Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

Prediction of creep rupture in 2.25Cr–1Mo notched bars

This paper re-examines the creep life methodology based on the continuum damage mechanics (CDM) of the Kachanov and Rabotnov theory. Uniaxial creep and multiaxial creep rupture formulations are presented taking into account the primary creep effect. The scalar damage parameter is computed up to time-to-rupture as a function of time and stress. The methodology implemented is based on the uniaxial time-to-rupture obtained experimentally. The times-to-rupture for bars with different notches are calculated. It is demonstrated that the use of the damage parameter is vital to indicate the critical damage location where failure occurs. Results are compared to those obtained experimentally. It is shown that the primary creep inclusion has a significant effect on the damage distribution zone.     

On the viscoelastic characterization of the Jeffreys–Lomnitz law of creep

In 1958, Jeffreys (Geophys J?R Astron Soc 1:92–95) proposed a power law of creep, generalizing the logarithmic law earlier introduced by Lomnitz, to broaden the geophysical applications to fluid-like materials including igneous rocks. This generalized law, however, can be applied also to solid-like viscoelastic materials. We revisit the Jeffreys–Lomnitz law of creep by allowing its power law exponent α, usually limited to the range 0?≤?α?≤?1 to all negative values. This is consistent with the linear theory of viscoelasticity because the creep function still remains a Bernstein function, that is positive with a completely monotone derivative, with a related spectrum of retardation times. The complete range α?≤?1 yields a continuous transition from a Hooke elastic solid with no creep $\left(\alpha \,\to\, -\infty\right)$ to a Maxwell fluid with linear creep $\left(\alpha \,=\,1\right)$ passing through the Lomnitz viscoelastic body with logarithmic creep $\left(\alpha\, =0\right)$ , which separates solid-like from fluid-like behaviors. Furthermore, we numerically compute the relaxation modulus and provide the analytical expression of the spectrum of retardation times corresponding to the Jeffreys–Lomnitz creep law extended to all α?≤?1.     

Influence of nanoprecipitates on the creep strength and ductility of a Fe–Ni–Al alloy

Creep studies of a duplex Fe–Ni–Al intermetallic alloy, in two microstructural states, have been carried out at temperatures between 725 and 800 °C (about 0.6 T_m). In the as-cast state, the alloy contains a large volume fraction of nanoprecipitates (50–100 nm) which confer a very high creep strength with a stress exponent of 3 and an activation energy of 280 kJ/mol. The different microstructure obtained in the second state of the alloy, obtained after annealing at 1000 °C for 24 h, leads to a much lower creep strength with a higher stress exponent as well as a large value of the apparent activation energy. While volume diffusion appears to control creep in the as-cast state, both thermal and athermal processes seem to contribute to the different creep rate of material in the annealed state. The latter also exhibits a much larger ductility (12%) relative to that observed in the as-cast material (3%), due to the presence of large numbers of interfaces between the two phases present where strain incompatibilities can be accommodated.     

Time-dependent thermoelastic creep analysis of rotating disk made of Al–SiC composite

Time-dependent creep stress redistribution analysis of rotating disk made of Al–SiC composite is investigated using Mendelson’s method of successive elastic solution. All mechanical and thermal properties except Poisson’s ratio are radial dependent based on volume fraction percent of SiC reinforcement. The material creep behavior is described by Sherby’s constitutive model using Pandey’s experimental results on Al–SiC composite. Loading is an inertia body force due to rotation and a distributed temperature field due to steady-state heat conduction from inner to outer surface of the disk. Using equations of equilibrium, stress strain, and strain displacement, a differential equation, containing creep strains, for displacement is obtained. History of stresses and deformations are calculated using method of successive elastic solution. It is concluded that the uniform distribution of SiC reinforcement does not considerably influence on stresses. However, the minimum and most uniform distribution of circumferential and effective thermoelastic stresses belongs to composite disk of aluminum with 0% SiC at inner surface and 40% SiC at outer surface. It has also been found that the stresses, displacement, and creep strains are changing with time at a decreasing rate so that after almost 50 years the solution approaches the steady-state condition.     

