ES 247 Fracture Mechanics Zhigang Suo

Transcription

1 ES 47 Frature Mehanis Zhigang Suo The Griffith Paper Readings. A.A. Griffith, The phenomena of rupture and flow in solids. Philosophial Transations of the Royal Soiety of London, Series A, Volume 1 (191) This paper initiated the theory of frature, and has foreshadowed muh of the subsequent development. Start to read the paper now, and return to it for illumination later in your areer. Qualitative ontent of the Griffith theory. After the atomisti nature of matter was onfirmed by many experimental observations, about a entury ago, it beame useful to relate marosopi phenomena to atomi proesses. In 191 the British engineer A.A. Griffith published a paper on one suh marosopi phenomenon: frature of a glass. The main puzzle had been that the glass usually breaks under a stress several orders of magnitude below the strength of atomi ohesion. Griffith took up the notion that a piee of glass is never perfet: small raks pre-exist in a body of the glass. The tip of suh a rak greatly onentrates stress. The intense stress at the tip of the rak breaks atomi bonds one by one, like opening a zipper. The rak advanes, leading to the frature of the body. This piture is easy to understand, but diffiult to quantify. The essential diffiulties are 1. The atomi behavior at the tip of the rak is nonlinear, and differs in details for different materials.. The shape of the tip is unknown, but the shape of the tip greatly affets the magnitude of stress, at least within the theory of linear elastiity. It remains unlear what permitted Inglis (1913) to haraterize the tip of a flaw by the radius of urvature, and then use a linear elasti solution. A diret approah to resolve these diffiulties would be to use a nonlinear material model, and to use more realisti geometry. For example, one may invoke atomisti simulations. Atomisti simulations of frature have been pursued sine the time of Inglis, but have not been widely used in engineering pratie to this day. Griffith took a less diret approah. Consider a pre-existing rak in a body subjet to an external fore. Regard the body and the external fore together as a thermodynami system, haraterized by two thermodynami variables: the area of the rak and the displaement of the loading grips. To fous on essential ideas, suppose that, after a ertain amount of displaement, the loading grips are held fixed, but the rak is allowed to advane. Beause the loading grips are held fixed, the external fore does no work. The energy of the system is the sum of the elasti energy in the body, and the surfae energy in the faes of the rak. The energy of the system is a funtion of a single thermodynami variable: the area of the rak. When the rak advanes, the stress in the sample is partially relieved, so that the elasti energy is redued. At the same time, the advaning rak reates more surfae area, so that the surfae energy inreases. Thermodynamis ditates that the proess should go in the diretion that redues the total free energy. If the derease in elasti energy prevails, the rak grows. If the inrease in surfae energy prevails, the rak heals. The nonlinear zone, loalized around the tip of the rak, remains invariant as the rak advanes. Consequently, the presene of the nonlinearity does not affet the aounting of energy. The Griffith approah irumvents the nonlinear rak-tip behavior by invoking one quantity: the surfae energy. The siene of frature was born. To sum up, the qualitative ontent of the Griffith theory is In a body of a glass raks pre-exist. The tip of suh a rak onentrates stress. The intense stress breaks atomi bonds one by one, like opening a zipper. As the rak advanes, fresh surfaes are reated. The surfae energy inreases, but the elasti energy dereases. The rak advanes if the advane redues the sum of the surfae energy and elasti energy. 1/8/10 1

