MODELLING OF TIME-DEPENDENT BEHAVIOUR OF PARTICULATE THERMOSET POLYMERS

A preliminary study of a numerical model describing the behaviour of polymer-based composites is presented. The numerical model consists of three main parts. The first is the microplane M4 model, which is the main part of the model and is used to simulate elastoplastic behaviour and damage. The second part consists of a generalized Maxwell model, which adds the effect of linear creep of the material to the calculation. The last part is a free volume model that extends the linear creep to the nonlinear creep. The creep is calculated on the deviatoric part of the normal stress of each microplane, which allows the model to capture the polymer behaviour adequately without adjusting the free volume of the model.

Mechanics and physics of a glass/particles photonic sponge

A glass containing mechanoluminescent crystalline particles behaves as a photonic sponge: that is to say it fills up with trapped electrons when exposed to UV light, and it emits light when submitted to a mechanical loading, similar to a sponge soaked with water that is wringed under mechanical action! A major finding of the present study is that the elasto-mechanoluminescence effect showing up on unloading is governed by the deviatoric part of the applied stress (no effect under hydrostatic pressure). Furthermore, the structural source for this phenomenon was elucidated by a detailed density functional theory analysis of the e− energetics at the possible oxygen vacancy sites within the crystalline phase. Both the e− trapping and detrapping processes under load could be explained. An analogy with hydraulic circuits and the rheology of viscoelastic media was successfully introduced to pave the way to a constitutive law for the mechano-optical coupling phenomenon.

Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on different representation formulae for objective and isotropic functions in general, as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

Viscoelastic resonant responses of shear deformable imperfect microbeams

A viscoelastic model for the nonlinear analysis of the coupled transverse, longitudinal, and rotational oscillations of an imperfect shear deformable microbeam is developed, for the first time, based on the modified couple stress theory. An energy dissipation mechanism is developed via use of the Kelvin–Voigt internal energy dissipation mechanism. For the stress and deviatoric part of the symmetric couple stress tensors, the viscous components along with the corresponding work terms are obtained. The size-dependent elastic energy along with the kinetic energy of the viscoelastic microsystem is formulated in terms of the displacement field together with system geometric and physical parameters. The internal energy dissipation is developed via the work done by the viscous components of the stress and the deviatoric part of the symmetric couple stress tensors by means of the Kelvin–Voigt mechanism. These work and energy terms are inserted into Hamilton’s principle together with the work due to an external force in order to obtain three viscoelastically coupled equations governing the transverse, longitudinal, and rotational motions with cubic and quadratic nonlinear terms. A high-dimensional Galerkin approximation method is applied for all the three equations, yielding three sets of second-order coupled ordinary differential equations with cubic and quadratic nonlinearities. Upon application of a transformation, a continuation technique along with the backward differentiation formula (BDF) is employed in order to obtain the time-variant response of the system subject to a harmonic load. Special attention is paid to the effect of the Kelvin–Voigt type viscoelasticity on the system response in the presence of the length-scale parameter.

An ellipticity domain for the distortional Hencky logarithmic strain energy

We describe ellipticity domains for the isochoric elastic energy F ↦ ∥ dev n log ⁡ U ∥ 2 = ∥ log ⁡ F T F ( det F ) 1 / n ∥ 2 = 1 4 ∥ log ⁡ C ( det C ) 1 / n ∥ 2 for n =2,3, where C = F T F for F ∈ GL + ( n ). Here, dev n log U = log U − ( 1 / n ) tr ( log U ) ⋅ 1 is the deviatoric part of the logarithmic strain tensor log ⁡ U . For n =2, we identify the maximal ellipticity domain, whereas for n =3, we show that the energy is Legendre–Hadamard (LH) elliptic in the set E 3 ( W H iso , LH , U , 2 3 ) := { U ∈ PSym ( 3 ) | ∥ dev 3 log ⁡ U ∥ 2 ≤ 2 3 } , which is similar to the von Mises–Huber–Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy W H ( F ) = μ ∥ dev 3 log ⁡ U ∥ 2 + ( κ / 2 ) [ tr ( log ⁡ U ) ] 2 , U = F T F with μ >0 and κ > 2 3 μ , previously obtained by Bruhns et al.

Three-Parameter Damage Evolution for Quasi-Brittle Solids

This paper aims to investigate the performance of a new three-parameter damage mechanics model which describes three basic damage mechanisms of quasi-brittle solids: tension, shear, and hydrostatic compression. The stress is first decomposed into its positive part and negative part, and then the latter is further decomposed into its deviatoric part and hydrostatic part, whereby a three-parameter damage description is formulated. Through matrix representation of the tensor formulation, specific forms of the three-parameter damage theory are illustrated in various stress states. It is found that the proposed framework of three-parameter damage theory can inherit the existing two-parameter models and extend them to a broader scope of application.

Partial regularity of solution to generalized Navier-Stokes problem

In the presented work, we study the regularity of solutions to the generalized Navier-Stokes problem up to a C 2 boundary in dimensions two and three. The point of our generalization is an assumption that a deviatoric part of a stress tensor depends on a shear rate and on a pressure. We focus on estimates of the Hausdorff measure of a singular set which is defined as a complement of a set where a solution is Hölder continuous. We use so-called indirect approach to show partial regularity, for dimension 2 we get even an empty set of singular points.

Mathematical and numerical model for nonlinear viscoplasticity

A macroscopic model describing elastic–plastic solids is derived in a special case of the internal specific energy taken in separable form: it is the sum of a hydrodynamic part depending only on the density and entropy, and a shear part depending on other invariants of the Finger tensor. In particular, the relaxation terms are constructed compatible with the von Mises yield criteria. In addition, Maxwell-type material behaviour is shown up: the deviatoric part of the stress tensor decays during plastic deformations. Numerical examples show the ability of this model to deal with real physical phenomena.

Thermodynamically Consistent Anisotropic Plasticity Model

The objective of the work presented in this paper is to consider the thermodynamically consistent anisotropic plasticity model based on full decomposition of stress tensor into generalised deviatoric part and generalised spherical part of stress tensor. Two fundamental tensors αij and βij which represents anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol which plays the main role in the theory of isotropic materials. Using two fundamental tensors, αij and βij, the concept of total generalised “pressure” and pressure corresponding to the volumetric deformation are redefined. It is shown that the formulation of anisotropic plasticity in the case of incompressible plastic flow must be considered independently from the generalised hydrostatic “pressure”. Accordingly, a modification to the anisotropic Hill criterion is introduced. Based on experimental research which has been published, the modified Hill (1948, 1950) criterion for anisotropic elasto-plasticity is validated. The results are presented, discussed and future studies are outlined.