A method for calculating the stress–strain state in the general boundary-value problem of metal forming—part 1

To simulate metal-forming processes, one has to calculate the stress–strain state of the metal, i.e. to solve the relevant boundary-value problems. Progress in the theory of plasticity in that respect is well known, for example, via the slip-line method, the finite element method, etc.) , yet many unsolved problems remain. It is well known that the slip-line method is scanty. In our opinion the finite element method has an essential drawback. (No one is against the idea of the discretization of the body being deformed and the approximation of the fields of mechanical variables.) The results of calculation of the stress state by the FEM do not satisfy Newtonian mechanics equations (these equations are said to be softened, i.e, satisfied approximately) and stress fields can be considered poor for solution of the subsequent fracture problem. We believe that it is preferable to construct an approximate solution by the FEM and soften the constitutive relations (not Newtonian mechanics equations) , especially as, in any event, they describe the rheology of actual deformable materials only approximately. We seem to have succeeded in finding the solution technique.Here we present some new results for solving rather general boundary-value problems which can be characterized by the following: the anisotropy of the materials handled; the heredity of their properties and compressibility; finite deformations; non-isothermal flow; rapid flow, with inertial forces; a non-stationary state; movable boundaries; alternating and non-classical boundary conditions, etc.Solution by the method proposed can be made in two stages: (1) integration in space with fixed time, with an accuracy in respect of some parameters; (2) integration in time of certain ordinary differential equations for these parameters.In the first stage the method is based on the principle of virtual velocities and stresses. It is proved that a solution does exist and that it is the only possible one. The approximate solution softens (approximately satisfies) the constitutive relations, all the rest of the equations of mechanics being satisfied precisely. The method is illustrated by some test examples.     

Identification of the parameters of the stress–strain problem for a multicomponent elastic body with an inclusion

Explicit expressions for residual functional gradients are derived. They are used to identify, using gradient methods, the parameters of elastic problems for multicomponent bodies. The method employs the solutions of conjugate problems in the theory (developed by the authors) of optimal control of distributed multicomponent systems     

Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

We consider the classic spring–mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data, we conduct a rigorous asymptotic analysis in terms of the Poicaré–Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

     

Discretization of the plane problem for a cracked body with nonlinear stress–strain diagram under tension

The plane problem for a cracked body with a piecewise-linear stress–strain diagram under tension is reduced by the Fourier transformation to a system of nonlinear algebraic equations. The system is numerically solved for plane strain and stress states of a perfect elastoplastic material to study plastic zones, stress and strain distributions, and displacements of crack faces     

Influence of conicity on the stress–strain state of a flexible orthotropic conical shell in a nonstationary magnetic field

A problem of magnetoelasticity for a flexible conical shell in a nonstationary magnetic field is solved. The effect of conicity on the stress–strain state of the shell is analyzed     

A Ritz’s method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space

In this article, Ritz’s method is used to calculate with unprecedented accuracy the displacements related to a deformable rectangular plate resting on the surface of an elastic quarter-space. To achieve this required three basic steps. The first step involved the study of Green’s function describing the vertical displacements of the surface of an elastic quarter-space due to vertical force applied on its surface. For this case, an explicit formula was obtained by analytically resolving a complicated integral that did not previously have an analytical solution. The second step involved the study of the coupled system of a plate and an elastic quarter-space. This portion focused on determining reactive forces in the contact zone based on Hetenyi’s solution. After determination of the reactive forces, certain features were attributed to the plate’s edges. The final step involved the application of Ritz’s method to determine the deflections of the plate resting on the surface of the quarter-space. Finally, an example calculation and validation of results are given. This is the first semi-analytical solution proposed for this type of contact problem.     

A high-order nonlinear Schrödinger equation as a variational problem for the averaged Lagrangian of the nonlinear Klein–Gordon equation

Nonlinear Dynamics - We use Whitham’s averaged Lagrangian method extended with the multiple-scale formalism to derive a sixth-order nonlinear Schrödinger equation for the complex...     

A method for the estimation of the film thickness and plate tilt angle in thin film misaligned plate–plate rheometry

This communication describes a general procedure for the estimation of the true gap and tilt angle in commercially available torsional flow plate–plate rheometers by simply measuring the torque and normal force acting on the plates when shearing a Newtonian fluid.     

A new method for solving Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system

A new method is developed to solve Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system. Based on the governing equations of Biot＇s consolidation and the technique of Laplace transform, Fourier expansions and Hankel transform with respect to time t, coordinate θ and coordinate r, respectively, a relationship of displacements, stresses, excess pore water pressure and flux is established between the ground surface （z = 0） and an arbitrary depth z in the Laplace and Hankel transform domain. By referring to proper boundary conditions of the finite soil layer, the solutions for displacements, stresses, excess pore water pressure and flux of any point in the transform domain can be obtained. The actual solutions in the physical domain can be acquired by inverting the Laplace and the Hankel transforms.     

A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.     

A study on the optimization of the angle of curvature for a Ranque–Hilsch vortex tube,using both experimental and full Reynolds stress turbulence numerical modelling

The working tube is a main part of vortex tube which the compressed fluid is injected into this part tangentially. An appropriate design of working tube geometry leads to better efficiency and performance of vortex tube. In the experimental investigation, the parameters are focused on the working tube angle, inlet pressure and number of nozzles. The effect of the working tube angle is investigated in the range of θ = 0–120°. The experimental tests show that we have an optimum model between θ = 0 and θ = 20°. The most objective of this investigation is the demonstration of the successful use of CFD in order to develop a design tool that can be utilized with confidence over a range of operating conditions and geometries, thereby providing a powerful tool that can be used to optimize vortex tube design as well as assess its utility in the field of new applications and industries. A computational fluid dynamics model was employed to predict the performances of the air flow inside the vortex tube. The numerical investigation was done by full 3D steady state CFD-simulation using FLUENT6.3.26. This model utilizes the Reynolds stress model to solve the flow equations. Experiments were also conducted to validate results obtained for the numerical simulation. First purpose of numerical study in this case was validation with experimental data to confirm these results and the second was the optimization of experimental model to achieve the highest efficiency.