Equilibrium Path Bifurcation Due to Strain-Softening Localization in Ellipsoidal Region

1990 ◽  
Vol 57 (4) ◽  
pp. 810-814 ◽  
Author(s):  
Z. P. Bazˇant
Keyword(s):  
Strain State ◽  
Strain Softening ◽  
Strain Diagram ◽  
Equilibrium Path ◽  
Loss Of Stability ◽  
Infinite Layer ◽  
Neutral Equilibrium ◽  
Homogeneous Strain ◽  
Preceding Study ◽  
Path Bifurcation

A preceding study of the loss of stability of a homogeneous strain state in infinite homogeneous solid due to localization of strain into an ellipsoidal region is complemented by determining the condition of bifurcation of equilibrium path due to ellipsoidal localization mode. The bifurcation occurs when the tangential moduli matrix becomes singular, which coincides with Hill’s classical bifurcation condition for localization into an infinite layer. The bifurcation is normally of Shanley type, occurring in absence of neutral equilibrium while the controlled displacements at infinity increase. During the loading process with displacement increase controlled at infinity, this type of bifurcation precedes the loss of stability of equilibrium due to an ellipsoidal localization mode, except when the tangential moduli change suddenly (which happens, e.g., when the slope of the stress-strain diagram is discontinuous, or when temperature is increased).

Softening Instability: Part I—Localization Into a Planar Band

1988 ◽  
Vol 55 (3) ◽  
pp. 517-522 ◽  
Author(s):  
Zdeneˇk P. Bazˇant
Keyword(s):  
Strain Softening ◽  
Geometrical Nonlinearity ◽  
Small Strain ◽  
Minimum Thickness ◽  
Infinite Layer ◽  
The Stability ◽  
Conditions Of Stability ◽  
Range Part ◽  
Homogeneous Strain ◽  
The Continuum

Distributed damage such as cracking in heterogeneous brittle materials may be approximately described by a strain-softening continuum. To make analytical solutions feasible, the continuum is assumed to be local but localization of softening strain into a region of vanishing volume is precluded by requiring that the softening region, assumed to be in a state of homogeneous strain, must have a certain minimum thickness which is a material property. Exact conditions of stability of an initially uniform strain field against strain localization are obtained for the case of an infinite layer in which the strain localizes into an infinite planar band. First, the problem is solved for small strain. Then a linearized incremental solution is obtained taking into account geometrical nonlinearity of strain. The stability condition is shown to depend on the ratio of the layer thickness to the softening band thickness. It is found that if this ratio is not too large compared to 1, the state of homogeneous strain may be stable well into the softening range. Part II of this study applies Eshelby’s theorem to determine the conditions of localization into ellipsoidal regions in infinite space, and also solves localization into circular or spherical regions in finite bodies.

Softening Instability: Part II—Localization Into Ellipsoidal Regions

1988 ◽  
Vol 55 (3) ◽  
pp. 523-529 ◽  
Author(s):  
Zdeneˇk P. Bazˇant
Keyword(s):  
Strain Softening ◽  
Finite Size ◽  
Elastic Solid ◽  
Two Dimensions ◽  
The Body ◽  
Three Dimensions ◽  
Strain Diagram ◽  
Equilibrium Path ◽  
Special Cases ◽  
Localization Instability

Extending the preceding study of exact solutions for finite-size strain-softening regions in layers and infinite space, exact solution of localization instability is obtained for the localization of strain into an ellipsoidal region in an infinite solid. The solution exploits Eshelby’s theorem for eigenstrains in elliptical inclusions in an infinite elastic solid. The special cases of localization of strain into a spherical region in three dimensions and into a circular region in two dimensions are further solved for finite solids — spheres in 3D and circles in 2D. The solutions show that even if the body is infinite the localization into finite regions of such shapes cannot take place at the start of strain-softening (a state corresponding to the peak of the stress-strain diagram) but at a finite strain-softening slope. If the size of the body relative to the size of the softening region is decreased and the boundary is restrained, homogeneous strain-softening remains stable into a larger strain. The results also can be used as checks for finite element programs for strain-softening. The present solutions determine only stability of equilibration states but not bifurcations of the equilibrium path.

Research of throwing areols of underground pipeline in permafrost

2021 ◽  
pp. 21-30
Author(s):  
T. S. Sultanmagomedov ◽  
◽  
R. N. Bakhtizin ◽  
S. M. Sultanmagomedov ◽  
T. M. Halikov ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Experimental Study ◽  
Finite Element ◽  
Strain State ◽  
Operating Temperature ◽  
Operating Conditions ◽  
Loss Of Stability ◽  
Underground Pipeline ◽  
The Finite Element Method ◽  
Permafrost Thawing

Study is due to the possibility of loss of stability of the pipeline in the process of pumping a product with a positive operating temperature and the formation of thawing halos. The article presents the ways of solving the thermomechanical problem of pipeline displacement due to thawing. The rate of formation of a thawing halo is investigated depending on the initial temperatures of the soil and the pumped product. The developed monitoring system makes it possible to study the rate of occurrence of thawing halos in the process of pumping the product. An experimental study on the formation of thawing halos around the pipeline was carried out on an experimental model. A thermophysical comparative calculation of temperatures around the pipeline on a model by the finite element method has been carried out. Keywords: underground pipeline; permafrost; thawing halo; monitoring; operating conditions; stress–strain state.

scholarly journals Stress-strain state and loss of stability of anisotropic thermal coating under thermal shock

