Deformation and Long-Term Strength of a Thick-Walled Tube of a Physically Non-Linear Viscoelastic Material under Constan
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RMATION AND FRACTURE MECHANICS
Deformation and Long-Term Strength of a Thick-Walled Tube of a Physically Non-Linear Viscoelastic Material under Constant Pressure A. V. Khokhlova, b, * aInstitute
of Mechanics, Moscow State University, Moscow, 119192 Russia AO Komposit, Korolev, Moscow oblast, 141070 Russia *e-mail: [email protected]
b
Received February 5, 2020; revised February 25, 2020; accepted February 28, 2020
Abstract—An exact solution is constructed for the problem of creep and fracture of a hollow cylinder made of a physically nonlinear rheonomic isotropic incompressible material, which obeys Rabotnov’s constitutive viscoelasticity relation with two arbitrary material functions, under the action of internal and external pressures. Сlosed form equations for long-term strength curves are derived using three versions of a deformation fracture criterion, and the strain intensity, the maximum shear strain, or the maximum tensile strain is chosen as the measure of damage. Their properties are analytically investigated for arbitrary material functions of the constitutive relation. Keywords: viscoelasticity, physical nonlinearity, Rabotnov’s constitutive relation, creep, time to failure, longterm strength curve DOI: 10.1134/S0036029520100122
1. INTRODUCTION The results of testing samples and structural elements show that, under a constant, even a sufficiently low load (causing stresses much lower than the ultimate strength), the strain increases in time (creep is observed) and fracture occurs after certain time t* after the application of the load (due to creep and related damage accumulation mechanisms). Lifetime t* depends on the load, the temperature, and other parameters and can be a few years. The dependence of t* on the load or the related stress in a sample t*(σ) (or the inverse dependence σ(t*)) is called the long-term strength curve of a material or structural element. The data of testing various viscoelastoplastic materials [1– 9] show that the t*(σ) dependences always decrease, t*(σ) → +∞ at σ → σ0 and t*(σ) → 0 at σ → σ*, were σ0 ≥ 0 is the (conventional) creep threshold and σ* > 0 is the instantaneous ultimate tensile strength σ*. To predict the lifetime during creep and to simulate the long-term strength of viscoelastic materials, a selected (or constructed) constitutive relation (CR) describing their deformation should be supplemented with a fracture criterion characterizing the fracture time t* when a critical measure of damage ω(t) (scalar, vector, or tensor [1–9]) is reached. The simplest type of fracture criteria is represented by the classical deformation fracture criteria postulating that ω(t) = Cε(t) and fracture occurs at time t = t* when a certain measure of strain ε(t, σ) reaches the limiting value ε* ,
ε(t*,σ) = ε*. They can describe the fracture of both a material and a structural element. It is important that a selected criterion and CR interact well with each other, i.e., make it possible to derive equations for theoretical creep curves ε = ε(t, σ, T) and a long-term strength cur
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