Incompressible Half-space Research Articles

MATHEMATICAL MODELLING OF DYNAMIC PROCESSES IN A LAYERED INCOMPRESSIBLE HALF-SPACE WITH INITIAL STRESSES UNDER THE ACTION OF A MOVING LOAD

This work is devoted to the study of the influence of the protective coating, initial stresses, mechanical characteristics of materials, movement parameters of the surface load on the stress-deformed state of the elastic base. The relevance of the research results is related to the possibility of their use in the creation of qualitatively new materials, structures and building structures. Two models of a layered incompressible half-space are considered and compared: 1) an elastic plate on an elastic half-space; 2) the top layer (protective coating) is modelled by concentrated masses. The concentrated force moves along the free surface of the protective layer at a constant speed at a certain angle to the surface of the half-space. The solution of the problem was obtained using the method of integral Fourier transformations. Analytical results are given in a general form for materials with an arbitrary elastic potential, for cases of unequal and equal roots of characteristic equations, for various conditions of combination of elements of a layered medium and for any speed of movement of the load. The material with the Bartenev-Khazanovich potential was considered for numerical analysis. The calculations were carried out within the framework of the theory of finite initial deformations. The impact of the moving load, initial stresses and mechanical parameters of the elements of the layered base on the main characteristics of its stress-strain state was studied.

The stress-strain state of an incompressible prestressed half-space with a protective coating under the action of a moving load

This work is devoted to the study of the influence of the protective coating, initial stresses, mechanical characteristics of materials, movement parameters of the surface load on the stress-deformed state of the elastic base. The relevance of the research results is related to the possibility of their use in the creation of qualitatively new materials, structures and building structures. A prestressed incompressible half-space with inhomogeneity in the form of a thin surface layer is considered. The concentrated force moves along the free surface of the protective layer at a constant speed at a certain angle to the surface of the half-space. The solution of the problem was obtained using the method of integral Fourier transformations. Analytical results are given in a general form for materials with an arbitrary elastic potential, for cases of unequal and equal roots of characteristic equations, for various conditions of combination of elements of a layered medium and for any speed of movement of the load. The material with the Bartenev-Khazanovich potential was considered for numerical analysis. The calculations were carried out within the framework of the theory of finite initial deformations. The impact of the moving load, initial stresses and mechanical parameters of the elements of the layered base on the main characteristics of its stress-strain state was studied.

Surface waves on a coated incompressible elastic half-space

Abstract Recently, Khajiyeva et al. (2018) studied the dynamics of surface wave propagation in a pre-stressed incompressible half-space when accompanied by both vertical and tangential loads. In this study, however, we extend the work of Khajiyeva et al. to include the presence of a coating layer above the half-space, upon which the loads are acted. Moreover, we prescribe sufficient perfect continuity conditions between the two layers, resulting in an inhomogeneous composite structure. Furthermore, related effective boundary conditions within the long-wave assumption are acquired for the model through the application of the asymptotic approximation method. Finally, approximate uncoupled pseudo-differential equations are derived on the surface, thereby admitting all the results of Khajiyeva et al. as limiting cases of concern.

Propagation of interfacial (Stoneley-type) waves at the joint boundary of two initially-stressed incompressible half-spaces

The presence of initial stress in solids changes the material’s response to finite and infinitesimal deformations. This paper presents a study on the effects of initial stress, finite deformation, and material properties on the speed of interfacial waves which are assumed to travel at the joint boundary of two incompressible half-spaces. The theory of infinitesimal deformations superimposed on finite deformations is used along with the theory of invariants. Boundary conditions at the interface result in an explicit secular equation, which is analyzed generally. For each half-space, a prototype strain–energy function is assumed, which is a function of the deformation gradient and initial stress tensors. The effect of different governing parameters is observed on the wave speed and is analyzed theoretically. Graphical illustrations are also presented to elaborate on the theoretical results. It is observed that for admissible values of parameters, a real wave speed exists and is bounded by values depending on the governing parameters. A comparison between Rayleigh and interfacial wave speeds is also made and supported with graphs of dimensionless wave speeds with respect to initial stresses.

Effect of Finite Initial Strains on the Wave Process in the System of an Incompressible Half-Space and an Ideal Liquid Layer*

Effect of Finite Initial Strains on the Wave Process in the System of an Incompressible Half-Space and an Ideal Liquid Layer*

The H/V ratio of Rayleigh waves in layered incompressible orthotropic half-spaces and the incompressible limit method

The main purpose of this paper is to derive exact formulas for the H/V ratio (the horizontal-to-vertical displacement ratio at the upper surface of the layer) of Rayleigh waves propagating in an orthotropic elastic half-space coated by an orthotropic elastic layer. The half-space and the layer are in welded contact and at least one of them is incompressible. These formulas are derived by employing the incompressible limit method. One of the obtained formulas recovers Love's H/V ratio formula. When the layer is absent, the obtained formulas are simplified to the formulas derived previously for the H/V ratio of Rayleigh waves propagating in traction-free orthotropic elastic half-spaces. It is numerically shown that the incompressibility of deposited layers has strong impact on the H/V ratio behavior. As the H/V ratio is a dimensionless quantity and it is easily experimentally measured, the obtained formulas will be a useful tool for nondestructively testing mechanical properties of deposited layers.

