Long-term strength of metals in complex stress state (a survey)

An analytic survey of experimental data and theoretical approaches characterizing the long-term strength of metals in complex stress state is given. In Sections 2 and 3, the results of plane stress tests (with opposite and equal signs of the nonzero principal stresses, respectively) are analyzed. In Section 4, the results of inhomogeneous stress tests (thick-walled tubes under the action of internal pressures and tensile forces) are considered. All known experimental data (35 test series) are analyzed by a criterion approach. An equivalent stress σ _e is introduced as a characteristic of the stress state. Attention is mainly paid to the dependence of σ _e on the principal stresses. Statistical methods are used to obtain an expression for σ _e, which can be used to study various types of the complex stress state. It is shown that for the long-term strength criterion one can use the power or power-fractional dependence of the time to rupture on the equivalent stress. The methods proposed to describe the test results give a good correspondence between the experimental and theoretical values of the time to rupture. In Section 5, the possibilities of complicating the expressions for σ _e by using additional material constants are considered.     

Constitutive equations for hot-working of metals

Elevated temperature deformation processing - “hot-working,” is an important step during the manufacturing of most metal products. Central to any successful analysis of a hot-working process is the use of appropriate rate and temperature-dependent constitutive equations for large, interrupted inelastic deformations, which can faithfully account for strain-hardening, the restoration processes of recovery and recrystallization and strain rate and temperature history effects. In this paper we develop a set of phenomenological, internal variable type constitutive equations describing the elevated temperature deformation of metals. We use a scalar and a symmetric, traceless, second-order tensor as internal variables which, in an average sense, represent an isotropic and an anisotropic resistance to plastic flow offered by the internal state of the material. In this theory, we consider small elastic stretches but large plastic deformations (within the limits of texturing) of isotropic materials. Special cases (within the constitutive framework developed here) which should be suitable for analyzing hot-working processes are indicated.     

Complete general solution of Stokes equations for plane boundaries

The completeness of a representation for a solution of Stokes equations is proved which is suitable for solving problems which involve plane boundaries. We discuss two theorems for Stokes flow past a plane boundary with different boundary conditions and illustrate them with examples.     

Complete list of first integrals for dynamic equations of motion of a solid body in a resisting medium with consideration of linear damping

A new case of integrability in the spatial problem of motion for a solid body with consideration of the nonconservative moment of forces is discussed. A nonconservative force field of action of the medium on the body is constructed. Contrary to some previous author??s works, a linear dependence of this field on the angular velocity is taken into account, although the introducing of this dependence into the components of such a field is not obvious in advance.     

Shock waves and equations of state of matter

The physical properties of hot dense matter over a broad domain of the phase diagram are of immediate interest in astrophysics, planetary physics, power engineering, controlled thermonuclear fusion, impulse technologies, enginery, and several special applications. The use of intense shock waves in dynamic physics has made the exotic high-energy density states of matter a subject of laboratory experiments and enabled advancing by many orders of magnitude along the pressure scale to range into the megabars and even gigabars. The present report reviews the contribution of shock-wave methods to the problem of the equation of state (EOS) at extreme conditions. Experimental techniques for high-energy density cumulation, the drivers of intense shock waves, and methods for the fast diagnostics of high-energy matter are considered. It is pointed out that the available high pressure and temperature information covers a broad range of the phase diagram, but only irregularly and, as a rule, is not thermodynamically complete; its generalization can be done only in the form of a thermodynamically complete EOS. As a practical example, construction of multi-phase EOS for iron is presented. The model’s results are shown for numerous shock-wave data, the high-pressure melting and evaporation regions and the critical point of iron.     

Hierarchial structure theory for basic equations of fluid mechanics (BEFM) and the simplified Navier-Stokes equations (SNSE)

A hierarchial structure for the basic equations of fluid mechanics (BEFM) is found through the analysis of scales of length and time that proves a measure of the rate of change of the quantities describing the motion of the fluid as well as an estimation of the order of magnitude of various terms included in BEFM. The hierarchial structure theory shows that if (1) the characteristic Reynolds numbersRe is larger than unity and (2) the length scale in one coordinate direction is larger than that in other coordinate directions. BEFM can be classified into some levels according to the estimation of the order of magnitude of various terms included in BEFM. The hierarchial structure of BEFM has two branches: one is from BLE- to BEFM inner hierarchy, the other is from EE- to BEFM outer hierarchy, where BLE and EE are abbreviations of the boundary-layer equations and of Euler equations, respectively. The relationship between the two branches of the hierarchial structure, the characteristics, subcharacteristics and mathematical properties of the hierarchial equations are studied. A comparison between the present hierarchial equations and the Simplified Navier-Stokes equations (SNSE) appeared in literatures is also made. BLE-, EE-and Inner-outer-matched (IOM) equations hierarchies are the most important and useful three levels for solving viscous flow-fields approximately.     

Constitutive equations of a tensorial model for ductile damage of metals

Based on a micromechanical concept of void growth and change in void shape, a dissipation potential and constitutive equations for ductile damage of metals are presented. Multiplicative decomposition of the metric transformation tensor and thermodynamic formulation of the constitutive equations lead to a symmetric second-order tensor of damage which is physically meaningful. Its first invariant defines the damage related to plastic dilatation of the material due to the void growth. The second invariant of the deviatoric tensor accounts for the damage associated with a change in the void shape. Two physically motivated normalized measures allow us to represent the kinetic process of strain-induced damage by using the equivalent parameter of damage including the limit conditions for the onset of void coalescence and ductile failure. An experimental analysis of the evolution of ductile damage is presented for the case of uniaxial tension of sheet steel specimens with artificial defects.     

A hierarchy of dynamic equations for solid isotropic circular cylinders

This work considers homogeneous isotropic circular cylinders adopting a power series expansion method in the radial coordinate. Equations of motion together with consistent sets of end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton’s principle. Time domain partial differential equations are obtained for longitudinal, torsional, and flexural modes, where these equations are asymptotically correct to all studied orders. Numerical examples are presented for different sorts of problems, using exact theory, the present series expansion theories of different order, and various classical theories. These results cover dispersion curves, eigenfrequencies and the corresponding displacement and stress distributions, as well as fix frequency motion due to prescribed end displacement or lateral distributed forces. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering cylinder equations using low order truncation.     

Long-term strength of metals under an equiaxial plane stress state

This paper reports test results suggesting a significant dependence of the long-term strength of metals on the form of the stress state and the method of short-term loading. The experimental data obtained are described using a modification of the kinetic theory of long-term strength containing a vector damage parameter and taking into account strength anisotropy and the damage due to short-term loading. It is shown that the experimental and theoretical values of the time to rupture are in good agreement. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 150–157, July–August, 2009.     

Stability of Schr(o)dinger-Poisson type equations

Variational methods are used to study the nonlinear Schr(o)dinger-Poisson type equations which model the electromagnetic wave propagating in the plasma in physics. By analyzing the Halniltonian property to construct a constrained variational problem, the existence of the ground state of the system is obtained. Furthermore, it is shown that the ground state is orbitally stable.