Rate-dependent Dissipation Research Articles

Balanced-Viscosity Solutions to Infinite-Dimensional Multi-Rate Systems

We consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable u having a viscous damping with relaxation time varepsilon ^alpha and an internal variable z with relaxation time varepsilon we obtain different limits for the three cases alpha in (0,1), alpha =1 and alpha >1. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.

Three-dimensional material point method modeling of runout behavior of the Hongshiyan landslide

This study presents a field-scale simulation of the Hongshiyan landslide in China. It uses an advanced numerical approach (material point method (MPM)) and a constitutive model (the Drucker–Prager model + μ(I) rheological relation) for the three-dimensional (3D) simulation. The performance of the developed MPM model is validated with laboratory-scale experimental data on granular collapse before being applied to field-scale analyses. ArcGIS data are used to create a 3D MPM model of the soil body with complicated geometry. Although the developed model can describe the multiple phases of granular flow, it focuses on the runout behavior of the landslide in this work. The landslide is assumed to have occurred suddenly due to an earthquake, and global sudden failure rather than progressive failure is modeled. The MPM simulation results match reasonably well with the measured post-earthquake topography (e.g., deposit height of about 120 m and stretch length of about 900 m in the river) and landslide duration of about 1 min. The velocity of the sliding mass increases rapidly during flow, especially in the first 20 s. The velocity profiles along the depth direction at different locations of the sliding body exhibit an exponential distribution similar to that of a Bagnold-type profile, indicating that the sliding body is fully mobilized. The rate-dependent dissipation parameter β used in the model significantly influences the runout behavior (e.g., flow speed, velocity distribution, and deposit shape).

Design and Experimental Validation of a Re-centring Viscous Dissipater

The design and experimental validation of a passive, hybrid self-centring supplemental damper for low-damage structures is presented. The device combines rate-dependent dissipation of a viscous damper with the rate-independent dissipation and re-centring restoring force of a ring-spring in a parallel fashion. Experimental proof-of-concept validation tests use sinusoidal displacement inputs consisting of [25, 30]mm amplitudes and [0.25–1.75]Hz frequencies, for two levels of ring-spring pre-load equal to 21% and 34% of the design force. Each component is tested individually to characterise and delineate their specific contributions to the overall device. The ring-spring has a design force capacity of 26kN and a return stiffness ratio of 34% that remain independent of the velocity input. The viscous device input velocities of [50–330]mm/s provide velocity dependent dissipative forces with a maximum of 22kN at the maximum input velocity. Individual results in summation match the hybrid device results for the same input, indicating no loss of efficiency in the hybrid device load transfer. Overall, these results offer a range of new, easily implemented options for energy dissipation in developing low-damage structures, which can also provide necessary re-centring capability within the same package. The overall method is readily generalised for a wide range of hybrid device force capacities and design requirements.

Laminate-based modelling of single and polycrystalline ferroelectric materials – application to tetragonal barium titanate

The present contribution deals with the development of a laminate-based model designed to study the single and polycrystalline tetragonal ferroelectric material behaviour. Laminate-based models are micromechanically motivated and consider the volume fraction of the distinct ferroelectric variants directly in their formulation. At first, a single crystal laminate-based model is established by considering the average strain and polarisation compatibility conditions. A suitable thermodynamic electric Gibbs energy and a rate-dependent dissipation equation are postulated to capture the dissipative hysteretic material response. The update of the inequality constrained volume fractions is solved by adopting a Fischer–Burmeister-type algorithm in combination with a Newton–Raphson scheme. Following the single crystal formulation, a homogenisation procedure based on random orientation of the individual grains in a polycrystalline aggregate is considered. The material properties and the polarisation switching response of the randomly oriented individual grains are averaged using a finite element framework in order to study the macroscopic polycrystalline behaviour. A parameter fitting procedure based on experimental data for single crystalline response, taken from the literature, is detailed and the material model as well as the algorithmic scheme are verified by solving representative boundary value problems. Moreover, the finite element based simulation results are compared with newly generated experimental hysteresis data for a barium titanate piezoceramic.

