Thread by @zhigangsuo, One hundred years of toughnessGriffith repays repeated study. Recently I noticed Section [...]

One hundred years of toughness

Griffith repays repeated study. Recently I noticed Section 11, in which he described an experiment on rupture of liquid.

Griffith 1921 the phenomena of rupture and flow in solids

⁦ @Junsoo_twit⁩ ⁦ @jasongsteck⁩ https://scholar.google.com/scholar?hl=en&as_sdt=0%2C22&q=Griffith+1921+the+phenomena+of+rupture+and+flow+in+solids&btnG=

Here are the notes I wrote on the Griffith paper for my graduate course on fracture mechanics. The notes connect his experiments to current practice.

When writing these notes, I did not notice his experiments on rupture of liquid.

http://imechanica.org/node/7470

To celebrate one hundred years of toughness, I will tweet a thread on landmark papers on the discovery of toughness that have influenced me. Think of your own favorite papers and we can compare notes.

@ProfZhaoMIT @ToLiTeng @lin_shaoting @jasongsteck @Junsoo_twit @Tengh_Yin

Stiffness is a property of a material in a state of thermodynamic equilibrium.

Toughness is a property of a material undergoing a process of thermodynamic dissipation.

@Junsoo_twit @MeixuanziS @SammyHassan_ @ToLiTeng @RuobingB @DrJiaweiYang @ZhaoXuanhe @darren_lipomi

Never run after a bus or woman or cosmological theory, because there will always be another one in a few minutes.

Attributed by @Horganism to a friend of Wheeler's at Yale.

To have equal opportunity, one may change woman to man, and cosmological to toughening.

1930 Obreimoff The Splitting strength of mica

Happy childhood of a subject! A delightful paper with no affectation.

G for a beam
Mica heals fully.
Splitting generates light and charge.
v-G curve
G depends on environment

@jasongsteck @Junsoo_twit

https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1930.0058

Obreimoff showed that split and heal can be reversible thermodynamic processes.

In hindsight, this observation relates to applications including

Adhesion
Wafer bonding
Sintering
Self-healing materials
Stress corrosion
Degradable materials

Don’t miss this paper.

Obreimoff did not mention Griffith, but wrote, "The present observations are connected with the well-known work of A. Joffe* on the the tensile strength of rock salt.

*Joffe, "Physics of Crystals," pp. 62-64.

Joffe did discuss Griffith's work.

https://archive.org/details/physicsofcrystal031376mbp/page/n77/mode/2up

I now return to the thread "One Hundred Years of Toughness".

1944 Orowan The fatigue of glass under stress

A glass can sustain a load for a long time and then breaks.

The phenomenon connects mechanics to chemistry.

https://scholar.google.com/scholar?hl=en&as_sdt=0%2C22&q=orowan+the+fatigue+of+glass+under+stress&btnG=

@MeixuanziS @jasongsteck @QuanJiao

After I started to teach, I read papers by Orowan.

Orowan was PhD advisor of Ali Argon. "He and I had a love-hate relation," Ali said to me, with smile. I don't recall his explanation, and cannot find it in Orowan's biographical memoir that Ali co-wrote.

https://en.wikipedia.org/wiki/Egon_Orowan

Fracture has been my main subject for decades. I have collected many books on the subject, but haven’t read any from cover to cover. Much of what I know has been through lectures, conversations, and papers. Perhaps this lack of book reading is typical for researchers today.

Griffith 1921 did not apply directly to steel, as noted by several authors in the late 1940s. A "fix" of the problem was stated clearly in Orowan 1948, in a "note added in proof", on page 214. This footnote marked a new beginning of toughness. A must read.
https://iopscience.iop.org/article/10.1088/0034-4885/12/1/309/meta?casa_token=YprZtMJHGl8AAAAA:lUNv5xXhvhg3WiWNpJZQrp6-hEYzF1AVeSy-6JPNrh0DrGe3fwbo_GBlbfvk5fMiS346i4DM

In the footnote, Orowan recalled the Griffith equation:

s ~ sqrt(pE/c)

s = strength
E = modulus
c = crack length
p = "plastic work per unit are", i.e., toughness

He estimated p = 1000-10,000 J/m2 for his steel.

A new star was born: toughness, distinct from surface energy.

Today, we justify Orowan's interpretation using two quantities:

(1) Energy release rate, which is a loading parameter.

(2) Toughness (also called fracture energy, etc.), which is a material property.

