Finite thermo-viscoplasticity of amorphous glassy polymers: Experiments,modeling and simulations

The contribution is concerned with experimental procedures, constitutive modeling and the numerical simulations of finite thermo-viscoplastic behavior of glassy polymers. The experimental study involves both homogeneous and inhomogeneous tests at different temperatures under isothermal conditions. The true stress-true strain curves obtained from compressive homogeneous uniaxial and plane strain experiments are employed in the identification of adjustable material parameters. In contrast to the existing kinematic approaches to finite plasticity of glassy polymers, we propose a distinct kinematic framework constructed in the logarithmic strain space. This leads us to an algorithmically very attractive, additive kinematic structure in R⁶ similar to the geometrically linear theory. The proposed three-dimensional model is implemented into a finite element code. The load-displacement curves acquired from inhomogeneous experiments are compared against the results obtained from finite element analyses where the material parameters identified from homogeneous experiments are used. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Temperature and Rate effects in Finite Viscoplasticity of Glassy Polymers at Different Deformation Modes: Experiments and Simulations

The non–linear inelastic response of glassy polymers is highly influenced by the three–dimensional deformation state, the temperature and the strain rate at which they are deformed. The contribution presents new experiments for different deformation modes which are carried out at different temperatures and rates on commercial bis–phenol A polycarbonate. Emphasis is put not only on the experimental results by themselves but also on the setup and the technique employed in the obtention of the data. The effect of temperature on the velocity with which the neck propagates along the gaged section of a flat specimen under tension is studied means a facility based on photogrammetry. From the homogeneous compression experiments a single set of material parameters appearing in a constitutive model based on the distributed free volume theory under the frame work of additive kinematics will be identified. The inhomogeneous experimental results serve then as a validation for 3–D simulations since the non–uniform strain distribution on the surfaces of both, simulations and experiments, can be compared. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the moisture dependent material behavior of adhesive bonds close to the glass transition temperature

In the presented work, a viscoelastic cross-linked polyurethane is investigated. Environmental influences lead to an inhomogeneous spatial distribution of the mechanical properties in polymer adhesives. Diffusive transport mechanisms transfer water from the environment into the polymer. Further effects like temperature, also have an influence on the mechanical behavior of adhesives. The model respects these influences and takes the incompressibility of the material into account. Viscoelastic behavior can be observed, especially close to the glass transition temperature [3]. Additional to these general effects on polymers, adhesive bonds show a dependency of the mechanical behavior on the thickness of the layer. For numerical investigations, all necessary balance and constitutive equations are implemented in the open-source C++ finite element code deal.II [1, 2]. With the help of this implementation and by comparing experimental results and results gained from simulations, material parameters of the used polymer can be identified. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of deformation and failure in rubber-toughened polymers – effect of distributed crazing

A continuum mechanical model for rubber-toughened polymers undergoing inelastic deformation solely by distributed crazing is introduced. Scaling relations with regard to microstructural parameters are derived analytically from a simple unit cell model. The constitutive model is calibrated from experimental data for a commercial ABS material and well captures various aspects of its deformation and failure behaviour. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Creep-fatigue analysis of the steel AISI type 316 component failure

The purpose of this work is to extend a typical creep-damage model in order to describe material behavior under variable thermal and mechanical loading in wide stress range. The model basis is creep constitutive law in form of hyperbolic sine stress response function proposed by Nadai. The constitutive law is extended to assume the damage process under creep and fatigue by the introduction of scalar damage parameters and appropriate evolution equations according to Kachanov-Rabotnov concept. The material constants for model are identified by fitting the experimental creep and low-cycle fatigue data for the steel AISI type 316 at the range of temperatures 500°C – 750°C. The development of such model is motivated by the well described failure case study of high-temperature components at unit 1 of Eddystone power plant, which have operated during 130520 hours under creep-fatigue interaction conditions. The main steam piping (MSP) from this power plant is selected for thermo-mechanical creep-fatigue analysis applying the proposed material model. The estimated values of damage parameters comply with the real location of the component failure and a scatter of experimental data on creep-fatigue interaction diagram. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resonant destabilization of a floating homogeneous ice layer

