Reconstruction of seismotectonic deformations of Asia in different layers of the Earth’s crust

The calculation of seismotectonic deformations for different depth levels 1–15, 16–35, 36–70 km was performed according to the data of 1819 mechanisms of earthquake foci that occurred in Central Asia (φ = 25° – 60° N, λ=60° – 115° E) for the period from 1976 to the end of July 2020 with M>4.7. The orientation of the main axes of the strain tensor reconstructed from the mechanisms of earthquake foci with M>4.7 coincide at different depth levels with mainly submeridional and north-eastern shortening and varying elongation from sublatitude to north-western and near-vertical. The consistency of the orientation of the main axes of shortening and elongation reconstructed from seismological materials and from the published results of calculating GPS observations, is traced.

Hadamard’s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Influence of stresses on deformation process under the irradiation creep and swelling

Approaches for describing the deformation of structural elements made from the material, in which radiation creep and swelling strains develop simultaneously, are discussed. The technique for description of irradiation swelling strains, which is used for calculational analysis of stress-strain state arising in structural elements under the joint action of irradiation and thermal-stress fields, is regarded. A complete system of equations of the boundary –initial value problem is presented, in which elastic and thermal strains, strains of radiation creep and swelling are taken into account. Numerical modelling was carried out using the specialized software FEM Creep, in which the boundary value problem is solved by the Finite Element Method, and the initial one is integrated in time by the difference predictor-corrector method. Two forms are given for the equation of state describing the radiation swelling strains: first is for the components of the strain tensor as well as second is prepared for their rates. The hypothesis about the linear correspondence of the received radiation dose and the deformation time, during which radiation swelling strains develop, are analyzed. A number of questions that require answers when using equations with a complex stress state in which the radiation swelling strains are directly depend on stresses, are discussed. Based on the processing of experimental data on the swelling of tubes made of steel 316Ti in the temperature range of 450-460 °С, a form of the equation for the radiation swelling strain rate is proposed, and the constants included in it are determined. Using the example of numerical modelling of the deformation of tubes were made of steel 316Ti and loaded by inner pressure, the applicability of the classical approach for the analysis of the stress-strain state in the presence of radiation swelling strains is shown.

Constitutive relations for materials with strain state dependent properties

Many materials demonstrate a dependence of mechanical properties on the type of stressed or deformed states. This is most noticeable in the dependence of the processes of shear and bulk deformation. Such materials include rocks, structural graphite, concrete, some grades of steel, cast iron, and aluminum. The main properties of these materials are an absence of a "single curve" relationship between the intensity of stresses and the intensity of deformations. Under shear conditions, bulk deformations can occur. Such materials can be described by constitutive equations that depend on the parameter of the type of a stress state, which is the ratio of the first invariant of the stress tensor to the stress intensity. Thus, these defining relations give the dependence of the strain tensor components on the stress tensor components. Such defining relations can be quite cumbersome, and therefore do not allow an analytical treatment to obtain defining relations that give the dependence of the components of the stress tensor on the components of the strain tensor. The paper proposes the constitutive relations obtained from the analysis of test results of various materials, which properties depend on the type of deformed state. Conditions are derived for material constants that ensure the uniqueness of the solution of boundary value problems. Based on experimental data obtained under the conditions of the proportional loading of various rocks: limestone and talcochlorite, as well as the results of mechanical tests of several grades of concrete, the constants of the mathematical model are determined. The results of the experimental studies are compared with theoretical dependencies predicted by the model. The limited applicability of the proposed constitutive relations is established.

Strain Response in the Facet Joint Capsule During Physiological Joint Rotation and Translation Following a Simulated Impact Exposure: an in Vitro Porcine Model

Low back pain (LBP) is frequently reported following rear impact collisions. Knowledge of how the facet joint capsule (FJC) mechanically behaves before and after rear impact collisions may help explain LBP development despite negative radiographic evidence of gross tissue failure. This study quantified the Green strain tensor in the facet joint capsule during rotation and translation range-of-motion tests completed before and following an in vitro simulation of a rear impact collision. Eight FSUs (4 C3-C4, 4 C5-C6) were tested. Following a preload test, FSUs were flexed and extended at 0.5 degrees/second until an ±8 Nm moment was achieved. Anterior and posterior joint translation was then applied at 0.2 mm/s until a target ±400 N shear load was imposed. Markers were drawn on the facet capsule surface and their coordinates were tracked during pre- and post-impact range-of-motion tests. Strain was defined as the change in point configuration relative to the determined neutral joint posture. There were no significant differences (p > 0.05) observed in all calculated FJC strain components in rotation and translation before and after the simulated impact. Our results suggest that LBP development resulting from the initiation of strain-induced mechanoreceptors and nociceptors with the facet joint capsule is unlikely following a severe rear impact collision within the boundaries of physiological joint motion.

Incompatible Deformation Model of Rocks with Defects around a Thick-Walled Cylinder

Before the excavation of underground engineering, joints, fissures, and voids already exist in the rock—that is, there are defects in the rock. Due to the existence of these defects, the rock produces plastic deformation, which can lead to incompatible deformation. Therefore, the classic continuum theory cannot accurately describe the deformation of the rock. In this paper, a relationship between the strain tensor and metric tensor was studied by analyzing the three states of elastic plastic deformation, and the elasto-plastic incompatible model was built. Additionally, the stress and deformation of a thick-walled cylinder under hydrostatic pressure was investigated by using a finite element program written in the FORTRAN language. The results show that the plastic strain is associated with not only deviator stress but also the distribution of defects (represented by the incompatible parameter R). With the value of R increasing, the defects in the rock increased, but the elastic plastic stiffness matrix decreased. Thus, as more rock enters the plastic state, the deformation of the surrounding rock is enlarged.