2 ES 47 Frature Mehanis Zhigang Suo I ll do two things in this leture. In formulating his theory, Griffith used this quantity surfae energy. For some of us, the last time we enountered the surfae energy was in kindergarten, when we blew soap bubbles. So I ll first outline aspets of the surfae energy. I ll then desribe the ontent of the Griffith paper. Surfae energy. An atom at the surfae of a body has a bonding environment different from that of an atom inside the body, so that the free energy per atom at the surfae is higher than that in the body. The exess defines the surfae energy. The above desription appeals to intuition, but is not operational. An atom does not have its private energy. An operational definition of the surfae energy goes like this. Imagine a U. body whose size is muh larger than an individual atom. Denote the energy of this body by 0 It inludes all the energy stored in eletron louds, or even the nulear energy stored in the atomi nulei. Next imagine that this body is split in two halves. The at of splitting may ause some damage of the material, e.g., introduing some disloations or some miroraks. Let s say we are very areful in the at of splitting, and anneal the samples afterwards, so that suh damage is eliminated. All atoms on the two surfaes relax to their equilibrium onfigurations. The energy in the two halves will be greater than U 0. Let s all the energy in the two halves U. The differene, U U 0, is defined as the surfae energy. Of ourse, this differene does not belong to a single layer of atoms on the surfae. It is a olletive effet. For this definition to be useful, we take advantage of a physial fat: atoms a few layer beneath the surfae reover the onfiguration of atoms in the bulk. Let be the surfae energy per unit area. The quantity is alled the surfae energy density, or surfae energy for brevity. The surfae energy density is independent of the size of the body, unless the body approahes the atomi dimension. Water: 0.08 J/m. Most other liquids have smaller values. Metals and eramis: ~1 J/m Polymers: J/m A.W. Adamson, Physial Chemistry of Surfaes. Wiley, New York (1990). Chapter 1 desribes many phenomena onerning the effet of surfae energy for liquids. P.G. de Gennes, F. Brohard-Wyart, and D. Quere, Capillarity and Wetting Phenomena, Springer (004). Measure the surfae energy of a liquid. A U-shaped rigid frame is fixed in spae. A liquid membrane lies in the area onfined by the frame and a slider. The slider an move without frition. The thikness of the membrane is muh larger than the dimension of the individual moleule of the liquid. As the slider moves to the right, the membrane beomes thinner and has a larger area. Moleules are drawn from the interior of the membrane to the surfaes. A fore F is applied on the slider. What is the fore needed to maintain equilibrium? When the slider moves by a distane x, the area of the membrane inreases by bx, and the surfae energy inreases by bx. You an always represent a onstant fore by a weight. When the slider moves by a distane x, the weight drops by the same distane, so that the gravitational energy dereases by F x. The net energy hange is the ombination of the two effets: U bx Fx. The two terms ompete. The surfae energy dereases when the slider moves to the left. On the other hand, the gravitational energy dereases when the slider moves to the right. 1/8/10

3 ES 47 Frature Mehanis Zhigang Suo The system reahes a state of equilibrium when the variation in the energy assoiated with the variation in the position of the slider vanishes, namely, when F b. By measuring the weight F that equilibrates the membrane, one determines the surfae energy density. The surfae energy density behaves like a fore per unit length. Perhaps for this reason the quantity is also known as the surfae tension. x b Liquid Membrane F A liquid droplet on a eiling: a ompetition between surfae energy and gravity. Let a be the radius of a water droplet on a eiling. The gravity tends to pull the droplet down, but the surfae tension tends to keep it up. In this ase, the surfae tension of the surfae moleules ats like a rubber bag, holding the water inside. Determine the maximum droplet size. Fore balane approah. 3 a ga. The ompetition between the gravity and the surfae energy defines a length sale: a g Estimation of the maximum size of a water droplet on the eiling. Take = 0.1 J/m, = 1000 kg/m 3, and g = 10 m/s. We find that a is of magnitude of m. Measuring the Surfae Energy for Solids Experimental determination of surfae energy for solids is hallenging. To measure the surfae energy, one has to observe a phenomenon in whih new surfae area is reated. For a liquid, the new surfae area is reated by flow, drawing moleules from the interior of the liquid to the surfae. For a solid, one has to do essentially the same thing. Heat the solid up so that it reeps (i.e., flows slowly). In the following words, Griffith desribed how he measured the surfae energy of glass. Between 730 C and 900 C the method desribed below was found to be pratiable. Fibers of glass about inhes long and from 0.00-inh to 0.01-inh diameter, with enlarged spherial ends, were prepared. These were supported horizontally in stout wire hooks and suitable weights were hung on their mid-points. The enlarged ends prevented any sagging exept that due to extension of the fibers. The whole was plaed in an eletri resistane furnae maintained at the desired temperature. Under these onditions visous strething of the fiber ourred until the suspended weight was just balaned by the vertial omponents of the tension in the fiber. The latter was entirely due, in the steady state, to the surfae tension of the glass, whose value ould therefore 1/8/10 3