We show that a homogeneous elastic ice layer of finite thickness and infinite horizontal extension floating on the surface of a homogeneous water layer of finite depth possesses a countable unbounded set of of resonant frequencies. The water is assumed to be compressible, the viscous effects are neglected in the model. Responses of this water-ice system to spatially localized harmonic in time perturbations with the resonant frequencies grow at least as ?t\sqrt{t} in the two-dimensional (2-D) case and at least as lnt in the three-dimensional (3-D) case, when time t?￥.t\to\infty. The analysis is based on treating the 3-D linear stability problem by applying the Laplace-Fourier transform and reducing the consideration to the 2-D case. The dispersion relation for the 2-D problem D(k,w) = 0,{D}(k,\omega) = 0, obtained previously by Brevdo and Il'ichev [10], is treated analytically and also computed numerically. Here k is a wavenumber, and w\omega is a frequency. It is proved that the system D(k,w) = 0, D_k(k,w) = 0{D}(k,\omega) = 0, {D}_k(k,\omega) = 0 possesses a countable unbounded set of roots (k, w) = (0,w_n), n ? \Bbb Z(k, \omega) = (0,\omega_n), n\in\Bbb Z with Im w_n = 0.\rm{Im}\ \omega_n = 0. Then the analysis of Brevdo [6], [7], [8], [9], which showed the existence of resonances in a homogeneous elastic waveguide, is applied to show that similar resonances exist in the present water-ice model. We propose a resonant mechanism for ice-breaking. It is based on destabilizing the floating ice layer by applying localized harmonic perturbations, with a moderate amplitude and at a resonant frequency.     

Embeddings of anisotropic Besov spaces into Sobolev spaces

We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix A into Sobolev spaces, with a focus on the influence of A on the embedding behavior. For a large range of parameters, we derive sharp characterizations of embeddings.     

Thermoviscoplasticity of Glassy Polymers in the Logarithmic Strain Space Based on the Free Volume Theory

In this contribution a new constitutive model of finite thermo-visco-plastic behavior of amorphous glassy polymers and details of its numerical implementation are outlined. In contrast to existing kinematical approaches to finite plasticity of glassy polymers, the formulation applies a plastic metric theory based on an additive split of Lagrangian Hencky-type strains into elastic and plastic parts [1, 3]. The characteristic strain hardening of the model is derived from a polymer network model, the thermo-visco-plastic flow rule in the logarithmic strain space uses structures of the free volume flow theory [4]. The integration of this micromechanically motivated approach in a three-dimensional computational model is the key novel aspect of this work. An important aspect of this work is the model validation based on experimental findings, whereas the excellent performance of the proposed formulation is demonstrated by means of numerical examples. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parameter identification based on multiple inhomogeneous experiments of practical relevance

Existing procedures to identify material parameters are based on experiments with simple specimens. Additionally, the load distribution is approximately homogeneous. But there are only few feasible experiments which produce homogeneous or almost homogeneous load distributions. Furthermore, deviations from homogeneity and their consequences cannot be avoided, but are often ignored. We present a solution algorithm which takes several different experimental results into account. Thereby, the experiments are performed on specimens which respond with inhomogeneous distributions of strains and stresses. The restriction to homogeneous loads is not necessary. Thus, it is possible to use different measured data of multiple load cases on one and the same test specimen. The component-oriented design of the specimen permits to consider the specific properties of product groups, the load types and the effect of the manufacturing process on the final material properties already during the identification process. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ruin problem for an inhomogeneous semicontinuous integer-valued process

For a process ξ(t = ξ₁(t)+χ(t), t≥0, ξ(0) = 0, inhomogeneous with respect to time, we investigate the ruin problem associated with the corresponding random walk in a finite interval, (here, ξ₁ (t) is a homogeneous Poisson process with positive integer-valued jumps and χ(t) is an inhomogeneous lower-semicontinuous process with integer-valued jumps ξ _n ≥-1).     

Uniqueness of weak solutions of time-dependent 3-D Ginzburg-Landau model for superconductivity

We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ ₀, A ₀) ∈ L ²(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L ³(Ω)) using the Lorentz gauge.      

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lamé’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.     

