Third Invariant
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Author(s):  
Jonathan Skipp ◽  
Sergey Nazarenko

Abstract We study the thermodynamic equilibrium spectra of the Charney- Hasegawa-Mima (CHM) equation in its weakly nonlinear limit. In this limit, the equation has three adiabatic invariants, in contrast to the two invariants of the 2D Euler or Gross-Pitaevskii equations, which are examples for comparison. We explore how the third invariant considerably enriches the variety of equilibrium spectra that the CHM system can access. In particular we characterise the singular limits of these spectra in which condensates occur, i.e. a single Fourier mode (or pair of modes) accumulate(s) a macroscopic fraction of the total invariants. We show that these equilibrium condensates provide a simple explanation for the characteristic structures observed in CHM systems of finite size: highly anisotropic zonal flows, large-scale isotropic vortices, and vortices at small scale. We show how these condensates are associated with combinations of negative thermodynamic potentials (e.g. temperature).


2021 ◽  
Vol 21 (1) ◽  
pp. 22-31
Author(s):  
A. M. Artemov ◽  
E. S. Baranovskii ◽  
A. A. Verlin ◽  
E. V. Syomka

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.


Author(s):  
Vasyl Muzychuk

The article considers the process of forming the inner slot profile on a pipe billet by the method of cold plastic deformation, by compressing them with a matrix on a profile slotted mandrel (by the method of "covering" drawing). A comprehensive study of the mechanics of shaping products to assess their quality and study the possibility of improving the process itself. In the case of three-dimensional molding, the surface of plasticity depends on the history of deformation, which is determined by the change in the stress state with increasing accumulated intensity of deformation. The surface of plasticity in this case is not fixed and can be constructed using the criterion of deformability, which provides the position of the point of the fracture surface, taking into account the history of deformation. The planes of deformation and boundary surfaces of plasticity are constructed, which showed a sufficient margin of plasticity for the process of forming the inner splined profile. It is substantiated that when constructing the load trajectory in the space of dimensionless indicators and its type is unambiguously determined by the conditions of formation characteristic of the studied process and practically does not depend on the mechanical properties of the deformed metal. The areas of deformation closest to the failure are determined by indicators that take into account the influence of the first and third invariants of the stress tensor (lateral region and area of depressions of the profile relative to the process of forming the internal splined profile), in which the used plasticity reaches values = 0,34 ... 0,4. From the point of view of providing a margin of safety, such calculations must be performed taking into account the indicator that takes into account the influence of the third invariant of the stress tensor.


2021 ◽  
Author(s):  
Jose Rodriguez-Martinez ◽  
Oana Cazacu ◽  
Nitin Chandola ◽  
Komi Espoir N'souglo

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.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Lin Cai ◽  
Xiaoyun Wan ◽  
Houtse Hsu ◽  
Jiangjun Ran ◽  
Xiangchao Meng ◽  
...  

AbstractDue to the independence of the gradiometer instrument’s orientation in space, the second invariant $$I_2$$ I 2 of gravity gradients in combination with individual gravity gradients are demonstrated to be valid for gravity field determination. In this contribution, we develop a novel gravity field model named I3GG, which is built mainly based on three novel elements: (1) proposing to utilize the third invariant $$I_3$$ I 3 of the gravity field and steady-state ocean circulation explorer (GOCE) gravity gradient tensor, instead of using the $$I_2$$ I 2 , similar to the previous studies; (2) applying an alternative two-dimensional fast fourier transform (2D FFT) method; (3) showing the advantages of $$I_3$$ I 3 over $$I_2$$ I 2 in the effect of measurement noise from the theoretical and practical computations. For the purpose of implementing the linearization of the third invariant, this study employs the theory of boundary value problems with sphere approximation at an accuracy level of $$O(J_2^2\cdot T_{ij})$$ O ( J 2 2 · T ij ) . In order to efficiently solve the boundary value problems, we proposed an alternative method of 2D FFT, which uses the coherent sampling theory to obtain the relationship between the 2D FFT and the third invariant measurements and uses the pseudo-inverse via QR factorization to transform the 2D Fourier coefficients to spherical harmonic ones. Based on the GOCE gravity gradient data of the nominal mission phase, a novel global gravity field model (I3GG) is derived up to maximum degree/order 240, corresponding to a spatial resolution of 83 km at the equator. Moreover, in order to investigate the differences of gravity field determination between $$I_3$$ I 3 with $$I_2$$ I 2 , we applied the same processing strategy on the second invariant measurements of the GOCE mission and we obtained another gravity field model (I2GG) with a maximum degree of 220, which is 20 degrees lower than that of I3GG. The root-mean-square (RMS) values of geoid differences indicates that the effects of measurement noise of I3GG is about 20% lower than that on I2GG when compared to the gravity field model EGM2008 (Earth Gravitational Model 2008) or EIGEN-5C (EIGEN: European Improved Gravity model of the Earth by New techniques). Then the accuracy of I3GG is evaluated independently by comparison the RMS differences between Global Navigation Satellite System (GNSS)/leveling data and the model-derived geoid heights. Meanwhile, the re-calibrated GOCE data released in 2018 is also dealt with and the corresponding result also shows the similar characteristics.


