For a crack in an elastic body subject to a load, the elastic energy stored in the body is a function of two independent variables: the displacement of the load, and the area of the crack. The energy release rate is defined by the partial derivative of the elastic energy of the body with respect to the area of the crack.

This definition of the energy release rate assumes that the body is elastic, but invokes no field theory. Indeed, the energy release rate can be determined experimentally by measuring the load-displacement curves of identically loaded bodies with different areas of the cracks. No field need be measured.

Due in class, Thursday, 1 April 2010

Fracture mechanics without invoking any field theory. In Lecture 1 on Fracture of Rubber, we considered the extension of a crack in an elastic body subject to a load. Following Rivlin and Thomas (1953), we regarded the elastic energy stored in the body as a function of two independent variables: the displacement of the load, and the area of the crack. The partial derivative of the elastic energy with respect to the area of the crack defined the energy release rate.

A rubber band can be stretched several times its original length. This large deformation may hide its brittleness: the strain to rupture can be markedly reduced by the presence of a crack. This lecture describes fracture mechanic of highly deformable materials, such as rubbers and gels.

Demonstrate in class the effect of a crack on a rubber band. Use a wide rubber band. Show the class that the rubber band can be stretched several times its original length. Then use scissors to cut a crack into the rubber band. Pull the rubber band to rupture. Note that the strain to rupture is markedly reduced by the crack. Pass the scissors and some rubber bands around. Invite every student to try.

A glass may withstand a static load for a long time (days, weeks, or years) and then, without warning, breaks suddenly. Here are salient empirical observations:

• The delay time depends on the magnitude of the load: The smaller the load, the longer the delay time.
• The phenomenon is environment-sensitive. Glass suffers delayed fracture in moisture, but not in vacuum. The lower the humidity, the longer the delay time.
• The phenomenon is thermally-activated. The lower the temperature, the longer the delay time.

The phenomenon occurs to all materials to some degree in some environments. The phenomenon is known variously as

Due in class, Thursday, 11 March 2010

Required reading. P.C. Paris, M.P. Gomez and W.E. Anderson, A rational analytic theory of fatigue. The Trend in Engineering 13, 9-14 (1961). I went online and found that the Trend in Engineering is the alumni newsletter of the College of engineering, of the University of Washington.  I could not find this paper online. John Hutchinson offered to write to Paul Paris for a copy of the paper, which Paris sent by airmail. A scanned copy of the paper is attached with this lecture. 

Due in class, Thursday, 4 March 2010

For a crack advancing in a body under a load, we can solve boundary-value problems of several types:

The square-root singular field is invalid in the fracture process zone, where inelasticity prevails, and is also invalid on the size scale of the external boundary, where the boundary conditions prevail. A careful interpretation of the square-root singular field clarifies why the Linear Elastic Fracture Mechanics works.

We have modeled a body by using the linear elastic theory. We have modeled a crack in the body by a flat plane, and the front of the crack by a straight line. Within this idealized model, the field around the front of the crack is singular. The singular field is clearly an artifact of the idealized model, but Irwin and others made the singular field a centerpiece of fracture mechanics.

The mathematics of this singular field had been known long before Irwin entered the field. We will focus on the mathematics in this lecture, and will describe Irwin’s way of using the singular field in the following lecture.

Due in class, Thursday, 18 February 2010

The qualitative picture of the fracture of a body may be well-understood, across disparate scales of length and time, from the distortion of electron clouds, to the jiggling of atoms, to the motion of dislocations, to the extension of the crack, to the drop of the load-carrying capacity of the body. This statement by itself, however, is of limited value: it offers little help to the engineer trying to prevent fracture of a structure. Hypes of multiscale computation aside, no reliable method exists today to predict fracture by computation alone.

Due in class, Thursday, 11 February 2010

Following Griffith, you perform the same fracture test using a steel rather than a glass. Using a diamond saw, you cut a crack into a body of a steel. You load the body in tension, and record the applied stress at fracture. You find that the Griffith theory agrees with one part of the experimental observation, but disagrees with the other. While other people complained about this large discrepancy, Irwin and Orowan did something about it: they invented a procedure to apply the Griffith theory to ductile materials such as steels.

After the atomistic nature of matter was confirmed by many experimental observations, about a century ago, it became useful to relate macroscopic phenomena to atomic processes. In 1921 the British engineer A.A. Griffith published a paper on one such macroscopic phenomenon: fracture of a glass. The main puzzle had been that the glass usually breaks under a stress several orders of magnitude below the strength of atomic cohesion.

Due in class, Thursday, 4 February 2010

A body is subject to a load. What is the magnitude of the load that will cause the body to fracture? Let us begin with a body made of a glass, which deforms elastically by small strains. A procedure you have been taught before probably goes as follows. You first determine the maximum stress in the body. You then determine the strength of the material. The body is supposed to fracture when the maximum stress in the body reaches the strength of the material.

I’ll first review this procedure, so that you and I agree exactly what this procedure is. I’ll then explain why this procedure is difficult to apply in practice

Spring 2010, Tuesday and Thursday 2:30 pm –400 pm, Maxwell Dworkin Room G-135. 

I’ve just come back from a Winter School on Dielectric Elastomer Transducers, held at Monte Verità, Ascona, Switzerland, 10-16 January 2010.  Lectures were given by various people, covering the theory of electromechanical interaction, design of devices, development of materials, and technologies of manufacturing.  I was asked to give three lectures on the theory.  I attach the slides of my lectures.

Recent experiments have shown that a voltage can induce a large deformation in an elastomer of interpenetrating networks. We describe a model of interpenetrating networks of long and short chains. As the voltage ramps up, the elastomer may undergo a snap-through instability. The network with long chains fills the space and keeps elastomer compliant at small to modest deformation. The network with short chains acts as a safety net that restrains the elastomer from thinning down excessively, averting electrical breakdown.  It appears possible to find a dielectric elastomer capable of giant deformation of actuation.  You can read the paper, or take a look at the slides posted here.

A growing number of mechanicians are entering the field of soft materials, such as polymers, gels, and tissues. While they interact with researchers in technical societies traditionally identified with the field, they also maintain connection with researchers in applied mechanics.

In response to this trend, the Applied Mechanics Division, of the ASME International, has recently created a new Technical Committee on Soft materials.  The new Technical Committee will serve as a home for this group of people, and provide links between applied mechanics and new applications.

With great pleasure I inform you that Jerrold E. Marsden, of Caltech, are selected to receive the 2010 Thomas K. Caughey Dynamics Award.

I am writing on behalf of the Thomas K. Caughey Dynamics Award Committee: Zhigang Suo (Chair), Tayfun E. Tezduyar, Ares J. Rosakis, Kenneth M. Liechti, Lawrence A. Bergman, Daniel J. Inman, Krishnaswamy Ravi-Chandar, Thomas N. Farris, Wing Kam Liu, Mary C. Boyce, Paul Jennings, Geof Tomlinson, and Ali Nayfeh.

With great pleasure I inform you that Yoichiro Matsumoto, of the University of Tokyo, is selected to receive the 2010 Ted Belytschko Applied Mechanics Award.

I am writing on behalf of the Ted Belytschko Applied Mechanics Award Committee: Zhigang Suo (Chair), Tayfun E. Tezduyar, Ares J. Rosakis, Kenneth M. Liechti, Lawrence A. Bergman, Daniel J. Inman, Krishnaswamy Ravi-Chandar, Thomas N. Farris, Wing Kam Liu, Mary C. Boyce, Choon Fong Shih, Oscar Dillon, Lewis Wheeler, Carl T. Herakovich, and Arthur W. Leissa.