Piecewise Homogeneous Body Model Research Articles

The axisymmetric lamb’s problem for a finite prestrained half-space covered with a finite prestretched layer

The piecewise-homogeneous body model and the three-dimensional linearized theory of elastic waves in prestressed bodies are used to solve the axisymmetric time-harmonic Lamb’s problem for a finite prestrained half-space covered with a finite prestretched layer. It is assumed that the half-space and layer are incompressible and their deformation is described by the Treloar potential. The normal stress at the interface is calculated

Surface undulation instability of the viscoelastic half-space covered with the stack of layers in bi-axial compression

The near-surface undulation stability loss problems of the viscoelastic half-space covered with the stack of layers in bi-axial compression in the layers’ plane are investigated within the framework of the piecewise homogeneous body model by employing the three-dimensional linearized theory of stability of deformable bodies (TLTSDB). The equations of the TLTSDB are obtained from the three-dimensional geometrically nonlinear exact equations of the theory of viscoelasticity by the use of the boundary-form perturbation technique. By employing Laplace transform, the method for the solution to these problems is developed. It is supposed that the layers in the stack have an insignificant initial imperfection and as a stability loss criterion the case is considered where this imperfection starts to increase and grows indefinitely. The developed method is employed for the investigation of concrete problems. As a result of these investigations the influence of the rheological parameters of the layers and half-space materials on the values of the critical time is studied. The viscoelasticity of the materials is described by the fractional–exponential operator of Rabotnov.

On the stress field in a half-plane covered by the pre-stretched layer under the action of arbitrary linearly located time-harmonic forces

Within the framework of the piecewise homogeneous body model with the use of the Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies the stress field in a half-plane covered by the pre-stretched layer under the action of the inclined linearly located time-harmonic forces is investigated. The algorithm for obtaining numerical results is proposed. Concrete numerical results are presented for the pair of materials Aluminium (layer) + Steel (half-plane) as well as for Steel (layer) + Aluminium (half-plane). These numerical results are involved in the normal stress acting on the interface plane. The dependencies between this stress and the frequency of the external force and the influence of the pre-stretching of the covering layer on these dependencies is analysed.

A dynamical (time-harmonic) axisymmetric stress field in a finitely prestretched multilayered slab on a rigid foundation

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the dynamical (time-harmonic) axisymmetric stress field in a finitely prestretched multilayered slab resting on a rigid foundation is studied. It is assumed that the slab consists of two-layer packets. The elasticity of layer materials is described by the Treloar potential. It is assumed that the material of the lower layer in the packets is more rigid than that of the upper one. Numerical results are presented for the cases where the number of layers (packets) in the slab is 2 (1), 4 (2), or 6 (3). These results concern the normal stresses acting on the interface between the layers of the first, upper packet and on the interface between the first and second packets. The influence of the number, prestretch level, and thickness of the layers on relation ships between the stresses and the frequency of the external force is analyzed.

Stress distribution in a composite materialwith a row of antiphase periodically curved fibers

The stress distribution in an infinite matrix containing a row of antiphase periodically curved fibers is studied using the piecewise-homogeneous body model and the exact three-dimensional equations of elasticity theory for anisotropic bodies. It is assumed that the midlines of the fibers are located in the same plane and the materials of the fibers are the same. Numerical results are presented for the case where the materials of the fibers and matrix are isotropic and homogeneous

On the dynamical axisymmetric stress field in a finite pre-stretched bilayered slab resting on a rigid foundation

The dynamic axisymmetric stress field in an initially finite pre-stretched bilayered slab resting on a rigid foundation is studied within the framework of the piecewise homogeneous body model. The three-dimensional linearized theory of elastic waves in initially stressed bodies is used. It is assumed that a time-harmonic point-located normal force acts on the free face plane of the slab. The considered problem is solved by employing the Hankel integral transformation. The materials of the layers are assumed to be incompressible, and the elastic relations are given through the Treloar potential. The formulation and solution to the problem coincide with the corresponding ones of classical linear theory of elasticity for an incompressible body in the case where there is no initial stretching in the layers. Numerical results are presented and these results involve stresses acting on the interface planes. In particular, it is established that stresses on the interface planes decrease as the pre-stretching is increased.

A problem of fracture mechanics for composite materials with locally curved layers

Based on a piecewise homogeneous body model, by using the exact equations of linear theory of elasticity, a method for calculating the stress intensity factor in composites with locally curved layers containing cracks parallel to the direction of external normal loads is worked out.

