Three-dimensional Elastic Medium Research Articles

Modeling of hydraulic fracture in a medium containing a hollow inclusion

In this work, the problem of the development of a hydraulic fracture crack near a circular cavity in a three-dimensional homogeneous elastic medium is considered. The results of numerical calculations performed using the extended finite element method (XFEM) implemented in the ABAQUS software package are presented. Various variants of the position of the initial disk crack relative to the cavity are considered. Some regularities of the development of a three-dimensional hydraulic fracture near the cavity have been found.

Computational modeling of cracks different forms in three-dimensional space

All celestial bodies have defects in the form of cracks. Some of the bodies are large. Colliding with such objects can be dangerous. Therefore, the problem of grinding such bodies remains very urgent. In order to avoid large financial costs, you need to know where to strike in order to achieve the greatest effect. This paper studies the behavior of internal defects under various loads. A three-dimensional elastic medium weakened by a system of plane parallel cracks was examined in this study. A method of boundary elements called the method of discontinuous displacements, was chosen as the numerical method. The advantage of this method is that only the crack boundary is divided into elements, which allows the use of a small (compared to other numerical methods) amount of computer memory. The code was implemented in C ++. The interaction of two circular cracks was investigated depending on the size of the cracks and their relative locations. Calculating the singular integral is required when solving problems involving one crack. In case of several cracks, such integrals are not analytically incalculable. In this study, we present a method that does not require the calculation of singular integrals.

The study of the strength of structures weakened by a system of cracks

Strength of Space vehicles is a crucial issue for Space flight safety. The presence of latent defects in the material has a significant effect on the strength at different loads: the speed and direction of crack growth. In this paper, we study a three-dimensional elastic medium weakened by a system of plane cracks and one crack with a bend. As a numerical method, the method of boundary elements was chosen, namely, the method of discontinuous displacements. The code is implemented in C ++. A comparison was made with known analytical results. Bend crack behavior is studied for various loads.

Three-dimensional thermal shock problem in the frame of memory-dependent generalized thermoelasticity

The Lord–Shulman theory of generalized thermoelasticity based on a memory-dependent derivative is employed to study the thermoelastic response in a three-dimensional elastic half-space medium whose free surface is assumed to be traction-free and subjected to a thermal shock. The numerical solutions for the field variables are presented graphically and analyzed for various values of the time delay parameter. The predictions of the new model are discussed and compared with the Lord–Shulman model (without memory-dependent derivative).

Hall viscosity and the acoustic Faraday effect

For more than 20 years, observation of the non-dissipative Hall viscosity in the quantum Hall effect has been impeded by the difficulty to probe directly the momentum of the two-dimensional electron gas. However, in three-dimensional systems such as superfluid ${}^{3}\mathrm{He}\!\!-\!\!\mathrm{B}$, the momentum density is readily probed through transverse acoustic waves. We show that in a three-dimensional elastic medium supporting transverse waves, a non-vanishing Hall viscosity induces circular birefringence. Such an effect has been observed in ${}^{3}\mathrm{He}\!\!-\!\!\mathrm{B}$ in the presence of a weak magnetic field, and is known as the acoustic Faraday effect. The acoustic Faraday effect has been understood in terms of the Zeeman splitting of the excited order parameter modes which support the transverse wave propagation in the superfluid. We show that the Zeeman effect can generically lead to a non-zero Hall viscosity coefficient, and confirm this prediction using a simple phenomenological model for the ${}^{3}\mathrm{He}\!\!-\!\!\mathrm{B}$ collective modes. Therefore, we claim that the observation of the acoustic Faraday effect can be leveraged to make a direct observation of the Hall viscosity in superfluid ${}^{3}\mathrm{He}\!\!-\!\!\mathrm{B}$ in a magnetic field and other systems such as the crystalline $\mathrm{Tb}_{3}\mathrm{Ga}_{5}\mathrm{O}_{12}$ material.

