scholarly journals Thick-walled spherical shell problem

2021 ◽  
Vol 21 (1) ◽  
pp. 22-31
Author(s):  
A. M. Artemov ◽  
E. S. Baranovskii ◽  
A. A. Verlin ◽  
E. V. Syomka
Keyword(s):  
Spherical Shell ◽  
Stress Tensor ◽  
Hollow Sphere ◽  
Maximum Shear Stress ◽  
Plastic Region ◽  
Plastic State ◽  
Plasticity Condition ◽  
Stress Deviator ◽  
The Third ◽  
Third Invariant

Introduction. Cylindrical and spherical shells are extensively used in engineering. They face internal and/or external pressure and heat. Stresses and strains distribution in elastoplastic shells has been studied by many scientists. Numerous works involve the use of the von Mises yield conditions, maximum shear stress, maximum reduced stress. These condi- tions do not include the dependence on the first invariant of the stress tensor and the sign of the third invariant of the stress deviator. In some cases, it is possible to obtain numerical-analytical solutions for stresses, displacements and de- formations for bodies with spherical and cylindrical symmetry under axisymmetric thermal and force action.Materials and Methods. The problem on the state of a thick-walled elastoplastic shell is solved within the framework of the theory of small deformations. A plasticity condition is proposed, which takes into account the dependence of the stress tensor on three independent invariants, and also considers the sign of the third invariant of the stress deviator and translational hardening of the material. A disconnected thermoelastoplastic problem is being solved. To estimate the stresses in the region of the elastic state of a spherical shell, an equivalent stress is introduced, which is similar to the selected plasticity function. The construction of the stress vector hodograph is used as a method for verification of the stress state.Results. The problem has an analytical solution for linear plasticity functions. A solution is obtained when the strength- ening of the material is taken into account. Analytical and graphical relationships between the parameters of external action for the elastic or elastoplastic states of the sphere are determined. For a combined load, variants are possible when the plastic region is generated at the inner and outer boundaries of the sphere or between these boundaries.Discussion and Conclusions. The calculation results have shown that taking into account the plastic compressibility and the dependence of the plastic limit on temperature can have a significant impact on the stress and strain state of a hollow sphere. In this case, taking into account the first invariant of the stress tensor under the plasticity condition leads to the fact that not only the pressure drop between the outer and inner boundaries of the spherical shell, but the pressure values at these boundaries, can vary within a limited range. In this formulation of the problem, when there is only thermal action, the hollow sphere does not completely pass into the plastic state. The research results provide predicting the behavior of an object (a hollow sphere) that experiences centrally symmetric distributed power and thermal external influences.

Study on Character of Triaxial Extension Strength of Compacted Clay

2011 ◽  
Vol 243-249 ◽  
pp. 2183-2187
Author(s):  
Jun Xin Liu ◽  
Zhong Fu Chen ◽  
Wei Fang Xu
Keyword(s):  
Stress Tensor ◽  
Triaxial Compression ◽  
Deviatoric Stress ◽  
Stress Deviator ◽  
Strength Ratio ◽  
Compacted Clay ◽  
The Third ◽  
Third Invariant ◽  
Coulomb Criterion ◽  
Triaxial Extension

For soils, failure occurs with lower deviatoric stress under the same pressure (the first invariant of stress tensor) in TXE compared with the strength of the triaxial compression, which is indicated that the strength of soils strongly depends on the third invariant of stress deviator; Although in the traditional Mohr-Coulomb criterion it can be reflected in difference of strength between triaxial extension and compression under the same pressure, it’s nothing to do with the pressure for the strength ratio between triaxial extension and compression. By TXC and TXE, changes of deviatoric stress and the ratio with the pressure were studied

An Extension of Burzyński Hypothesis of Material Effort Accounting for the Third Invariant of Stress Tensor

2011 ◽  
Vol 56 (2) ◽  
pp. 503-508 ◽  
Author(s):  
R. Pęcherski ◽  
P. Szeptyński ◽  
M. Nowak
Keyword(s):  
Stress Tensor ◽  
Elastic Energy ◽  
Yield Condition ◽  
The Third ◽  
Third Invariant ◽  
Lode Angle

An Extension of Burzyński Hypothesis of Material Effort Accounting for the Third Invariant of Stress Tensor The aim of the paper is to propose an extension of the Burzyński hypothesis of material effort to account for the influence of the third invariant of stress tensor deviator. In the proposed formulation the contribution of the density of elastic energy of distortion in material effort is controlled by Lode angle. The resulted yield condition is analyzed and possible applications and comparison with the results known in the literature are discussed.

Effects of Third Invariant of Deviatoric Stress Tensor on Transient Creep of Thick-Walled Tubes

1974 ◽  
Vol 96 (3) ◽  
pp. 207-213 ◽  
Author(s):  
S. Murakami ◽  
Y. Yamada
Keyword(s):  
Mild Steel ◽  
Stress Tensor ◽  
Equivalent Stress ◽  
Transient Creep ◽  
Deviatoric Stress ◽  
The Third ◽  
Third Invariant ◽  
Stress Functions ◽  
Thin Walled ◽  
Deviatoric Stress Tensor

Creep theories with the effect of the third invariant of the deviatoric stress tensor and their accuracy as applied to practical problems are discussed. Constitutive equations for transient creep are first formulated by assuming creep potentials of the Prager-Drucker and the Bailey-Davis type together with the associated equivalent stress functions. Strain-hardening and time-hardening hypotheses are assumed. Experimental results hitherto reported for thin-walled tubes are discussed according to these equations. Then, the creep of a thick-walled tube of mild steel is analyzed and compared with experiments.

scholarly journals Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

2021 ◽  
Author(s):  
Jose Rodriguez-Martinez ◽  
Oana Cazacu ◽  
Nitin Chandola ◽  
Komi Espoir N'souglo
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Yield Criterion ◽  
Equivalent Strain ◽  
Stress Deviator ◽  
Plane Strain Tension ◽  
The Third ◽  
Third Invariant

In this paper, we have investigated the effect of the third invariant of the stress deviator on the formation of necking instabilities in isotropic metallic plates subjected to plane strain tension. For that purpose, we have performed finite element calculations and linear stability analysis for initial equivalent strain rates ranging from 10^−4 s−1 to 8 · 10^4 s−1. The plastic behavior of the material has been escribed with the isotropic Drucker yield criterion [11], which depends on both the second and third invariant of the stress deviator, and a parameter c which determines the ratio between the yield stresses in uniaxial tension and in pure shear \sigma_T /\tau_Y . For c = 0, Drucker yield criterion [11] reduces to the von Mises yield criterion [32] while for c = 81/66, the Hershey-Hosford (m = 6) yield criterion [19, 22] is recovered. The results obtained with both finite element calculations and linear stability analysis show the same overall trends and there is also quantitative agreement for most of the loading rates considered. In the quasi-static regime, while the specimen elongation when necking occurs is virtually insensitive to the value of the parameter c, both finite element results and analytical calculations using Considère criterion [10] show that the necking strain increases as the parameter c decreases, bringing out the effect of the third invariant of the stress deviator on the formation of quasi-static necks. In contrast, at high initial equivalent strain rates, when the influence of inertia on the necking process becomes important, both finite element simulations and linear stability analysis show that the effect of the third invariant is reversed, notably for long necking wavelengths, with the specimen elongation when necking occurs increasing as the parameter c increases, and the necking strain decreasing as the parameter c decreases.

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