Outline.

1. From stresses to molar energy.
2. Physical meaning of molar local energy.
3. Molar energy of the strained rigid body.
4. From the structure-energy ideal model to the state equation of the real strained rigid body.
5. Theses of the structure-energy model of rigid body strength, fracture and straining.
6. Definitions. Dependences of properties of the strained rigid body, gained with use of physical molar characteristics of the body state with strength quasi-particles.
7. Examples of calculations of parameters of strength, fatigue, etc. Calculation of molar characteristics of Д16Т alloy.

1. From stresses to molar energy.

The Zhurkov's formula of the kinetic concept of solids strength is:

\(\begin{equation} \tau_{*}=\tau_{o} \exp \frac{{U_{o}-\gamma_{o}\sigma}}{RT} \tag{1} \end{equation}\)

where

\(\tau_{*}\) - is durability, interval of time before fracture of the sample;

\(\tau_{o}, U_{o}\) are parameters of monatomic level;

\(\gamma_{o}\) is a structural parameter of a material;

\(T\) is temperature;

\(R\) is gas constant;

\(\sigma\) is constant tension stresses.

In the kinetic concept the structural parameter \(\gamma_o, \: m^3/mole \), is considered as a value of activation volume which controls overstresses on the atomic bonds in a rigid body.

The concept lacks the dependences revealing the physical gist of “stresses on atoms”, on belief of authors of the concept, these dependences are hidden in the parameter \(\gamma_{o}\)  [1, 2]. Process of atomic bonds fracture in the course of time, at constant test parameters, is observed during experiments [2]. But in the concept there are no analytical dependences of atomic bonds energy or structure parameter   from the time period of macroscopic stresses effect.

The formula (1) inherently represents a probabilistic process of emersion of destroying fluctuations of atoms energy in a molar volume of the strained rigid body [3]. Process of irreversible diminution of amount of strong bonds in a structurally-heterogeneous conglomeration of the stressed material in the course of time is experimentally confirmed at the paper [4].

From general statements of the physical theory it is known that atomic bonds work by quantum mechanics principles, laws of microscopic random statistical processes, but  in the concept there is no clear physical explanation of such bonds and these processes. However, the concept includes the fundamental experimental results [4], allowing to determine analytical relation for modification of structural parameter \(\gamma\) in the course of time under the influence of stresses and temperature, and to evaluate the rate of irreversible fracture of the strong bonds.

The concept of mechanical stresses originated in the theory of elasticity, where the hypothetical ideal elastic medium – continuum is considered, in which there is no monatomic microstructure, and processes are reversible. An elastic stress presumes reversible processes in the strained rigid body, just as in thermodynamics of gases. Various limit stresses (of fluidity, strength, etc.) are the empirical characteristics which do not have the grounds in the theory of elasticity, without unique physical substantiation.

Further let's pass on to the statistical physics and consider microscopic irreversible (dissipative) energy processes of elastic energy dissipation in the solids.

Рассмотрим, что представляют в формуле (1) основные параметры \(U_o\), \(\gamma_o\). in the formula (1). Relying on the acknowledged experimental results and fundamental theory, we determine the physical sense of these parameters, having the dimension of energy and volume, relating to a gram-molecule of some states of microscopic level [5, 6]. In particular, we are interested in the \(U_o, \: J/mole\) energy of fracture activation per mole, \(\gamma_o, \: m^3/mole\) – the material structural parameter with dimension of molar volume, and product \(\gamma_o \sigma\) with dimension of molar density of energy, J/mole. What is the physical meaning of the molar density of energy of the strained rigid body in the kinetic concept?

Let's term this value briefly as local molar energy and designate it as follows:

\(\begin{equation} W_L=\gamma_{o} \sigma, \: J/mole \tag{2} \end{equation}\)

2. Physical meaning of molar local energy.

(Remark: the gram-molecule has been acknowledged as the physical quantity in 1971, and thus the explanation of this concept has been dilated).

