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The Griffith Paper

Zhigang Suo's picture

I wrote these notes on the Griffith (1921) paper for a graduate course on fracture mechanics taught in 2010.  The notes were updated when I taught the course in 2014, and were discussed in a new thread titled Inglis (1913) vs. Griffith (1921).

 

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Mike Ciavarella's picture

 

Alan Arnold Griffith

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Alan Arnold Griffith (13 June 1893 – 13 Oct 1963) was an English engineer, who, among many other contributions, is best known for his work on stress and fracture in metals that is now known as metal fatigue, as well as being one of the first to develop a strong theoretical basis for the jet engine.

Contents[hide]

[edit] Early work

A. A. Griffith took a first in mechanical engineering, followed by a Master’s Degree and a Doctorate from Liverpool University. In 1915 he was accepted by the Royal Aircraft Factory as a trainee, before joining the Physics and Instrument Department the following year in what was now been renamed as the Royal Aircraft Establishment (RAE).

Some of Griffith's earlier works remain in widespread use today. In 1917 he and G. I. Taylor suggested the use of soap films as a way of studying stress problems. Using this method a soap bubble is stretched out between several strings representing the edges of the object under study, and the coloration of the film shows the patterns of stress. This method, and similar ones, were used well into the 1990s when computer power became generally available that could do the same experiment numerically.

[edit] Metal fatigue

Griffith is more famous for a theoretical study on the nature of stress and failure in metals. At the time it was generally taken that the strength of a material was E/10, where E was the Young's modulus for that material. However it was well known that those materials would often fail at 1000 times less than this predicted value. Griffith discovered that there were many microscopic cracks in every material, and hypothesized that these cracks lowered the overall strength of the material. This was because any void in a solid concentrates stress, a fact already well known to machinists at the time. This concentration would allow the stress to reach E/10 at the head of the crack long before it would seem to for the material as a whole.

From this work Griffith formulated his own theory of brittle fracture, using elastic strain energy concepts. His theory described the behavior of crack propagation of an elliptical nature by considering the energy involved. The equation basically states that when a crack is able to propagate enough to fracture a material, that the gain in the surface energy is equal to the loss of strain energy, and is considered to be the primary equation to describe brittle fracture. Because the strain energy released is directly proportional to the square of the crack length, it is only when the crack is relatively short that its energy requirement for propagation exceeds the strain energy available to it. Beyond the critical Griffith crack length, the crack becomes dangerous.

The work, published in 1920 ("The phenomenon of rupture and flow in solids"[1]), resulted in sweeping changes in many industries. Suddenly the "hardening" of materials due to processes such as cold rolling were no longer mysterious. Aircraft designers immediately understood why their designs had failed even though they were built much stronger than was thought necessary at the time, and soon turned to polishing their metals in order to remove cracks. The result was a series of particularly beautiful designs in the 1930s, such as the Boeing 247. This work was later generalized by G. R. Irwin, in the 1950s, applying it to almost all materials, not just rigid ones.

[edit] Turbine engines

In 1926 he published a seminal paper, An Aerodynamic Theory of Turbine Design. He demonstrated that the woeful performance of existing turbines was due to a flaw in their design which meant the blades were "flying stalled", and proposed a modern airfoil shape for the blades that would dramatically improve their performance. The paper went on to describe an engine using an axial compressor and two-stage turbine, the first stage driving the compressor, the second a power-take-off shaft that would be used to power a propeller. This early design was a forerunner of the turboprop engine. As a result of the paper, the Aeronautical Research Committee supported a small-scale experiment with a single-stage axial compressor and single-stage axial turbine. Work was completed in 1928 with a working testbed design, and from this a a series of designs was built to test various concepts.

At about this time Frank Whittle wrote his thesis on turbine engines, using a centrifugal compressor and single-stage turbine, the leftover power in the exhaust being used to power the aircraft directly. Whittle sent his paper to the Air Ministry in 1930, who passed it on to Griffith for comment. After pointing out an error in Whittle's calculations, he stated that the large frontal size of the compressor would make it impractical for aircraft use, and that the exhaust itself would provide little thrust. The Air Ministry replied to Whittle saying they were not interested in the design. Whittle was crestfallen, but was convinced by friends in the RAF to pursue the idea anyway. Luckily for all involved, Whittle patented his design in 1930 and was able to start Power Jets in 1935 to develop it.

