Explicit-Implicit Schemes for Calculating the Dynamics of Layered Media with Nonlinear Conditions at Contact Boundaries

In this paper we consider the problem of dynamic loading of a deformable solid medium con- taining slip planes with nonlinear slip conditions on them. An explicit-implicit scheme was constructed for the numerical solution of the constitutive system of equations, which exactly reduces to correcting the stress tensor values after performing the elastic step. An implicit approximation of the constitutive relations containing a small parameter in the denominator of the nonlinear free term was used with the second order of the approximation. The correction procedure is applicable for those cases when the viscosity parameter of interlayers providing the sliding mode of the contact boundaries is not small. The solution of the problem of the seismic waves propagation in an inhomogeneous fractured geological massif in a two-dimensional case was obtained numerically

Do It by Yourself: An Instructional Derivation of the Laplacian Operator in Spherical Polar Coordinates

For scientists and engineers, the Laplacian operator is a fundamental tool that has made it possible to carry out important frontier studies involving wave propagation, potential theory, heat conduction, the distribution of stresses in a deformable solid and quantum mechanics. Knowing, understanding, and manipulating the Laplacian operator allows us to tackle complex and exciting physics, chemistry, and engineering problems. In this paper, contained in the Special Issue “Mathematics as the M in STEM Education”, we present an instructional derivation of the Laplacian operator in spherical coordinates. Our derivation is self-contained and employs well-known mathematical concepts used in all science, technology, engineering, and mathematics (STEM) disciplines. Our lengthy but straightforward procedure shows that this fundamental tool in mathematics is not intractable but accessible to anyone who studies any of the STEM disciplines. We consider that this work may be helpful for students and teachers who wish to discuss the derivation of this vital tool from an elementary approach in their courses.

Purely elastic instabilities in the airways and oral area

The present study considers a shear-thinning viscoelastic liquid layer sheared by the air and flowing past a deformable-solid layer in the presence of a surfactant at the air–liquid interface to model the airflow in the oral area and airways. The stability analysis reveals the existence of purely elastic and unconditionally unstable ‘liquid elastic’ and ‘solid elastic’ modes. The mechanism responsible for the destabilisation of the solid elastic mode is the shear stresses exerted by the air on the liquid and by the liquid on the deformable solid while for the liquid elastic mode, the mechanism is the first normal stress difference across the air–liquid interface. The liquid and solid elastic modes undergo resonance, resulting in the ‘resonance mode’ of instability. The resonance mode exhibits a much higher growth rate than the liquid and solid elastic modes. The shear-thinning characteristic of the liquid and presence of the surfactant leads to enhancement in the growth rate of the resonance mode. An estimate shows a good correlation between the exhaled fluid particle (i.e. droplets and aerosols) diameters and the wavelength of the perturbations with maximum growth rate. In essence, the present analysis predicts that the airflow in the airways and oral area could lead to an elastic instability arising due to the elastic nature of the saliva, mucus and underlying muscle layers.

