ES 247 Fracture Mechanics Zhigang Suo. Applications of Fracture Mechanics

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1 Appliation of Frature Mehani Many appliation of frature mehani are baed on the equation σ a Γ = β. E Young modulu i uually known. Of the other four quantitie, if three are known, the equation predit the fourth. If you jut read thi equation, frature mehani ound like a illy tautology. It i not really o illy if you think through eah appliation. Some quantitie are eay to meaure. Other quantitie are eay to ompute. One an make real predition. After finding out that I am a frature expert, people often ak me if a raked wall i dangerou, or how muh time a rak will grow aro a pavement. After ome embarrament, I realize I often annot anwer their quetion, beaue I haven t done alulation and obtained experimental data. Appliation 1. Meaure the frature energy. Know β, σ, a. Determine Γ. The experiment follow that of Griffith. Start with a body of a material. Cut a rak of a known ize a uing a aw. Load the ample with an inreaing tre, and reord the tre at frature, σ. Separately find the elatiity olution for the energy releae rate, βσ a/ E. Convert the ritial tre to the frature energy, Γ = βσ a/ E The meaured frature energy i ued to (a) rank material, (b) tudy the effet of variou parameter (e.g., loading rate, temperature, heat treatment) on frature reitane, () deign a truture to avoid frature. Appliation. Predit ritial load. Know β, a, Γ. Determine σ. The body i given. The frature energy of the material ha been meaured. The rak ize a ha been meaured. Find the elatiity olution for the energy releae rate βσ a / E. Thi appliation require one to determine the rak ize. A large rak ize i determined by viual inpetion. A mall rak an be determined by the x-ray or aouti wave (Nondetrutive Evaluation, or NDE). If the meaurement tehnique annot find any rak, imply put the mallet rak ize an be deteted by the tehnique (i.e., the reolution) into the equation, and predit a lower bound of the ritial load. A rak in a truture may inreae lowly over time. Inpet the truture periodially to monitor the rak ize. Retire or repair the truture before the rak i too large. Appliation. Etimate flaw ize from experimentally meaured breaking tre. Know β, Γ, σ. Determine a. /18/14 1

2 Meaure the frature load σ. Independently meaure the frature energy of the material. Approximate the energy releae rate by that of a Griffith rak, πσ a/ E. The flaw ize a i etimated by a = ΓE /πσ. Appliation 4. The proof tet Load a ample to a given load σ, and the ample doe not frature. Independently meaure the frature energy of the material. Approximate the energy releae rate by that of a Griffith rak, πσ a/ E. The flaw ize a i bonded by a < ΓE /πσ. Heart valve. Eah proof-teted individually. Make people feel good at heart Cerami tile on reentry vehile. Eah tile i proof-teted individually. Appliation 5. Deign a truture to avert frature. If a material i given, and the load level i preribed, one an deign a truture to avoid frature. One alo need to know the poible flaw ize. P B Δ H Double-antilever beam. To determine the energy releae rate for a given raked body, one ha to olve a boundary-value problem. Thi require ome work. People ue handbook or finite element pakage. The following example i one of a few that an be olved in the laroom. The double-antilever beam are ommonly ued in frature tet. The elati field an be determined by uing the beam theory. Treat eah beam a a antilever. One end i lamped, and the other end i pulled by a fore P. The opening diplaement i Δ. Aording to the beam theory, the defletion of eah beam i given by Δ P =. EI P /18/14