One parameter model for creep in a whisker reinforced anisotropic rotating disc of Al–SiCw composite

Steady state creep in a rotating disc of anisotropic aluminum silicon carbide whisker composite has been studied in the present study. The creep behavior is described by Norton's power law. Stress and strain rate distributions for anisotropic discs have been calculated and compared with those obtained for isotropic disc. It is concluded that the radial strain rate which always remained compressive for the isotropic composite ($\alpha =1.0$) and anisotropic disc ($\alpha =1.3$), becomes tensile in the middle region of the disc when the anisotropy parameter $\alpha =0.7$. Also if α is reduced from 1.3 to 0.7 the variation of tensile strain rate in the tangential direction remains similar but the magnitude reduces by five orders of magnitude. The study revealed that anisotropy introduces significant change in the strain rates although its effect on the resulting stress distribution may be relatively small.     

Mechano-sorptive creep under compressive loading – A micromechanical model

The creep of paper is accelerated by moisture cycling, an effect known as mechano-sorptive creep. It has also been observed that the mechano-sorptive effects are larger in compression than in tension. In this paper a simplified network model for mechano-sorptive creep is presented. It is assumed that the anisotropic hygroexpansion of the fibres leads to large stresses at the fibre–fibre bonds when the moisture content changes. The resulting stress state will accelerate creep if the fibre material obeys constitutive laws that are non-linear in stress. Geometrical fibre effects are included in the model in order to capture experimental observations of the differences between paper loaded in tension and compression. Theoretical predictions based on the developed model are compared to experimental results for paper both under tensile and compressive loading at varying moisture content. The important features in the experiments are captured by the model, i.e. the creep is accelerated by the moisture cycling and the mechano-sorptive effects are larger in compression than in tension.     

Approximate estimates of the high‐temperature creep of structural elements

The creep and longterm strength of structural elements under temperatures much higher than operational temperatures are studied. The actual nonuniform stress–strain state is reduced to a fictitious uniform state by averaging the specific dissipated power over the volume of the body. This allows one to estimate the intensity of creep and longterm strength of structural elements in terms of volumeaveraged energy parameters by using methods of ideal plasticity and considering statically possible stress fields and kinematically possible velocity and strain rate fields.     

Effect of the silicon-carbide micro- and nanoparticle size on the thermo-elastic and time-dependent creep response of a rotating Al–SiC composite cylinder

The history of stresses and creep strains of a rotating composite cylinder made of an aluminum matrix reinforced by silicon carbide particles is investigated. The effect of uniformly distributed SiC micro- and nanoparticles on the initial thermo-elastic and time-dependent creep deformation is studied. The material creep behavior is described by Sherby’s constitutive model where the creep parameters are functions of temperature and the particle sizes vary from 50 nm to 45.9 μm. Loading is composed of a temperature field due to outward steady-state heat conduction and an inertia body force due to cylinder rotation. Based on the equilibrium equation and also stress-strain and strain-displacement relations, a constitutive second-order differential equation for displacements with variable and time-dependent coefficients is obtained. By solving this differential equation together with the Prandtl–Reuss relation and the material creep constitutive model, the history of stresses and creep strains is obtained. It is found that the minimum effective stresses are reached in a material reinforced by uniformly distributed SiC particles with the volume fraction of 20% and particle size of 50 nm. It is also found that the effective and tangential stresses increase with time at the inner surface of the composite cylinder; however, their variation at the outer surface is insignificant.     

Two‐dimensional problems of beam forming under conditions of creep

Direct and inverse problems of forming of longlength profiles with double curvature and a given angle of twisting under conditions of creep are considered. A finitedifference scheme for the numerical solution is proposed. Examples of solving problems with different types of external actions for a profile with a rectangular cross section are given. Experimental and numerical data are compared for twisting of beams with square and circular cross sections in the regime of creep at temperatures of 725 and 740° C for St. 45 steel.     

Hadamard instability analysis of “negative creep” in coupled chemo-thermo-mechanical systems

The coupled system of nonlinear partial differential equations for momentum, diffusion and energy is examined in terms of Hadamard instability, which in a unified way provides the conditions of both “negative creep” and “spinodal decomposition” (loss of convexity of thermodynamic functions) (Markenscoff in Quart Appl Math 59:147–151, 2001; Quart Appl Math 59:471–477, 2001) by balancing terms of different orders in the eigenvalue equation. It is shown here that instabilities of “negative creep” occur in both infinite and finite domains.     