Griffith went on to become the principal scientific officer in charge of the new Air Ministry Laboratory in South Kensington. There he invented the contraflow gas turbine, which used two sets of compressor disks rotating in opposite directions, one "inside" the other. This is as opposed to the more normal design in which the compressors blow the air against a stator, essentially a non-moving compressor disk. The effect on compression efficiency was noticeable, but so was the effect on complexity of the engine. In 1931 he returned to the RAE to take charge of engine research, but it was not until 1938, when he became head of the Engine Department, that work on developing an axial-flow engine actually started. Joined by Hayne Constant, they started work on Griffith's original non-contraflow design, working with steam turbine manufacturer Metropolitan-Vickers (Metrovick).

After a short period Whittle's work at Power Jets started to make major progress and Griffith was forced to re-evaluate his stance on using the jet directly for propulsion. A quick redesign in early 1940 resulted in the Metrovick F.2, which ran for the first time later that year. The F.2 was ready for flight tests in 1943 with a thrust of 2,150 lbf, and flew as replacement engines on a Gloster Meteor, the F.2/40 in November. The smaller engine resulted in a design that looked considerably more like the Me 262, and had improved performance. Nevertheless the engine was considered too complex, and not put into production.

Griffith's original rejection of Whittle's concepts has long been commented on. It certainly set back development of the jet engine in England. His motivations have long been the topic of curiosity, with many people suggesting that his endless quest for perfectionism was the main reason he didn't like Whittle's "ugly" little engine, or perhaps the belief that "his" design was innately superior.

Griffith joined Rolls-Royce in 1939, working there until 1960. He designed the AJ.65 axial turbojet which led to the development of the Avon engine, the company's first production axial turbojet. He then turned to the turbofan (known as "bypass" in England) design and was instrumental in introducing the Rolls-Royce Conway. Griffith carried out pioneering research into vertical take-off and landing (VTOL) technology, culminating in the development of the Rolls-Royce Thrust Measuring Rig.

[edit] References

  1. ^ "The phenomenon of rupture and flow in solids", Philosophical Transactions of the Royal Society, Vol. A221 pp.163-98

[edit] External links

"Suddenly the "hardening" of materials due to processes such as cold rolling were no longer mysterious."

Griffith did not explain strain hardening as suggested here.

Mike Ciavarella's picture

famous Liverpool alumnus, A. A. Griffith FRS.  Before joining the Royal Aircraft Factory in 1915, Alan Arnold Griffith studied Mechanical Engineering at the University of Liverpool, and went on to be awarded an M.Eng and then a D.Eng, both from Liverpool.  Griffith developed fundamental theories in fracture mechanics and metal fatigue and, with his 1926 paper ‘An Aerodynamic Theory of Turbine Design’, laid the foundation for the development of the jet engine.  Griffith joined Rolls Royce in 1939 and made major contributions to the design of turbo-jet and turbo-fan engines until he retired in 1960.  Emeritus Professor Norman Jones FREng occupied the Griffith Chair from 1979 until his retirement in 2005 and the legacies from his impact research continue to shape the research within the Structural Materials and Mechanics (SM&M) research group, now led by Professor Wesley Cantwell. 

Konstantin Volokh's picture

Dear Zhigang,

 

What an interesting topic!

 

Let me start with two objections to the concept of the surface energy for solids.

 

1. Intuitively, in fluids molecules are more bonded and less mobile in the boundary layer as compared to the bulk. The later gives strength to the liquid surfaces. In solids molecules are less bonded and more mobile in the boundary layer as compared to the bulk. There is no additional strength due to the surface layer.

 

2. The surface effects are sound in fluids when one of the characteristic lengths is small: bubbles, capillary effects etc. The effect of the surface strength is not a universal one.