From the perfect gas law

\(\begin{equation} pV_\mu=RT, \: J/mole, \: T=const \tag{3} \end{equation}\)

it is easy to obtain the value of the elementary volume which incorporates one ideal corpuscle:

\(\begin{equation} \mathrm{d} v = \bar{v}_\mu = \frac{V_\mu}{N_A}, \: m^3/unit \tag{3.1} \end{equation}\)

The elementary volume contains the elementary average portion of energy of ideal gas: 

\(\begin{equation} \mathrm{d} \bar{w}=\bar{w}_\mu = \frac{pV_\mu}{N_A}=\frac{RT}{N_A}=\frac{kN_AT}{N_A}=kT, \: J/unit \\ \mathrm{d} \bar{w}=\bar{w}_\mu=kT, \: J/unit \\ \mathrm{d} \bar{w} \approx \bar{w}_\mu = \mathrm{d}w_x = \mathrm{d} w_y = \mathrm{d} w_z \\ \tag{3.2} \end{equation}\)

where, \(V_\mu, \: m^3/mole\), is molar volume of gas, \(N_A=6.022 \cdot 10^{23}, \: unit/mole \) is Avogadro number. Relying on basic postulates of the kinetic and wave field theory, at the paper [7] it is shown that the gram-molecule is an amount of the elementary masses and simultaneously of the elementary portions of wave energy (energy quasi-particles) of kinetic motion of the elementary microscopic physical states in the elementary molar volumes of the examined substance. There is formulated the extended definition of gram-molecule of steady physical structure states of energy of microscopic motion in the elementary molar volumes, which follows from the wave equation of energy continuity and the state equation of gram-molecule of ideal gas. The amount of these states as portions of this energy is equal to the number of molecules, atoms, ions, etc. in the viewed molar macroscopic volume. 

According to the kinetic theory and the field theory in the elementary volume there are two colliding processes of liberation and absorption of microscopic energy. Within the bounds of fundamentals of the gas kinetic theory each elementary portion of microscopic energy in the elementary molar volume \(\bar{v}_{\mu}\) could be considered in the linear approximation as a quasi-particle or a wave of energy of the microscopic heat motion, produced by periodic fluctuation of thermal kinetic energy in the elementary volume. For the characteristic period of fluctuation the elementary energy originates and is absorbed in the elementary volume under the linear law, ensuring a  state of thermodynamic balance.

Further we consider this elementary portion of microscopic motion energy as a result of fluctuation, as the quasi-particle of energy originating with a certain period (frequency) as a result of violation (fluctuation) of the microscopic balance of energy in the elementary volume. At the same time, the macroscopic amount of the elementary volumes \(V_\mu=N_A \bar{v}_\mu\) is characterized by the average parameters \(p, \: T\) of the thermodynamic balance. 

The quasi-particle of strength is a part of the whole energy of idealized microscopic motion which we formally attribute to the elementary volume of the whole macroscopic volume of a gram-molecule of states.

The quasi-particle energy appears and disappears in the volume element with the characteristic period of fluctuations \(\Delta \tau_\mu\) The average level of the microscopic energy of fluctuation is \(\bar{w}_\mu\).

Using such approach as a first approximation, we get a linear time-dependent function of the elementary microscopic energy local potential of the kinetic motion in the elementary volume and average state variables of one gram-molecule of idealized gas corpuscles being in a thermodynamic balance. It is possible to term these idealized corpuscles (waves) as quasi-particles, as we deal with probability of founding of portions of mass and energy in the elementary volume simultaneously. Let's consider the divergence flux of local molar energy of microscopic motion of quasi-particles for volume of one gram-molecule [7]: 

\(\begin{equation} \bar{W}_{pL}(t)=\frac{pV_\mu}{\Delta \tau_\mu}t, \: J/mole \tag{4} \end{equation}\)

where \(\bar{W}_{pL}(t)\) is the integral potential of density of local molar energy of one gram-molecule of theoretical corpuscles (energy quasi-particles) of ideal gas. 