Finite-time scaling at the Anderson transition for vibrations in solids

A model in which a three-dimensional elastic medium is represented by a network of identical masses connected by springs of random strengths and allowed to vibrate only along a selected axis of the reference frame, exhibits an Anderson localization transition. To study this transition, we assume that the dynamical matrix of the network is given by a product of a sparse random matrix with real, independent, Gaussian-distributed non-zero entries and its transpose. A finite-time scaling analysis of system's response to an initial excitation allows us to estimate the critical parameters of the localization transition. The critical exponent is found to be $\nu = 1.57 \pm 0.02$ in agreement with previous studies of Anderson transition belonging to the three-dimensional orthogonal universality class.

Dynamic behavior of rectangular crack in three-dimensional orthotropic elastic medium by means of non-local theory

The dynamic behavior of a rectangular crack in a three-dimensional (3D) orthotropic elastic medium is investigated under a harmonic stress wave based on the non-local theory. The two-dimensional (2D) Fourier transform is applied, and the mixed-boundary value problems are converted into three pairs of dual integral equations with the unknown variables being the displacement jumps across the crack surfaces. The effects of the geometric shape of the rectangular crack, the circular frequency of the incident waves, and the lattice parameter of the orthotropic elastic medium on the dynamic stress field near the crack edges are analyzed. The present solution exhibits no stress singularity at the rectangular crack edges, and the dynamic stress field near the rectangular crack edges is finite.

Interval modulation of recurrent slow slip events by two types of earthquake loading

Geodetic studies have discovered recurrent spontaneous slow slip events (SSEs) at major faults. The SSE recurrence intervals should reflect stress states at the faults, including load effects of large earthquakes in neighboring areas. Here, we focus on temporal changes of the SSE recurrence intervals. We perform numerical model experiments with the rate- and state-dependent friction in a three-dimensional elastic medium to simulate the SSE recurrence interval changes by the earthquake loading effects. One result is gradual shortening of the SSE recurrence intervals owing to a nucleation process of a nearby large earthquake, as revealed by several previous studies. This effect reflects magnitude of the elastic interaction between the SSE and earthquake areas. As an example, when the distance between the SSE and earthquake areas is almost zero, a short-term rapid decrease of the SSE recurrence intervals precedes the earthquake occurrence (approximately by a decade). The other result is that external stress perturbation, as large as 0.1 MPa, can reduce the SSE recurrence intervals to a similar extent. Furthermore, the interval modulation by the stress perturbation continues for a prolonged period until the occurrence of the adjacent earthquake. Both effects may be observable, as is advancing at the Boso zone, Japan, but their separation is difficult under the present circumstances.

FEM-SGBEM coupling for modeling of mode-I planar cracks in three-dimensional elastic media with residual surface tension effects

A computationally efficient numerical technique capable of modeling mode-I planar cracks in three-dimensional linear elastic media by taking the influence of the residual surface tension into account, is presented in this paper. The elastic medium (i.e., the bulk material) is modeled by the classical theory of linear elasticity, whereas the crack surface is treated as a zero-thickness layer perfectly bonded to the bulk material with its behavior governed by the special case of Gurtin–Murdoch surface elasticity model. Governing equations of the bulk material are formulated in terms of weakly singular, weak-form boundary integral equations, whereas those of the surface are cast in a weak form using a weighted residual technique. The solution of the final coupled system of governing equations is subsequently accomplished by using a numerical procedure based primarily on a coupling between standard finite element technique and a weakly singular, symmetric Galerkin boundary element method. Extensive numerical simulations are conducted and the results are compared with available benchmark solutions to verify the formulation and numerical implementation. Applications of the technique to the analysis of nano-crack problems are presented for some selected cases, to study nano-scale influence and size-dependency behavior.

Asymptotic distribution of the eigenvalues and eigenfunctions in basic boundary value oscillation problems in hemitropic elasticity

The basic boundary value oscillation problems for a three-dimensional elastic medium bounded by a closed surface are considered. Asymptotic formulas are derived for the eigenvalue and eigenfunction distributions in the problems.