See my course notes on fracture mechanics:

https://imechanica.org/node/7448

Toughness is a material property. Questions:
(1) How is toughness defined?
(2) How is toughness measured?
(3) How is toughness used to design structures?
(4) How to create materials for toughness?

This thread focuses on (4).

@ToLiTeng @ProfZhaoMIT @ProfGraceGu @ProfBuehlerMIT

Here are two other papers in the 1940s that noted the "fix" of the Griffith 1921.

Hollomon and Zener 1946
https://doi.org/10.1063/1.1707697

Irwin 1948
https://drive.google.com/file/d/0B8UBSuBlYFWjZ3YyUXRuT2NiVGM/view?usp=sharing

More papers on this fix are likely. Let me know if you have found others. "Fixing Griffith" was in the air!

Problems in Fracture of Metals

In this report a review is made of the possible types of future research which might lead to an understanding of the fracture of metals, and hence to an increase of the strength level to which meta...

https://doi.org/10.1063/1.1707697

People do not use words consistently. For example, Wikipedia defines toughness by the area under the stress-strain curve. The quantity has the unit of energy/volume. I call it the work of fracture.

For me, toughness has the unit of energy/area.

https://en.wikipedia.org/wiki/Toughness

For a brittle material, such as silica, the strength varies by orders of magnitude, from sample to sample. Thus, the experimental value of work of fracture is not a material property, and is defect sensitive.

By contrast, toughness is a material property.

Here is a sloppy and slick definition of toughness:

toughness is the energy needed to advance a crack by a unit area.

The definition appeals to intuition, and can be made operational.

See my course notes for detail: http://imechanica.org/node/7507

Fracture is an inelastic process, breaking bonds to create the crack and causing inelastic deformation beneath the surafaces of the crack.

Often, inelasticity is localized within thin layers.

This condition is noted in Orowan 1945, and is called small scale inelasticity today.

Orowan 1945/46 is refreshing.

p.190 "back reflection of X-ray...blurring"

"after a layer of about 0.5 mm depth was removed by etching. The sharpness of the reflections shows that the plastic deformation was confined to a thin layer."

https://drive.google.com/file/d/1lX6MQYJvCyBSVc5rd3hui9vyj2kRpZXY/view?usp=sharing

Why did many people work on fracture of steel in the 1940s?

The work was connected with the Liberty Ships, cargo ships in service during the World War II. Some broke in halves and sank.

See the story on Wikipedia.

https://en.wikipedia.org/wiki/Liberty_ship

In the story of Liberty ships, Constance Tipper stood out. She made a central discovery: a notched steel plate used by the ships underwent a ductile-to-brittle transition as temperature dropped.

She lived to 101. Her own life is a story. https://link.springer.com/article/10.1007/s11837-015-1697-9

Constance Tipper Cracks the Case of the Liberty Ships

https://link.springer.com/article/10.1007/s11837-015-1697-9

SS Schenectady was an oil tanker, one of over 6000 ships built by the United States during World War II. It broke in half on 16 January 1943.

During the war, there were 1500 cases of brittle fracture in ship hulls and decks.

https://en.wikipedia.org/wiki/SS_Schenectady

$SS Schenectady was an oil tanker, one of over 6000 ships built by the United States during World War II. It broke in half on 16 January 1943. During the war, there were 1500 cases of brittle fracture in ship hulls and decks. https://en.wikipedia.org/wiki/SS_Schenectady$

Around 1950, toughness emerged as a link between two sciences: mechanics and materials. It has since forged a division of labor.

1. Measure toughness for various materials.
2. Use toughness to predict load-carrying capacity of structures.
3. Relate toughness to microstructures.

I had been looking for an electronic copy of Tipper's book "The Brittle Fracture Story". @zhirenz pointed out that I could read the book on https://hathitrust.org , because my university is a partner institution.

What a story! A treat for every student of fracture.

A crosslink between two threads:

a. One hundred years of toughness
b. Fractocohesive length

An unnotched sample shows ductility, but a notched sample shows little plasticity.

Mystery no more.

This steel had a fractocohesive length, say, R = 1 mm. https://twitter.com/zhigangsuo/status/1298943288012165121

https://twitter.com/zhigangsuo/status/1298943288012165121

Continue the thread of "One Hundred Years of Toughness"

1953 Rivlin and Thomas, Rupture of Rubber.

The second paper that should be read by all students of fracture, the first being Griffith 1921.
@CretonCost @aljcrosby @ProfZhaoMIT @RuobingB https://doi.org/10.1002/pol.1953.120100303

Rupture of rubber. I. Characteristic energy for tearing

A criterion for tearing of test‐pieces cut from thin sheets of a natural rubber vulcanizate, similar in form to the Griffith criterion for spreading of a crack, is formulated. This criterion involv...

https://doi.org/10.1002/pol.1953.120100303

1953 Rivlin and Thomas published, for the first time, the practice of fracture mechanics:

Toughness is a material constant, independent of sample geometry.