 -Kosta

Zhigang Suo's picture

Dear Kosta:  Good to hear from you!  But I don't understand your objections to the concept of surface energy for solids.  I have read you comments several times, and still do not know what you are trying to say. 

If you go by the definition of the surface energy given in the notes posted in this entry, which aspects do you disagree with?

Konstantin Volokh's picture

Dear Zhigang, sorry for being unclear. I've read your notes.

The surface energy was introduced for solids by analogy with liquids. I do not find the analogy reasonable because in liquids the surface layer is stronger (more bonds) than the bulk while in solids the surface layer is weaker (less bonds) than the bulk. That was my first point.

Moreover, the very concept of the strong surface, when the energy of the surface layer is sound, is restrictive even for liquids. The concept works for thin membranes/films, small capillaries etc. That is the surface strength and energy are appealing concepts for the liquid bodies with a small characteristic length. If you consider water in a big bath then the surface tension and energy are negligible. That was my second point.

Please, ask me specific questions if my points are still vague.

 -Kosta

Zhigang Suo's picture

Dear Kosta:  Thank you for expanding on your previous comments.  I believe that surface energy is a well defined quantity for solids.  Here are specific responses to your comments.

  1. I agree with you that using atomic bonds to define surface energy, while intuitively appealing, is a paractice that ultimately leads to confusion.  As discussed in my notes, each atom does not have its private energy.  The notion of bonds is too vague to serve to define surface energy.  Rather, one defines the surface energy as the difference in the free energy between two bodies.  The two bodies have the same number of atoms, but one body has a larger area of surface than the other body.  This definition works for both liquids and solids, and is essentially how one might calculate surface energy from atomistic simulations.
  2. As already noted in Griffith's 1921 paper, a comparision of surface energy and elastic energy introduces a length scale.  If this length scale is much smaller than the length characteritic of a phenomenon, then surface energy is unimportant to the phenomenon.  One can obtain other length scales by compairing surface energy with other types of energy.  A familiar example is to compare surface energy to gravity, as described in my notes
Konstantin Volokh's picture

Zhigang, you write that "the free energy per atom at the surface is higher than in body." (paragraph Surface Energy) Is that due to the additional mobility of the atoms at the surface?

If the answer is yes (I guess) then the excessive energy has nothing in common with the essentially elastic/bond energy which provides the response of the liquid membrane which you consider in the subsequent paragraph. This surface tension is due to additional bonds rather than the additional mobility of atoms.

We will probably need a few more iterations to understand each other Innocent

Zhigang Suo's picture

Dear Kosta:  The sentence that you quoted is indeed confusing.  That is why I have indicated in the notes that we should abandon such a definition.  An operational definition of surface energy appears in the paragraph right after the sentece you have just quoted.

Surface energy is a thermodynamic quantity, rather than a kinetic quantity.  You and I agree on this point, I think.

Konstantin Volokh's picture

Dear Zhigang, the sentence is not confusing it is good. Without the sentence the paragraph after it gets vague.

In solids, the surface energy is related to the increased mobility of the surface atoms. In fluids, the surface energy/tension is related to the decreased mobility of the surface atoms. These phenomena are opposite!

It is crucial for us that both considered phenomena have no relation to cracks considered by Griffith. The energy related to cracking is the energy dissipated during the bond rupture like in cohesive zone models. It is true that new surfaces are created in fracture. The latter has nothing to do with the surface energy, however.

Mike Ciavarella's picture

 

Surface energy

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Contact angle measurements can be used to determine the surface energy of a material. Here, a drop of water on glass.

Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favorable than the bulk of a material, otherwise there would be a driving force for surfaces to be created, removing the bulk of the material (see sublimation). The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk.

For a liquid, the surface tension (force per unit length) and the surface energy density are identical. Water has a surface energy density of 0.072 J/m2 and a surface tension of 0.072 N/m.

Cutting a solid body into pieces disrupts its bonds, and therefore consumes energy. If the cutting is done reversibly (see reversible), then conservation of energy means that the energy consumed by the cutting process will be equal to the energy inherent in the two new surfaces created. The unit surface energy of a material would therefore be half of its energy of cohesion, all other things being equal; in practice, this is true only for a surface freshly prepared in vacuum. Surfaces often change their form away from the simple "cleaved bond" model just implied above. They are found to be highly dynamic regions, which readily rearrange or react, so that energy is often reduced by such processes as passivation or adsorption.