100-years’ story of success: photoelasticity — LSR–MICE – MSUCE

Introduction. The 100-years’ history of MICE – MSUCE is also a history of establishment and flourishing of one of the world’s most powerful experimental school of development and application of the photoelasticity method in the investigation of the stress state of civil and special structures. Reliable forecast of the stress state of buildings and structures under static, thermal and dynamic impacts is an indispensable prerequisite of safe operation provision of civil structures during the construction and the operation. The photoelasticity method, allows, along with the analytical methods of the mechanical science, for continual study of the stress state of the object under test, being an experimental method efficiently used both in “standalone” mode and in conjunction with numerical methods. The goal and the objectives hereof: a retrospective development review of the photoelasticity method, in particular, of the contribution of LSR of MICE, and the assessment of the contemporary state of the method in terms of its actuality, novelty and practical significance of mechanical problem solutions, of stress-strain state studies of structures and buildings. Materials and methods. The subject hereof is the polarized light method or the photoelasticity method: development, problematics, solution methods of mechanical problems and studies of structures, applications. Results. The paper describes solution examples of vital problems by the photoelasticity method: stress analysis in the wedge-shaped notch zone of the area boundary, stress state problem under forced deformation in the block-basement contact area, stress state study of underground hydropower plant tunnel with the weir erected by pinpoint blasting. The performed analysis of the photoelasticity method and its contemporary level illustrates the opportunities provided by the method in solving of mechanical problems and in structural studies. Conclusions. Is the experimental school of the photoelasticity method of LSR–MICE – MSUCE really the sunk Atlantis? The development of the experimental mechanics of deformable solid bodies is characterized by progressive proximity of experimental and numerical analysis research methods. The physical modelling, in particular, the polarized light method, allows for obtaining stress-strain state diagrams of structures, or solutions of mechanical problems, feasible for assessment or for calculation result corrections, that is why the polarized light method retains actuality and practical significance. The development of the deformable solid body mechanics determines the significance of the physical modelling, including the polarized light method. The benefit of the polarized light method is the continuity of mechanical problem solutions which is especially important when local areas with specific features of the stress strain behaviour are considered. Modern technologies of material property studies, invention of new materials lead to creation of new mechanical models, for adequacy assessment of which physical modelling methods are required, in particular, the polarized light method.

MODEL AND METHOD FOR CALCULATING THE RESOURCE OF REINFORCED CONCRETE ELEMENTS AND STRUCTURES

The article classifies, identifies and analyzes in detail the main disadvantages of existing models and methods for calculating the resource of building structure elements. A universal model and method for calculating the general and residual resources of reinforced concrete elements and structures that are under prolonged influence to operational loads are proposed. The generalized deformation-force model of the reinforced concrete elements and structures resistance to force effects is represented by an extended system of equations of the deformable solid mechanics. It is shown that the most important force and deformation parameters of the reinforced concrete elements state diagrams at all stages are functionally interconnected not only by rigidity, but by the potential energy of deformation. Therefore, due to the application of the hypothesis of invariability in a unit of volume and independence from the loading mode of the potential energy of their limiting deformation, this model has been developed to the energy level. The main advantages of the developed model of the reinforced concrete elements to force effects resistance in comparison with the existing force and deformation models in determining the resource of such elements are demonstrated. The methodology for calculating the general and residual life of reinforced concrete elements and structures is proposed to be built according to the deflections directly measured during field surveys or the step and width of the opening of normal cracks. In practice, they can be determined by geodetic, photogrammetric or any other means. The combination of the deformation-force model and the energy criterion makes it possible to calculate the general and residual resources of reinforced concrete elements and structures from a unified methodological standpoint. Therefore, the proposed "energy" hypothesis is recommended to be used as a universal energy criterion not only for the bearing strength exhaustion of the reinforced concrete elements, but also for limiting their deflections, as well as the width of the normal cracks opening under the action of any duration loads.

Physics of martial arts: Incorporation of angular momentum to model body motion and strikes

We develop a physics-based kinematic model of martial arts movements incorporating rotation and angular momentum, extending prior analyses. Here, our approach is designed for a classroom environment; we begin with a warm-up exercise introducing counter-intuitive aspects of rotational motion before proceeding to a set of model collision problems that are applied to martial arts movements. Finally, we develop a deformable solid-body mechanics model of a martial arts practitioner suitable for an intermediate mechanics course. We provide evidence for our improved model based on calculations from biomechanical data obtained from prior reports as well as time-lapse images of several different kicks. In addition to incorporating angular motion, our model explicitly makes reference to friction between foot and ground as an action-reaction pair, showing that this interaction provides the motive force/torque for nearly all martial arts movements. Moment-of-inertia tensors are developed to describe kicking movements and show that kicks aimed high, towards the head, transfer more momentum to the target than kicks aimed lower, e.g. towards the body.

