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This paper covers deployable systems assembled from trapezium plates. When the plate package is unwrapped, a net shell with six loop cells is formed. It is proved that additional degrees of freedom appear in case of certain correlation between the sizes of the six loop faces. When thin plates were used, the continuum approximation of the deployed net could be interpreted as a shell with a wide variety of local curvatures. Kinematics of the continuum model is analyzed by the method of Cartan moving hedron. Mechanical behavior of continuum nets is studied when cylindrical hinges between the plates are completed of shape memory plastic materials. The paper researches into shell transformations from one stable form to the other. Various practical applications of the continuum nets are demonstrated.

The paper demonstrates a fractal system of thin plates connected with hinges. The system can be studied using the methods of mechanics of solids with internal degrees of freedom. The structure is deployable — initially it is close to a small diameter one-dimensional manifold that occupies significant volume after deployment. The geometry of solids is studied using the method of the moving hedron. The relations enabling to define the geometry of the introduced manifolds are derived based on the Cartan structure equations. The proof substantially makes use of the fact that the fractal consists of thin plates that are not long compared to the sizes of the system. The mechanics is described for the solids with rigid plastic hinges between the plates, when the hinges are made of shape memory material. Based on the ultimate load theorems, estimates are performed to specify internal pressure that is required to deploy the package into a three-dimensional structure, and heat input needed to return the system into its initial state.

Under increasing loading and at a constant temperature shape memory solids become deformed in an ideal elastic plastic way as other metals, and the maximum elastic strains are much less than the ultimate plastic ones. The shape is restored at the elevated temperature and low stress level. Phenomenologically, the «reverse» deformation is equivalent to the change in shape under active loading up to sign. Plastic deformation plays a leading role in a non-elastic process; thus, the mechanical behavior should be analyzed within the ideal rigid-plastic model with two loading surfaces. In this model two physical states of the material correspond to the loading surfaces: plastic flow under high stresses and melting at a relatively low temperature. The second section poses a problem of deformation of rigid-plastic bodies at the constant temperature in two forms: as a principle of virtual velocities with the von Mises yield condition and as a requirement of the minimum dissipative functionаl. The equivalence of the accepted definitions and the existence of the generalized solutions is proved for both principles. The third section studies the rigid-plastic model of the solid at the variable temperature with two loading surfaces. For the assumed model two optimal principles are defined that link the external loads and the displacement velocities of the solid points both under active loading and in the process of shape restoration under heating. The existence of generalized velocities is proved for the wide variety of 3D domains. The connection between the variational principles and the variable temperature is ensured by inclusion of the first and second principles of thermodynamics in the calculation model. It is essential that only the phenomenological description of the phenomenon is used in the proving process. The austenite-tomartensite transformations of alloys, which are often the key elements in explanations of the mechanical behavior of shape memory materials, are not used here. The fourth section includes the definition of the shape memory materials as solids with two loading surfaces and proves the existence of solutions within the accepted restrictions. The adequacy of the model and the experiments on deformation of shape memory materials is demonstrated. In the conclusion mathematical problems that could be interesting for future research are defined.

The paper reviews the existing approaches to calculating the destruction of solids. The main attention is paid to algorithms using a unified approach to the calculation of deformation both for nondestructive and for the destroyed states of the material. The thermodynamic derivation of the unified rheological relationships taking into account the elastic, viscous and plastic properties of materials and describing the loss of the deformation resistance ability with the accumulation of microdamages is presented. It is shown that the mathematical model under consideration provides a continuous dependence of the solution on input parameters (parameters of the material medium, initial and boundary conditions, discretization parameters) with softening of the material.

Explicit and implicit non-matrix algorithms for calculating the evolution of deformation and fracture development are presented. Non-explicit schemes are implemented using iterations of the conjugate gradient method, with the calculation of each iteration exactly coinciding with the calculation of the time step for two-layer explicit schemes. So, the solution algorithms are very simple.

The results of solving typical problems of destruction of solid deformable bodies for slow (quasistatic) and fast (dynamic) deformation processes are presented. Based on the experience of calculations, recommendations are given for modeling the processes of destruction and ensuring the reliability of numerical solutions.