Contents[hide]

[edit] Measuring the surface energy of a liquid

As first described by Thomas Young in 1805 in the Philosophical Transactions of the Royal Society of London, it is the interaction between the forces of cohesion and the forces of adhesion which determines whether or not wetting, the spreading of a liquid over a surface, occurs. If complete wetting does not occur, then a bead of liquid will form, with a contact angle which is a function of the surface energies of the system.

Contact angle and surface energy measurements can be made using a contact angle goniometer.

Surface energy is most commonly quantified using a contact angle goniometer and a number of different methods.

Thomas Young described surface energy as the interaction between the forces of cohesion and the forces of adhesion which, in turn, dictate if wetting occurs. If wetting occurs, the drop will spread out flat. In most cases, however, the drop will bead to some extent and by measuring the contact angle formed where the drop makes contact with the solid the surface energies of the system can be measured.

[edit] Young's equation

Young established the well-regarded Young's Equation which defines the balances of forces caused by a wet drop on a dry surface. If the surface is hydrophobic then the contact angle of a drop of water will be larger. Hydrophilicity is indicated by smaller contact angles and higher surface energy. (Water has rather high surface energy by nature; it is polar and forms hydrogen bonds). The Young equation gives the following relation,

\gamma_{\mathrm{SL}}+\gamma_{\mathrm{LV}}\cos{\theta_\mathrm{c}}=\gamma_{\mathrm{SV}}\,

where γSL, γLV, and γSV are the interfacial tensions between the solid and the liquid, the liquid and the vapor, and the solid and the vapor, respectively. The equilibrium contact angle that the drop makes with the surface is denoted by θc. To derive the Young equation, normally the interfacial tensions are described as forces per unit length and from the one-dimensional force balance along the x axis Young equation is obtained.

The Young equation assumes a perfectly flat surface, and in many cases surface roughness and impurities cause a deviation in the equilibrium contact angle from the contact angle predicted by Young's equation. Even in a perfectly smooth surface a drop will assume a wide spectrum of contact angles ranging from the so called advancing contact angle, θA, to the so called receding contact angle, θR. The equilibrium contact angle (θc) can be calculated from θA and θR as was shown by Tadmor [1] as,

<br />
\theta_\mathrm{c}=\arccos\left(\frac{r_\mathrm{A}\cos{\theta_\mathrm{A}}+r_\mathrm{R}\cos{\theta_\mathrm{R}}}{r_\mathrm{A}+r_\mathrm{R}}\right)<br />

where

<br />
r_\mathrm{A}=\left(\frac{\sin^3{\theta_\mathrm{A}}}{2-3\cos{\theta_\mathrm{A}}+\cos^3{\theta_\mathrm{A}}}\right)^{1/3}<br />
~;~~<br />
r_\mathrm{R}=\left(\frac{\sin^3{\theta_\mathrm{R}}}{2-3\cos{\theta_\mathrm{R}}+\cos^3{\theta_\mathrm{R}}}\right)^{1/3}<br />

In the case of "dry wetting", one can use the Young-Dupré equation which is expressed by the work of adhesion. This method accounts for the surface pressure of the liquid vapor which can be significant. Pierre-Gilles de Gennes, a Nobel Prize Laureate in Physics, describes wet and dry wetting and how the difference between the two relate to whether or not the vapor is saturated [2].

[edit] Measuring the surface energy of a solid

The surface energy of a liquid may be measured by stretching a liquid membrane (which increases the surface area and hence the surface energy density). In that case, in order to increase the surface area of a mass of liquid by an amount, δA, a quantity of work, γδA, is needed (where γ is the surface energy density of the liquid). However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy.