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Assume that(X,d,μ) is a space of homogeneous type in the sense of Coifman and Weiss(1971,1977). In this article, motivated by the breakthrough work of Auscher and Hyt(o|¨)nen(2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay(for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al.(2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on(X, d,μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure,not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.     

Wellposedness for anisotropic rotating fluid equations

The wellposedness problem for an anisotropic incompressible viscous fluid in R3,rotating around a vector B(t,x):=(b1(t,x),b2(t,x),b3(t,x)),is studied.The global wellposedness in the homogeneous case (B...     

Two-scale simulation of damage in unidirectionally fibre reinforced composite materials

The finite element analysis of engineering structures usually assumes a homogeneous as well as a continuous medium. The heterogeneity of matter, which is always found on a sufficiently small length scale is neglected by replacing the inhomogeneous medium through a model of a mathematically homogenized material. The macroscopic constitutive behaviour is derived from volume averaging procedures that smear the microscopic heterogeneities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element formulation based on the theory of a Cosserat point

The theory of Cosserat points is the basis of a 3D finite element formulation allowing for large deformations in structural mechanics, that recently was presented by [1]. First attempts have revealed, that this formulation is free of showing undesired locking or hourglassing-phenomena. It additionally shows excellent behaviour for any type of incompressible material, for large deformations and sensitive structures such as plates or shells. Within the theory of Cosserat points, the position vectors X and x , are described through director vectors D _i and d _i by use of trilinear shape functions Nⁱ for an 8-node brick element. The special choice of shape functions Nⁱ allows for director vectors with which the deformation can be split into a homogeneous and an inhomogeneous part. This split enables the use of stiffnesses that correspond to different deformation modes. Analytical solutions to the inhomogeneous deformation modes are incorporated in the formulation and avoid the undesired phenomena. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

$\boldsymbol{\mathcal{G}-}$Inhomogeneous Markov Systems of High Order

In the present, we introduce and study the G-\mathcal{G-}inhomogeneous Markov system of high order, which is a more general in many respects stochastic process than the known inhomogeneous Markov system. We define the inhomogeneous superficial razor cut mixture transition distribution model extending for the homogeneous case the idea of the mixture transition model. With the introduction of the appropriate vector stochastic process and the establishment of relationships among them, we study the asymptotic behaviour of the G-\mathcal{G-}inhomogeneous Markov system of high order. In the form of two theorems, the asymptotic behaviour of the inherent G-\mathcal{G-}inhomogeneous Markov chain and the expected and relative expected population structure of the G-\mathcal{G-}inhomogeneous Markov system of high order, are provided under assumptions easily met in practice. Finally, we provide an illustration of the present results in a manpower system.     

Simultaneous inhomogeneous diophantine approximation on manifolds

In 1998, Kleinbock and Margulis proved Sprindzuk’s conjecture pertaining to metrical Diophantine approximation (and indeed the stronger Baker–Sprindzuk conjecture). In essence, the conjecture stated that the simultaneous homogeneous Diophantine exponent w ₀(x) = 1/n for almost every point x on a nondegenerate submanifold M \mathcal{M} of \mathbbRⁿ {\mathbb{R}^n} . In this paper, the simultaneous inhomogeneous analogue of Sprindzuk’s conjecture is established. More precisely, for any “inhomogeneous” vector θ ∈ \mathbbRⁿ {\mathbb{R}^n} we prove that the simultaneous inhomogeneous Diophantine exponent w ₀(x , θ) is 1/n for almost every point x on M \mathcal{M} . The key result is an inhomogeneous transference principle which enables us to deduce that the homogeneous exponent w ₀(x) is 1/n for almost all x ∈ M \mathcal{M} if and only if, for any θ ∈ \mathbbRⁿ {\mathbb{R}^n} , the inhomogeneous exponent w ₀(x , θ) = 1/n for almost all x ∈ M \mathcal{M} . The inhomogeneous transference principle introduced in this paper is an extremely simplified version of that recently discovered by us. Nevertheless, it should be emphasised that the simplified version has the great advantage of bringing to the forefront the main ideas while omitting the abstract and technical notions that come with describing the inhomogeneous transference principle in all its glory.