4 ES 47 Frature Mehanis Zhigang Suo be alulated from the observed sag of the fiber. In the experiments the angle of sag was observed through a window in the furnae by means of a telesope with a rotating ross wire. If w is the suspended weight, d the diameter of the fiber, T the surfae tension, and the angle at the point of suspension between the two halves of the fiber, then, evidently, dtsin w. The Griffith theory. A large sheet of a glass is under stress. The sheet has unit thikness. For the time being, assume that the loading grips are rigidly held, so that the displaement is fixed, and the loading devie does no additional work after a fixed displaement is applied. The state of referene is a stressed sheet with no rak. The state that interests us is the sheet with a rak of length a. We now alulate the differene in energy between the two states. The surfae energy inreases by 4 a. The elasti energy redues. To determine the amount of the redution, one has to solve the boundary-value problem. This tough elastiity problem is for professional elastitians. Look how ompliated the stress field must be near the rak. Griffith used the elastiity solution of Inglis, beause a rak is just a speial ase of an ellipse when b / a 0. This part of the Griffith paper is diffiult to read, and is not very interesting. In the end he made small errors. An alternative approah is to invoke linearity and dimensional onsiderations. For a linearly elasti problem, the stress field is linearly proportional to the applied stress. The elasti energy per unit volume is proportional to / E. The elasti energy in an infinite sheet is infinite. However, we are interested in the differene in elasti energy between the raked sheet and the unraked sheet. Note that the rak length a is the only length sale in the boundary-value problem. Consequently, the differene in elasti energy between the two sheets takes the form a, E where is a numerial value. Thus, from very basi onsiderations, we get nearly everything exept for a pure number. This number must be determined by solving the elastiity boundary value problem. The solution turns out to be. You an find the solution of the full problem in Timoshenko and Goodier. Relative to the unraked sheet, in the raked sheet the ombined surfae energy and elasti energy is Ua 4a a. E The rak length, a, is the thermodynami variable. The surfae energy density and the applied stress are taken to be onstant for the time being. As expeted, when the rak length inreases, the surfae energy inreases, but the elasti energy dereases. Critial rak size. Plot the free energy as a funtion of the rak length. The free energy first goes up, reahes a peak, and then goes down. Beause there is no minimum free energy, the rak annot reah equilibrium. The free energy reahes the peak at the rak length E a*. Let us examine the physial signifiane of this partiular rak length. Let a be the length of the rak pre-existing in the sheet. Distinguish two situations. Crak healing. If a a*, the surfae energy prevails over the elasti energy. To redue the free energy, the rak length must derease. The rak does so by healing, i.e., 1/8/10 4

5 ES 47 Frature Mehanis Zhigang Suo forming atomi bonds one by one, like losing a zipper. In reality rak healing is not often observed. This is not beause the thermodynamis is wrong, but beause surfaes are not flat to the atomi dimension, so that atoms annot meet aross the gap and form bonds. Several examples show that rak healing happens. Adhesives. Soft material an heal readily by flows. Wafer bonding. If the surfaes are indeed made flat, they will join. Sintering. At elevated temperatures, atoms an diffuse, so that the two surfaes hange shape and an join. Crak growth. If a a*, the elasti energy prevails over the surfae energy. To redue the free energy, the rak must grows. The rak does so by breaking atomi bonds one by one, like opening a zipper. This is the situation studied in this ourse. U a* a Crak Length, a mm Measured Strength, MPa a MPa m sample 1 sample sample 3 sample as (Data from the Griffith experiment) The Griffith experiments. The main predition of the Griffith theory an be written E a. He performed several experiments to asertain various parts of the equation. Experiment 1. Confirm that a on sta n, t independent of the size of the rak. Start with several glass sheets (large spherial bulbs atually). Introdue a rak in eah sheet. Measure the strength of eah sheet. Two important points: (1) The rak introdued is in the mm to m range, muh longer than any natural flaws in the sheets, so that the natural flaws are negligible. In this way Griffith irumvented the unertainties assoiated with the natural flaws. 1/8/10 5

6 ES 47 Frature Mehanis Zhigang Suo () The introdued raks in different sheets have different lengths, and the measured strengths are also different. His data onfirmed that a on sta n. t Experiment. Confirm that the onstant is indeed E /. Young s modulus for the glass used by Griffith was E = 6 GPa. The surfae energy inferred from the measured breaking strength is 1.75J/m. Griffith needed an independent measurement of the surfae energy. He did the reeping fiber experiment. The value he obtained was 0.54J/m. The agreement was fair. Experiment 3. Measure strengths of glass fibers. Experimental strength. For a fixed pre-existing rak size a, there is a ritial stress: E. a This is the stress needed to frature the sample. This relation shows that the frature strength depends on the rak size. Beause different samples have different rak sizes, the frature strength is not a material property. The measured strength has large satter. Take representative values = 1 J/m, E = N/m, a = 10-6 m, the strength is 50 MPa. This orresponds to the experimental strength. Theoretial strength. Assume that the solid is flawless. There is no stress onentration. The solid breaks when the applied stress is so high to break atomi bonds. If we put a m (atomi dimension) into the above formula, we obtain an estimate of the theoretial strength ~10 GPa. Alternatively, a ommonly quoted rough estimate of the theoretial strength is E S th. 10 Today the theoretial strength an be alulated by atomisti simulations. Griffith measured the strength of glass fibers of diameters between 107 m and 3.3 m. The data sattered, but over trend was that the strength inreased as the fiber diameter dereased. He reported a strength of 171 MPa for a fiber of diameter 107 m, and a strength of 3.4 GPa for a fiber of diameter 3.3 m. The theoretial strength of the glass orresponds to a fiber of the smallest possible (moleular) diameter. He plotted his data, and extrapolated to the moleular diameter, and estimated the theoretial strength to be 1 GPa. 1/8/10 6