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The toughest hydrogel in the world

Zhigang Suo's picture

The class started today.  I'll be teaching fracture mechanics this semester.  I'll be mostly using the class notes I wrote in 2010, but will post updated ones. 

In today's class I covered "Trouble with linear elastic theory of strength."  I have just posted updated notes of the lecture.  The new notes begin with the follwoing paragraphs.

The toughest hydrogel in the world.  We reported an exceptionally tough hydrogel:  Jeong-Yun Sun, Xuanhe Zhao, Widusha R.K. Illeperuma, Kyu Hwan Oh, David J. Mooney, Joost J. Vlassak, Zhigang Suo. Highly stretchable and tough hydrogels. Nature 489, 133-136 (2012).  A hydrogel is a three-dimensional polymer network swollen with water.  Familiar examples include jello and contact lenses.  Our hydrogel achieved a toughness of ~9,000 J/m2.  This statement raises several questions.

What is toughness?  Toughness is the ability of a material to resist the growth of crack.  Understanding toughness is a main object of this course.

What does the value 9,000 J/m2 mean?  We will talk about how to measure toughness later in the course.  For now, you can have some feel for orders of magnitude.  Jello and tofu have toughness ~10 J/m2.  Contact lenses have toughness ~100 J/m2.  Cartilage has toughness ~1,000 J/m2.  Natural ruber has toughness ~10,000 J/m2.  Our tough gel contains about 90% water, yet its toughness approaches that of the natural rubber.    

How can our hydrogel be so tough?  For now let’s have a qualitative picture.  A window glass is brittle, but a steal is tough.  That is, the ability of a window glass to resist the growth of a crack is much less than the ability of a steel to resist the growth of a crack.  This is everyday experience, but why?  The answer to such question sooner or later leads to an atomistic picture. 

You prepare a sheet of glass.  To study the growth of the crack, you cut a crack into the glass with a diamond saw.  You then pull the glass to cause the crack to grow.  The crack grows by breaking atomic bonds.  The tip of the crack concentrates stress, so that the atomic bonds at the tip of crack break.  As the tip the crack advances, a plane of atomic bonds unzips.  Atomic bonds off the plane of the crack remain elastic, and do not participate in resisting the growth of the crack.  The elasticity is visible:  after fracture, the two halves the glass fit together nicely.

Now you prepare a sheet of steel with a pre-cut crack.  You then pull the steel to cause the crack to grow.  The crack grows by breaking atomic bonds, but something else happens.  Atomic bonds off the plane of the crack no longer remain elastic:  they change neighbors.  That is, the steel off the crack plane undergoes plastic deformation.  The amount of material involved in plastic deformation is much, much more than the two planes of atoms on the surfaces of the crack.  The plastic deformation enables the steel to resist the growth of the crack much more than breaking a plane of atomic bonds.  The plasticity is visible:  after fracture, the two halves the steel do not fit together nicely.

Our hydrogel is tough because the growth of a crack does more than breaking a single layer of polymer chains.  The polymer off the plane of the crack undergoes deformation similar to plastic deformation in the steel.

We will make this picture precise as we go along in this course.  But if you absolutely cannot wait to know how our gel works, I’d be delighted if you read the paper now.   

Comments

Konstantin Volokh's picture

Dear Zhigang

Welcome back to physics Wink

Your attack on linear elasticity is incomplete:

1. Linear elasticity is a very successful computational approximation
rather than a physical theory – it prescribes stressing under rigid body
motion.

2. Linear elasticity produces singularities which do not
really exist. The whole body of linear elastic fracture mechanics is built upon
these imaginary singularities. Should we call this theology? Cool

Strength is a separate issue. I do think that strength
exists because various samples of the same material show similar critical
stresses in tests.

I agree completely that material failure must be a part of
the constitutive description.

Kosta

Zhigang Suo's picture

Dear Kosta:  Thank you very much for the comments.  Quick responses to your comments.