Effect of rigid boundary on Rayleigh wave in an incompressible heterogeneous medium over an incompressible half-space

In the present problem, an attempt has been made to study the propagation of Rayleigh waves in an incompressible medium with polynomial variation (m) of rigidity over an incompressible half-space under rigid layer. Instead of using the Whittaker function, the expansion formula proposed by Newlands has been used for a better result in shallow depth. The velocity equation has been calculated and the results are shown in figures. The study in the assumed medium, the authors obtained that the phase velocity of Rayleigh waves increases except for the polynomial variation of rigidity m = 1, 2 and 3.

Axisymmetric JKR-type adhesive contact under equibiaxial stretching

ABSTRACT An axisymmetric frictionless adhesive contact problem for a spherical indenter pressed against an isotropic elastic incompressible half-space under equibiaxial stretching is studied in the framework of the generalized Johnson–Kendall–Roberts (JKR) theory, which accounts for the effect of weak coupling between fracture modes I and II by means of a phenomenological mode-mixity function. The model predicts that contact area can withstand a larger level of the substrate stretch under moderate pre-pulling force. We have provided simple formulas to evaluate the pull-off force and the critical contact radius at the detachment point.

Reflection of plane waves from the boundary of an initially stressed incompressible half-space

In this paper, nonlinear theory of elasticity is used to study the effect of initial stress on plane waves in an incompressible material. For this problem, the initial stress is not associated with a finite elastic deformation and the material is assumed to be isotropic in the absence of the initial stress. The theory of superposition of infinitesimal deformations on finite deformation is applied to a problem of plane incremental motions in an initially stressed incompressible homogeneous elastic half-space. The general formulation of the problem is presented first and then specialized using a prototype strain energy function. Homogeneous plane waves are considered and the analysis is carried out for incompressible materials in both the deformed and the undeformed reference configurations. In addition to this, problems for the reflection of small amplitude homogeneous waves from the plane boundary of an initially stressed half-space are also considered and graphical results are included, which show the effect of initial stress on reflection. It is noted that the reflection coefficients in this case behave in a similar fashion when the initial stress is a pre-stress.

Bending a Plate on Prestressed Elastic Foundation Under Live Static Load

The results of studying the effect of initial stress on the bending moment in a plate laying on a prestressed half-space under live static load are presented. The equation of motion of the plate is written considering the shear and inertia of rotation. The solution is obtained in the transform domain for the general case of compressible and incompressible half-spaces, different contact conditions, and load velocities. The numerical analysis is carried out for a compressible material with harmonic potential and an incompressible material with Bartenev-Khazanovich potential. Numerical results are analyzed (diagrams and tables)

On Rayleigh Waves in Self-reinforced Layer Embedded over an Incompressible Half-space with Varying Rigidity and Density

In this paper a mathematical model to study the Rayleigh wave propagation in a self-reinforced layer over an incompressible inhomogeneous elastic half-space with quadratically varying rigidity and linearly varying density has been represented. The displacement components for both the media have been deduced separately. Under suitable boundary conditions the secular equation has been established in closed form. Numerical results analyzing the equation are discussed and presented graphically. It is quite ocular that the reinforcement in the upper medium and variations in density and rigidity related to the lower semi-infinite medium has remarkable effect on the Rayleigh wave propagation. Due to the importance of self-reinforced material, the above problem has certain realistic applications in geophysics and civil engineering.

Impact of quadratically varying rigidity and linearly varying density on the Rayleigh wave propagation: An analytic solution

In this paper, a mathematical model of Rayleigh wave propagation in a sandy earth crust has been investigated under the influence of quadratically varying rigidity and linearly varying density in the incompressible half space. The upper boundary plane of the sandy crust has been assumed to be rigid. The displacement of the wave in two medium have been deduced separately and the dispersion equation has been found in the closed form. Numerical results analyzing the dispersion equation are discussed and presented graphically. It has been observed that the inhomogeneity parameter has a direct bearing on the phase velocity while it is also affected by the sandy parameter associated with the earth crust.

The effect of initial stress on the propagation of surface waves in a layered half-space

In this paper the propagation of small amplitude surface waves guided by a layer with a finite thickness on an incompressible half-space is studied. The layer and half-space are both assumed to be initially stressed. The combined effect of initial stress and finite deformation on the speed of Rayleigh waves is analyzed and illustrated graphically. With a suitable simple choice of constitutive law that includes initial stress, it is shown that in many cases, as is to be expected, the effect of a finite deformation (with an associated pre-stress) is very similar to that of an initial stress (without an accompanying finite deformation). However, by contrast, when the finite deformation and initial stress are considered together independently with a judicious choice of material parameters different features are found that do not appear in the separate finite deformation or initial stress situations on their own.

