At the invitation of Yonggang Huang, I’ll give 4-hour lectures at the NSF Summer Institute Course on the Mechanics of Soft Materials.   I attach the slides of the lectures, to be given on Monday, 10 May 2010.  An abstract of the lectures follows.

Theory of dielectric elastomers capable of giant deformation of actuation
Xuanhe Zhao, Zhigang Suo
Physical Review Letters, 104, 178302 (2010)

The deformation of a dielectric induced by voltage is limited by electrical breakdown if the dielectric is stiff, and by electromechanical instability if the dielectric is compliant. The interplay of the two modes of instability is analyzed for a dielectric elastomer, which is compliant at a small stretch, but stiffens steeply. The theory is illustrated with recent experiments of interpenetrating networks, and with a model of swollen elastomers. The theory predicts that, for an elastomer with a stress-stretch curve of a desirable form, the voltage can induce giant deformation.

When an indenter is pressed into a gel to a fixed depth, the solvent in the gel migrates, and the force on the indenter relaxes. Within the theory of poroelasticity, the force relaxation curves for indenters of several types are obtained in a simple form, enabling indentation to be used with ease as a method for determining the elastic constants and permeability of the gel. The method is demonstrated with a conical indenter on an alginate hydrogel.

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.

In May 2008, I posted 3 lectures on Soft Active Materials given at UCSB.  I have since given similar lectures on other occasions, but never all three at the same place.  The field has been active.  The lectures have been updated with new items.  I’m now posting the “2nd edition” of these lectures.

• Dielectric elastomers
• Neutral gels
• Polyelectrolyte gels
• pH-sensitive gels 

The slides are posted as delivered.  No effort is made to eliminate repeating slides. 

We perform uniaxial tensile tests on polyimide-supported copper films with a strong (111) fiber texture and with thicknesses varying from 50 nm to 1 μm. Films with thicknesses below 200 nm fail by intergranular fracture at elongations of only a few percent. Thicker films rupture by ductile transgranular fracture and local debonding from the substrate. The failure strain for transgranular fracture exhibits a maximum for film thicknesses around 500 nm. The transgranular failure mechanism is elucidated by performing finite element simulations that incorporate a cohesive zone along the film/substrate interface.

Many engineering devices and natural phenomena involve gels that swell under the constraint of hard materials. The constraint causes a field of stress in a gel, and often makes the swelling inhomogeneous even when the gel reaches a state of equilibrium. To analyze inhomogeneous swelling of a pH-sensitive gel, we implement a finite element method in the commercial software ABAQUS.  The program is attached here.  Contact Shenqiang Cai (shqcai@gmail.com) for a description of the program.

When a block of an elastomer is bent, the compressed surface may form a crease. This paper analyzes the critical condition for creasing by comparing the elastic energy in a creased body and that in a smooth body. This difference in energy is expressed by a scaling relation. Critical conditions for creasing are determined for elastomers subject to general loads and gels swelling under constraint. The theoretical results are compared with existing experimental observations.

Immersed in an ionic solution, a network of polyelectrolyte polymers imbibes the solution and swells, resulting in a polyelectrolyte gel. The swelling is reversible, and is regulated by ionic concentrations, mechanical forces, and electric potentials. This paper develops a field theory to couple large deformation and electrochemistry. A specific material model is described, including the effects of stretching the network, mixing the polymers with the solvent and ions, and polarizing the gel. We show that the notion of osmotic pressure in a gel has no experimental significance in general, but acquires a physical interpretation within the specific material model.

For a flexible electronic device integrating inorganic materials on a polymer substrate, the polymer can deform substantially, but the inorganic materials usually fracture at small strains.  This paper describes an approach to make such a device highly stretchable.  A polyimide substrate is first coated with a thin layer of an elastomer, on top of which SiNx islands are fabricated.  When the substrate is stretched to a large strain, the SiNx islands remain intact.  Calculations confirm that the elastomer reduces the strain in the SiNx islands by orders of magnitude.

In a lithium-ion battery, both electrodes are atomic frameworks that host mobile lithium ions. When the battery is being charged or discharged, lithium ions diffuse from one electrode to the other. Such an insertion reaction deforms the electrodes, and may cause the electrodes to crack. This paper uses fracture mechanics to determine the critical conditions to avert cracking. The method is applied to cracks induced by the mismatch between phases in crystalline particles of LiFePO4

Electromechanical instability in semicrystalline polymers
Xuanhe Zhao , Zhigang Suo
Abstract
When a layer of a semicrystalline polymer is subject to a tensile force in its plane and a voltage through its thickness, the deformation of the layer is initially homogeneous,but then localizes.