Friedrichs systems equivalent to the systems of wave equations

Description up to equivalence transformations is given for all evolutionary symmetric t-hyperbolic Friedrichs systems equivalent to the systems of two- and three-dimensional wave equations. We obtain the evolutionary symmetric t-hyperbolic Friedrichs systems that describe shear waves in a three-dimensional isotropic elastic medium both in the presence and absence of mass forces. Studying the group properties of some of these systems, we point out a system equivalent to the Maxwell equations for the electromagnetic field in vacuum that consists of the equations in involution under charge conservation of an evolutionary symmetric t-hyperbolic Friedrichs system and the Lorentz condition.

Homogenized Interface Model Describing Inhomogeneities Located on a Surface

We study the influence of heterogeneities located near a planar surface on the elastic response of a three-dimensional elastic medium. These heterogeneities can be either reinforcements, like steel reinforcements in concrete, or defects, like micro-cracks periodically distributed. We prove that their influence is of the second order from an energetic viewpoint. Then, we propose an “up to second order effective model” in which the influence of the heterogeneities is given by a surface energy contribution involving both the jump of displacement across the surface and the tangential strain components on the surface. The effective coefficients entering in the definition of the surface energy are obtained by solving “elementary” elastic problems formulated on an infinite representative cell containing the defects. We analyze this model, in particular the properties of the effective surface coefficients, and establish its coherence with limit models previously described in the literature for stiff or soft interfaces. This approach is finally applied to several kinds of heterogeneities.

An integrated extended Kalman filter–implicit level set algorithm for monitoring planar hydraulic fractures

We describe a novel approach to the inversion of elasto-static tiltmeter measurements to monitor planar hydraulic fractures propagating within three-dimensional elastic media. The technique combines the extended Kalman filter (EKF), which predicts and updates state estimates using tiltmeter measurement time-series, with a novel implicit level set algorithm (ILSA), which solves the coupled elasto-hydrodynamic equations. The EKF and ILSA are integrated to produce an algorithm to locate the unknown fracture-free boundary. A scaling argument is used to derive a strategy to tune the algorithm parameters to enable measurement information to compensate for unmodeled dynamics. Synthetic tiltmeter data for three numerical experiments are generated by introducing significant changes to the fracture geometry by altering the confining geological stress field. Even though there is no confining stress field in the dynamic model used by the new EKF-ILSA scheme, it is able to use synthetic data to arrive at remarkably accurate predictions of the fracture widths and footprints. These experiments also explore the robustness of the algorithm to noise and to placement of tiltmeter arrays operating in the near-field and far-field regimes. In these experiments, the appropriate parameter choices and strategies to improve the robustness of the algorithm to significant measurement noise are explored.

Assessment of the finite element solutions for 3D spontaneous rupture using GeoFEM

Numerical simulations of spontaneous shear rupture on a planar fault using a slip-weakening model in a three-dimensional uniform elastic medium were conducted using a parallel FE code, GeoFEM, which was originally developed for the solid-Earth simulations on the Earth Simulator. The aim was to evaluate the accuracy and applicability of GeoFEM to earthquake rupture. The present numerical results are compared with published results obtained using a finite-difference (FD) method and a boundary integral (BI) method for rupture times and time functions of slip, slip rate, and shear stress at two particular points on the fault plane. The effects of mesh size and damping on the attenuation of spurious oscillations were also examined in a range of simulations. The appropriate degree of damping depends on mesh size and must be introduced in order to obtain reliable numerical solutions; weak damping leads to significant oscillation and strong damping to artificially low rupture speeds and low slip rates. Our results indicate that mesh size should be sufficiently small to allow the inclusion of a few grids in the cohesive zone, as shown in other numerical methods. The solutions by GeoFEM that have the appropriate mesh size and damping are quite similar to those obtained using the FD and BI methods, the difference between them being generally less than 2% in terms of the rupture time and 5.2% in terms of the peak slip rate.

