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.

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

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 COLD PLASTIC DEFORMATION OF PROCESSES IN CONDITIONS OF BORDER FORMATION

2021 ◽  
pp. 15-22
Author(s):  
Vasyl Muzychuk
Keyword(s):  
Plastic Deformation ◽  
Stress Tensor ◽  
Three Dimensional ◽  
Point Of View ◽  
Cold Plastic Deformation ◽  
The Third ◽  
Third Invariant ◽  
History Of ◽  
Margin Of Safety ◽  
Comprehensive Study

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.

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.

scholarly journals The ‘ I 3 ’ generalization of the Galileo–Rankine tension criterion

2014 ◽  
Vol 470 (2172) ◽  
pp. 20140568 ◽  
Author(s):  
R. Lagioia ◽  
A. Panteghini ◽  
A. M. Puzrin
Keyword(s):  
Stress Tensor ◽  
Structural Engineering ◽  
Constitutive Models ◽  
Physical Significance ◽  
The Third ◽  
Third Invariant ◽  
Engineering Problems ◽  
Principal Stress Space ◽  
Rankine Criterion ◽  
New Criterion

The paper presents a new tension failure criterion which generalizes the so-called Galileo–Rankine formulation. The criterion can be used as a component of the so-called perfectly no-tension model for masonry and cements as well as for establishing a tension cut-off in complex constitutive models for soils, granular materials and powders. The criterion is described by means of a very concise equation based on the third invariant of the stress tensor, approximating the boundaries of the compressive octant of the principal stress space. This sheds new light on the physical significance of the third invariant of the stress tensor. The new criterion has been validated against two known analytical solutions for no-tension materials and also effectively applied for solving two geotechnical and structural engineering problems. The proposed formulation allows for an efficient implementation in finite-element programmes, removing some of the numerical difficulties associated with the Galileo–Rankine criterion.

On the Development and Identification of Phenomenological Damage Models - Application to Industrial Wire Drawing and Rolling Processes

2013 ◽  
Vol 554-557 ◽  
pp. 213-226 ◽  
Author(s):  
Trong Son Cao ◽  
Christian Bobadilla ◽  
Pierre Montmitonnet ◽  
Pierre Olivier Bouchard
Keyword(s):  
Stress Tensor ◽  
Damage Model ◽  
Stress Triaxiality ◽  
Wire Drawing ◽  
Damage Models ◽  
The Third ◽  
Third Stress Invariant ◽  
Lode Angle ◽  
Stress Invariant ◽  
Angle Parameter

The continuum thermodynamics-based Lemaitre damage model is nowadays widely used to deal with coupled damage analyses for various mechanical applications (e.g. forming process simulations). However, such a model, which only accounts for the stress triaxiality (the ratio between the first and the second invariants of stress tensor) has been found to give incorrect results under shear dominated loading (in terms of damage location as well as risk of crack). Several recent studies have demonstrated the importance of the third stress invariant in damage prediction; the Lode angle parameter is generally used to include it. The idea is to describe completely the stress state in damage model’s formulations, which is defined by the equivalent stress, the stress triaxiality ratio and the Lode angle parameter. This later parameter has proved to have an important influence on ductile damage under low stress triaxiality. Xue’s coupled damage model accounts for the third invariant of the deviatoric stress tensor, allowing a better balance between respective effects of shear and elongation on damage. Some extensions of more physically based damage models, such as the Gurson-Tvergaard-Needleman model, have also been presented to account for this influence of the third stress invariant. In the present work, the phenomenological damage models have been implemented in Forge® Finite Element (FE) software to investigate ductile damage occurring during industrial forming processes. This paper presents the comparative study of Xue’s model and Lemaitre’s model. A complete procedure is detailed to identify the material and damage parameters from experimental mechanical tests on high carbon steel. This identification process has been carried out both for Lemaitre’s coupled damage model and Xue’s coupled damage model. Application to wire drawing followed by flat rolling shows that in such shear-inducing processes, these models predict damage at different locations, due to their different emphasis on shear with respect to elongational strain damage.

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