Three-dimensional Linearized Theory Of Stability Research Articles

Stability analysis of double-walled and triple-walled carbon nanotubes having local curvature

In this study, stability loss research has been made for composite material containing a locally curved one double-walled carbon nanotube and one triple-walled carbon nanotube separately. The research has been made within the scope of piecewise-homogenous body model by the use of three-dimensional linearized theory of stability (TDLTS). The carbon nanotubes are resistant to tension, but as they contain large gaps in their structure, their resistance to compression is very low. For this reason, searching the behaviors of nanotubes under compression is very important. The van der Waals forces between the walls of the carbon nanotube have been considered. It has been thought that the referred composite material is under the effect of uniformly distributed normal forces in the direction of carbon nanotube at infinity. In the model, ideal contact conditions have been used between the nanotube and the matrix. In addition, the effect of increasing the number of layers of the nanotube on the critical load has been investigated. The results obtained from this study will be able to guide in the implementations relevant to modeling of mechanical behaviors of the addressed composite material. Thus, the limits required to be considered in the production of composite material containing double-walled and triple-walled nanotubes with local curvature have been obtained.

Covering stack of layers consisting of a single layer: Local near-surface buckling instability

The local near-surface buckling of a material system consisting of a half-space, which is covered by the single layer and half-space materials is elastic within the framework of a three-dimensional linearized theory of stability (TLTS). The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary form perturbations technique. By employing the Laplace and Fourier transform, a method for solving the problem is developed. Numerical results on the critical compressive forces and the critical times are presented. Key words: Buckling instability, curved-layer, critical time, local near-surface buckling, stability, viscoelastic layer.

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate, a viscoelastic (elastic) bond layer, and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator.

Determination of critical loads in the three-dimensional stability problem of thin-walled bar of angular profile

The paper considers a plane stability problem of thin-walled bar of angular profile made of an isotropic linearly elastic material. The critical parameters were determined with the aid of the second version of three-dimensional linearized stability theory and the ground state from equations of the linear theory of elasticity. The approximate solution of the stability problem was sought by the net method. The dependence of the critical parameters on the thin-walledness parameter of structure has been examined, and the errors of the applied stability theory, which is used for the design of frameworks, have been determined.

The loss of stability analyses of an elastic and viscoelastic composite circular plate in the framework of three-dimensional linearized theory

In the present paper, in the framework of three-dimensional linearized theory of stability (TDLTS) the statement of the problem of stability loss of a circular plate made from a viscoelastic composite material is suggested. The method for solution to these problems is developed by employing Laplace transform and FEM. It is assumed that the plate has an insignificant initial rotationally symmetrical imperfection. Stability is assumed to be lost when the imperfection starts to increase and grow indefinitely. Numerical results obtained by TDLTS are compared with corresponding results obtained in the framework of the third order refined plate theory.

On the three-dimensional stability loss problem of the viscoelastic composite plate

The three-dimensional linearized theory of stability (TDLTS) of deformable solid body mechanics is developed for the investigation of the three-dimensional stability loss problems of elements of constructions fabricated from the time-dependent composite materials. All procedures are made on the plate from the composite material that is modelled as homogeneous anisotropic linear viscoelastic material with normalised mechanical properties. As a stability loss criterion the values of time (critical time) for which the size of the initial imperfection starts to increase and grows indefinitely, is selected. By employing semi-analytical FEM modelling and Laplace transformation the numerical investigations are made. From the analyses of these results it is concluded that they can be significantly different from those obtained in the framework of the third-order refined plate theory.

Effect of geometric and mechanical characteristics of antifriction coatings on their stability

We consider the problem of the stability of layered antifriction coatings on the basis of three-dimensional linearized stability theory, within the framework of a piecewise-homogeneous medium subjected to small precritical strains. We have formulated the problem and constructed the basic characteristic equations. We have done a numerical study of the effect of the basic mechanical and geometric characteristics of the layers on the surface instability of the coating, and we give recommendations for designing coatings with optimal parameters.

