eithe the displacement, o the taction If these equations look vague to you, ijn j please eview the notes on the elements of linea elasticity http://imechanicaog/node/05) The singula field aound the font of a cack While a body is in a thee-dimensional space, the singula field aound the font of a cack is nealy two-dimensional Let us claify this eduction in dimension In the thee- /1/14 3
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo dimensional space, we model a body by a volume, a cack by a smooth suface, and the font of the cack by a smooth cuve Fo any point on the font, we use the point as the oigin to set up local coodinates, with the axis x pointing in the diection of popagation of the font, y nomal to the plane of the cack, and z tangent to the font Because the font is assumed to be a smooth cuve, the field aound the font is singula in x and y, but smooth in z That is, fo any component of the field, f x, y, z), the deivative f / z is small compaed to the deivatives f / x and f / y Consequently, we may dop all patial deivatives with espect to z in the govening equations Consequently, the singula field aound the font is locally chaacteized by a field of the fom u x, y), v x, y), w x, y) We futhe assume that the elastic behavio of the mateial is isotopic The field decouples into two types: Plane-stain defomation: u x, y) 0, v x, y) 0, but w x, y) 0 Anti-plane defomation: u x, y) 0, v x, y) 0, but w x, y) 0 The plane-stain defomation descibes mode I and mode II cacks, and the antiplane defomation descibes mode III cacks In class we will mostly talk about mode I cacks Plane-stain conditions To analyze this locally plane-stain field, we focus on an actual plane-stain poblem: The body is unde the plane-stain conditions The cack is a flat plane The font of the cack is a staight line The field is elastic all the way to the font The fields educed to Displacement field: u x, y) and v x, y) Note that w 0 Stain field:,, ae functions of x and y Othe stain components vanish Stess field:,,, ae functions of x and y Othe stess zz components vanish Stain-displacement elations: u v 1 u v,, x y y x Balance of foces: 0, 0 x y x y Hooke s law: ν ν ) 1, zz /1/14 4
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo ν ν ) 1, zz ν ν ) zz 1, zz ν ) 1, Unde the plane stain conditions, zz 0, so that ν zz ) Substituting this elation to Hooke s law, we obtain that 1 ν ν, 1 ν 1 ν ν, 1 ν ν ) 1 The quantity, 1 ν is known as the plane-stain modulus Unde the plane stain conditions, the elastic field is epesented by displacements, 3 stains, and 4 stesses The 9 functions ae govened by 9 field equations 3 stain-displacement elations, 3 stess-stain elations, equations to balance foces, and 1 elation between,, If you have not studied plane-stain poblems befoe, please eview the notes http://imechanicaog/node/319 Aiy s function We now have 9 equations fo 9 functions As usual we can eliminate some functions by combining equations Many appoaches of elimination have been devised Hee we will follow the appoach due to Aiy 1863), who educed the system of equations to one equation fo one function Recall a theoem in calculus If functions f x, y) and g x, y) satisfy the following elation f g, x y then a function α x,y) exists, such that α α f, g y x Accoding to this theoem, one equation of foce balance, z x y /1/14 5
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo, x y implies that a function α x,y) exists, such that α α, y x The othe equation of foce balance,, x y implies that a function β x,y) exists, such that β β, x y In the above, we have expessed the shea stess quating the two expessions, we obtain that α β x y in two ways Accoding to the theoem in calculus, this equation implies that a function x,y) exists, such that α, β y x Summing up, we can expess the thee stesses in tems of one function:,, y x The function x,y) is known as Aiy s function Stesses expessed by Aiy s function satisfy the equations of foce balance Stains in tems of Aiy s function Using Hooke s law, we expess the stains in tems of Aiy s function: 1 ν ν, y 1 ν x 1 ν ν, x 1 ν y 1 ν ) The equation of compatibility liminate the displacements fom the stain-displacement elations, and we obtain that /1/14 6
