Vol 24, No 3 (2020)

The present paper is devoted to the problem of boundary conditions formulation in the growing micropolar solid mechanics. The static equations of the micropolar continuum in terms of relative tensors (pseudotensors) are derived due to virtual work principle for a solid of constant staff. The constitutive quadratic form of the elastic potential (treated as an absolute scalar) for a linear hemitropic micropolar solid is presented and discussed. The constitutive equations for symmetric and antisymmetric parts of force and couple stress tensors are given. The final forms of the static equations for the hemitropic micropolar continuum in terms of displacements and microrotations rates are obtained including the case of growing processes. A transformation of the equilibrium equations is proposed to obtain boundary conditions on the propagating growing surface in terms of relative tensors in the form of differential constraints. Those are valid for a wide range of materials and metamaterials. The algebra of rational relative invariants is intensively used for deriving the constitutive relations on the growing surface. Systems of joint algebraic rational relative invariants for force, couple stress tensors and also unit normal and tangent vectors to propagating growing surface are obtained, including systems of invariants sensitive to mirror reflections and 3D-space inversions.

Strain rate sensitivity of stress-strain curves family generated by the Boltzmann–Volterra linear viscoelasticity constitutive equation (with an arbitrary relaxation modulus) under uni-axial loadings at constant strain rates is studied analytically as the function of strain and strain rate. The general expression for strain rate sensitivity index is derived and analyzed assuming relaxation modulus being arbitrary. Dependence of the strain rate sensitivity index on strain and strain rate and on relaxation modulus qualitative characteristics is examined, conditions for its monotonicity and for existence of extrema, the lower and the upper bounds and the limit values of the strain rate sensitivity as strain rate tends to zero or to infinity are studied. It is found out that (within the framework of the linear viscoelasticity) the strain rate sensitivity index which is, generally speaking, the function of two independent variables (namely strain and strain rate), depends on the single argument only that is the ratio of strain to strain rate. So defined function of one real variable is termed the strain rate sensitivity function and it may be regarded as a material function. The explicit integral expression (and the two-sided bound) for relaxation modulus in terms of strain rate sensitivity function is derived which enables one to restore relaxation modulus assuming a strain rate sensitivity function is given. The strain rate sensitivity function is represented as a linear function of ratio of tangent modulus to secant modulus of a stress-strain curve at any fixed constant strain rate and can be evaluated in such a way using experimental data. It is proved that the strain rate sensitivity value is confined in the interval from zero to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) whatever strain and strain rate magnitudes. It is found out that the linear theory can reproduce increasing or decreasing or non-monotone dependences of strain rate sensitivity on strain rate (for any fixed strain) and it can provide existence of local maximum or minimum or several extrema as well without any complex restrictions on the relaxation modulus. General properties and peculiarities of the theoretic strain rate sensitivity function are illustrated by the examination of the classical regular and singular rheological models (consisting of two, three or four spring and dashpot elements) and fractional models. Namely, the Maxwell, Kelvin–Voigt, standard linear solid, Zener, anti-Zener, Burgers, anti-Burgers, Scott–Blair, fractional Kelvin–Voigt models and their parallel connections are considered. The carried out analysis let us to conclude that the linear viscoelasticity theory (supplied with common relaxation function which are non-exotic from any point of view) is able to produce high values of strain rate sensitivity index close to unity (the upper bound of strain rate sensitivity index for pseudoplastic media) and to provide existence of the strain rate sensitivity index maximum with respect to strain rate. Thus, it is able to simulate qualitatively existence of a flexure point on log-log graph of stress dependence on strain rate and its sigmoid shape which is one of the most distinctive features of superplastic deformation regime observed in numerous materials tests.

The paper discusses a class of exact solutions of the Oberbeck–Boussinesq equations suitable for describing three-dimensional nonlinear layered flows of a vertically swirling viscous incompressible fluid. An inhomogeneous distribution of the velocity field (there is a dependence of the field components on the horizontal coordinates) generates a vertical swirl in the fluid without external rotation (excluding Coriolis acceleration). Setting the linearly distributed heat field and the field of shear stresses at the boundaries of the flow region is one of the reasons inducing convection in a viscous incompressible fluid. The main attention is paid to the study of the properties of the temperature field. The effect of vertical twist on the distribution of isolines of this field is studied. It is shown that the homogeneous component of the temperature field can be stratified into several zones relative to the reference value, and the number of such zones does not exceed nine. The inclusion of inhomogeneous components of the temperature field can only decrease this number. It is also demonstrated that the class discussed in the paper allows one to generalize the previously obtained results on modeling convective flows of viscous incompressible fluids.

We consider the modeling multidimensional hyperbolic-parabolic equation in the cylindrical area of Euclidean space and formulate the mixed problem with non-homogeneous boundary conditions for it. We show the unique solvability of the problem for the class of continuously differentiable functions and give a way to construct its explicit classical solution.

A rod system under the action of a quasi-statically increasing tensile tension is considered. The load is carried out according to soft and hard schemes. One of the rods of the system has the property of deformation softening, that is, its tension diagram has a branch falling to zero. As a result, the equilibrium equations do not satisfy the Hadamard conditions. The system has several equilibrium positions, including unstable ones. The application of the simple iterations method is shown to determine the parameters of all possible equilibrium positions and their stability when solving these equations that do not satisfy the Hadamard conditions.