\(\begin{equation} \bar{G}_{L}(t)=\frac{RT}{\Delta \tau_\mu}t, \: J/mole \tag{4.1} \end{equation}\)

where \(\bar{G}_{L}(t)\) is a function of the average molar idealized elementary flux potential (the divergence flux) of the local molar energy of the tridimensional vector field of a gram-molecule of the elementary volumes of energy quasi-particles. It is easy to see that the partial derivatives of functions (4), (4.1) are the average rates of changing the local microscopic potential during \(\Delta \tau_\mu\), for the stationary equilibrium state of gas under condition of \(T=const\). 

In terms of the kinetic theory of ideal gas, the wave and vector field theory for one component (coordinate) of tridimensional volume:

\(W_{pL}=pV_\mu, \: J/mole\), is an average molar density of energy of theoretical corpuscles (quasi-particles);

\(G_L = RT, \: J/mole\), is an average divergence flux of molar volume of energy corpuscles or flux of the kinetic energy of corpuscles (quasi-particles) [7].

From the theorem of Gauss – Ostrogradsky follows, that the rate of change of the molar density of energy of corpuscles is equal to the divergence of energy flux of corpuscles. According to the ideal gas kinetic theory we have a flux of the kinetic energy of thermal microscopic motion of mass corpuscles, or quasi-particles, of the molar energy. From this reasoning we gain the equation of relationship between the molar energy and the state variables of gas for one gram-molecule of idealized quasi-particles for one of tridimensional coordinates [7, 8]:

\(\begin{equation} \frac{\partial W_{pL}}{\partial t}=\frac{2RT}{\Delta \tau_\mu}, \: J/mole, \: T=const \tag{5} \end{equation}\)

where \(\Delta \tau_\mu\) is the specific period of occurrence of fluctuation of the kinetic energy of corpuscles \(\bar{w}_\mu\) in the elementary associated molar volume of one of assemblages of such ideal corpuscles (3.2), generally it is a function of time and system parameters.

It is easy to show that for a steady state of thermodynamic balance the equation (5) could be transformed into the equation for ideal gas (2). The equation (5) allows to specify an ideal gas state with additional molar characteristics: 

\(\frac{\partial W_{pL}}{\partial t}\) - a rate of change of the molar density of microscopic energy;

\(\frac{2RT}{\Delta \tau_\mu}\) - an average rate of change of the local flux, a local acceleration of the divergence flux of the molar microscopic energy of ideal corpuscles (of the kinetic flux) or energy quasi-particles of gas.

Thus, the perfect gas law can be gained from the wave energy continuity equation for volume of one gram-molecule of ideal elementary mass particles, or quasi-particles of energy of the thermal kinetic motion; the particles state of thermodynamic balance is characterized by periodic thermal fluctuations.

The main deduction of these reasoning is that a gram-molecule is an amount of the elementary portions of mass, and also idealized portions (waves) or energy quasi-particles \(\mathrm{d}\bar{w}\) of microscopic kinetic motion in the elementary molar volumes \(\mathrm{d}v = \bar{v}_\mu\)(3.1), from which the volume of one gram-molecule of ideal gas is organized. On the surface of volume of one gram-molecule of corpuscles or quasi-particles, according to the vector field theory, there exists a flux of energy divergence (the kinetic flux). The average value of flux of the local molar energy of quasi-particles in the conditions of the thermodynamic balance along one coordinate axis is equal to \(RT\).

\(R=kN_A,\: J/mole \cdot deg\) is the universal gas constant, \(N_A\) is the Avogadro number, \(k, \: J/deg \cdot unit\) is the Boltzmann constant. This is the necessary and sufficient physical condition of presence at one gram-molecule volume of steady microscopic states of thermal motion of the given substance. Hence, it is possible to say that the increase of temperature \(T+\Delta T\) of system from ideal corpuscles (for example, gas) means the growth of the molar energy density and growth of the molar flux of ideal elementary portions of energy by value \(\Delta Tk\), where \(k\) is the elementary minimum ideal physical portion (quantum) of energy of microscopic thermal motion of medium.