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo /1/14 7 y x x y This equation is known as the equation of compatibility The compatibility equation can be expessed in tems of Aiy s function: 0 4 4 4 4 4 y y x x This equation is known as the bi-hamonic equation Accoding to Meleshko, this equation was fist deived by Maxwell when asked by Stokes to eview the pape by Aiy The plane-stain poblem is govened by the bi-hamonic equation and the bounday conditions Once is solved, one can detemine the stesses, stains, and displacements The bihamonic equation is often witten as 0 y x y x Pola coodinates When Aiy s function is expessed as a function of the pola coodinates, ),, the pola components of the stess ae expessed as,, The bi-hamonic equation is 0 The Williams expansion Within the linea elastic theoy, the field in a body is detemined by a bounday-value poblem The field depends on the bounday conditions, namely, the size and the shape of the cack and the body, as well as the magnitude and the distibution of the load Some such boundayvalue poblems had been solved befoe Williams enteed the field Williams took a diffeent appoach Instead of solving individual bounday-value poblems, he focused on the singula field aound the tip of the cack, in a zone so small that the cack can be assumed to be semi-infinite, and the bounday of the body infinitely fa away He discoveed that the fom of the singula field is univesal, independent of the shape of the body and the cack Let ), be the pola coodinates, centeed at a paticula point on the font of the cack The cack popagates in the diection 0, and the two faces of the cack coincide with π ± The two faces of the cack ae taction-fee We solve the bihamonic equation using the method of sepaation of vaiables ach tem in the bi-hamonic equation has the same dimension in Fo such an equi-dimensional equation, the solution is to some powe Wite the solution in the fom ) ) λ f 1,,
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo f ae to be detemined Inset this fom into the bihamonic equation, and we obtain an odinay diffeential equation OD): d d whee the constant λ and the function ) λ 1) λ 1) f ) 0 d d This is an OD with constant coefficients The solution is of the fom f ) exp b ), whee b is to be detemined Inseting this fom into the OD, we find that b λ 1) ) b λ 1) ) 0 This is a fouth-ode algebaic equation fo b The fou oots ae b ± λ 1 i, ± λ 1, ) )i whee i 1 The geneal solution to the OD is f ) Acos λ 1) Bcos λ 1) C sin λ 1) Dsin λ 1), whee A, B, C and D ae constants Fo the mode I cack, the field is symmetic with espect to the x-axis, so that C D 0 Summaizing, we find that the solution fo a mode I cack is λ 1, ) [ Acos λ 1) Bcos λ 1) ], whee λ, A and B ae constants to be detemined by using the bounday conditions The stesses ae λ 1 λ A λ 1 cos λ 1 B λ 3 cos λ 1 [ ) ) ) ) ] λ 1 λ 1) λ [ Acos λ 1) Bcos λ 1) ] λ 1 λ 1) [ A λ 1) sin λ 1) B λ 1) sin λ ) ] 1 At π, both components of the taction vanish, 0, namely, λ 1) λ cos λπ λ 1) λ cos λπ A 0 ) ) λ 1 sin λπ λ 1 sin λπ B 0 This pai of linea algebaic equations fo A and B fom an eigenvalue poblem To have a solution such that A and B ae not both zeo, the deteminant must vanish, namely, λ 1) sin λπ cos πλ 0 λ The solutions ae 1 1 3 λ 1,,0,,1,, Consequently, the field of stess takes the fom of an expansion: ij am m m/ f m ) ) ij /1/14 8