Dynamical (time-harmonic) axisymmetric interface stress field in the finite pre-strained half-space covered with the finite pre-stretched layer

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), the dynamical (time-harmonic) stress field in the initially finite strained half-space covered with an initially and finitely stretched layer is investigated. It is assumed that on the upper free face of the covering layer the point located force which acts is harmonic with respect to time. The corresponding boundary contact problem is solved by employing the Hankel integral transformation. Moreover, it is assumed that the material of the layer and half-space are incompressible and elastic relations for those are given through the Treloar’s potential. In the case where the initial strains are absent in the layer and half-space the considered problem formulation and solution to that coincide with the corresponding ones of the classical linear theory of elasticity for an incompressible body. The algorithm for obtaining numerical results is proposed. The numerical results regarding the stresses acting on the interface plane are presented. These results are obtained for the case where the stiffness (distortion wave velocity) of the covering layer material is less (greater) than that for the half-space material. In this case, the main attention is focused on the dependencies between the values of the stresses and frequency of the external force and also the influence of the initial strains on these dependencies. In particular, it is established that the “resonance” values of the frequency of the external force increase, but the absolute maximum values of the stresses decrease significantly with the amount of the initial tension of the covering layer.

The influence of the third order elastic constants on the dynamical interface stress field in a half-space covered with a pre-stretched layer

The influence of the third order elastic constants on the dynamical (time-harmonic) axisymmetric interface stress field in the system which comprises the half-space and the pre-stretched covering layer is investigated within the framework of the piecewise homogeneous body model by employing the three-dimensional linearized theory of elastic waves in initially stressed bodies. The elasticity relations for the layer and half-space materials are given through the Murnaghan potential. It is assumed that the force acting on the free face plane of the covering layer is a time-harmonic point-located normal force. The influences due to both of the quantities of the pre-stretching of the layer and the third order elastic constants of the layer material on the interface normal stress are analysed. The numerical results are presented for concrete selected materials such as steel, aluminium and acrylic plastic.

Stress Distribution in an Infinite Elastic Body with a Locally Curved Fiber in a Geometrically Nonlinear Statement

Within the framework of a piecewise homogeneous body model, with the use of three-dimensional geometrically nonlinear exact equations of elasticity theory, a method for determining the stress—strain state in unidirectional fibrous composites with locally curved fibers is developed for the case where the interaction between the fibers is neglected. All the investigations are carried out for an infinite elastic body containing a single locally curved fiber. Numerical results illustrating the effect of geometrical nonlinearity on the distribution of the self-balanced normal and shear stresses acting on the interface and arising as a result of local curving of the fiber are presented.

The Lamb's problem for a half-space covered with the pre-stretched layer

The Lamb's problem for the half-space covered with the pre-stretching layer is studied within the framework of the piecewise homogeneous body model. The three-dimensional linearized theory of elastic waves in initially stressed bodies is used. It is assumed that a time-harmonic point-located normal force acts on the free face plane of the covering layer. A numerical algorithm is also developed. Numerical results are presented for two cases of material pairs: rubber (layer)+aluminum (half-space); and aluminum (layer)+rubber (half-space). These results involve stresses acting on the interface plane and in the covering layer. The influence of the harmonic force frequency and the pre-stretching of the covering layer on the distribution of stresses is analyzed. In particular, it is established that stresses on the interface plane are decreased as the pre-stretching is increased.

Forced Vibration of a Prestretched Two-Layer Slab on a Rigid Foundation

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the forced vibration of a prestretched two-layer slab resting on a rigid foundation is studied. To the upper plane of the slab, a harmonic point force is applied. It is assumed that the layer materials are incompressible, and their elastic properties are characterized by Treloar’s potential. Numerical results are presented for the case where the material stiffness of the lower layer is greater than that of the upper one. The influence of prestretching the layers on the frequency dependences of the normal stresses operating on the interface between the layers and between the slab and the rigid foundation are analyzed.

Dynamical (harmonic) interface stress field in the half-plane covered by the prestretched layer under a strip load

Within the framework of the piecewise homogeneous body model, by employing the three-dimensional linearized theory of elastic waves in initially stressed bodies the dynamical problem of the stress distribution in a half-plane covered with a prestretched layer is investigated. It is assumed that the free face plane of the covered layer is subjected to a uniformly distributed harmonic load acting on a strip extending to infinity in the x3 direction, which is perpendicular to the x1-x2 plane and is of width 2a in the x1 direction. The plane-strain state in the x1-x2 plane is analysed. The corresponding boundary-value problems are investigated by employing the exponential Fourier integral transformation. The numerical results regarding the interface normal stress distribution are presented. The influences of the problem parameters and pre-stretching of the covered layer on this distribution are analysed. Practical engineering application fields of the results are suggested.