Do you know any earlier publication?

For rubbers that they tested, toughness was in the range 1000-20,000 J/m2.

Some principles of toughness are applicable to diverse materials, but historically they were discovered in individual materials:

Glass
Steel
Rubber
Composite
Plastic
hydrogel

This thread will approach the principles historically.

Let's make the practice of fracture mechanics explicit.

Fracture always involves inelasticity.

Let us be frank and imagine a fake elastic body, in which

elasticity is everywhere,

crack surfaces are coincident planes in the reference state,

crack front is a straight line.

U = elastic energy in the fake elastic body
A = area of the crack in the reference state
P = applied force
D = displacement such as PdD is the applied work

U(D,A) is a function of two variables.

@Baohong_Chen @jasongsteck

Define the energy release rate G

dU = PdD - GdA

According to calculus, P is the partial derivative of U(D,A) with respect to D, and -G is the partial derivative of U(D,A) with respect to A.

Textbooks avoid math and define G verbally. That's mouthful and confusing.

Define toughness Gc by

PdD = dU + GcdA

PdD = work done on the real body
dU = change in energy in the fake body

GcdA, by definition, is the work on real body minus the change in the energy of the fake body.

A definition of virtual reality!

@CretonCost @EnduricaWill

Now we have defined two quantities, G and Gc, using two equations:

dU = PdD - GdA
PdD = dU + GcdA

A comparison of the two equations says that the crack advances when

G = Gc.

@srledm87 @SammyHassan_

In the definitions of G and Gc, something is unclear. Is PdD fake work or real work?

Let us avoid this question by considering two special cases.

First case, fix D. The two equations become

dU = -GdA
0 = dU + GcdA

Crack advances when

G = Gc.

Second case, fix P. A fake system includes the fake body and P, pictured as a weight. The free energy of the fake system is U - PD. Both U and D are fake, only P is real.

Define G by d(U - PD) = -GdA.
Define Gc by 0 = d(U - PD) +GcdA.

Crack advances when

G = Gc.

A strategy in thermodynamics: when confused, isolate. Complete isolation is inconvenient, so we consider an isothermal system.

F = Helmholtz function of the fake system.
F(A) is a function of a single variable.

Define G by dF = -GdA
Define Gc by 0 = dF + GcdA.

Energy release rate G is defined by

G = -dF(A)/dA
A = Area of the crack in the reference state
F = Helmholtz function of the fake system

G can be calculated from the elastic boundary-value problem of the fake system. No inelasticity. No fracture.

https://onlinelibrary.wiley.com/doi/abs/10.1002/pol.1953.120100303

$Energy release rate G is defined byG = -dF(A)/dAA = Area of the crack in the reference stateF = Helmholtz function of the fake systemG can be calculated from the elastic boundary-value problem of the fake system. No inelasticity. No fracture. https://onlinelibrary.wiley.com/doi/abs/10.1002/pol.1953.120100303$

Any geometry containing a crack can, in principle, serves as a specimen to measure toughness.

Examples:
1921 Griffith: a crack in an infinite sheet subject to remote tensile stress.
1930 Obreimoff: A flake from a block.
1953 Rivlin-Thomas: Several geometries for rubber.

Griffith assumed thermodynamic equilibrium.

The Helmholtz function includes both elastic energy and surface energy.

The internal variable is the crack length.

The crack is in thermodynamic equilibrium if the Helmholtz function is stationary when the crack length changes.

The Griffith theory should not even apply to his own experiment.

His theory assumed thermodynamic equilibrium, involving elastic energy and surface energy.

His experiment must also generate sound from the propagating crack.

What happened to the energy of the sound?

Today we do not invoke thermodynamic equilibrium.

Define the energy release rate using a fake system:

G = - dF(A)/dA.

Define the toughness Gc by the value of G when the crack grows in the real body.

@Baohong_Chen @SammyHassan_

This modern definition of toughness may have started in a footnote in Orowan 1946, but was in full display in Rivlin-Thomas 1953.

Rivlin and Thomas 1953 measured toughness of a rubber using samples of various shapes, and found that

1. Gc is a constant independent of the shapes of sample.
2. Gc is orders of magnitude larger than surface energy.

They interpreted 2 by inelasticity.
Rivlin was surprised by 1.