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo The functions m ) ) f ae detemined by the eigenvalue poblem The ij amplitudes a, howeve, ae not detemined by the eigenvalue poblem, and m should be detemined by the full bounday-value poblem The squae-oot singulaity ach tem in the Williams expansion coesponds to a solution to the bihamonic equation Which tem should we choose? Note that ~ λ 1, ~ λ 1, λ u ~ Recall the stess concentation fo the ellipse Requie that the displacement to be bounded, so that λ > 0 Requie the stess to be singula, so that λ < 1 The two equiements foce us to choose 1 λ Substituting this value to the algebaic equations, we find that B 3A In paticula, we find the hoop stess is B 3, ) cos The above justifications fo the choice of the squae-oot singulaity ae flimsy The significance of this choice has to be undestood late, when we see how this singulaity is used in pactice Fo a discussion, see CY Hui and Andy Ruina, Why K? High ode singulaities and small scale yielding, Intenational Jounal of Factue 7, 97-10 1995) Stess intensity facto Williams solved an eigenvalue poblem because no load was specified: both the field equations and the bounday conditions ae homogeneous Like any othe eigenvalue poblem, this eigenvalue poblem leaves the amplitude undetemined In this case, all the field is detemined up to the constant B By convention, the constant B is witten as B K / π Consequently, the stess at a distance diectly ahead the cack tip is K, 0) π The legend has it that Iwin chose the lette K afte JA Kies, one of his cowokes The field of stess aound the tip of the cack is given by K cos 1 sin π K 3 cos π /1/14 9
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo K cos sin π One can also detemine the displacement components In paticula, the cack opening displacement a distance behind the cack tip is 8K δ, π whee / 1 ν ) unde the plane stain conditions, and unde the plane stess conditions Notes K is called the stess intensity facto Its magnitude is undetemined in the eigenvalue poblem K is the amplitude of the field aound the tip of the cack The facto π ) 1/ is set by convention The and dependence ae independent of the extenal bounday conditions By modeling the cack font as a mathematical cuve, the linea elastic theoy does not account fo any pocess of factue Detemine the stess intensity facto by solving a boundayvalue poblem Fo a given bounday-value poblem, K can be detemined Fo example, conside the Giffith cack, a cack of length a in an infinite plate, subject to emote stess The bounday-value poblem was solved by Inglis 1913) The field in the body is expessed in analytical tems Fo example, the stess ahead of the cack is given by x x > a x a, The distance of a point x ahead of the cack tip is given by x a Replace x by, and we obtain that a) a) When the tip of the cack is appoached, «a, we have a This cack-tip field is obtained fom the bounday-value poblem, and it ecoves the fom detemined by the eigenvalue poblem, namely, K, π A compaison of the two expessions gives K πa /1/14 10
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo This is the stess intensity facto fo the Giffith cack Altenatively, one can detemine the stess intensity facto by using the displacement field The bounday-value poblem solved by Inglis also gives the opening displacement of the cack: x ) 4 δ a x, x < a The distance of a point x behind the cack tip is given by a x Replace x by, and we obtain that δ x) 4 a) When the tip of the cack is appoached, «a, we have 4 δ x) a This cack-tip field is obtained fom the bounday-value poblem, and it ecoves the fom detemined by the eigenvalue poblem, namely, 8K δ π A compaison of the two expessions gives K πa Handbooks fo K Stess intensity facto fo a given cacked body is detemined by solving a bounday-value poblem Many configuations of cacked bodies have been solved The esults ae collected in handbooks eg, H Tada, PC Pais and GR Iwin, The Stess Analysis of Cacks Handbook, Del Reseach, St Louis, MO, 1995) In geneal, the stess intensity facto takes the fom K Y a, whee is an applied stess, a is a length scale chaacteize the cack geomety, and Y is a dimensionless numbe Two examples follow Fo a cack at the edge of a semi-infinite plane, the stess intensity facto is K 1115 πa Fo a penny-shaped cack in an infinite body, the stess intensity facto is K πa π Fo the two examples, can you explain why the fist example has a lage stess intensity facto than the Giffith cack, and the second example has a smalle one? In any event, the diffeence in the stess intensity factos fo the thee cases is small Thus, fo a small cack in a lage body, the stess intensity facto is detemined by the size of the cack, but is insensitive to the shape of the cack /1/14 11