The surface energy of a solid is usually measured at high temperatures. At such temperatures the solid creeps and even though the surface area changes, the volume remains approximately constant. If γ is the surface energy density of a cylindrical rod of radius r and length l at high temperature and a constant uniaxial tension P, then at equilibrium, the variation of the total Gibbs free energy vanishes and we have

<br />
\delta G = -P~\delta l + \gamma~\delta A = 0  \qquad \implies \qquad \gamma = P\cfrac{\delta l}{\delta A}<br />

where G is the Gibbs free energy and A is the surface area of the rod:

<br />
A = 2\pi r^2 + 2\pi r l \qquad \implies \qquad \delta A = 4\pi r\delta r + 2\pi l\delta r + 2\pi r\delta l<br />

Also, since the volume (V) of the rod remains constant, the variation (δV) of the volume is zero, i.e.,

<br />
V = \pi r^2 l = \text{constant} \qquad \implies \qquad \delta V = 2\pi r l \delta r + \pi r^2 \delta l = 0 \implies \delta r = -\cfrac{r}{2l}\delta l ~.<br />

Therefore, the surface energy density can be expressed as

<br />
\gamma = \cfrac{Pl}{\pi r(l-2r)} ~.<br />

The surface energy density of the solid can be computed by measuring P, r, and l at equilibrium.

[edit] See also

[edit] References

  1. ^ Rafael Tadmor (2004). "Line energy and the relation between advancing, receding and Young contact angles". Langmuir, 20, 7659-7664, (2004). ISBN 0743-7463. 
  2. ^ Pierre-Gilles de Gennes, Françoise Brochard-Wyart, David Quéré (2002). Capillary and Wetting Phenomena -- Drops, Bubbles, Pearls, Waves. Springer. ISBN 0-387-00592-7. 

[edit] External links

Retrieved from "http://en.wikipedia.org/wiki/Surface_energy"

Kejie Zhao's picture

Hi Mike,

A quick question, it is mentioned the stretch method for liquid is not applicable for solid case, so in the uniaxial tension measurement, how to exclude elastic energy from the total Gibbs free energy? Thanks

Kejie

Zhigang Suo's picture

Dear Kejie:  In the first few pages of these slides I try to show how one differentiates surface energy from volume energy.  As they say, a picture is worth a thousand words.  But without words, the picture probably means nothing.  If you are ever in the mood of reading words, Section 2 of the following paper discusses these ideas.

Z. Suo and W. Lu, "Composition modulation and nanophase separation in a binary epilayer". J. Mech. Phys. Solids.,48, 211-232 (2000).

Kejie Zhao's picture

Dear Zhigang, Thanks for the clarification. It recalls me a question raised in Dr. Aziz's course, how to differentiate surface tension, surface energy and surface stress. People dont agree with each other on these names (and sometimes are not so careful in using these names which makes more confusion), but they do have the same common conceptual features I think.

Here is my answer according to Mullins. It's a long paragragh, just to be clear. 

All three quantities, surface tension, surface stress and surface
energy, have the units of energy per area or force per length. According
to Mullins, the surface tension is defined as the reversible-work
required to create a unit area of the surface at constant temperature,
volume and chemical potential. For surface energy, let’s consider an
n-component system π, consisting of a large
block of single-phase material surrounded by a fluid phase in a
container whose volume is constant. The system is supposed to be in
contact with a thermal reservoir at constant temperature T, and with
chemical reservoirs, for each component, to maintain the specific
values. Assume an ideal mechanical device for reversely separating the
material along a specified planar surface S,
but without sensibly changing the state of the bulk phase. We further
assume the material has centrally symmetric structure that the energy
required to create an opposing surface is equivalent. Then the surface
tension of the newly created surface is defined as γ
= dw / dA
. The creation of the new
area will, in general, be attended by a flow from the chemical reservoir
into system π, since the component will in
general preferentially concentrate on the fresh surface. So the change
of the Helmholtz free energy consists of two parts: the isothermal work,
γdA, and the contribution
brought by each molecule flowed into the system:

 dF_{\pi}=\gamma dA + \sum_{i=1}^N<br />
\mu_i \Gamma_i dA

or

 f_{\pi}=\frac{<br />
dF_{\pi}}{dA}=\gamma  + \sum_{i=1}^N \mu_i \Gamma_i

Here fπ is defined to
be the surface energy per unit area or the specific surface free energy,
where defines the surface excess, dNi
/ dA
. From this equation we can see that the
surface energy equals to the surface tension only if the system has one
component.