It is possible to say that in the direction of each coordinate axis (tensor) we have a flux of quasi-particles of gas kinetic energy, as in the elementary volume of ideal gas there is no classical mechanical mass or an energy corpuscle in the literal (material) sense. In the elementary volume there is a probability and the relevant frequency of occurrence of mean values of the indicated physical microscopic properties.

Further we will use the new physical concepts of the molar density and the flux of energy quasi-particles of the joint kinetic structure-energy idealized model of microscopic processes of heat-mechanical motion of atoms (the elementary units) that takes place at the irreversible (plastic) volume forming at fracture of strong atomic bonds of the real structurally heterogeneous strained rigid body.

3. Molar energy of the strained rigid body. 

At the paper [8], using the Zhurkov's formula of durability  (1) and the experimental relationship of the kinetic concept [4], the analytic function of structural parameter of a material \(\gamma(t)\), has been gained, that was shown as dependency of base (minimum theoretical) molar volume of strength quasi-particles from time and the prespecified constant state variables of strained rigid body \(T=const\), \(\sigma=const\):

\(\begin{equation} \gamma(t)=\frac{1}{\sigma}\left [ U_o - RT \cdot ln(\frac{\tau_{*o}-t}{\tau_o}) \right ], \: m^3/mole \tag{6} \end{equation}\)

Using Taylor's series development of dependency (6), at the work [8] there has been specified the rate of change of the base molar volume at constant stresses:

\(\begin{equation} \frac{\mathrm{d} \gamma}{\mathrm{d} t}=\frac{RT}{\tau_o \sigma} exp\frac{\gamma_o \sigma - U_o}{RT}, \: m^3/mole \\ T=const, \: \sigma=const, \: |\sigma|>0 \tag{6.1} \end{equation}\)

Using (1), from (6.1), transforming by multiplying on \(\sigma\), we gain the molar power or rate of irreversible change of the molar energy density, similarly to the dependency for ideal gas (5): 

\(\begin{equation} \frac{\mathrm{d} W_L}{\mathrm{d} t}=\frac{RT}{\tau_*} \\ T=const, \: \sigma=const \tag{6.2} \end{equation}\)

where \(\tau_*\) - is a latency period of the associated fluctuation of fracture of atomic bonds (electron-pair bindings) in the macroscopic volume, and also the durability from (1).

Proceeding from physical analogy of kinetic processes described in (5), we have as a first approximation from (6.2) physically similar dependences which represent two characteristics of gram-molecule:

\(\frac{\mathrm{d} W_L}{\mathrm{d} t}\) - a rate of change of the molar density of quasi-particles energy as a result of irreversible fracture of atomic bonds; 

\(\frac{RT}{\tau_*}\) - a local speed of the molar flux of the quasi-particles kinetic energy. 

 

4. From the structure-energy ideal model to the state equation of the real strained rigid body.

At papers [8, 9], using new physical concepts of quasi-particles molar volume and quasi-particles gram-molecule energy, there has been revealed the relationship between the density of elastic deformations energy \(W_\sigma\), the minimum (base) molar volume of quasi-particles \(\gamma(t)\) and the molar energy of strength quasi-particles \(\bar{W}_L\). The strength quasi-particles are considered as microscopic waves of kinetic energy in the small elementary volumes.

The elementary volume is equal to the volume of an ideal structural fragment (ISF) of the strained rigid body which includes one ideal strong bond of monatomic level. Quasi-particles of strength appear as a result of fracture of some quantity of actual atomic bonds, from temperature fluctuations of kinetic energy of the elementary components in the real structurally-heterogeneous strained rigid body. Energy of quasi-particle is an energy of destruction of one idealized (abstract) strong associated structure bond in an ideal structural fragment (ISF).

Physical analytical dependences of the molar characteristics and the basic physical properties of state of the strained rigid body from time and constant tension stresses has been gained in relations (6), (6.1), (6.2). These dependences result from immediate experimental data of the strength kinetic concept [4].