2014 ◽  
Author(s):  
B. A. Lyukshin ◽  
P. A. Lyukshin ◽  
S. A. Bochkareva ◽  
N. Yu. Matolygina ◽  
S. V. Panin
Keyword(s):  
Thermal Shock ◽  
Strain State ◽  
Loss Of Stability ◽  
Stress Strain ◽  
Stress Strain State ◽  
Thermal Coating

Stability predictions through a succession of folds

1979 ◽  
Vol 292 (1386) ◽  
pp. 1-23 ◽  
Keyword(s):  
Control Parameter ◽  
Direct Analysis ◽  
Gradient System ◽  
Equilibrium Path ◽  
Loss Of Stability ◽  
Physical Sciences ◽  
Cold Star ◽  
Degree Of Instability ◽  
Rigid Loading

For a discrete, or discretized, conservative gradient system, such as is envisaged in catastrophe theory and arises throughout the physical sciences, it is often necessary to assess the stable regions of an equilibrium path that exhibits a succession of folds. At each fold the degree of instability changes by one, so that as the system evolves from a region of known stability the first fold must represent a loss of stability. At a second fold, however, it is not clear whether the system is suffering a further loss, as we shall see in some examples, or is regaining its original stability as is more common in elasticity. A new theorem involving a conjugate parameter allows all such stability changes to be readily assessed on the basis of the form of the equilibrium paths themselves. The application of the general theory to the external and internal stabilities of an elastic structure under dead and rigid loading is demonstrated. Under the former, the load is the control parameter and the corresponding deflection plays the role of the conjugate parameter, while in a direct analysis of rigid loading these roles are reversed. A supplementary study of rigid loading which uses Lagrange multipliers supplies further theorems relating the dual concepts of external and internal stability. The use of the theorems is demonstrated in the buckling of elastic arches and shallow domes, and in the incipient gravitational collapse of a massive cold star. The possible stabilization of bifurcations by rigid loading is examined, and shows how the results can also be of value in bifurcational instabilities.

scholarly journals Determining the moment of destruction of protective coatings on pipes and spherical vessels

2020 ◽  
pp. 6-18
Author(s):  
V. V. Struzhanov ◽  
◽  
A. E. Chaikin ◽  
Keyword(s):  
Stress State ◽  
Deformation Process ◽  
Mathematical Theory ◽  
Strain Softening ◽  
Protective Coatings ◽  
Plane Stress State ◽  
Coating Material ◽  
Loss Of Stability ◽  
Thin Coatings ◽  
The Moment

An analytical method is developed to determine the moment of destruction of thin coatings on pipes and spherical vessels. The coating material works at the stage of elasticity; it has the property of strain softening, that is, destruction with increasing deformation occurs in the process of stress drop. The properties of the coating material are described by convex-concave potentials both under uniaxial tension and in a plane stress state. To determine the moment of destruction, methods of the mathematical theory of catastrophes are applied, which allow one to find all the equilibrium positions of systems and the point of loss of stability of the deformation process.

scholarly journals Plastic Buckling of Initially Imperfect Stiffened Cylinders in Axial Compression

1983 ◽  
Vol 50 (1) ◽  
pp. 88-94
Author(s):  
G. A. Duffett ◽  
B. D. Reddy
Keyword(s):  
Deformation Theory ◽  
Virtual Work ◽  
Compressive Load ◽  
Initial Imperfection ◽  
Complete Information ◽  
Equilibrium Path ◽  
Algebraic Equations ◽  
Buckling Mode ◽  
Nonlinear Algebraic Equations ◽  
Path Bifurcation

The behavior in the plastic range of axially compressed stringer-stiffened cylinders is investigated. The shell under consideration is assumed to have an initial imperfection in the form of sinusoidal deviation both axially and circumferentially. The constitutive relation employed here is J2 deformation theory of plasticity. This relation, as well as kinematic assumptions regarding the behavior of the panels and stiffeners that constitute the stiffened shell, is used in the principle of virtual work to obtain a set of nonlinear algebraic equations whose solution provides complete information about the prebuckling equilibrium path. Bifurcation from the primary path is examined by making use of a functional whose first variation is zero when two solutions to the problem are possible. This leads to an eigenvalue problem, the eigenvalue being the critical compressive load and the eigenfunction being the corresponding buckling mode. Results are presented for shells of different geometries and material properties, and a comparison of results is made with results obtained by others. The imperfect shells analyzed all exhibit stable behavior, with sufficiently large imperfections having a beneficial effect. Results for bifurcation from these paths are also discussed.

scholarly journals Algorithm of DRM with Kinetic Damping for Finite Element Static Solution of Strain-Softening Structures

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Wei Wang ◽  
Xiaozu Su
Keyword(s):  
Finite Element ◽  
Static Analysis ◽  
Strain Softening ◽  
Computational Cost ◽  
Relaxation Method ◽  
Static Solution ◽  
Solution Procedure ◽  
Static Equilibrium ◽  
Equilibrium Path ◽  
Fe Model

In order to deal with the divergence and instability due to the ill-posedness of the nonlinear finite element (FE) model of strain-softening structure in implicit static analysis, the dynamic relaxation method (DRM) was used with kinetic damping to solve the static increments in the incremental solution procedure so that the problem becomes well-posed. Moreover, in DRM there is no need to assemble and inverse the stiffness matrix as in implicit static analysis such that the associated computational cost is avoided. The ascending branch of static equilibrium path was solved by load increments, while the peak point and the descending branch were solved by displacement increments. Two numerical examples illustrated the effectiveness of such application of DRM in the FE analysis of static equilibrium path of strain-softening structures.

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