Torsional Creep Research Articles

Torsional creep problems involving Grushin-type operators

In this paper we study the asymptotic behaviour of solutions for a family of torsional creep problems involving the Grushin p-Laplacian. Our results complement some earlier works on the topic by Payne and Philippin (1977), Kawohl (1990) and Bhattacharya et al. (1991).

A monotonicity property of the [formula omitted]-torsional rigidity

For a bounded domain Ω⊂RN,N≥2, and a real number p>1, we denote by up the p-torsion function on Ω, that is the solution of the torsional creep problemΔpu=−1 in Ω, u=0 on ∂Ω, where Δpu≔div(∇up−2∇u) is the p-Laplace operator. Our aim is to investigate some monotonicity properties for the p-torsional rigidity on Ω, defined as TpΩ≔∫Ωupdx. More precisely, we first show that there exists T∈0,1 such that for each open, bounded, convex domain Ω⊂RN, with smooth boundary and δΩ≤T, where δΩ represents the average integral on Ω of the distance function to the boundary of Ω, the function p→Tp;Ω≔Ωp−1TpΩ1−p is increasing on 1,∞. Moreover, we also show that for any real number s>T, there exists an open, bounded, convex domain Ω⊂RN, with smooth boundary and δΩ=s, such that the function p→Tp;Ω is not a monotone function of p∈(1,∞). Finally, we use this result to get a new variational characterization of T(p;Ω), in the case when δΩ is small enough.

On a family of torsional creep problems in Finsler metrics

AbstractThe asymptotic behavior of solutions to a family of Dirichlet boundary value problems, involving differential operators in divergence form, on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the distance function to the boundary of the domain, which takes into account the Finsler norm involved in the equation. This implies that a well-known result in the analysis of problems modeling torsional creep continues to hold in this more general setting.

Anisotropic torsional creep problems involving rapidly growing differential operators

The asymptotic behaviour of the sequence of positive solutions for a family of torsional creep-type problems involving anisotropic rapidly growing differential operators is studied in a bounded domain from the Euclidian space RN. We prove that the sequence of solutions converges uniformly on the domain to a certain distance function defined in accordance with the anisotropy of the problem.

On a family of torsional creep problems involving rapidly growing operators in divergence form

Let Ω⊂ℝN (N≥2) be a bounded domain with smooth boundary and {pn} be a sequence of real numbers converging to+∞ as n→∞. For each integer n>1, we define the function $\varphi_{n}(t)=p_{n} \vert t \vert^{p_{n}-2}te^{ \vert t \vert^{p_{n}}}$, for all t∈ℝ, and we prove the existence of a unique nonnegative variational solution for the problem−div(((φn(|∇ u(x)|))/(|∇ u(x)|))∇ u(x))=φn(1), when x∈Ω, subject to the homogeneous Dirichlet boundary condition. Next, we establish the uniform convergence in Ω of the sequence of solutions for the above family of equations to the distance function to the boundary of Ω. Our result complements the earlier developments on the topic obtained by Payne and Philippin [26], Kawohl [21], Bhattacharya, DiBenedetto and Manfredi [2], Perez-Llanos and Rossi [27] and Bocea and Mihăilescu [4].

Physical aging of shape memory polymers based upon epoxy-thiol “click” systems

The physical aging behaviour of two epoxy-thiol shape memory polymers has been studied: diglycidyl ether of bisphenol-A cross-linked with pentaerythritol tetrakis, denoted 4-thiol, with a calorimetric glass transition temperature, Tg, of 51 °C; and the same system modified with tri (2,3-epoxypropyl)isocyanurate in a 30 wt% proportion, denoted 4-thiol-30%iso, with Tg = 63 °C. The aging of both polymers has been measured by torsional creep for aging temperatures between room temperature and the Tg of each polymer. The creep behaviour of each polymer is characterised by a discrete distribution of relaxation times, which shifts with aging time according to a double logarithmic aging rate, μ, and with aging temperature according to an activation energy. The distribution of creep relaxation times for 4-thiol is rather symmetrical, and slightly narrower than that for 4-thiol-30%iso, which is also skewed to longer times. The dependence of μ on temperature displays a peak as the aging temperature reduces below Tg, and is sharper for 4-thiol, in accordance with the narrower distribution, as predicted by a theoretical model based upon structural relaxation kinetics. The other parameters defining the aging behaviour, namely the reduced activation energy (Δh*/R = 92 kK for 4-thiol and 82 kK for 4-thiol-30%iso) and the non-linearity parameter of structural relaxation (x = 0.35 for 4-thiol and 0.25 for 4-thiol-30%iso), have been determined experimentally and are compared with the predictions of the theoretical model. These parameters can be used to predict the effects of physical aging on the shape memory response.