On the three-dimensional stability loss problems of elements of structures of viscoelastic composite materials

Up to now, numerous problems of the stability loss for elements of structures made from composite materials have been investigated in the framework of the three-dimensional linearized theory of stability (TDLTS). It follows from the analysis of these investigations that the TDLTS was mainly applied to the design of elements of structures made from time-independent materials. For the solution of these problems for viscoelastic materials in the framework of the TDLTS, the dynamic investigation method and the critical deformation method are recommended in many references. However, it is known that a very reliable and frequently used approach for viscoelastic materials is the approach based on the study of the growth of insignificant initial imperfections in elements of structures with time. Taking into account the above-mentioned, an approach based on the growth of the initial imperfection for the investigation of the stability loss problems of elements of structures made from viscoelastic composite materials in the framework of TDLTS is proposed in the present paper. The composite material is modeled as an anisotropic, viscoelastic solid with averaged mechanical properties and all investigations are made on the strip simply supported at the ends.

Fracture of an infinite incompressible hyperelastic material under compression along a cylindrical crack

For problems concerning fracture of materials under compression along crack, we will use the fracture criteria proposed earlier in [2], based on a mechanism for loss of stability near defects of the crack type in the initial stage of fracture. A quantitative expression for the indicated criterion is the critical values of the loading parameters (contractions along the axes, compressive stresses), character~ing the local instability near cracks. In this case, the critical loads are determined by using the relations of the three-dimensional linearized stability theory in [4]. In [5], a general approach is suggested for such problems. This approach to investigation of the initial stage of fracture within the linearized theory of deformable bodies currently seems to be most acceptable both because of its sufficient rigor and because of its generality for different models of materials. In [6, 7], a review is given of papers in which problems have been solved in the linearized formulation for different geometric schemes for the location of cracks, and values have been obtained for the breaking loads for different models of the materials. However, we should note that within the indicated approach, up to now only those problems have been investigated in which cracks in the material are located exclusively in planes along which compression of the material occurs [8]. In this paper, we investigate the question of fracture of an infinite material containing a crack of finite dimensions, located on a circular cylindrical surf.ace, under axial compression conditions. As an example, we consider such a crack in an incompressible hyperelastic body, i.e., in a body for which there exists an elastic potential which is specified as a function of the invariants 9 (11, 12) = ~[ll(Ai, Az), I2(A,, A2)] =" W(At, Az) [3]. We will model the crack, as in fracture mechanics, as a mathematical discontinuity. Furthermore, we will consider only the initial stage of fracture, and contact of the crack edges is not considered here. We also note that relatively few investigations have been carried out for the considered crack geometry even in the linear formulation [10-14]. 1. We will use a cylindrical coordinate system (r, 0, x3). In this coordinate system, let a circular cylindrical crack of radius b and length 2a occupy the region {r = b, 0 < 0 < 2x, a ~ x3 < a}. In the case of compression of the material along the directrixes of the cylindrical crack, a homogeneous subcritical stress-strain state arises in the material [4]

Computational schemes in three-dimensional stability theory (the piecewise-homogeneous model of a medium) for composites with cracks between layers

At the present time there exists a large amount of studies devoted to the stability problem of composite materials and structural elements. Most often it is assumed in them that the structural element of the material are rigidly attached, i.e., the continuity conditions of stresses and displacements are satisfied at the boundary between structural elements. In real composite materials the usual concept of ideal contact between structural elements does not always correspond to reality. Different cases of reduced adhesion (interphase contact) between components of the composite are possible due to the manufacturing technology or the operating conditions. This can include cleavage, cracks, non-adhesion, slippage zones, and other imperfections. In this case the assumption of discontinuity of the stress and displacement vectors at the boundary between structural elements is valid. In investigating the stability of a stratified composite with cracks at the separation boundary between media one often uses various approximate theories (such as continuum theories) and applied computational schemes (rods, shells, and others). Their analysis, however, is not the object of the present study. Most accurate results can be obtained within the three-dimensional linearized stability theory of deformable bodies (TLSTDB) [i]. Application of the model of a piecewise-homogeneous medium and of the fundamental relations of TLSTDB makes it possible to eliminate the restrictions imposed on using the applied theories and approximate computational schemes, along with the inaccuracies they involve. Thus, of the presently known approaches to the study of stability of composites with cracks between layers, the most accurate is the model of a piecewise-homogeneous medium along with the fundamental TLSTDB relations. The present study is devoted to analyzing various computational schemes for stability loss of stratified composite materials with cracks at the separation boundary of media within the exact statement mentioned.