Application of the Cercignani–Lampis scattering kernel to calculations of rarefied gas flows. III. Poiseuille flow and thermal creep through a long tube

The Cercignani–Lampis scattering kernel of the gas–surface interaction is applied to numerical calculations of the Poiseuille flow and thermal creep through a long tube. The S model of the Boltzmann equation was numerically solved by the discrete velocity method. The calculations have been carried out in the wide ranges of the rarefaction parameter and of the accommodation coefficients. Comparing the present results with experimental data the values of the accommodation coefficients have been calculated.     

Rabotnov’s two-layer model of a shell and critical time of shell buckling during creep

The main propositions of Rabotnov’s two-layer model of a shell are given. It is shown that the Rabotnov’s functional can be obtained from the mixed variational principle of creep theory. The notion of critical time is introduced and a procedure for obtaining an explicit formula for it using a variational equation is described.     

Vlasov–Maxwell–Boltzmann Diffusive Limit

Inspired by the work (Bastea et al. in J Stat Phys 1011087–1136, 2000) for binary fluids, we study the diffusive expansion for solutions around Maxwellian equilibrium and in a periodic box to the Vlasov–Maxwell–Boltzmann system, the most fundamental model for an ensemble of charged particles. Such an expansion yields a set of dissipative new macroscopic PDEs, the incompressible Vlasov–Navier–Stokes–Fourier system and its higher order corrections for describing a charged fluid, where the self-consistent electromagnetic field is present. The uniform estimate on the remainders is established via a unified nonlinear energy method and it guarantees the global in time validity of such an expansion up to any order.     

Yielding and intrinsic plasticity of Ti–Zr–Ni–Cu–Be bulk metallic glass

Bulk metallic glass with composition Ti₄₀Zr₂₅Ni₈Cu₉Be₁₈ exhibits considerably high compressive yield stress, significant plasticity (with a concomitant vein-like fracture morphology) and relatively low density. Yielding and intrinsic plasticity of this alloy are discussed in terms of its thermal and elastic properties. An influence of normal stresses acting on the shear plane is evidenced by: (i) the fracture angle (<45°) and (ii) finite-element simulations of nanoindentation curves, which require the use of a specific yield criterion, sensitive to local normal stresses acting on the shear plane, to properly match the experimental data. The ratio between hardness and compressive yield strength (constraint factor) is analyzed in terms of several models and is best adjusted using a modified expanding cavity model incorporating a pressure-sensitivity index defined by the Drucker–Prager yield criterion. Furthermore, comparative results from compression tests and nanoindentation reveal that deformation also causes strain softening, a phenomenon which is accompanied with the occurrence of serrated plastic flow and results in a so-called indentation size effect (ISE). A new approach to model the ISE of this metallic glass using the free volume concept is presented.     

An example of stick–slip and stick–slip–separation waves

The dynamical problem of a brake-like mechanical system composed of an elastic cylindrical tube with Coulomb's friction in contact with a rigid and rotating cylinder is considered. This model problem enables us to give an example of non-trivial periodic solutions in the form of stick–slip or stick–slip–separation waves propagating on the contact surface. A semi-analytical analysis of stick–slip waves is obtained when the system of governing equations is reduced by condensation to a simpler system involving only the contact displacements. This reduced system, of only one space variable in addition to time, can be solved almost analytically and gives some interesting informations on the existence and the characteristics of stick–slip waves such as the wave numbers on the circumference, stick and slip proportions, wave celerities, tangential and normal forces. It is shown in particular that the stick–slip–separation solutions would occur for small normal pressures or high rotational speeds. Since the analytical discussion becomes cumbersome in this case, a second approach based on numerical analysis by the finite element method is performed. The existence and the characteristics of stick–slip and stick–slip–separation waves are discussed numerically.     

Ginzburg–Landau Vortices Driven by the Landau–Lifshitz–Gilbert Equation

A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg–Landau vortices for sphere-valued maps. In particular, we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization are ruled by the Landau–Lifshitz–Gilbert equation, which combines characteristic properties of a nonlinear Schrödinger equation and a gradient flow. This paper studies the motion of the vortex centers under this evolution equation.     

Weak–Strong Uniqueness Property for the Full Navier–Stokes–Fourier System

The Navier–Stokes–Fourier system describing the motion of a compressible, viscous and heat conducting fluid is known to possess global-in-time weak solutions for any initial data of finite energy. We show that a weak solution coincides with the strong solution, emanating from the same initial data, as long as the latter exists. In particular, strong solutions are unique within the class of weak solutions.