Yuri Lapusta

A theory of constrained swelling of a pH-sensitive hydrogel†‡

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. This paper develops a theory of constrained swelling of a pH-sensitive hydrogel, a network of polymers bearing acidic groups, in equilibrium with an aqueous solution and mechanical forces. The condition of equilibrium is expressed as a variational statement of the inhomogeneous field. A free-energy function accounts for the stretching of the network, mixing of the network with the solution, and dissociation of the acidic groups. Within a Legendre transformation, the condition of equilibrium for the pH-sensitive hydrogel is equivalent to that for a hyperelastic solid. The theory is first used to compare several cases of homogenous swelling: a free gel, a gel attached to a rigid substrate, and a gel confined in three directions. To analyze inhomogeneous swelling, we implement a finite element method in the commercial software ABAQUS, and illustrate the method with a layer of the gel coated on a spherical rigid particle, and a pH-sensitive valve in microfluidics.

On designing dielectric elastomer actuators

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 with 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. We show that, for the spring-roll actuator, loss of tension in the axial direction will always precede electromechanical instability. We then describe a procedure to maximize the range of actuation by choosing the parameters of design, such as the prestretch of the elastomer and the stiffness of the spring.

An Electrically Charged Crack in a Piezoelectric Material Under Remote Electromechanical Loading

The problem of electrically charged crack in a piezoelectric material under the action of remote electromechanical load is considered. The material is polarized in the direction orthogonal to the crack faces. Assuming that all fields are independent of the coordinate co-directed with the crack front and using the special presentations of electromechanical quantities via sectionally-analytic function, the problem is reduced to the vector Hilbert problem and solved exactly. All required electromechanical quantities are presented by means of simple formulas. The behavior of stresses, electric field and electric displacement jump are analyzed. It is shown that the electric charge can eliminate the field singularities at one of the two crack tips. The proposed solution, together with the presented table of parameters for some widely used piezoceramics materials, like PZT-4 and many others, can be easily used by the reader for an engineering calculation of the near-crack-tip electromechanical fields in these materials.

Interface cracks in piezoelectric materials

Due to their intrinsic electromechanical coupling behavior, piezoelectric materials are widely used in sensors, actuators and other modern technologies. It is well known that piezoelectric ceramics are very brittle and susceptible to fracture. In many cases, fracture occurs at interfaces as debonding and cracks. This leads to an undesired degradation of electrical and mechanical performance. Because of the practical and fundamental importance of the problem, interface cracks in piezoelectric materials have been actively studied in the last few decades. This review provides a comprehensive survey of recent works on cracks situated at the interface of two materials, at least one of which has piezoelectric or piezoelectromagnetic properties. Different electric boundary conditions along the crack faces are discussed. The oscillating and contact zone models for in-plane straight interface cracks between two dissimilar piezoelectric materials or between piezoelectric and non-piezoelectric ones are reviewed. Different peculiarities related to the investigation of interface cracks in piezoelectric materials for the anti-plane case, for functionally graded and thermopiezoelectric materials are presented. Papers related to magnetoelectroelastic bimaterials, to steady state motion of interface cracks in piezoelectric bimaterials and to circular arc-cracks at the interface of piezoelectric materials are reviewed, and various methods used to address these problems are discussed. The review concludes with an outlook on future research directions.

Effective Piezoelectric Coefficients of Ferroelectric Thin Films on Elastic Substrates

In micro-electromechanical systems consisting of a piezoelectric thin film on a substrate, due to the clamping by the substrate, the effective piezoelectric film properties are different from the bulk material behavior. However, it is of particular difficulty to determine the transverse piezoelectric parameters for such a system. A simple theoretical model by Muralt et al. (1996) allows calculation of the transverse piezoelectric coefficients in terms of the bulk parameters of the piezoelectric material. Relying on the assumption of a rigid wafer this model is a reasonable first approximation, but on the other hand, for the high accuracy needed in technical applications, it may not always be sufficient. Therefore, a more complex theoretical model for a piezoelectric thin film on an elastic substrate was derived, which delivers more realistic results for the transverse piezoelectric coefficients. This model takes the elasticity of the substrate into account, while the PZT layer is fully covered by an electrode and the vertical displacements are suppressed at the bottom of the substrate. In this way, the model represents a system of infinite lateral extent with no overall bending. As the next step, finite element simulations were carried out to verify the simple theoretical model and the new developed model, and there was a good agreement between the theoretical models and the numerical results. Furthermore, a parametric study was performed considering the influence of various bulk material parameters. Finally, the newly proposed theoretical model was compared to more realistic models where the PZT layer possesses an isolated electrode spot instead of being covered fully by an electrode. Different options were used for the boundary conditions at the bottom of the substrate. First, the same boundary conditions as for the new theoretical model were chosen (suppression of the vertical displacements at the bottom of the substrate). Second, the bottom of the substrate was free to move such that overall bending was no longer prevented. As the main result, the comparison to the new theoretical model taking into account the elastic substrate showed only negligible differences, and, thus, it is suggested for the determination of the effective piezoelectric parameters of piezoelectric layers on elastic substrates.

Assessment of the Probability of Failure of Reactor Vessels After Warm Pre-stressing Using Monte Carlo Similations

The probability of failure of reactor vessel with semi–elliptic crack after warm pre-stressing was assessed taken into account the scatter of mechanical properties and loading parameters by Monte–Carlo with importance sampling and first–order reliability method based on limit–state functions from FITNET procedure. The failure assessment diagrams were obtained by means of Monte–Carlo simulations for different crack depth and variable internal pressure.

Stress Intensity Factors for Viscoelastic Axisymmetric Problems Applied to Wood

Many materials used in engineering applications obey to time-dependent behaviours and the mechanical fields are affected by the time effects. As a result, the evolution of the stresses and strains in these materials appear still very complex and difficult to study. Among such cases is the situation when the material has an axisymmetric shape and when it is submitted to a complex fracture loading. In this paper, the creep loading is applied on an axisymmetric viscoelastic orthotropic material and the stress intensity factors are computed in the opening mode, in the shear mode and in the mixed mode using to a finite element approach. The uncoupling method is based on M integral, combining the virtual and real mechanical fields. In the same time, the viscoelastic effects are introduced according to the generalized Kelvin-Voigt model composed by four branches. The numerical solution is obtained with an incremental viscoelastic formulation in the time domain. Using a Compact Tension Shear (CTS) specimen, the evolutions of stress intensity factor versus time are posted in each fracture mode configuration. The obtained results demonstrate the efficiency of the proposed model.

Linear theory of electroelasticity

This chapter is devoted to the presentation of the fundamental concepts of the theory of linear electroelasticity. At the beginning a typical unit cell of a crystalline piezoelectric material is illustrated and the conditions for the occurrence of the piezoelectric effect are discussed.

Electrically impermeable interface cracks in piezoelectric materials

This chapter is devoted to the consideration of electrically impermeable interface cracks between two piezoelectric materials. The presentations of the electromechanical quantities at the interface via sectionally-analytic vector functions, which are convenient for the analysis of the mentioned kind of electrical conditions at the crack faces, are obtained.

A Crack with Electromechanical Pre-fracture Zones

This chapter is devoted to the consideration of cracks in piezoelectric materials with zones of mechanical yielding and electrical saturation.