3 The form of thi expreion an be undertood if you remember bai of the beam theory. The diplaement hould be linear in the fore, and hould be inverely proportional to the bending rigidity EI, where the eond moment of the ro etion i I = BH /1. A dimenional onideration how the ubi dependene on the length. The elati energy tored in the two arm i U = PΔ /. The rak area i A = B. Expre the energy a a funtion of the diplaement Δ and the rak area A: EIB Δ U ( Δ, A) =. 4A The energy releae rate i given by partial differentiation with repet to the rak area: U ( Δ, A) / A 9EIΔ. 4 4B Thi expree the energy releae rate in term of the opening diplaement Δ. Alternatively, one an expre the energy releae rate in term of the load: 1 P. EH B J.W. Obreimoff, The plitting trength of mia, Proeeding of Royal Soiety of London A17, (190). S.M. Spearing and A.G. Evan, The role of fiber bridging in the delamination reitane of fiber-reinfored ompoite. Ata Metallurgia Materilia 40, (199). Q.Y. Tong and U. Goele, Semiondutor wafer bonding: Reent development. Material Chemitry and Phyi 7, (1994). J. Bio, B. Roman, L. Moulin, A. Boudaoud, Elatoapillary oaleene in wet hair. Nature 4, 690 (004). N.J. Glamaker, A. Jagota, C.-Y. Hui, J. Kim, Deign of biomimeti fibrillar interfae: 1. Making ontat. Journal of Royal Soiety Interfae 1, - (004). Stability of a growing rak. When the double-antilever beam i plit by inerting a wedge of height Δ, the energy releae rate i 9EIΔ. 4 4B To keep the rak growing, the energy releae rate need to be at the level of the frature energy, namely, Γ. Beaue G dereae with the length of the rak, the rak will arret. When the double-antilever beam i plit by hanging a weight P, the energy releae rate i /18/14

4 1 P EH B Beaue G inreae with the length of the rak, the rak will not arret one it tart to grow. Although the two expreion of the energy releae rate are equivalent, how we plit the beam an make a huge differene to how the rak will grow. We will return for a fuller diuion of the tability of a growing rak later in the la. A more aurate expreion for energy releae rate. The end of eah arm i not really lamped, but an have ome rotation. The beam theory itelf i an approximation of the elatiity theory, and neglet the effet of hear. Both error are mall when the beam are long, namely when / H i large. The above reult i an exat aymptote when / H. For a finite value of / H, numerial analyi ha given a better approximation: 1 P H EH B For example, ee G. Bao, S. Ho, Z. Suo, and B. Fan, The role of material orthotropy in frature peimen for ompoite. International Journal of Solid and Struture 9, ( Channel rak in a thin film bonded to a ubtrate. Compare two onfiguration. Firt onider a rak of length a, in a freetanding heet of thikne h, i ubjet to a tenile tre σ remote from the rak. Aume that the diplaement at the load point i fixed, o that the load doe no work when the rak extend. When the rak i introdued, the tre near the rak fae i partially relieved. The volume in whih the tre relaxe ale a a h, o that relative to the unraked, treed heet, the elati energy in the raked heet hange by Δ U ~ a hσ / E. Conequently, the energy releae rate for a rak in a freetanding heet i G ~ aσ / E. The energy releae rate of a rak in a freetanding heet inreae with the length of the rak. Next onider a thin elati film bonded to an elati ubtrate. When the length of the rak a i muh larger than the thikne of the film h, the tre field in the wake of the rak beome invariant a the rak extend. The volume in whih the tre relaxe ale a ah, o that the introdution of the rak hange the elati energy by Δ U ~ ah σ / E. Conequently, the energy releae rate for a hannel rak in a film i G ~ σ h/ E. Compare the above two ituation. /18/14 4