Thermomechanical effects in the torsional actuation of twisted nylon 6 fiber

ABSTRACTThermally induced torsional and tensile actuators based on twisted polymeric fibers have opened new opportunities for the application of artificial muscles. These newly developed actuators show significant torsional deformations when subjected to temperature changes, and this torsional actuation is the defining mechanism for tensile actuation of twisted and coiled fibers. To date it has been found that these actuators require multiple heat/cool cycles (referred to as “training” cycles) prior to obtaining a fully reversible actuation response. Herein, the effect of annealing conditions applied to twisted nylon 6 monofilament is investigated and it is shown that annealing at 200 °C eliminates the need for the training cycles. Furthermore, the effect of an applied external torque on the torsional actuation is also investigated and torsional creep is shown to be affected by the temperature and load. © 2017 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2017, 134, 45529.

On the convergence of the sequence of solutions for a family of eigenvalue problems

The asymptotic behavior of the sequence {un} of positive first eigenfunctions for a class of eigenvalue problems is studied in a bounded domain with smooth boundary ∂Ω. We prove , where δ is the distance function to ∂Ω. Our study complements some earlier results by Payne and Philippin, Bhattacharya, DiBenedetto, and Manfredi, and Kawohl obtained in relation with the “torsional creep problem.”

On a family of inhomogeneous torsional creep problems

On a family of inhomogeneous torsional creep problems

Relaxation and creep in twist and flexure

Purpose– The purpose of this paper is to derive the exact analytical expressions for torsion and bending creep of rods with the Norton-Bailey, Garofalo and Naumenko-Altenbach-Gorash constitutive models. These simple constitutive models, for example, the time- and strain-hardening constitutive equations, were based on adaptations for time-varying stress of equally simple models for the secondary creep stage from constant load/stress uniaxial tests where minimum creep rate is constant. The analytical solution is studied for Norton-Bailey and Garofalo laws in uniaxial states of stress.Design/methodology/approach– The creep component of strain rate is defined by material-specific creep law. In this paper the authors adopt, following the common procedure Betten, an isotropic stress function. The paper derives the expressions for strain rate for uniaxial and shear stress states for the definite representations of stress function. First, in this paper the authors investigate the creep for the total deformation that remains constant in time.Findings– The exact analytical expressions giving the torque and bending moment as a function of the time were derived.Research limitations/implications– The material isotropy and homogeneity preimposed. The secondary creep phase is considered.Practical implications– The results of creep simulation are applied to practically important problem of engineering, namely for simulation of creep and relaxation of helical and disk springs.Originality/value– The new, closed form solutions with commonly accepted creep models allow a deeper understanding of such a constitutive model's effect on stress and deformation and the implications for high temperature design. The application of the original solutions allows accurate analytic description of creep and relaxation of practically important problems in mechanical engineering. Following the procedure the paper establishes closed form solutions for creep and relaxation in helical, leaf and disk springs.

Stress relaxation of nickel-based superalloy helical springs at high temperatures

The creep resistance of materials in spring applications is generally acknowledged to be well below that observed in other applications. Helical springs formed from three candidate nickel-based superalloys, Nimonic 90, René 41 and Haynes 282, have been tested under compression in order to gain some insight into this phenomenon. Stress relaxation tests conducted at 600–700°C found that, under constant displacement, the degradation of the spring force is one to three orders of magnitude faster than would be predicted from creep data from extruded samples under equivalent tensile loading. An analytical model for torsional creep in helical springs is derived from a modified version of the Dyson creep model. The effects of various microstructural features on the deformation rate are considered. Effects such as the coarsening of the precipitate-strengthening gamma-prime phase, tertiary creep due to dislocation multiplication, damage evolution and hardening due to transfer of the stress to the particles from the matrix are concluded to make negligible contributions. It is predicted that the poor performance of the springs is due to the very high population of geometrically necessary dislocations that result from the bending and twisting of the wire into a helical coil. It is expected that these dislocations are resistant to conventional heat treatments, resulting in a persistent residual stress field and a large number of dislocations to facilitate the creep process. In some cases, the stress relaxation is found to be so fast that the precipitate hardening of the alloy is too slow to prevent significant initial degradation of the spring.