Three-dimensional linearized stability theory for stratified rock (continuum theory and model of a piecewise-homogeneous medium)

Thus, the probability of applying rigorous three-dimensional linearized stability theory to stability problems of stratified rock has been demonstrated in the present work. New qualitative and quantative results have been obtained.

Stability of Laminated Composites

The characteristic special feature of deformation behavior of modern laminated composite materials and structural elements fabricated from these materials, at current levels of loading and operating conditions is the occurrence of the purely three-dimensional stress-deformed state. In this process some specific mechanical phenomena and effects may occur, which is impossible to describe within the framework of applied or approximate approaches existing currently in deformable solid body mechanics. The structure of massive laminated materials may be included in this class of phenomena when the critical parameters of the problem depend only on the ratio between mechanical and geometrical characteristics of single layers and are independent of the dimensions and the form of the total laminated body as a whole. Since this phenomenon may be the beginning of the process of fracture of these materials, and the loss of the load-carrying capacity of structure elements fabricated from them, we consider below, in three-dimensional formulation, the problem of the surface and internal instability in laminated composite materials under compressive surface loads. The classification of the existing types of stability problems is presented for laminated materials and approaches for their solution presented in the literature. On the basis of three-dimensional linearized stability theory, within the framework of the piecewise-homogeneous media model, the general formulation of the most characteristic classes of stability problems of laminated materials is given in Lagrangian coordinates at small and finite, homogeneous and inhomogeneous precritical deformation. Analytic and variational methods of investigation of formulated problems are given with application to various models of laminated bodies models, in accordance with accepted stability criteria. The accuracy of these models is evaluated, based on the example of the solution of certain model problems; the range of application is given for existing applied and approximate approaches to the analysis of formulated problems. Results of solution of specific problems are given; specific mechanical effects, characteristic for the phenomenon considered, are found. Recommendations are formulated for engineering design and computational methods. New promising areas of research are pointed out for the problem of laminated composite materials stability.

Fracture of materials under compression along a periodic system of cracks under plane strain conditions

Elastic deformations in materials are considered within the framework of a three-dimensional linearized stability theory. On the basis of a fracture criterion proposed earlier /1, 2/ critical values are specified for the compression strain and stress under plane strain corresponding to the beginning of material fracture under compression along a periodic system of cracks.

Three-dimensional stability theory of deformable bodies. Stability of construction elements

In [8, 9] and in the present paper we analyzed the possibilities of using the approximate approach [15, 18] in the three-dimensional stability theory of deformable bodies as applied to effects of internal and surface instability and to stability of thinwalled structural elements. The analysis mentioned has been performed by comparing for standard problems the results obtained by the approximate approach [15, 18] with the results for the similar problems, obtained within the three-dimensional linearized stability theory of deformable bodies (for example [2–5, 7, 10, 19]), constructed with the accuracy usually adopted in mechanics. The following conclusions are drawn as a result of the analysis.

Symmetric failure of the halfspace with penny-shaped cracks in compression

The three-dimensional axisymmetric problem is investigated for a halfspace with penny-shaped crack parallel to the free surface. A uniform compression is applied parallel to the crack plane. The Griffith-Irwin theory is not applicable to this configuration of load and crack geometry since all the stress intensity factors are zero. Instead, a stability criterion will be invoked within the framework of the three-dimensional linearized stability theory. Reference can be made to previous works [5,6] involving compressible and incompressible elastic bodies. Use was made of an arbitrary form of the elastic potential for high subcritical deformations and for two variants of the theory of small subcritical deformations that involved equal and unequal roots of the characteristics equation [6]. It was found that consideration of the mutual influence of the subsurface crack and the free surface results in a considerable reduction of the theoretical strength limit. This was previously derived for infinite material with a crack. Examples of potentials with equal roots are discussed: the Bartenev-Khazanovich (incompressible bodies) and harmonic potential (compressible bodies).