Mode I fracture of a biopolymer gel: Rate-dependent dissipation and large deformations disentangled

We have designed a new experimental setup able to investigate fracture of soft materials at small scales. At high crack velocity, where energy is mostly dissipated through viscoelastic processes, we observe an increasingly large high strain domain in the crack tip vicinity. Taking advantage of our ability to determine where linear elasticity breaks down, we derive a simple prediction for the evolution of the energy release rate with the crack velocity.

On rate-dependent dissipation effects in electro-elasticity

This paper deals with the mathematical modelling of large strain electro-viscoelastic deformations in electro-active polymers. Energy dissipation is assumed to occur due to mechanical viscoelasticity of the polymer as well as due to time-dependent effective polarisation of the material. Additive decomposition of the electric field E=Ee+Ev and multiplicative decomposition of the deformation gradient F=FeFv are proposed to model the internal dissipation mechanisms. The theory is illustrated with some numerical examples in the end.

A VANISHING VISCOSITY APPROACH TO A RATE-INDEPENDENT DAMAGE MODEL

We analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arclength reparametrization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.

From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results

We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models.

Rate‐dependent incremental variational formulation of ferroelectricity

AbstractThe paper presents continuous and discrete variational formulations for the treatment of the non‐linear response of piezoceramics under electrical loading. The point of departure is a general internal variable formulation that determines the hysteretic response of the material as a generalized standard medium in terms of an energy storage and a rate–dependent dissipation function. Consistent with this type of standard dissipative continua, we develop an incremental variational formulation of the coupled electromechanical boundary value problem. We specify the variational formulation for a setting based on a smooth rate–dependent dissipation function which governs the hysteretic response. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A theoretical study of the hyperelasticity of electro‐gels

The continuum theory of electro‐elasticity is used in order to describe the large deformations observed in gels endowed with electric properties when they are placed in electric fields. The analytical solution of the properly constructed boundary‐value problem agrees quantitatively with available experimental data.

Crack blunting and the strength of soft elastic solids

When a material is so soft that the cohesive strength (or adhesive strength, in the case of interfacial fracture) exceeds the elastic modulus of the material, we show that a crack will blunt instead of propagating. Large–deformation finite–element model (FEM) simulations of crack initiation, in which the debonding processes are quantified using a cohesive zone model, are used to support this hypothesis. An approximate analytic solution, which agrees well with the FEM simulation, gives additional insight into the blunting process. The consequence of this result on the strength of soft, rubbery materials is the main topic of this paper. We propose two mechanisms by which crack growth can occur in such blunted regions. We have also performed experiments on two different elastomers to demonstrate elastic blunting. In one system, we present some details on a void growth mechanism for ultimate failure, post–blunting. Finally, we demonstrate how crack blunting can shed light on some long–standing problems in the area of adhesion and fracture of elastomers.

Adhesion and Friction of Thin Films

Polymers are routinely placed between solid walls to provide lubrication or adhesion. Their function in these roles depends critically on the degree of dissipation within the polymer film and at the film/wall interface as the film shears or ruptures. Good lubrication is achieved by minimizing frictional dissipation while dissipation increases the strength of adhesive bonds.Gent and Schultz suggested a direct link between frictional losses and the adhesive performance of polymers. This correspondence has been supported by recent experiments and by some of the molecular-dynamics simulations to be described. However we find that the correspondence breaks down when the molecular motion producing dissipation occurs at different locations during shear and rupture. In the following sections, we discuss the types of rate-dependent dissipation observed in thin films and the different factors that control whether dissipation occurs within the polymer or at the wall/film interface. The results suggest an origin for interesting memory effects observed in surface-force-apparatus (SFA) experiments on thin films and expose the atomic-scale processes that produce dissipation during internal rupture of a thin film.