Some basic solutions for nonlinear creep

The aim of the paper is to derive the exact analytical expressions for torsion and bending creep of rods that obey the Norton–Bailey, Prandtl–Garofalo and Naumenko–Altenbach–Gorash constitutive models. The common secondary creep constitutive model is the Norton–Bailey law which gives a power law relationship between creep rate and stress. The closed form solutions for fractional Norton–Bailey creep law are derived. The analytical formulas express the torque and bending moment as functions of the time for the period of relaxation. Other formulas express the twist rate and curvature as functions of the time for the duration of engineering creep experiment. The derived formulas are suitable for the practically important problems of machinery. Namely, the formulas are relevant for calculation of hereditary effects for helical, leaf and disk springs and twisted shafts.

Creep strength of Mod.9Cr1Mo steel under biaxial stress states

This paper describes the development of tension, torsion, and combined tension-torsion creep testing machine with hydro static thrust bearing, criterion on creep rupture strength under biaxial stress states was examined. Tension, torsion, and combined tension-torsion creep tests were carried out at 873 K in air employing the developed creep testing machine and hollow cylindrical specimen of Mod.9Cr1Mo steel. Ratio of rupture time tR/tR* increased with the decreasing principal stress ratio λ at constant Mises' equivalent stress, where tR, and tR* denote rupture times for biaxial and tension creep tests, and λ expresses ratio of minimum principal stress to maximum principal stress. Hayhurst's stress successfully correlated with tension, torsion, and combined tension-torsion loading creep lives but Mises' type equivalent stress and maximum principal stress were inadequate to predict creep lives of Mod.9Cr1Mo steel. Oxide film and micro cracks on specimen surfaces were observed using SEM. Micro cracks and oxide film cracks initiated to the perpendicular direction to maximum principal stress axis for -1≦λ≦0. Main crack grew in direction perpendicular to specimen axis independent on the maximum stress ratio λ. The residual lives under biaxial stress states were estimated using Omega method. The Omega method was able to estimate the residual lives within a factor of 4 under biaxial stress states.

Orthogonal Dynamic Model of Viscoelastic Deformation Under Variable Concentration and Loading

Abstract In the present work, a generalized mathematical model is considered for describing the process of viscoelastic deformation of poly- mer materials. The model is applied for describing the creep of poly- ethylene in the conditions of diffusion in liquid environment interacting chemically with the polymer and variable loading. The orthogonal expansions method is used and is extended for a process with integrating properties (without self-settling). In this case, it is necessary to separate the integral component, which is also presented as parameter-dependent on loading and saturation. The output experiments are executed with samples which possess reached beforehand different levels of saturation with cyclohexan and have been afterwards submitted to mechanical testing with brief creep in pure torsion at various constant levels of loading.

Comparison of microstructures in superplastically deformed synthetic materials and natural mylonites: Mineral aggregation via grain boundary sliding

We conducted compressional, tensile, and torsional creep experiments on fine-grained forsterite plus Ca-bearing pyroxene aggregates. A distinct microstructure with aggregation of the same phase in the direction of compression was formed in our samples after all the experiments. The stress–strain rate relationship, grain-size dependent flow strength, and the achievement of large tensile strain all indicate that samples underwent creep due to grain boundary sliding (GBS). As a result of GBS, grain-switching events allow dispersed phases to contact grains of the same phase and orient in the direction of compression. We identify similar aggregated microstructures in previously reported micrographs of polymineralic granite-origin ultramylonites. Mineral phase mixing through GBS, which helps to retain fine grain size in rocks due to grain boundary pinning, has been speculated to occur during formation of mylonites. However, our results contradict this hypothesis because mineral aggregation through GBS promotes demixing rather than mixing of the mineral phases. GBS processes alone will not promote a transformation of well-developed monomineralic bands to polymineralic bands during mylonitization.