"Although this work was not published until 1953, it was substantially complete by the summer of 1950, and I lectured on it in George Irwin's department at the Naval Research Laboratory in the fall of 1950"

R.S. Rivlin https://www.amazon.com/Collected-Papers-R-S-Rivlin-II/dp/0387948252/ref=sr_1_1?dchild=1&keywords=rivlin+collected+papers&qid=1598714255&s=books&sr=1-1

Collected Papers of R.S. Rivlin: Volume I and II (v. 1 & v. 2)

Collected Papers of R.S. Rivlin: Volume I and II (v. 1 & v. 2)

https://www.amazon.com/Collected-Papers-R-S-Rivlin-II/dp/0387948252/ref=sr_1_1?dchild=1&keywords=rivlin+collected+papers&qid=1598714255&s=books&sr=1-1

Watch out: People use one word, toughness, to mean several quantities:

1. Gc, used in this thread

2. Kc, introduced by Irwin

3. Area under stress-strain curve, called the work of fracture in this thread

4. Result of Izod test

5. Result of Charpy test

The Rivlin-Thomas 1953 paper was part I of a series. I have found 11 parts, and placed them in this folder for my students.

https://drive.google.com/drive/folders/1QikT7JxdfwTx283SqwvTPI4D-O58FkQI?usp=sharing
The 11 papers were written by researchers from British Rubber Products Research Association.
https://en.wikipedia.org/wiki/Tun_Abdul_Razak_Research_Centre
@EnduricaWill

In tracing one hundred years of toughness, or history of any idea, I would like to stick to primary sources, i.e., published papers. Secondary sources (reviews, textbooks, personal recounts) provide hindsight and perspective, but are unreliable so far as history is concerned.

The need for a fake elastic body is implicit in Griffith 1921. He never mentioned any fake body, but watch his action. He assumed continuum elasticity holds in the body.

Like Gibbs (page 219), Griffith defined surface energy as excess over a fake body https://www.google.com/books/edition/Scientific_Papers_of_J_Willard_Gibbs/8RhWAAAAMAAJ?hl=en&gbpv=1&dq=gibbs+the+scientific+papers&printsec=frontcover

Scientific Papers of J. Willard Gibbs

https://www.google.com/books/edition/Scientific_Papers_of_J_Willard_Gibbs/8RhWAAAAMAAJ?hl=en&gbpv=1&dq=gibbs+the+scientific+papers&printsec=frontcover

Neither Orowan 1946 nor Rivlin-Thomas 1953 mentioned any fake elastic body. But their actions implied the use of a fake elastic body. The elasticity solution would be wrong in detail given the inelasticity in steel and rubber, just as it was wrong in glass for Gibbs 1921.

The moral: when words conflict with actions in a paper, follow the actions.

On p. 291 of Rivlin-Thomas, "the energy W stored elastically in the test piece" is an undefined quantity.

According to his autobiography, Rivlin did not expect that Griffith 1921 would work for rubber.

p.293 of Rivlin-Thomas 1953:

"If a cut is made with a razor blade in a test-piece..., then it is seen under a low power microscope that even quite small forces cause a noticeable tearing to occur at the tip of the cut."

The basis of R-curve: G as a function of cut length.

I learned the following analogy from Tony Evans. The analogy has helped many students in my class over the years.

Stress is a loading parameter.
Strength is a material parameter.

Energy release rate is a loading parameter.
Toughness is a material parameter.

Rupture of rubber II (Thomas 1955) reported a relation:

Gc = DWc

Wc = work of fracture
D = diameter of an incision made in a sample
Gc = critical energy release rate of a sample with the incision.

Wc is a material property, but D and Gc are not.

https://drive.google.com/drive/folders/1QikT7JxdfwTx283SqwvTPI4D-O58FkQI?usp=sharing

The theory and experiment described in Thomas 1955 are no longer practiced today. But the last sentences of the paper are intriguing. They foreshadowed the fractocohesion length.

See the thread on fractocohesive length https://twitter.com/zhigangsuo/status/1298310178153140225

@EnduricaWill @CretonCost

https://twitter.com/zhigangsuo/status/1298310178153140225

To flow or to fracture, that is the question.

At room temperature, water flows, glass fractures.

Below the melting point, ice fractures.

At an elevated temperature, glass flows.

When pulled, a gold wire flows into a neck, and necks down to a point at atomic scale.

In a flow, atoms change neighbors, but do not separate.