Effect of surface tension on the relaxation of a viscoelastic half‐space perturbed by a point load

ABSTRACTWe study the effect of surface tension on the flattening of a surface perturbed by a point load subsequent to its removal. The surface bounds an infinite isotropic linear viscoelastic incompressible half space. The point load is initially applied for a sufficiently long time so that the half space is fully relaxed before the load removal. An exact solution is obtained assuming small deformation. We then specialize our theory to the case of a standard viscoelastic solid. There is an initial reduction of the surface displacement immediately after load removal that is found to be directly proportional to the ratio of applied load to surface tension. This is followed by a temporal decay of the surface profile that depends only on the relaxation time and the long and short time moduli of the viscoelastic solid. Our work also provides the Green's function for a suddenly applied point load on the surface of a viscoelastic half space. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016, 54, 274–280

The Asymptotic Analysis of the Nonstationary Problem of Variable Shear Loading on the Boundary of an Incompressible Solid

The loading process, in which the shear action on the boundary plane changes both the intensity and direction, is considered by the example of a one-dimensional plane problem for non-linear elastic incompressible half-space. It is shown that the solution in the front region of the shock wave is determined by a nonlinear evolution equations system in the space regions where the nonlinearity of the medium becomes a significant factor. The general solution of the evolution system was obtained. The particular solution of the evolution system for one of the most simple boundary conditions was considered as an example.

On the impact deformation of an incompressible half-space under the action of a shear load of variable direction

Using the example of a one-dimensional planar problem for a nonlinear elastic incompressible half-space, the loading process is considered in which the shear action on the boundary plane changes with respect to both intensity and direction. It is shown that, in the regions of the space where the nonlinearity of the medium becomes a significant factor, the solution in the nearfront region of the shock wave is determined by a system of nonlinear evolution equations. The general solution of the evolution system is obtained. As an example, a partial solution of the system is considered for one of the simplest boundary conditions. A parametric method is presented for determining the displacements on the basis of solution of the evolution system.

Specific features of the stress behavior in an incompressible half-space with internal V-shaped crack lying in a plane perpendicular to the surface of the half-space with a tip reaching the surface

We consider the problem of equilibrium of an incompressible half-space weakened by an internal V - shaped crack lying in a plane perpendicular to the surface of the half-space, and its tip reaches the surface. By using Mehler–Fock-type integral transformation and Wiener–Hopf method, we construct homogeneous solutions of the problem and, on the basis of them, perform an analysis of specific features of the behavior of discontinuous stresses in the vicinity of the tip of the crack and analyze the process of its propagation.

Equilibrium of an incompressible half-space weakened by an internal V-shaped symmetric crack with a tip reaching the surface

We consider the problem of equilibrium of an incompressible half-space with an internal symmetric V-shaped crack, whose tip reaches the half-space surface. Using the found homogeneous solutions of this problem, we determined the character of stress behavior in approaching the crack tip.

Estimating Young's modulus of zona pellucida by micropipette aspiration in combination with theoretical models of ovum.

The zona pellucida (ZP) is the spherical layer that surrounds the mammalian oocyte. The physical hardness of this layer plays a crucial role in fertilization and is largely unknown because of the lack of appropriate measuring and modelling methods. The aim of this study is to measure the biomechanical properties of the ZP of human/mouse ovum and to test the hypothesis that Young's modulus of the ZP varies with fertilization. Young's moduli of ZP are determined before and after fertilization by using the micropipette aspiration technique, coupled with theoretical models of the oocyte as an elastic incompressible half-space (half-space model), an elastic compressible bilayer (layered model) or an elastic compressible shell (shell model). Comparison of the models shows that incorporation of the layered geometry of the ovum and the compressibility of the ZP in the layered and shell models may provide a means of more accurately characterizing ZP elasticity. Evaluation of results shows that although the results of the models are different, all confirm that the hardening of ZP will increase following fertilization. As can be seen, different choices of models and experimental parameters can affect the interpretation of experimental data and lead to differing mechanical properties.

On the stress field singularity in an incompressible half-space weakened by two near-surface wedge-like cracks

We consider the equilibrium problem for an elastic incompressible half-space weakened by two near-surface wedge-like cracks, whose lie in the same plane perpendicular to the half-space surface and have a common vertex. We use the Papkovich-Neuber representation to reduce the problem to finding two harmonic functions satisfying the mixed boundary conditions. These functions are constructed in spherical coordinates by using a Mehler-Fock type integral representation in Legendre functions. The analytic solution thus obtained permits finding the character of the stress distribution near the common tip of the cracks.