Strength quasi-particles are considered at papers [9, 10] as energy of microscopic elastic waves in the actual structurally-non-uniform strained rigid body. As a result of fluctuations of energy of microscopic thermal motion a state of thermodynamic balance is violated, which also causes the destroying of the mechanical balance of microscopic energy fluxes that represent the strong energy bonds in the particular elementary volume. Thus, in a small volume the thermomechanical equilibrium (thermal and mechanical balance of microscopic energy fluxes) is destroyed. The quasi-equilibrium condition of atoms energy interchanging in the elementary volume is upset. There arise waves or microscopic fluxes of impulses of the structurally-oriented microscopic kinetic motion of elementary components (experimentally observable microscopic detrusion-displacement of structural components) which we regard in the linear approach as quasi-particles or energy waves, that are moving in three orthogonal directions (three degrees of freedom). By their physical wave nature the quasi-particles of the strength energy originating from fracture of strong atomic bonds are similar to excitons of Frenkel, excitons of Wannier – Mott, phonons of thermal conductivity [11].

The basic physical molar properties of rigid body in the structure-energy strength theory are considered at papers [9, 10, 12]. There were yielded the definitions of generic concepts of the real rigid body (RB), the strength quasi-particle, the gram-molecule of strength quasi-particles, and was built the idealized model of the real strained rigid body as assemblage of ISF, etc.

Let's have a look at the basic dependences taken as the principles of the theory.

At paper [8], using the concepts of gram-molecule and local molar energy of strength quasi-particles of the strained rigid body, there was offered the analytical dependence linking the elastic energy of the strained rigid body and the local molar energy of quasi-particles originating at fracture by fluctuations of ideal strong bonds in the strained rigid body. 

\(\begin{equation} \bar{W}_L=W_\sigma \cdot Sh, \: J/mole \tag{7} \end{equation}\)

where \(Sh(t), \: m^3/mole\) - is a function of the strained rigid body volume containing one gram-molecule of idealized elementary portions or quasi-particles of the microscopic kinetic energy. The strength quasi-particles characterize the microscopic waves of heat-mechanical energy at fracture of some quantity of base strong bonds of elastic interacting at monatomic level between base units (a lattice fragment, a molecule segment, a cluster) in a rigid body conglomeration, which originate on the remained base bonds of strength between interconnected elementary components of the rigid body, which at the same time are parts of the mentioned base units.

\(\begin{equation} W_\sigma = \frac{\sigma^2}{2E}, \: J/m^3 \tag{7.1} \end{equation}\)

where \(E\) is a macroscopic elastic modulus. In dependence (7) the effect of one component of density of energy of elastic deformations of the particular volume is considered.

From (7), (7.1), after elementary transformations and new designations we gain:

\(\begin{equation} \bar{W}_L=\frac{\sigma^2}{2E} Sh = \frac{\sigma}{2E}\sigma Sh = \frac{\sigma}{2E} \overleftrightarrow{Gr} =\frac{Gr}{E} \sigma=\gamma_r \sigma \\ \sigma \cdot Sh = \overleftrightarrow{Gr}, \: J/mole \\ 2 |Gr|=\overleftrightarrow{Gr} \tag{8} \end{equation}\)

where \(Gr(t), \: J/mole\), is a new physical quantity, a material structure-energy state variable or a potential of energy of one-way flux of the molar energy of microscopic heat-mechanical motion of the elementary components, presented by a flux of ideal quasi-particles of strength in the conditions of particular quasi-equilibrium state of the strained rigid body. Dependency (8) is the structure-energy law of the strained rigid body state [9]; it is similar by physical meaning to the Boyle's gas law. In the rigid body model it is supposed that the boundary of ISF segments is intercrossed by two equal in magnitude counterflows (incoming and outgoing) of the kinetic motion of corpuscles - \(Gr(t)\) [9]. The integral value of the molar energy flux through the boundary is equal to \(\overleftrightarrow{Gr}\) (two-way flux).

The initial value of the potential function is \(Gr_o=Gr(t=0)\). 