1.  What's wrong with using linear elasticity in design for strength?  Indeed, linear elasticity is an approximate theory, and has its merits and failings.  The question in this opening lecture of fracture mechanics  is specific: does linear elasticity work in design for strength.  I outline the commonly used procedure, and discuss why the procedure does not work.  For a brittle material like silica glass, the stress predicted from linear elasticity is accurate all the way close to atomic scale.  The growth of a crack corresponds to unzipping a plane of atomic bonds.  The tip of the crack concentrates stress.  The stress at the tip of the crack is sensitive to flaws.  Experimentally measured strength of silica glass ranges from less than 10 MPa to higher than 5 GPa.  

2.  Is fracture mechanics a theology?  No.  Linear elasticity does predict singular stress field at a sharp tip of a crack.  Irwin and others, however, devised a way to use this singular crack-tip field.  The outcome is linear elastic fracture mechanics.  I will describe Irwin's idea later in the course.  Section 1.2 of the following paper gives a brief description.

G. Bao and Z. Suo, " Remarks on crack-bridging concepts,Applied Mechanics Review45, 355-366 (1992).   

Konstantin Volokh's picture

Dear Zhigang,

1. I agree with you that “theoretical strength” based on
idealized crystal models does not exist for bulk material, which has a lot of imperfections
on various length scales. However, the samples of the same bulk material are “flawed”
similarly and, thus, can exhibit similar rupture strain/stress. For my students
I buy a set of rubber bands and stretch them up to rupture (can be painful).
The rupture occurs normally when the length of the band is about seven times its
initial length. Try yourself and you’ll see that strength for such imperfect
material exists. Your example of silica glass which is very sensitive to
imperfections is probably more an exception than a rule. I guess most engineering
materials are not sensitive to small imperfections (I am not talking about
visible cracks, notches etc).

2. People say that LEFM is useful for engineers. May be. For
my taste, the theory is highly controversial. You know, they say that all
models are wrong but some are useful. May be LEFM is a nice illustration of
this wise saying Smile.

Zhigang Suo's picture

I agree with you on both.  Later in the course I'll talk about notch-sensitivity.  Materials like silica glass and most ceramics are extremely notch sensitive.  Materials like metals and natural rubber are less notch-sensitive.  Fracture mechanics can quantify notch-sensitivity.

For a visual illustration of notch-sensitivity of a soft elastomer, see Fig. 1 in the following paper:

Matt Pharr, Jeong-Yun Sun, and Zhigang Suo. Rupture of a highly stretchable acrylic dielectric elastomer. Journal of Applied Physics 111, 104114 (2012) 

By coincidence, a student here has been looking at fracture of natural rubber, and has had the same experience that you mentioned. 

Dear Kosta,

Zhigang is not taking your bait, but I particularly liked much of what I was reading in this thread anyway (Cool). So, let me go a step further. Or, two. (May be three.)

1. Borrowing terminology from this physics-related blog [^] (HT to Hanning Dekant [^]), if a crack-tip in a (real) material quacks like a singularity, then...

2. But, I still do maintain that any theory that is crucially based on a singularity is theology.

3. No. No. No. LEFM is not a computational approximation---it's an analytical theoretical one.

--Ajit

- - - - -
[E&OE]

Konstantin Volokh's picture

Dear Ajit,

1-2 Nothing to argue about

3 LEFM gives indeed a short analytical description. Short formulas are so attractive that people (including myself) tend to compromise on their meaning. (Needless to say, only short formulas are good for gravestones Cool)

Zhigang Suo's picture

It will be hard for me to support the statement made in the title, because I don't really know what theology is.  But whatever theology is, as specified in Wikipedia, fracture mechanics is not theology.  Fracture mechanics makes predictions that you and I can do experiments to verify.

Also, singularity is just an appearence.  Take the familiar example of Newton's law of gravitation:  Force between two massive bodies is proportional to  (distance)^-2.  This expression is singuar if you let distance approach zero.  But we don't!

Similarly, in linear elastic fracture mechanics, the stress scales as (distance)^-1/2.  But we don't let distance approach zero in using this result.  The following paper summarizes this aspect of linear fracture mechanics in Section 1.2. 

G. Bao and Z. Suo, " Remarks on crack-bridging concepts,Applied Mechanics Review45, 355-366 (1992).    

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