5 Take σ = 10 9 Pa, = E Pa, and Γ = 10J/m. Equating the energy releae rate G to the frature energy Γ, we find a ritial film thikne h = 0.5µ m. Channel rak an propagate in film thiker than h, but not in film thinner than h. A very thin film an utain a very large tre without raking. Calulate the energy releae rate of the hannel rak. The bottom of the hannel i bloked by the interfae. The front of the hannel i urved, and extend in the film. We want to find the energy releae rate at the hannel front. Thi i a three dimenional problem. When the hannel length exeed everal time the film thikne, the hannel approahe a teady tate. That i, the elati energy redution U aoiated with the hannel extending per unit length approahe a ontant, independent of the hannel length. Thi energy redution an be alulated a follow. The extenion of the hannel by a unit ditane i equivalent to removing a lie of material of a unit thikne far ahead of the hannel, and then appending a lie of material far behind the hannel. Let σ ( x) be the tre prior to the introdution of the rak (the firt lie), and δ ( x) be the opening diplaement after the introdution of the rak (the eond lie). The two quantitie, σ ( x) and δ ( x), an be determined by olving the two plane train elatiity boundary value problem. The elati energy differene between the two lie i 1 U = σ ( x) δ ( x) dx The integral extend over the length of the rak. The energy releae rate at the hannel front i the energy redution aoiated with the rak extending per unit area. Thu, U / h, giving 1 σ ( x) δ ( x) dx. h In ummary, follow the tep below to determine the teady tate hannel energy releae rate. Solve the plane train problem without rak, and obtain σ( x). Solve the plane train problem with rak, and obtain δ( x). Calulate the integral to obtain the energy releae rate G. The firt two tep are uually arried out by uing the finite element method. The third tep i arried out by numerial integration. J.W. Huthinon and Z. Suo, Mixed-mode raking in layered material, Advane in Applied Mehani 9, (199). Meauring frature energy of a thin film. For a brittle olid, toughne i independent of ample ize, o that one may expet to extrapolate toughne meaured uing bulk ample to thin film. However, thin film ued in the interonnet truture are proeed under very different ondition from /18/14 5

6 bulk material, or are unavailable in bulk form at all. Conequently, it i neeary to develop tehnique to meaure thin film toughne. Suh a tehnique i attrative if it give reliable reult, and i ompatible with the interonnet fabriation proe. Ma et al. (1998) have developed uh a tehnique, on the bai of hannel rak in a thin film bonded to a ilion ubtrate. The tet proedure onit of two tep: (a) generating pre-rak, and (b) propagating the rak uing ontrolled tre. A onvenient way of generating pre-rak i by rathing the urfae uing a harp objet. A gentle rath uually generate multiple rak on the two ide of the rath. Care mut be taken to generate rak jut in the film, but not in the ubtrate. Ue a bending fixture to load the ample, and a digital amera to reord the rak growth event. After a rak propagate ome ditane away from the rath, the rak grow at a teady veloity. By reording rak growth at lightly different bending load, one an meaure the rak veloity a a funtion of the tre. The teady veloity i very enitive to the applied tre. Conequently, the ritial tre i aurately meaured by ontrolling the veloity within a ertain range that i onvenient for the experiment. The ritial tre then determine the frature energy aording to the mehani reult. The tehnique ha been ued to meaure the frature energy of ilia ( Γ = 16.5J/m ) and ilion-nitride ( Γ = 8.7J/m ) (Ma et al., 1998). After depoition, the film ha a reidual tre, σ R. The reidual tre an be meaured by the wafer urvature method. When the truture i bent by a moment M (per unit thikne), the film tre beome 6E f M σ = σ R +, E H where H i the thikne of the ubtrate, and E and f E are the plane train modulu of the film and the ubtrate. The bending moment alo aue a tenile tre in the ubtrate, σ = 6M / H. To meaure the toughne of the film, one ha to propagate the hannel rak in the film without fraturing the ubtrate. Thi, in turn, require that the ubtrate hould have very mall flaw ize, and that the rath hould not produe flaw in the ubtrate. When properly ut, the ilion ubtrate an utain tenile tree well over 1 GPa. The tehnique i inappliable when the film ha a very large frature energy, large reidual ompreive tre, mall thikne, or low modulu. In the experiment of Ma et al. (1998), a thin metal layer i depoited on the ilion ubtrate, and the brittle film i depoited on the metal. The metal layer erve a a barrier preventing the rak from entering the ubtrate. The metal layer, upon yielding, alo inreae the energy releae rate for a given bending moment. Q. Ma, J. Xie, S. Chao, S. El-Many, R. MFadden, H. Fujimoto. Channel raking tehnique for toughne meaurement of brittle dieletri thin film on ilion ubtrate. Mater. Re. So. Symp. Pro. 516, 1-6 (1998). /18/14 6