Three-dimensional theory of stability of elastic-viscous-plastic bodies

The main results of the advances in the three-dimensional linearized theory of stability of elastic--viscous--plastic bodies made in the last few years are suammrized in [4, 5] together with other results on inelastic stability. The results for elastic-viscous-plastic bodies were obtained by using the fundamental relations for this model proposed in [6] and also discussed in [7]. The fundamental relations [6, 7] were found by introducing a separate loading function for each of the inelastic strain components. These were basically []0] the loading function corresponding to the Mises yield condition as applied to translational hardening. In this approach linearization and stability criteria of elastic--viscous--plastic bodies (a type of quasistatic criterion [9, I, 3] and various generalizations of it) are used to obtain fundamental results with the three-dimensional stability theory of elastic-viscous-plastic bodies. The most complete discussion of these results is given in [I0]. The fundamental relations of elastic--viscous-plastic bodies are usually given in differential form, with the strain increments expressed in terms of the stress increments for a given level of stresses, plastic deformations, and their rates. The main results [10] in the threedimensional linearized stability theory of elastic-viscous--plastic bodies were found by using equations in the displacements which were obtained by inverting the fundamental linearized relations of elastic--viscous--plastic bodies, i.e., by expressing the stress increments in terms of the strain increments and their rates. By using certain stability criteria, the linearized fundamental relations [6, 7] for homogeneous subcritical states were inverted ~n [10], where mainly only problems for homogeneous subcritical states were considered. In the general case of arbitrary subcritical states, the linearized relations [6, 7] as applied to stability problems were inverted in [2]; these same results were also given in [3]. We note that the fundamental relations [11] for elastic, viscous-p-plastic bodies, the generalized linear relations for a Maxwell body, as applied to stability problems were inverted in [3]. Thus, the inversion of the linearized fundamental equations for an elastic--viscous-plastic body by using stability criteria is a necessary step in formulating the equations in the displacements of the three-dimensional linearizedstability theory; the main results were derived in [|0] by using the equations in the displacements; in the general case of an arbitrary suberitical state (within the framework of the theory of small subcritical deformations) this inversion was found in [$] for the fundamental relations proposed in [6, ]I]. In thepresent paper we consider the fundamental equations for an isotropic elastic-viscousplastic body in which the elastic strains are calculated from Hooke's law relations for a linearly elastic compressible body, the increments of plastic deformations by the relations of the Prandtl--Reuss plasticity theory, and the rate of viscous deformations (creep) are expressed in terms of the strain rate potential. This model of an isotropic elastic--viscous-plastic body is rather general; it differs from the model in [6, 11], and has apparently not been used in three-dimenslonal linearized stability theory; an analogous model as applied to the description of the unsteady creep of metals was considered in [9]. For this model the fundamental relations are linearized, and by using certain stability criteria the fundamental relations are inverted for an arbitrary (inhomogeneous) subcritical state. By using these relations the fundamental equations in the displacements of the three-dimenslonal linearized theory of stability of elastic--viscous--plastic bodies were derived. The results described for the model under consideration make it possible to investigate a number of classes of problems to the same extent as in []0] by using the fundamental equations [6]. The study was performed within the framework of the theory of small subcritical deformations, using the components of the metric tensor in the natural (undeformed) state.

Mechanics of composite-material failure under axial compression (plastic failure)

It follows from the results outlined above that, in constructing the mechanics of composite-material failure under compression on the basis of the concept of stability loss in the structure, it is expedient to adopt approaches associated with the use of three-dimensional linearized stability theory. In this case, reliable results corresponding to the given phenomena may be obtained. In many cases, it is possible here to use the continuum approach of [9] or—for more rigorous solution—the approach of [10], within the framework of the model of a piecewise-homogeneous medium.