In general, there are special forces acting within the surface of
a material, which we shall refer to surface stresses. The relationship
of surface energy and surface stress are connected that solid surfaces
can change their energy in two ways: (1): By increasing or decreasing
physical area of the surface with the arrangement identical with those
in the bulk. (2): By changing the arrangement of atoms at surface such
as relaxation or reconstruction. To create a surface we have to do work
that involves breaking bonds and removing the neighboring atoms. Under
equilibrium conditions with constant T and P, the reversible work
required to increase the surface area in one-component system is given
by γdA. In the case of
solids, elastic deformation of the materials can lead to the new
surface. The elastic deformation of a solid surface can be expressed in
terms of surface strain tensor εij.
Consider a reversible process that causes a small variation in the area
through an infinitesimal elastic strain deij.
We can define a surface stress tensor fij
which relates to the work associated with the variation in γdA :

dA) = Afijdeij

Therefore

 f_{ij}=\gamma \delta_{ij}<br />
+\frac{dr} {de_{ij}}

In contrast to the surface tension γ
which is a scalar, the surface stress is a second rank tensor. For
high-symmetry surface which is isotropic, the surface stress can be
taken as a scalar quantity:

 f=\gamma  +\frac{dr} {de}

It can be seen the difference between the surface stress and stress
tension is equal to the change of surface tension per unit change in
elastic strain. Note that only when dγ / de
= 0
the surface stress equals to the specific surface energy, as
in the case of liquids. For most solid, the surface stress can be
either negative or positive, whereas the surface tension is always
positive for a clean surface. And the surface tension does not
necessarily lead to any surface stress whatsoever. The quantity can also
vary as a function of the unit vector along the surface normal
direction.

The thermal creep method to measure the surface stress/tension is
standard and with simple principle, but it has the drawback of
neglecting the effects due to the crystalline anisotropy: the surface
energy per unit area is assumed to be constant.
Let’s consider a thin crystalline wire with bamboo structure of radius r and length L,
the number of grain boundary is n .
The wire is suspended with the external weight m
. If the external stress σm
> σsur + σgrain

, the structure will extend, otherwise it contracts under the effect of
the surface tension. Under equilibrium, we have:  mgdL =
γsvd(2πrL + 2πr2)
+ γgraind(nπr2)

where γsv is
the specific free surface energy and γgrain
represents the grain boundary interface energy. The right hand is the
variation of the potential energy in the gravity field, and the left
hand is the variation of the surface energy. Under creep the volume of
the structure does not change, Therefore:

 mg=\gamma_{sv} \pi (r- \frac<br />
{2r^2} {L})  - \gamma_{grain} \frac {n \pi r^2}{L}

If we assume the interface energy of grain boundaries is negligible
compared with the free surface energy, and L
> > r
, then simply we have:

 \gamma_{sv} = \frac {mg} { \pi r}

The obtained result represents the surface stress. According to the
argument in part I, it is clear to see that the key point that
determines whether the surface stress reduces to the surface tension, is
whether the state of the surface is sensibly changed by the stretching
process. So in the thermal creep approach, the measured quantity is the
surface stress.

Kejie

Mike Ciavarella's picture

Zhigang, 

Some papers now speak of capillary adhesion... Can you help me please?

I think capillary attraction is beyond me. My school physics lessons happily found the rise of water in a capillary
tube by taking the radius of curvature of the water column to be the
radius of the tube; and I've never quite recovered from the discovery
of the Kelvin radius (which really had been discovered before my school
lessons!). This is complicated by the belief that the reciprocal of
Kelvin radius rK should actually equal the sum of the two principal
curvatures..and one of these is the curvature seen in the elevation
view (which these papers seem to deal with), and the other that in the
plan view...where it ought to be related to 1/(asperity radius)....and
so for small scale contacts will be enormous..and greater than 1/rK.
And there I stop! [And one of Maugis papers says it all depends on the
volume of water present when unloading starts, so the geometry is
simply a matter of conservation of volume (and no Kelvin radius!}

Mike

Could you please give some tutorial refernces to the ideas discussed here, say the Kelvin radius etc. -Rajesh

Mike Ciavarella's picture

 

Here in fact new surfaces are created, but also fluid is inserted.  How would you treat this case?