The quasistatic deformation problem for shape memory alloys is reviewed within the phenomenological mechanics of solids without microphysics analysis. The phenomenological approach is based on comparison of two material deformation diagrams. The first diagram corresponds to the active proportional loading when the alloy behaves as an ideal elastoplastic material; the residual strain is observed after unloading. The second diagram is relevant to the case when the deformed sample is heated to a certain temperature for each alloy. The initial shape is restored: the reverse distortion matches deformations on the first diagram, except for the sign. Because the first step of distortion can be described with the variational principle, for which the existence of the generalized solutions is proved under arbitrary loading, it becomes clear how to explain the reverse distortion within the slightly modified theory of plasticity. The simply connected surface of loading needs to be replaced with the doubly connected one, and the variational principle needs to be updated with two laws of thermodynamics and the principle of orthogonality for thermodynamic forces and streams. In this case it is not difficult to prove the existence of solutions either. The successful application of the theory of plasticity under the constant temperature causes the need to obtain a similar result for a more general case of variable external forces and temperatures. The paper studies the ideal elastoplastic von Mises model at linear strain rates. Taking into account hardening and arbitrary loading surface does not cause any additional difficulties.

The extended variational principle of the Reissner type is defined. Together with the laws of thermal plasticity it enables to prove the existence of the generalized solutions for three-dimensional bodies made of shape memory materials. The main issue to resolve is a challenge to choose a functional space for the rates and deformations of the continuum points. The space of bounded deformation, which is the main instrument of the mathematical theory of plasticity, serves this purpose in the paper. The proving process shows that the choice of the functional spaces used in the paper is not the only one. The study of other possible problem settings for the extended variational principle and search for regularity of generalized solutions seem an interesting challenge for future research.

This paper studies solids with internal degrees of freedom using the method of Cartan moving hedron. Strain compatibility conditions are derived in the form of structure equations for manifolds. Constitutive relations are reviewed and ultimate load theorems are proved for rigid plastic solids with internal degrees of freedom. It is demonstrated how the above theorems can be applied in behavior analysis of rigid plastic continual shells of shape memory materials. The ultimate loads are estimated for rotating shells under external forces and in case of shape recovery from heating.

Technical carbon (soot) is a product obtained by thermal decomposition (pyrolysis) of hydrocarbons (usually oil) in a stream of heat carrier gas. Technical carbon is widely used as a reinforcing component in the production of rubber and plastic masses. Tire production uses 70% of all carbon produced. In furnace carbon production, the liquid hydrocarbon feedstock is injected into the natural gas combustion product stream through nozzles. The raw material is atomized and vaporized with further pyrolysis. It is important for the raw material to be completely evaporated before the pyrolysis process starts, otherwise coke, that contaminates the product, will be produced. It is impossible to operate without mathematical modeling of the process itself in order to improve the carbon production technology, in particular, to provide the complete evaporation of the raw material prior to the pyrolysis process. Mathematical modelling is the most important way to obtain the most complete and detailed information about the peculiarities of reactor operation.

A three-dimensional mathematical model and calculation method for raw material atomization and evaporation in the thermal gas flow are being developed in the FlowVision software package PC. Water is selected as a raw material to work out the modeling technique. The working substances in the reactor chamber are the combustion products of natural gas. The motion of raw material droplets and evaporation in the gas stream are modeled in the framework of the Eulerian approach of interaction between dispersed and continuous media. The simulation results of raw materials atomization and evaporation in a real reactor for technical carbon production are presented. Numerical method allows to determine an important atomization characteristic: average Sauter diameter. That parameter could be defined from distribution of droplets of raw material at each time of spray forming.

Superplastic forming of Ni and Ti based alloys is widely used in aerospace industry. The main advantage of using the effect of superplasticity in sheet metal forming processes is a feasibility of forming materials with a high amount of plastic strain in conditions of prevailing tensile stresses. This article is dedicated to study commercial FEM software SFTC DEFORM application for prediction thickness deviation during low temperature superplastic forming of UFG Ti-6-4 alloy. Experimentally, thickness deviation during superplastic forming can be observed in the local area of plastic deformation and this process is aggravated by local softening of the metal and this is stipulated by microstructure coarsening. The theoretical model was prepared to analyze experimentally observed metal flow. Two approaches have been used for that. The first one is the using of integrated creep rheology model in DEFORM. As superplastic effect is observed only in materials with fine and ultrafine grain sizes the second approach is carried out using own user procedures for rheology model which is based on microstructure evolution equations. These equations have been implemented into DEFORM via Fortran user’s solver subroutines. Using of FEM simulation for this type of forming allows tracking a strain rate in different parts of a workpiece during a process, which is crucial for maintaining the superplastic conditions. Comparison of these approaches allows us to make conclusions about effect of microstructure evolution on metal flow during superplastic deformation. The results of the FEM analysis and theoretical conclusions have been approved by results of the conducted Erichsen test. The main issues of this study are as follows: a) the DEFORM software allows an engineer to predict formation of metal shape under the condition of low-temperature superplasticity; b) in order to augment the accuracy of the prediction of local deformations, the effect of the microstructure state of an alloy having sub-microcristalline structure should be taken into account in the course of calculations in the DEFORM software.