In a fracture, atoms separate.

A glass wire pulls uniformly, but a gold wire pulls into a neck.

Rupture of Rubber III (Greensmith-Thomas 1955)

When a machine pulls the two arms of a sample at a constant speed, the recorded force can be either constant or fluctuate.

The two types of behavior are called steady-state tear and stick-slip tear.

https://drive.google.com/drive/folders/1QikT7JxdfwTx283SqwvTPI4D-O58FkQI?usp=sharing

Time dependent crack growth was reported in Obreimoff 1930 for mica and in Orowan 1944 for glass.

Greensmith and Thomas 1955 may well be the first paper that plotted the relation between energy release rate and crack velocity.

The relation may not be monotonic.

Rupture of rubber IV (Greensmith 1956) studies the tear of carbon-filled elastomers

When the machine pulls at a constant speed and the elastomer stick-slip tears, the tear may appear "knotty".

Rupture of rubber V (Thomas 1958) marks the beginning of two great developments.

(1) R-curve
A specimen with a precut starts to tear at a small load, and tears further only if the load increases. Energy release rate is a function of the crack length.

https://drive.google.com/drive/folders/1QikT7JxdfwTx283SqwvTPI4D-O58FkQI?usp=sharing

Rupture of rubber V (Thomas 1958) marks the beginning of two great developments.

(2) Fatigue crack growth
A precut specimen tears under a cyclic load of small amplitude. Plot the extension per cycle as a function of the amplitude of energy release rate.

https://drive.google.com/drive/folders/1QikT7JxdfwTx283SqwvTPI4D-O58FkQI?usp=sharing

Note an analogy:
Load is driving force for displacement.
G is driving force for crack length c.

Apply G(t) and measure c(t).

Three kinds of G(t)
Apply constant G and measure crack velocity.
Apply monotonic G and measure crack length.
Apply cyclic G and measure crack length.

"Do you have the initial paper on fatigue crack growth by Paris?" I asked John Hutchinson.

"I don't, but I'll ask him for a copy," said John.

The hard copy of the paper came in the mail, with great man's signature.

I posted the paper on iMechanica: https://imechanica.org/node/7705

Consider a fake body.

Everywhere is elastic.
Crack is a half plane.
Crack front is a straight line.
Plane strain deformation

G has unit J/m2.
The energy per volume W is a field.

The elasticity problem has no length scale. Thus,

W ~ G/R.

R is distance from the crack front.

W~G/R is the crack tip field in a fake body.

The material is specified by elastic energy per unit volume as a function of deformation gradient, W(F).

In the undeformed state, the crack is a half plane, and the front is a line.

R is measured in the undeformed state.

Thus, the 1/R singularity prevails in the fake body, of any model of elasticity, deforming by any magnitude.

The scaling W~G/R comes from dimensional considerations.

The dimensionless prefactor is a function of polar angle and dimensionless parameters of the elastic model.

A real body is inelastic, and the crack deviates from being plane.

Let inelasticity and geometric irregularity be confined in a layer of thickness Rc.

Let the representive size of the sample be a.

The condition of small scale inelasticity holds if

Rc << a.

The crack tip field W ~ G/R holds in a fake body: an elastic, infinite body.

In a real body under SSI, Rc << a, the field W ~ G/R prevails in an annulus

Rc << R << a.

Across this annulus, external load communicates to crack tip via a single messenger, G.

SSI. G-annulus.

Small scale inelasticity (SSI) explains an important discovery of Rivlin-Thomas 1953:

Toughness is a material constant, independent of the shape of test piece.

Under SSI, R-curve, G-v curve, G-dc/dn curve are all material properties, independent of the shape of test piece.

Small scale inelasticity also explains fractocohesive length: https://twitter.com/zhigangsuo/status/1298310179902062594

https://twitter.com/zhigangsuo/status/1298310179902062594

Like Griffith, Irwin 1957 studied a fake body of linear elasticity.

Result 1
He used an existing solution to show that the stress field near a crack tip is square-root singular.

The result is readily seen from W~G/R, although not seen at the time.

https://imechanica.org/node/7579

Result 2
The full solution gives the prefactor. In particular,

stress directly ahead the crack = (G/2piRE)^1/2.

pi = 3.14...
E = Young’s modulus

Irwin called the quantity K = (G/E)^1/2 the stress intensity factor.

K can be converted to G and will not concern this thread.

You can follow @zhigangsuo.

Tip: mention @twtextapp on a Twitter thread with the keyword “unroll” to get a link to it.

Latest Threads Unrolled: