Normal Component Of Displacement Research Articles

Mechanics of Indentation for an Elastic Half-Space by Punches

The dynamic and static problems of finding stress components under four moving punches (a≤|X|≤b,c≤|X|≤d), located close to each other over an elastic half-plane (Y=0), are solved. Employing the Fourier integral transform, the problem is reduced to a set of integral equations in both cases. Using the Hilbert transform technique, the integral equations are solved to obtain the stress and displacement components. Finally, exact expressions for the stress components under the punches and the normal displacement component in the region outside the punches have been derived. Numerical results showing the variations in stress intensity factors (SIF) at the punch ends, and the absolute value of torque applied over the contact regions with different values of the parameters used in the problems have been presented in the form of graphs.

Measurement of relative motion of the brain and skull in the mini-pig in-vivo

The mechanical role of the skull-brain interface is critical to the pathology of concussion and traumatic brain injury (TBI) and may evolve with age. Here we characterize the skull-brain interface in juvenile, female Yucatan mini-pigs from 3 to 6 months old using techniques from magnetic resonance elastography (MRE). The displacements of the skull and brain were measured by a motion-sensitive MR imaging sequence during low-amplitude harmonic motion of the head. Each animal was scanned four times at 1-month intervals. Harmonic motion at 100 Hz was excited by three different configurations of a jaw actuator in order to vary the direction of loading. Rigid-body linear motions of the brain and skull were similar, although brain rotations were consistently smaller than corresponding skull rotations. Relative displacements between the brain and skull were estimated for voxels on the surface of the brain. Amplitudes of relative displacements between skull and brain were 1–3 μm, approximately 25–50% of corresponding skull displacements. Maps of relative displacement showed variations by anatomical region, and the normal component of relative displacement was consistently 25–50% of the tangential component. These results illuminate the mechanics of the skull-brain interface in a gyrencephalic animal model relevant to human brain injury and development.

A nonsmooth optimization approach for time-dependent hemivariational inequalities

This paper is devoted to the study of time-dependent hemivariational inequality. We prove the existence and uniqueness of its solution, provide a fully discrete scheme, and reformulate this scheme as a series of nonsmooth optimization problems. The introduced theory is later applied to a sample quasistatic contact problem that describes a viscoelastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on the normal component of displacement and the tangential component of velocity. Finally, computational simulations are performed to illustrate the obtained results.

Nonlocal thermodynamical vibrations in a rotating magneto-thermoelastic medium based on modified Ohm's law with temperature-dependent properties

PurposeThe purpose of this paper is to study the transient thermoelastic interactions in a nonlocal rotating magneto-thermoelastic medium with temperature-dependent properties. Three-phase-lag (TPL) model of generalized thermoelasticity is employed to study the problem. An initial magnetic field with constant intensity acts parallel to the bounding plane. Therefore, Maxwell's theory of electrodynamics has been effectively introduced and the expression for Lorentz's force is obtained with the help of modified Ohm's law.Design/methodology/approachThe normal mode technique has been adopted to solve the resulting non-dimensional coupled field equations to obtain the expressions of physical field variables.FindingsFor uniformly distributed thermal load, normal displacement, temperature distribution and stress components are calculated numerically with the help of MATLAB software for a copper material and the results are illustrated graphically. Some particular cases of interest are also deduced from the present study.Originality/valueInfluences of nonlocal parameter, rotation, temperature-dependent properties, magnetic field and time are carefully analyzed for mechanically stress free boundary and uniformly distributed thermal load. The present work is useful and valuable for analysis of problem involving thermal shock, nonlocal parameter, temperature-dependent elastic and thermal moduli.

An Elastic Half Space with a Moving Punch

The dynamic problem of a punch with rounded tips moving in an elastic half-space in a fixed direction has been considered. The static problem of determining stress component under the contact region of a punch has also been solved. Fourier integral transform has been employed to reduce the problems in solving dual integral equations. These integral equations have been solved using Cooke’s [1] result (1970) to obtain the stress component. Finally, exact expressions for stress components under the punch and the normal displacement component in the region outside the punch have been derived. Numerical results for stress intensity factor at the punch end and torque applied over the contact region have been presented in the form of graph.

Capillary-induced deformation of an initially stressed neoHookean solid: A sessile liquid droplet

The development in lab-on-a-chip and flexible electronics has attracted great interest to understand the fluid-structure interaction. In this work, we analyze the capillary-induced deformation of an initially stressed neoHookean solid with an axisymmetric, sessile droplet on the surface in the framework of the incremental deformation theory. The neoHookean material is subjected initially to uniformly biaxial loading (stretching or compressing). Using the incremental deformation theory and the concept of breadth, we derive an explicit solution of the normal displacement component of the surface of the initially stressed neoHookean solid, which consists of a scaling factor and a shape function. The scaling factor decreases with the increase of the biaxial stretch, leading to the decrease in the deformation induced by the sessile droplet. The initially biaxial compression of the neoHookean solid increases the scaling factor. There is a critical-initial compression loading, at which the surface deformation changes from convex shape to concave shape. Increasing the ratio of the breadth to the radius of the contact zone between the sessile droplet and the initially stressed neoHookean solid increases the maximum value of the shape function.

Generalized photo-thermo-microstretch elastic solid semiconductor medium due to the excitation process

A novel model in the photo-thermoelasticity theory with microstretch properties is studied. The plasma-elastic-thermal plane waves are propagated in a linear isotropic generalized photo-thermo-microstretch elastic semiconductor solid medium. The photothermal excitation occurs in the context of the microinertia of the microelement process during two-dimensional (2D) deformation. The harmonic wave techniques are used to get the solutions for the basic variables. The analytical solution of the main physical fields; carrier intensity, normal displacement components, temperature, stress load force, microstress and tangential coupled stress can be obtained. Some graphics illustrated when using the plasma, thermal and mechanical load boundary conditions are applied at the outer free surface of the elastic medium. Some semiconductor materials, as silicon (Si) and Germanium (Ge), are used to make the numerical simulation and some comparisons in different thermal memories are made. The main physical variables with new parameters are discussed theoretically and shown graphically.

Effect of Impedance Boundary on the Reflection of Plane Waves in Fraction-Order Thermoelasticity in an Initially Stressed Rotating Half-Space with a Magnetic Field

In this research article, the propagation of the plane waves in an initially stressed rotating magneto-thermoelastic solid half-space in the context of fractional-order derivative thermoelasticity is studied. The governing equations in the x–z plane are formulated and solved to obtain a cubic velocity equation that indicates the existence of three coupled plane waves. A reflection phenomenon for the incidence of a coupled plane wave for thermally insulated/isothermal surface is studied. The plane surface of the half-space is subjected to impedance boundary conditions, where normal and tangential tractions are proportional to the product of normal and tangential displacement components and frequency, respectively. The reflection coefficients and energy ratios of various reflected waves are computed numerically for a particular material and the effects of rotation, initial stress, magnetic field, fractional-order, and impedance parameters on the reflection coefficients and energy ratios are shown graphically.

Fracture mechanics approach to the problem of subsidence induced by resource extraction

A new area of application of Fracture Mechanics methods is proposed: the determination of surface deformation induced by internal volume change (e.g. due to underground hydrocarbon extraction). The finite layer-like zone of volume alteration is modelled as a crack, that is as a continuous distribution of dislocation loops. This approach allows the determination of surface deformation and can serve as the basis for monitoring of the volume alteration from surface measurements. In a limiting case of nearly uniform deformation of the reservoir boundaries the model consists of a single dislocation loop (an inserted cylindrical body of a constant height) in half-space and results in simple approximate expressions for the normal and tangential displacement components of the free surface. For the example of subsidence in the Groningen gas field the proposed method produces more realistic results describing reality better than the traditional models based on representing the reservoir as a collection of independent dilation centres. Also, analytical expressions are proposed for the nuclei – the basic Green’s function type expressions that allow representation of a planar horizontal reservoir layer of an arbitrary shape as a distribution of infinitesimal dislocation loops. For the case of a cylindrical reservoir, the nuclei integration recovers the exact distribution of the surface displacements.

Multi-field variational formulations and mixed finite element approximations for electrostatics and magnetostatics

In this paper we propose different multi-field variational formulations for electrostatics and magnetostatics, which can provide optimal discrete approximation of any particular vector field. The proposed formulations are constructed by appealing to mechanics point of view amenable to using general constitutive equations, which is quite different from electrostatics and magnetostatics formulations typical of physics and electrical engineering focusing on the corresponding global form suitable only for linear case. In particular, the formulations we propose can be combined with mixed discrete approximations that can ensure the continuity of tangential component of electric or magnetic field and normal component of electric displacement and magnetic flux even for low order interpolations. The choice of this kind is quite different from currently favorite choice of high order finite element interpolations used for coupling electromagnetism with mechanics. The discrete approximation is based upon Whitney’s interpolations representing the vector fields in terms of corresponding differential forms, with electric and magnetic fields as one-form and electric displacement and magnetic flux as two-form. The implementation of interpolations of this kind is made for 3D tetrahedron elements with non-standard approximation parameters defined not only at vertices (for zero-form), but at edges (for one-form) and at facets (for two-form). The results of several numerical simulations are presented to illustrate the performance of different formulations proposed herein.

A Nonsmooth Optimization Approach for Hemivariational Inequalities with Applications to Contact Mechanics

In this paper we introduce an abstract nonsmooth optimization problem and prove existence and uniqueness of its solution. We present a numerical scheme to approximate this solution. The theory is later applied to a sample static contact problem describing an elastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on normal and tangential components of displacement. Finally, computational simulations are performed to illustrate obtained results.

Buckling Analysis of Toroidal Shell by Rayleigh-Ritz Method

The energy approach is used to analyze the buckling stability of toroidal shells. A closed and an open toroidal shell, as well as a shell segment are considered. Linear strain energy and nonlinear strain energy due to a uniform external pressure are formulated. Variations of the in-surface and normal displacement components in the circumferential and meridional directions are assumed in the form of a double Fourier series. The eigenvalue problem for the determination of the critical pressure is formulated by the Rayleigh–Ritz method (RRM). The proposed procedure is evaluated by numerical examples: one for a closed and another one for a simply supported open toroidal shell. The obtained results are validated by a comparison with results obtained by the finite strip method (FSM) and the finite element method (FEM), which shows a very good agreement.

REFLECTION OF PLANE WAVES FROM SURFACE OF A THERMOELASTIC SATURATED POROUS SOLID HALF-SPACE WITH IMPEDANCE BOUNDARY CONDITIONS

In the present paper, a problem on reflection of plane waves from a plane thermally insulated as well as isothermal surfaces of a half-space is studied in the context of Lord-Shulman (LS) and Green-Naghdi (GN) theories of generalized thermoelasticity of a saturated porous medium. The mechanical boundary conditions at the surface of the half-space are considered as impedance boundary conditions, where the normal and shear force tractions are assumed to vary linearly with the normal and tangential displacement components multiplied by the frequency, respectively. The impedance corresponds to the constants of proportionality. The governing equations are specialized in the x-z plane. The plane wave solution of these equations shows the existence of three coupled longitudinal waves and a shear vertical wave in a generalized thermoelastic saturated porous medium. For an incident plane wave (longitudinal or shear), four reflected waves will propagate in the medium. The appropriate potentials for incident and reflected waves in a half-space are formulated, which satisfy the required boundary conditions with the help of Snell's law. Finally, a nonhomogeneous system of four equations is obtained in the amplitude ratios, where the ratios depend on the material parameters, impedance parameter, angle of incidence, thermal relaxation, and speeds of plane waves. For relevant material parameters, the amplitude ratios are computed numerically for certain ranges of impedance parameters and the angle of incidence. For the purpose of validation, the numerical results are compared with those without thermal and porous parameters.

Three-Dimensional Thermoelastic Contact Model of Coated Solids with Frictional Heat Partition Considered

In this paper, a three-dimensional thermoelastic contact model of coated solids with the frictional heat partition considered is developed by introducing a frictional heat partition model. The influence coefficients of the temperature rise, normal displacement and stress components in the three-dimensional thermoelastic contact model are converted from their corresponding frequency response functions (FRFs) with a conversion method based on the fast Fourier transform (FFT), and the FRFs of solids coated with a homogeneous coating subjected to a coupled action of the mechanical loading and the frictional heat flux on its surface are deduced in the frequency domain by introducing a two-dimensional Fourier integral transform. The contact pressure and the frictional heat partition between the two bodies are solved by employing a fast numerical algorithm based on the conjugate gradient method (CGM) and a discrete convolution fast Fourier transformation (DC-FFT). Comparison between the solutions of the present model and those of a thermoelastic contact model in literature is conducted in order to validate the present model. Several specific conclusions on the effect of the sliding speed, thermoelastic properties and thickness of the coating are drawn based on the result of numerical investigation by utilizing the present model.

Solution of the Nonclassical Problems of Stationary Thermoelastic Oscillation

We consider the stationary oscillation case of the theory of linear thermoelasticity with microtemperatures of materials. The boundary-value problem of oscillation is investigated when the normal components of displacement and the microtemperature vectors and tangent components of rotation vectors are given on spherical surfaces. Uniqueness theorems are proved. We construct explicit solutions in the form of absolutely and uniformly convergent series.