I think HD Bui solved first this problem with an unfortunate student who liked mountaneering, but was killed in an incident.  He had noted that rocks cracked because of liquid transforming into ice giving huge pressure.  I am not sure it was never published properly as the student of Ecole Polytechnique died.  I will ask my friends there more infos.

 

References
D Garagash, E Detournat - Comptes Rendus de l'Academie des Sciences …, 1998 - Elsevier

Similarity solution of a semi-infinite fluid-driven fracture in a linear elastic solid

 

RomeS.A. Kristianovic, Zheltov, Formation of vertical fractures by means of highly viscous fluids, Proc. 4th World Petroleum Congress Vol. II (1955), pp. 579–586.

 

Geertsma and De Klerk F., A rapid method of predicting width and extent of hydraulic induced fractures, JPT 246 (1969), pp. 1571–1581.

 

H.D. Bui and R. Parnes, A reexamination of the pressure at the tip of a fluid-filled crack, Int. J. Engng. Sci. 20 (1982) (11), pp. 1215–1220. Abstract | PDF (372 K)
| View Record in Scopus | Cited By in Scopus (2)

 

W.L. Medlin and L. Masse, Laboratory experiments in fracture propagation, Soc. Pet. Eng. J. (1984), pp. 256–268. View Record in Scopus | Cited By in Scopus (10)

 

J.R. Lister, Bouyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors, J. Fluid Mech. 210 (1990), pp. 263–280. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (59)

 

A.M. Rubin, Tensile fracture of rock at high confining pressure: implications for dike propagation, J. Geophy. Res. 98 (1993), pp. 15919–15935. Full Text via CrossRef

 

J.
Desroches, E. Detournay, B. Lenoach, P. Papanastasiou, J.R.A. Pearson,
M. Thiercelin and A.H.D. Cheng, The crack tip region in hydraulic
fracturing, Phil. Trans. Roy. Soc. London Ser. A A447 (1994), pp. 39–48. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (37)

 

S.H.
Advani, T.S. Lee, R.H. Dean, C.K. Pak and J.M. Avasthi, Consequences of
fluid lag in three-dimensional hydraulic fractures, Int. J. Numer. Anal. Methods Geomech. 21 (1997), pp. 229–240. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (9)

 

Mike Ciavarella's picture

 

We will keep you posted!  Anybody interested of the presentations please write me at mciava@poliba.it

Mike Ciavarella's picture

 

The book is not that simple, but coincise to a very surprising extent, from an Academician of Science in France.

 

Fracture Mechanics: Inverse Problems and Solutions (Hardcover)

by H.D. Bui (Author)

  • Hardcover: 375 pages
  • Publisher: Springer; 1 edition (Sep 14 2006)
  • Language: English
  • ISBN-10: 140204836X
  • ISBN-13: 978-1402048364
  • Dear Zhigang,

    this message is just a short and very minor comment on the following phrase of your very interesting blog:

    "" The shape of the tip is unknown, but the shape of the tip greatly affects the magnitude of stress, at least within the theory of linear elasticity""

    I apologize in advance if my comment is not appropriate.

    Of course in general we do not know, for example in a glass, which is the shape of any individual crack.

    However, attempts were made in finding and describing the shape of the tip of fluid-filled cracks at the mesoscale.

    I had the possibility to study many natural sheet injections (magmatic dykes) in the field. Rarely a tip is exposed and preseved. As a minor argument in my PhD thesis, I proposed a very preliminary morphological classification of dyke tips. Unfortunately, I had no possibility to continue those studies and to publish.

     

    Dear Dr. Suo,

    I was read the paper of the father of the fracture mechanisms Prof. Griffith 1920. Now, I wish to read completely the paper of  Irwin in 1948 ''Fracture Dynamics, . In Fracturing of Metals, 147-166. 1947 ASM Symposium '' that it was to extend the  Griffith  approach to metals by including the energy dissipated by local plastic zone. Have you this paper?