Author(s):  
G. V. Kozhevnikova ◽  
A. A. Abramov ◽  
K. A. Klimov

A method of computer prediction of the size of metal grains, their disorientation, grain boundaries and dislocation density, depending on the modes of cross-wedge rolling, is considered. The regularities of the formation of the parameters of the metal structure depending on the stress state are revealed by methods of computer simulation. The stress state is described by two parameters: the average stress and the parameter of the third invariant of the stress deviator. The effect of the stress state in the deformation zone on the metal structure parameters was determined for the first time. The new method allows improvement of the quality of products by computer optimization of rolling modes. The results of determining the metal structure and parameters of the stress-strain state in the deformation zone during hot rolling of the water pump shaft of steel 45 are presented. The verification and analysis of the data of virtual experiments on the formation of the structure of structural steels in the processes of cross-wedge rolling are carried out. To analyze the output data of the simulation, the parameters for predicting the calculation of grain boundaries and grain size were used. The created computer model for predicting the characteristics of metal structures, depending on the modes of plastic deformation, provides, at minimal cost and without carrying out field experiments, finding the optimal thermodynamic and stress-strain modes of plastic flow of metal, which guarantee the highest operational properties of the products obtained.


Author(s):  
Mostafa E. Mobasher ◽  
Juan G. Londono ◽  
Pawel B. Woelke

Abstract We present VistaDam, a physics-based ductile fracture material model that is tailored to predict failure in thin metal sheets. VistaDam is based on a three invariant plasticity model in which metal fracture is dependent on the combined evolution of the triaxial stresses as well as the third invariant of deviatoric stress. Thus, VistaDam can predict damage due to combined volumetric void growth and shear dilation; which provides VistaDam with a superior capability to describe and predict fracture in a wide range of loading ranges. VistaDam relies on three independent material parameters that can be calibrated from experimental data at different triaxiality. Calibration is achieved through the automated calibration tool VistaCal. The calibrated VistaDam material card can be readily used in explicit FEM packages such as Abaqus and LS-DYNA. In addition, the calibrated VistaDam model can be used as a virtual testing platform that can generate data required by data-driven models such as GISSMO and Johnson-Cook. This process is currently automated within VistaCal’s graphical user interface. VistaDam and VistaCal have been developed for Navy applications and have been deployed successfully to predict pressurized pipes and vessel deformation and fracture under extreme loading conditions.


Author(s):  
A. V. Romanov ◽  
P. V. Fernati

The problem on the influence of stressed state on the process of long-term deformation of nonlinear viscoelastic materials under the simple and quasi-simple modes of loading by introduction of the function with the parameter of Lode angle into the defining equations is considered. The mentioned function is determined by analysis of base experimental data obtained from the base experiments on axial tension and pure torsion. Physical and mechanical properties of nonlinear viscoelastic solids are defined by the correspondence between the invariants of deformation tensors and tensions according to the modified nonlinear Rabotnov’s model for viscoelasticity. The heredity kernels are given by the fractional-exponential function. The constructed defining equations are verified experimentally for the problems of determination of nonlinear creep deformations under combined loading applied to the thin-walled tubular elements made of polyethylene of high density and low pressure polyethylene. As a result of juxtaposition of experimental data and calculations it is a stated that allowing for the type of stressed state improves their agreement qualitatively and quantitatively.


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