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo Linea supeposition lasticity poblem is linea Fo a given body with a given cack, if a foce P causes the stess intensity facto K αp, and foce Q causes the stess intensity facto K βq The combined action of the foces P and Q causes the stess intensity facto K αp βq Finite element method to detemine K Fo a complicated stuctue with a cack, once can detemine the elastic field using the finite element method, and then extact fom the field the stess intensity facto A bute foce method is that you use the finite element method to detemine the displacement field, and then fit the cack opening to 8K δ, π with K as the fitting paamete Thee ae a numbe of moe cleve methods We ll mention them late at suitable points Iwin s G-K elation Conside a cack in an elastic body subject to a load The enegy elease ate G is the decease in the elastic enegy associated with the cack advancing by a unit aea, while the load is igidly held The stess intensity facto K is the amplitude of the field aound the tip of the cack Iwin discoveed the following elation between the stess intensity facto and the enegy elease ate: K G Mode I, plane stain) Poof Conside two bodies, 1 and Both have the same configuation: a sheet of unit thickness, and a semi-infinite cack The cack in Body is longe than that in Body 1 by a length b The displacement of the applied load is fixed, so that the applied foce does no wok when the cack extends Let U 1 ) and U ) be the stain enegy stoed in the two bodies, espectively By definition, the enegy elease ate is Let K 1 ) field ahead the cack in Body 1 is G U 1 ) U ) b be the stess intensity facto of the cack in Body 1 The stess 1 ) K 1 ) π x Let K ) be the stess intensity facto of the cack in Body displacement of the cack in Body is ) 8K ) δ b x π The opening /1/14 1
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo The diffeence in the enegy in the two bodied is due to the wok done by the closing taction: U 1 b ) U ) 1 1 ) δ ) dx 0 This gives G 1 b K 1 ) 8K ) b x b π x π dx K 1 ) K ) 1 1 t dt K 1 ) K ) 0 π 0 t The change in the length of the cack b is small compaed to the size of body o the total length of the cack a As b / a 0, the stess intensity factos of the cacks in the two bodies appoach each othe, K 1 ) K ) K Thus, we each Iwin s G-K elation K G Fo some poblems, the enegy elease ate is easie to detemine than the stess intensity facto We have seen two examples: the double-cantileve beam, and the channel cack Once the enegy elease ate is detemined, one can obtain the stess intensity facto by using Iwin s elation Mode II Diectly ahead the cack tip π Behind the cack tip, the sliding displacement between the two cack faces is δ 8K II II π The enegy elease ate elates to the stess intensity facto as K II G K II Mode III Diectly ahead the cack tip z π Behind the cack tip, the teaing displacement between the two cack faces is K III δ 4 K, µ III III µ π 1 ν ) The enegy elease ate elates to the stess intensity facto as K III G µ /1/14 13
S 47 Factue Mechanics http://imechanicaog/node/7448 Zhigang Suo Histoical Notes Fo a vey inteesting histoical account involving Aiy, Stokes and Maxwell, see VV Meleshko, Selected topics in the histoy of the two-dimensional bihamonic poblem, Applied Mechanics Review 56, 33-85 003) The squae-oot singulaity nea the tip of a cack was known befoe Iwin enteed the field Hee ae two papes that clealy displayed the squae-oot singulaity HM Westegaad, Beaing pessues and cacks Jounal of Applied Mechanics 6, A49-A53 1939) Westegaad was a pofesso at Havad IN Sneddon, The distibution of stess in the neighbohood of a cack in an elastic solid Poceedings of the Royal Society of London A 187, 9-60 1946) ML Williams, On the stess distibution at the base of a stationay cack Jounal of Applied Mechanics 4, 109-115 1957) GR Iwin, Analysis of stesses and stains nea the end of a cack tavesing a plate, Jounal of Applied Mechanics 4, 361-364 1957) /1/14 14