Probabilistic model of surface layer removal when grinding brittle non-metallic materials

Introduction. The final quality of products is formed during finishing operations, which include the grinding process. It is known that when grinding brittle materials, the cost of grinding work increases significantly. It is possible to reduce the scatter of product quality indicators when grinding brittle materials, as well as to increase the reliability and efficiency of the operation, by choosing the optimal parameters of the technological system based on dynamic models of the process. However, to describe the regularities of the removal of particles of a brittle non-metallic material and the wear of the surface of the grinding wheel in the contact zone, the known models do not allow taking into account the peculiarities of the process in which micro-cutting and brittle chipping of the material are combined. Purpose of the work: to create a new probabilistic model for removing the surface layer when grinding brittle non-metallic materials. The task is to study the laws governing the removal of particles of brittle non-metallic material in the contact zone. In this work, the removal of material in the contact zone as a result of microcutting and brittle chipping is considered as a random event. The research methods are mathematical and physical simulation using the basic provisions of the theory of probability, the laws of distribution of random variables, as well as the theory of cutting and the theory of a deformable solid. Results and discussion. The developed mathematical models make it possible to trace the effect on material removal of the overlap of single cuts on each other when grinding holes in ceramic materials. The proposed dependences show the regularity of stock removal within the arc of contact of the grinding wheel with the workpiece. The considered features of the change in the probability of material removal upon contact of the treated surface with an abrasive tool and the proposed analytical dependences are valid for a wide range of grinding modes, wheel characteristics and a number of other technological factors. The obtained expressions make it possible to find the amount of material removal also for schemes of end, flat and circular external grinding, for which it is necessary to know the amount of removal increment due to brittle fracture during the development of microcracks in the surface layer. One of the ways to determine the magnitude of this increment is to simulate the crack formation process using a computer. The presented results confirm the prospects of the developed approach to simulate the processes of mechanical processing of brittle non-metallic materials.

Simulation of the stock removal in the contact zone during internal grinding of brittle non-metallic materials

Introduction. Finishing operations, in particular, cylindrical grinding, essentially form the quality parameters of products, its performance characteristics and functional suitability. At the same time, the cost of grinding work increases significantly in comparison with grinding metals, reaching an average of 20 ... 28% of the total cost of manufacturing products. The selection of the optimal parameters of the technological system based on the process simulation can improve the reliability, productivity and economic efficiency. To describe the processing of brittle nonmetallic materials, empirical dependences are mainly used, and the existing analytical models do not take into account the stochastic nature of the grinding operation and the combination of microcutting and brittle chipping when removing particles of brittle nonmetallic material and wear of the surface of the grinding tool. Purpose of the work: simulation of stock removal in the contact zone during internal grinding of brittle non-metallic materials. The task is to study the features and patterns of change in the probability of material removal when the treated surface comes into contact with an abrasive tool. In the work, the theoretical and probabilistic models are obtained, allowing to reveal the patterns of material removal in the contact zone. The models make it possible to trace the regularities of the interaction of cutting and piercing grains on the surface of the workpiece and the process of removing the allowance in the contact zone due to a combination of the phenomena of microcutting and brittle chipping, considered as a random event. The research methods are mathematical and physical simulation using the basic provisions of the theory of probability, the laws of distribution of random variables, as well as the theory of cutting and the theory of a deformable solid. Results and discussion. Data are obtained that provide a clear illustration of the patterns of material removal along the contact zone at various levels. Analysis of the results obtained shows that the peripheral speed of the tool and the rotation speed of the workpiece, which are directly included in the equation for calculating the probability of material removal, significantly affect the rate of material removal. The cross feed also has a significant effect on stock removal. A qualitative picture of the change in the probability of material removal in the contact zone during grinding of holes in brittle nonmetallic materials is obtained. The obtained patterns of change in the probability of material removal when the machined surface is in contact with an abrasive tool and analytical dependences are valid for a wide range of grinding modes, tool characteristics and other technological factors.