    Best wishes,

    Hkm

     

    Zhigang Suo's picture

    Zhigang Suo's picture

    at a thread titled Inglis (1913) vs. Griffith (1921).

    Prof. Suo

    In your notes the condition of fixed grip has been assumed so that no work is done by the external force. This gives an expression for U. This is differentiated to find the critical stress σc. It is assumed that that the stress σ is constant. But in a fixed grip situation won't σ change as the crack grows?

    Zhigang Suo's picture

    Thank you very much for the question.  For a Griffith cack, the crack is small compared to the sample size.  Consequently, the stress remote from the crack changes negligibly when the crack extends.  

    The distinction between load control and displacement control is indeed important.  A fuller treatment on this topic is in the notes on energy release rate.

    Thanks a lot for this clarification. This is not stated expicitly in textbooks. May be your notes should include it. So if I understand correctly, Griffiths equation can be derived either under load control as in Lawn; or displacement control  as in your notes. In the former stress is exactly constant while in the latter although stress is not really constant but it is approximately so for very large plate.

    Is there no stiffness in the wire which can carry part of the load in the zero creep experiment?

    Zhigang Suo's picture

    In this test and his analysis, elasticity of the wire was neglected.

    Is this not, then, equivalent to assuming that glass is behaving like a liquid at the temperature? And if so, then the droplet method should work fine for it. But I think Griffith chose to use the zero creep method only because at lower tempeartures droplet method was not working, in other words glass was behaving more like a solid and not liquid. 

    Zhigang Suo's picture

    Indeed, in the creep test, the glass is considered a liquid, a very viscous liquid.  In experiment, you can also estimate the elongation due to elastic deformation, and make sure it is small compared to creep deformation.

    Griffith uses π d T sin(θ/2) = w to calculate the surface surface tension T. But with θ defined as "angle at the point of suspension between the two halves of the fibre" the trigonometric factor should be cos(θ/2) and not sin(θ/2). his quoted "angle of of sag' of 18.25 degrees also appears not be between the two halves of the fibre but between fibre and the horizontal. If the angle θ is defined from the horizontal then still the factor should be sin(θ) and not sin(θ/2). Or am i making some silly misinterpretation here.

    Further, as you have explained in the notes, it should be the radius r and not the diameter d which should be used for equilibrium. Are Griffith's estimates, then, less by a factor of 2 than what his experiments indicate?

    Sorry to have troubled you about factor 2,  I was missing the fact that there are two fibres which are carrying the load.

    Zhigang Suo's picture

    Indeed his wroding is confusion.  The angle θ referes to the angle of rotation of one half relative to the other half.   Thus, the angle from the horizontal line to either half wire is θ/2.

    Here is the plot of surface tension measured by zero creep method in Griffith's Table I. The 1110 C data has been excluded as it is measured by a different method (Quincke's drop method).

    The data appears to have too much scatter. So I wonder whether Griffith's claim "So far as they go, these figures confirm the deduction that the surface tension of glass is approximately a linear function of temperature." But of course, if we include the isolated data for 1110 C, then his claim is probably more justified.

    Interestingly, there is hardly any change in the fitted line whether we include or exclude the 1110 data point.

    Please don't misunderstand me, I am not crticising Griffith's method. I am only trying to understand and appreciate it better.

    There is confusion in the literature regarding the year of publication of Griffith's classic. Some authors (AEH Love, TL Anderson, RW Hertzberg) call it 1920 while others maintain 1921. Interestingly, the source of the confusion is right at the source. The Philosophical Transactions of Royal Societ Website labels the publication as vol A221 (1921) whereas the very first page of the actual paper has a footnote "Published October 21, 1920"!

    Zhigang Suo's picture

    Thank you for this note.  It will be interesting to look into papers related to this landmark paper.  The Internet now allows us, collctively, to post our findings in reading obscure papers, in any language.  The invention of fracture mechanics is one of the most significat development in mechanics.

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