This paper studies the dynamic behavior of a dielectric elastomer balloon subject to a combination of pressure and voltage.  When the pressure and voltage are static, the balloon may reach a state of equilibrium.  We determine the stability of the state of equilibrium, and calculate the natural frequency of the small-amplitude oscillation around the state of equilibrium.  When the voltage is sinusoidal, the balloon resonates at multiple frequencies of excitation, giving rise to superharmonic, harmonic, and subharmonic responses.  When the frequency of excitation varies continuously, the oscillating amplitude of the balloon may jump, exhibiting hyste

Mechanical energy can be converted to electrical energy by using a dielectric elastomer generator.  The elastomer is susceptible to various modes of failure, including electrical breakdown, electromechanical instability, loss of tension, and rupture by stretch.  The modes of failure define a cycle of maximal energy that can be converted.  This cycle is represented on planes of work-conjugate coordinates, and may be used to guide the design of practical cycles.

An electronic device integrates diverse materials, and inevitably contains sharp features, such as interfaces and corners. When the device is subject to thermal and mechanical loads, the corners develop intense stress and are vulnerable sites to initiate failure. This paper analyzes stress fields at corners in flip-chip packages. The stress at a corner is a linear superposition of two modes of singular fields, with one mode being more singular than the other. The amplitudes of the two modes are represented by two stress intensity factors of dissimilar dimensions. To determine the stress intensity factors, we analyze the flip-chip structures under two loading conditions: stretching of the substrate and bending of the substrate.

As a complement to the current issue of journal club, I would like to bring your attention to our current work on dielectric elastomers. 

Xuanhe Zhao and Zhigang Suo A layer of a dielectric elastomer expands its area when a voltage is applied across its thickness.  The layer can be programmed to deform in three dimensions by using patterned prestretches, electrodes, and stiffeners. 

Dielectric elastomers are capable of large deformation subject to an electric voltage, and are promising for uses as actuators, sensors and generators. Because of large deformation, nonlinear equations of state, and diverse modes of failure, modeling the process of electromechanical transduction has been challenging.  This paper studies a membrane of a dielectric elastomer deformed into an out-of-plane, axisymmetric shape, a configuration used in a family of commercial devices known as the Universal Muscle Actuators. 

A gel is an aggregate of polymers and solvent molecules.  The polymers crosslink into a three-dimensional network by strong chemical bonds, and enable the gel to retain its shape after a large deformation.  The solvent molecules, however, interact among themselves and with the network by weak physical bonds, and enable the gel to be a conduit of mass transport.  The time-dependent concurrent process of large deformation and mass transport is studied by developing a finite element method. 

Xuanhe Zhao and Zhigang Suo  We develop a thermodynamic model of electrostriction for elastic dielectrics capable of large deformation. The model reproduces the classical equations of state for dielectrics at small deformation, but shows that some electrostrictive effects negligible at small deformation may become pronounced at large deformation.

Subject to a voltage, a dielectric elastomer can deform substantially, making it a desirable material for actuators. Designing such an actuator, however, has been challenging due to nonlinear equations of state, as well as multiple modes of failure, parameters of design, and measures of performance. This paper explores these issues, using a spring-roll actuator as an example. We formulate the equations of state of two degrees of freedom, and describe the constraints due to several modes of failure of the elastomer, including electrical breakdown, electromechanical instability, loss of tension, and tensile rupture. Also included is the compressive limit of the spring.

In a previous paper , we have demonstrated that a microcrystalline copper film well bonded to a polymer substrate can be stretched beyond 50% without cracking. The film eventually fails through the co-evolution of necking and debonding from the substrate. Here we report much lower strains to failure (around 10%) for polymer-supported nanocrystalline metal films, whose microstructure is revealed to be unstable under mechanical loading.

At the invitation of David Clarke on behalf of the UCSB/Los Alamos Institute of Multiscale Materials and Structures, I gave the following three lectures:

1. Large deformation and instability in dielectric elastomers
2. Large deformation and instability in swelling polymeric gels
3. Mechanics and electrochemistry of polyelectrolyte gels

The abstracts follow, and the slides are attached at the end of this post.

A network of polymers can imbibe a large quantity of a solvent and swell, resulting in a gel.  The swelling process can be markedly influenced by a mechanical load and geometric constraint.  When the network, solvent, and mechanical load equilibrate, the gel usually swells by a field of inhomogeneous and anisotropic deformation.  We show that this field in the swollen gel is equivalent to that in a hyperelastic solid.  We implement this theory in the finite-element package, ABAQUS, and analyze examples of swelling-induced deformation, contact, and bifurcation.  Because commercial software like ABAQUS is widely available, this work may provide a powerful tool to study complex phenomena in gels.