Propagation of Rayleigh waves in an incompressible rotating orthotropic elastic solid half-space with impedance boundary conditions

AbstractIn this paper, the governing equations of an incompressible rotating orthotropic elastic medium are formulated and are solved to obtain Rayleigh surface wave solutions in a particular half-space. The surface of half-space is subjected to impedance boundary conditions, in which normal and tangential stresses are proportional to frequency times normal and tangential displacement components, respectively. A secular equation for Rayleigh surface wave is obtained. With the help of MATLAB, the secular equation is solved numerically to obtain non-dimensional wave speed. The dependence of non-dimensional wave speed on non-dimensional material constant, rotation parameter and impedance parameters is shown graphically.

Stress state reconstruction and tectonic evolution of the northern slope of the Baikit anteclise, Siberian Craton, based on 3D seismic data

In this work, we consider application of an original method for determining the indicators of the tectonic stress fields in the northern Baikit anteclise based on 3D seismic data for further reconstruction of the stress state parameters when analyzing structural maps of seismic horizons and corresponded faults. The stress state parameters are determined by the orientations of the main stress axes and shape of the stress ellipsoid. To calculate the stress state parameters from data on the spatial orientations of faults and slip vectors, we used the algorithms from quasiprimary stress computation methods and cataclastic analysis, implemented in the software products FaultKinWin and StressGeol, respectively. The results of this work show that kinematic characteristics of faults regularly change toward the top of succession and that the stress state parameters are characterized by different values of the Lode–Nadai coefficient. Faults are presented as strike-slip faults with normal or reverse component of displacement. Three stages of formation of the faults are revealed: (1) partial inversion of ancient normal faults, (2) the most intense stage with the predominance of thrust and strike-slip faults at north-northeast orientation of an axis of the main compression, and (3) strike-slip faults at the west-northwest orientation of an axis of the main compression. The second and third stages are pre-Vendian in age and correlate to tectonic events that took place during the evolution of the active southwestern margin of the Siberian Craton.

Effect of Impedance Boundary on Reflection of Plane Waves from free Surface of a Rotating Thermoelastic Solid Half Space

The coupled partial differential equations governing a rotating thermoelastic medium in context of Lord and Shulman theory are solved for plane wave solutions. A cubic velocity equation is obtained, which correspond to the speeds of propagation of three coupled plane waves. A reflection phenomenon is considered in a rotating thermoelastic solid half-space for incidence of a coupled plane wave. The plane surface of the half-space is subjected to impedance boundary conditions, where normal and tangential tractions are proportional to normal and tangential displacement components time frequency, respectively. The expressions for energy ratios of all reflected waves are obtained and computed numerically for a particular material representing the medium. The dependence of energy ratios on rotation parameter, impedance parameters and angle of incidence is shown graphically.

Coupled electro-elastic analysis of functionally graded piezoelectric material plates

A unified formulation of finite layer methods (FLMs), based on the Reissner mixed variational theorem (RMVT), is developed for the three-dimensional (3D) coupled electro-elastic analysis of simply-supported, functionally graded piezoelectric material (FGPM) plates with open- and closed-circuit surface conditions and under electro-mechanical loads. In this formulation, the material properties of the plate are assumed to obey an exponent-law varying exponentially through the thickness coordinate, and the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the primary field variables of each individual layer, respectively, such as the elastic displacement, transverse shear and normal stress, electric potential, and normal electric displacement components. The relevant orders used for expanding these variables in the thickness coordinate can be freely chosen as the linear, quadratic and cubic orders. Four different mechanical/electrical loading conditions applied on the top and bottom surfaces of the plate are considered, and the corresponding coupled electro-elastic analysis of the loaded FGPM plates is undertaken. The accuracy and convergence rate of the RMVT-based FLMs are assessed by comparing their solutions with the exact 3D piezoelectricity ones available in the literature.

면외변형하의 압전재료에 대한 침투 쐐기균열

In this study, we analyze the problem of wedge cracks, which are geometrically unsymmetrical in transversely piezoelectric materials. A single concentrated antiplane mechanical load and inplane electrical load are applied at the point of the wedge surface, while one concentrated antiplane load is applied at the crack surface. The crack surfaces are considered as permeable thin slits, where both the normal component of electric displacement and the electric potential are assumed to be continuous across these slits. Using Mellin transform method, the problem is formulated and the Wiener-Hopf equation is derived. By solving the equation, the solution is obtained in a closed form. The intensity factors of the stress and the electric displacement are obtained for any crack length as well as inclined and wedge angles. Based on the results, the intensity factors are independent of the applied electric loads. The electric displacement intensity factor is always dependent on that of stress intensity factor, while the electric field intensity factor is zero. In addition, the energy release rate is computed. These solutions can be used as fundamental solutions which can be superposed to arbitrary electromechanical loadings.