Creep behaviour of ice single crystals loaded in torsion explained by dislocation cross-slip

Torsion creep experiments are carried out in order to understand the physics of ice plasticity. A dislocation spreading mechanism based on double cross-slip of basal dislocations is proposed to explain the strong plastic anisotropy and the power law relationship between stress and strain rates. The scenario is tested using three-dimensional dislocation dynamics simulations. Numerical investigations give a stress exponent n = 2.3 in agreement with experimental measurements. This dislocation spreading mechanism sheds a new light on the interpretation of former experimental observations.

Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions

We introduce an iterative method for computing the first eigenpair (λ p ,e p ) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (μ q, u q ) as q→p −, where u q is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with the same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of u q to e p is in the C 1-norm and the rate of convergence of μ q to λ p is at least O(p−q). Numerical evidence is presented.

Development and Application of Test Equipment for Torsion Strength of Non-Metal Materials

In order to determine the torsion strength of non-metal materials at ambient and high temperature, a kind of equipment with inductive heating, infrared thermometer and torque loading by mechanical electrical rider was development. This equipment has advantages such as quick heating, accurate temperature test and controlling, easy controlling and continues of torque loading and test atmosphere can be controlled. The torsion strength of fireclay bricks with sample size of 40 mm × 40 mm × 230 mm, were tested separately at room temperature, 800°C, 1000°C, 1100°C and 1200°C, using this equipment. Results indicated that for the same batch of samples, the torsion strength determined by this equipment has good consistency, and mean while, it was found that the torsion strength decreased with the increase of test temperature obviously. In additional, developed equipment could be used for the determination of torsion creep at high temperature, the highest temperature of fracture under certain torque during the heating process, torque cycle fatigue failure of materials and so on.

Solutions of the Cheeger problem via torsion functions

The Cheeger problem for a bounded domain Ω ⊂ R N , N > 1 consists in minimizing the quotients | ∂ E | / | E | among all smooth subdomains E ⊂ Ω and the Cheeger constant h ( Ω ) is the minimum of these quotients. Let ϕ p ∈ C 1 , α ( Ω ¯ ) be the p-torsion function, that is, the solution of torsional creep problem − Δ p ϕ p = 1 in Ω, ϕ p = 0 on ∂ Ω, where Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplacian operator, p > 1 . The paper emphasizes the connection between these problems. We prove that lim p → 1 + ( ‖ ϕ p ‖ L ∞ ( Ω ) ) 1 − p = h ( Ω ) = lim p → 1 + ( ‖ ϕ p ‖ L 1 ( Ω ) ) 1 − p . Moreover, we deduce the relation lim p → 1 + ‖ ϕ p ‖ L 1 ( Ω ) ⩾ C N lim p → 1 + ‖ ϕ p ‖ L ∞ ( Ω ) where C N is a constant depending only of N and h ( Ω ) , explicitely given in the paper. An eigenfunction u ∈ BV ( Ω ) ∩ L ∞ ( Ω ) of the Dirichlet 1-Laplacian is obtained as the strong L 1 limit, as p → 1 + , of a subsequence of the family { ϕ p / ‖ ϕ p ‖ L 1 ( Ω ) } p > 1 . Almost all t-level sets E t of u are Cheeger sets and our estimates of u on the Cheeger set | E 0 | yield | B 1 | h ( B 1 ) N ⩽ | E 0 | h ( Ω ) N , where B 1 is the unit ball in R N . For Ω convex we obtain u = | E 0 | − 1 χ E 0 .

The Influence of Different Sterilization Techniques on the Time-Dependent Behavior of Polyamides

For this investigation conventional polyamide 6 with monomodal molecular mass distribution, and the newly developed bimodal one were used. Conventional polyamide 6 was used as a reference material in order to emphasize prospects of using bimodal material for medical applications from the point of view of sterilization resistance and improved creep behavior. Time-dependent mechanical properties of testing samples were characterized by torsional creep measurements in non-sterilized state and after sterilization with three different techniques: with autoclave, ethylene oxide, and hydrogen peroxide plasma. Results show that the two materials exhibit pronounced difference in morphology and consequently, mechanical properties. Both of them were not significantly affected by any of used sterilization techniques. However, bimodal material, originally being noticeably more time-stable in comparison to monomodal one, retains these preferences also post sterilization.