A Study of the Anisotropic Static Elasticity System in Thin Domain

We study the asymptotic behavior of solutions of the anisotropic heterogeneous linearized elasticity system in thin domain of ℝ 3 which has a fixed cross-section in the ℝ 2 plane with Tresca friction condition. The novelty here is that stress tensor has given by the most general form of Hooke’s law for anisotropic materials. We prove the convergence theorems for the transition 3D-2D when one dimension of the domain tends to zero. The necessary mathematical framework and (2D) equation model with a specific weak form of the Reynolds equation are determined. Finally, the properties of solution of the limit problem are given, in which it is confirmed that the limit problem is well defined.

NUMERICAL SOLUTION OF GEOMECHANICAL PROBLEMS (THE CASE OF THE BALTIC SEA SHELF FIELD)

Geomechanical modeling aims at solving problems associated with ensuring accident-free well drilling. The paper deals with building a numerical 3D geomechanical model for a studied field with a further production well stability computation. The region of operations is located in the Baltic Sea shelf. Apart from the field summary, the study contains results of acquisition and audit of initial data for modeling. The method is discussed for unidimensional geomechanical modeling on key wells, including determination of dynamic and static elasticity and strength of the rock varieties, computations of pore pressure, vertical and horizontal stresses. Well stability computations have been obtained and analyzed on the basis of the findings of 1D geomechanical modeling. A further analysis deals with the results of 3D geomechanical modeling, i.e. determination of boundaries and building a structural framework of the model, geometry testing, filling the grid with mechanical properties, and computation of complete stress tensor using the finite element method (FEM). Results of the 1D- and 3D-modeling have been compared. Thus, a numerical 3D geomechanical model has been built for the field under study. The following stage of works was focused on the wellbore stability computation for the planned wells. Additionally, computation was performed for drilling mud absorption pressure gradient cubes, caving pressure, and rock hydraulic fracturing pressure at different inclination angles and drilling azimuths. Recommendations were developed for accident-free construction of wells in the field under study, including upkeep and update of the geomechanical model in real time during drilling of wells. The obtained results and techniques can be used in design and construction of wells in other fields though taking into account regional specifics.

Application of free software FreeFem++/Gmsh and FreeCAD/CalculiX for simulation of static elasticity problems

The paper discusses the stages of computer numerical simulation of engineering problems and ways to improve the accuracy of simulation; provides a brief overview of free software for simulation elasticity problems by the finite element method, as well as trends in the development of free CAD and CAE software. For a successful engineering study, it is necessary to choose a convenient tool that takes into account all the features of the problem being solved. Based on the solution of a test static problem of linear elasticity, two approaches to engineering modeling were demonstrated. The first approach requires programming skills - the full modeling cycle was written in the programming language of the FreeFem++ software. Additionally, the method mesh generating in the Gmsh program with subsequent use in the FreeFem++ program is shown. In the second approach, the full cycle of modeling is carried out through the interface of the FreeCAD program with the built-in CalculiX solver, which does not require programming skills. A way to parameterize the task using the Python interpreter built into FreeCAD is also proposed. The simulation results obtained using both approaches are compared for an object to which an external action is applied, determined by the Dirichlet or Neumann boundary conditions, and two types of object fastening are analyzed: rigid embedding and limitation by a plane with zero friction. The analysis of the use of computing resources by various direct and iterative methods is carried out. Within the framework of the considered test problem of static linear elasticity, the most optimal method in FreeFem++ is the iterative method of conjugate gradients CG both in terms of computation time and in terms of the memory used. The highest speed of calculations is provided by the Cholesky iterative method with conditioning by the incomplete Cholesky expansion in the CalculiX program.

A computational method for earthquake cycles within anisotropic media

SUMMARY We present a numerical method for the simulation of earthquake cycles on a 1-D fault interface embedded in a 2-D homogeneous, anisotropic elastic solid. The fault is governed by an experimentally motivated friction law known as rate-and-state friction which furnishes a set of ordinary differential equations which couple the interface to the surrounding volume. Time enters the problem through the evolution of the ordinary differential equations along the fault and provides boundary conditions for the volume, which is governed by quasi-static elasticity. We develop a time-stepping method which accounts for the interface/volume coupling and requires solving an elliptic partial differential equation for the volume response at each time step. The 2-D volume is discretized with a second-order accurate finite difference method satisfying the summation-by-parts property, with boundary and fault interface conditions enforced weakly. This framework leads to a provably stable semi-discretization. To mimic slow tectonic loading, the remote side-boundaries are displaced at a slow rate, which eventually leads to earthquake nucleation at the fault. Time stepping is based on an adaptive, fourth-order Runge–Kutta method and captures the highly varying timescales present. The method is verified with convergence tests for both the orthotropic and fully anisotropic cases. An initial parameter study reveals regions of parameter space where the systems experience a bifurcation from period one to period two behaviour. Additionally, we find that anisotropy influences the recurrence interval between earthquakes, as well as the emergence of aseismic transients and the nucleation zone size and depth of earthquakes.

EXPERIMENTAL STUDYING OF MECHANICAL-AND- PHYSICAL PROPERTIES OF RUBBER DURING AGEING

Elastomeric materials, both natural rubber and synthetic, are widely used in industry and civil engineering, due to their unique properties: high elasticity, low volume compressibility, capability to absorb and dissipate input energy, a linear relationship between stress and strain up to strain of 15% ÷ 20%, resistance to aggressive environmental factors. Different kind of compensation devices, vibration dampers, shock absorbers are fabricated from rubber materials.At the same time the elastomeric materials nonreversible change their properties over time, this disadvantage is called ageing.In given paper the results of experimental studying of the influence of aging on the physical-and-mechanical properties of polyurethane rubber is presented. The samples of cylindrical form were prepared from soft flexible polyurethane rubber Xenias PX30 and subjected to the artificial ageing. Accelerated aging of samples was fulfilled in accordance with European standard ISO 188:2011 (Rubber, vulcanized or thermoplastic - Accelerated ageing and heat resistance tests).The changing of volume, Shore A hardness, elastic rebound coefficient and static elasticity modulus under compression were investigated. Experiments showed the volume decrease, hardness shore increasing, elastic rebound increase and compression modulus under static loading increasing. This data are necessary for correct designing of the compensation devices to provide their working properties during all service life.

An overview of 38 least squares–based frameworks for structural damage tomography

The ability to reliably detect damage and intercept deleterious processes, such as cracking, corrosion, and plasticity are central themes in structural health monitoring. The importance of detecting such processes early on lies in the realization that delays may decrease safety, increase long-term repair/retrofit costs, and degrade the overall user experience of civil infrastructure. Since real structures exist in more than one dimension, the detection of distributed damage processes also generally requires input data from more than one dimension. Often, however, interpretation of distributed data—alone—offers insufficient information. For this reason, engineers and researchers have become interested in stationary inverse methods, for example, utilizing distributed data from stationary or quasi-stationary measurements for tomographic imaging structures. Presently, however, there are barriers in implementing stationary inverse methods at the scale of built civil structures. Of these barriers, a lack of available straightforward inverse algorithms is at the forefront. To address this, we provide 38 least-squares frameworks encompassing single-state, two-state, and joint tomographic imaging of structural damage. These regimes are then applied to two emerging structural health monitoring imaging modalities: Electrical Resistance Tomography and Quasi-Static Elasticity Imaging. The feasibility of the regimes are then demonstrated using simulated and experimental data.