The Effective Behavior of Elastic Bodies Containing Microcracks or Microholes Localized on a Surface

We propose a two-scale method to find the effective behavior of a three-dimensional linear elastic medium containing a series of microcracks or microholes located on a surface. The obtained effective behavior is that of a homogeneous body with, in place of the actual microdefects, a surface across which the displacements and the stresses suffer jump discontinuities. The transmission conditions are in general of Ventcel's type. The coefficients entering in these jump conditions are obtained by solving six elastic problems posed on an infinite representative cell. The theoretical analysis is illustrated by a few examples.

Friedrichs systems for systems of wave equations and shear waves in a three-dimensional elastic medium

This paper considers a two-dimensional hydraulic model which describes gas or oil flow to a horizontal well with hydraulic fractures and takes into account the reservoir geometry and fluid flow between the reservoir and the well. A computational algorithm is proposed, and calculations for gas and oil reservoirs are performed. A comparison of the calculation results and the solutions of the corresponding problems in a three-dimensional formulation show that the calculations using the approximate hydraulic model yield reasonably accurate results.

Study on in-situ test on mechanical properties of the particle improved roadbed in permafrost

The adoption of particle improved roadbed (PIR) is one of the proactive measures to protect permafrost from warming and thawing in cold regions. The field plate loading test was performed to study the mechanical parameters of fillings after a freeze–thaw cycle, including bearing capacity, modulus of deformation, etc. In addition, the deformation features of the roadbed fillings were analyzed and discussed. The results indicated that the allowable bearing capacity of the roadbed fillings was 418.78 kPa, and the deformation modulus and shear modulus decreased with the increase of the loads. It is reasonable to consider the roadbed fillings as a three-dimensional elastic continuous medium to establish a model to study deformation of permafrost. The calculated deformation agreed well with the measured. Therefore, the model can provide reference for the deformation predication of other roadbed fillings in permafrost.

Group foliation of the Lamé equations of the classical dynamical theory of elasticity

We perform the group foliation of the system of Lame equations of the classical dynamical theory of elasticity for an infinite subgroup contained in a normal divisor of the main group. The resolving system of this foliation includes the following two classical systems of mathematical physics: the system of equations of vortex-free acoustics and the system of Maxwell equations, which allows one to use wider groups to obtain exact solutions of the Lame equations. We obtain a first-order conformal-invariant system, which describes shear waves in a three-dimensional elastic medium. We also give examples of partially invariant solutions.

The effect of visco-elastic core thickness on modal loss factors of a thick three-layer cylinder

Free vibrations of damped three-layer sandwich cylinders with thick layers are considered. In particular, the effect of the different thicknesses for the middle layer on the overall natural frequencies and modal damping factors are studied. The constrained-layer damping is accomplished by sandwiching a linear visco-elastic material between two isotropic elastic cylinders with the same properties. The governing equations are derived analytically by applying Newton's second law of motion for a three-dimensional elastic medium, and by employing complex elastic moduli for the sandwiched layer. Dimensionless natural frequencies and modal loss factors for the first three thickness modes associated with wave numbers of n = 0, 1, 2, 3, and 4 are tabulated for a range of thicknesses for the middle visco-elastic layer while keeping the thicknesses of inner and outer layers unchanged.

An improved higher order zigzag theory for the static analysis of laminated sandwich plate with soft core

An improved higher order zigzag theory is proposed for the static analysis of laminated sandwich plate with soft compressible core. The variation of in-plane displacements is assumed to be cubic for both the face sheets and the core and transverse displacement is assumed to vary quadratically within the core while it remains constant through the faces. The core is considered to behave as a three-dimensional elastic medium to incorporate the effect of transverse normal deformation. A computationally efficient C 0 finite element is also proposed for this model. Numerical examples of laminated composite and sandwich plate are provided for different thickness ratios and aspect ratio to illustrate the accuracy of the present formulation by comparing the present results with the three-dimensional elasticity solutions. Some new results are also presented. The performance of the present model is excellent in calculating displacements and stresses for a wide range of sandwich plate problems with transversely flexible core.