Mechanics of Curved Composites and Some Related Problems for Structral Members

ABSTRACT A review is made of investigations of the mechanics of curved composites. By curved composites we mean unidirectional fibrous and layered composites whose reinforcing fibers or layers have an initial curving or bending. The investigations carried out within the framework of the continuum theories and within the framework of the piecewise-homogeneous body model are considered separately. The corresponding theories, formulation of the problems, and solution methods for these problems are considered and some typical results on the influence of the curvature on the mechanical behavior of the composite are analyzed. Subjects for future investigations are presented.

Internal stability loss of two neighbouring fibers in a viscoelastic matrix

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional geometrically nonlinear exact equations of the theory of viscoelasticity the approach for the investigation of the internal stability loss (microbuckling) in the structure of the viscoelastic unidirected fibrous composites under compression along the fibers is developed in this work. This development concerns mainly the cases where the interaction between the fibers is taken into account and all investigations are carried out for the infinite viscoelastic matrix containing two neighbouring fibers. As a microbuckling criterion the initial imperfection criterion is used and co-phase microbuckling mode in plane and co-phase microbuckling mode out of plane are considered. The numerical results illustrating the influence of the interaction between the fibers on the values of the critical time are presented.

Lamb Problem for a Half-Space Covered with a Two-Axially Prestretched Layer

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, an approach for investigating the Lamb problem for a half-space covered with a layer is developed. It is assumed that the half-space and the cover layer have two-axial initial stresses operating parallel to the layer plane. To the free face of the layer, a normal point force changing harmonically with time is applied. For solving the corresponding boundary value problems, the double-exponential Fourier transformation is employed. An algorithm for obtaining numerical results is proposed, which is examined in particular cases. Numerical results for the influence of prestretching the cover layer on the interfacial stresses are presented.

Stress Distribution in an Elastic Body with a Periodically Curved Row of Fibers

Within the framework of a piecewise homogeneous body model, with the use of exact three-dimensional equations of elasticity theory for anisotropic bodies, a method is developed for investigating the stress distribution in an infinite elastic matrix containing a periodically curved row of cophasal fibers. It is assumed that fiber materials are the same and fiber midlines lie in the same plane. The self-balanced stresses arising in the interphase in uniaxial loading the composite along the fibers are investigated. The influences of problem parameters on these stresses are analyzed. The corresponding numerical results are presented.

Axisymmetric longitudinal wave propagation in pre-stressed compound circular cylinders

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed body the axisymmetric longitudinal wave propagation in the pre-stretched compound cylinder is investigated. It is assumed that the elasticity relations of the materials of the components of the cylinder are given through the Murnagan potential. It is established that the influence of the initial stretching of the cylinders on the longitudinal wave dispersion has not only a quantitative, but also a qualitative character. The concrete numerical investigation were made for steel, tungsten (W) (as a material for the solid cylinder) and aluminium (Al) (as a material for hollow cylinder). A lot of numerical results on the wave dispersion in the compound cylinders from tungsten (W) and aluminium (Al) are presented.

The influence of the third order elastic constants to the generalized Rayleigh wave dispersion in a pre-stressed stratified half-plane

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elasticity the influence of the third order elastic constants on the velocity of the generalized Rayleigh wave propagation in a pre-stressed stratified half-plane is investigated. Between the layer and half-plane the complete contact conditions are satisfied. The concrete numerical investigations are made on the various materials for which the values of the third order elastic constants are known. The initial stresses are determined within the framework of the classical linear theory of elasticity.

Stress distribution caused by anti-phase periodical curving of two neighbouring fibers in a composite material

Abstract In the present paper the stress distribution in an infinite elastic body containing two neighbouring fibers is studied. It is assumed that these fibers are located along two parallel lines and each of them has a periodical curving with the same period. Moreover, it is assumed that the curving of each fiber is out of phase with the other. At infinity uniformly distributed normal forces act in the direction of the fibers , location. The investigations are carried out in the framework of the piecewise homogeneous body model with the use of the exact three-dimensional linear theory of elasticity. The normal and shear self-equilibrated stresses arising as a result of the fiber curving are analysed. In particular, the influence of the interaction between the fibers on the distribution of these stresses is studied. A lot of numerical results related to this interaction are presented.