In the same way the new values are denoted: 

\(\begin{equation} \gamma_r = \frac{\overleftrightarrow{Gr}}{2E}=\frac{Gr}{E}, \: m^3/mole \tag{9} \end{equation}\)

Finally from (7), (8), (9) we gain dependency for the molar density of energy (2) which is present at the Zhurkov's formula of durability (1): 

\(\bar{W}_L=\gamma_r \cdot \sigma\)

Thus, \(Gr\) is an average energy potential or parameter of the microscopic energy flow of the kinetic motion (a flux of impulses of the kinetic motion) from irreversible destruction of microscopic energy fluxes that form atomic bonds, determined in a direction of one component of the tensor of principal stresses of the particular quasi-balance structurally-physical state of the strained solid material.

\(\gamma_r(t), \: m^3/mole\), is a structure-sensitive state parameter of the material, that is equal to the minimum theoretical (base) volume of a gram-molecule of the elementary bundles of quasi-particles energy that are generated at fracture of atomic bonds of the elementary structure units (ESU) of the strained rigid body in the current structurally-physical state of medium at the stress \(\sigma=E\). 

Proceeding from stated, we gain the initial value of structural coefficient of a material in the formula (1) or the minimum theoretical base molar volume of quasi-particles:

\(\gamma_o=\gamma_r(t),\: t = 0\) 

From (7), (8) follows:

\(\begin{equation} \gamma_o=Sh_o=Sh(t=0,\sigma=E)=\frac{Gr_o}{E} \tag{10} \end{equation}\)

Volume\(\gamma_o\) is determined experimentally by procedures of the kinetic concept of strength, under conditions of \(T=const\), \(\sigma=const\).

Dependences (7) and (8) follow from the experimental property of constancy of structural parameter \(\gamma_o\) (invariancy to stress) in the Zhurkov's formula of durability, the physical property of the molar energy that follows from the kinetic theory of ideal gas, the structure-energy strength theory and the idealized rigid body model.

From the dependence (10) follows that the initial structural parameter \(\gamma_o\) in the Zhurkov's formula of durability (1) is equal to the initial minimum value of molar volume of strength quasi-particles in the rigid body \(Sh_o\) at the stress of \(\sigma=E\). It is easy to show that \(\gamma_o\) is a coefficient of proportionality between the molar energy density \(\bar{W}_L\) and acting stresses \(\sigma\) in the formula (1).

In the paper [9] it is shown that the dependency of the molar local power of destruction of strength quasi-particles, fracture of the strong atomic bonds of the strained rigid body is based upon the theoretical dependence for the continuance of fluctuation of energy necessary for fracture of atomic bonds at detachment from a crystal lattice surface, described in the kinetic theory of crystals by Y.I. Fraenkel:

\(\Delta \tau_r=\tau_o exp\frac{U_R}{RT}\)

where \(U_R=U_o-\bar{W}_L,\: U_R\) is an activation energy of fracture of bindings with surface of SU (base structure unit) for one gram-molecule of quasi-particles (the elementary atomic bonds) characterizing bond of atom (ESU) with a boundary surface of the elementary segment (a crystal lattice, a molecule, a cluster) of an ideal base unit of a conglomeration of the strained rigid body. 

\(\tau_o = 1 \cdot 10^{-13} \: s\) is the specific period of thermal collective oscillations of atoms in the body volume by Debye [13]. In the idealized model of the strained rigid body the state of the strength quasi-particles is considered like ideal gas and is characterized by the corresponding equation of state [9, 10].

With use of the gas kinetic theory, the crystalline solid kinetic theory, the vector field theory and the gained analytical results for conditions of \(T=const\), \(\sigma=const\), at papers [9, 10] the generalized idealized structure-energy model and the structure-energy law (8) for the strained rigid body were formulated. The equation of state of the strained rigid body for the given arbitrary function of stresses and multi-axial stress condition [10] was gained.

For one component of the principal stresses the equation of state of the strained rigid body looks like: 

\(\begin{equation} \frac{\mathrm{d} \bar{W}_L}{\mathrm{d} t}-\frac{RT}{\tau_o}exp\frac{W_L(t)-U_o}{RT}=0 \\ T=const, \: |\sigma|>0 \tag{11} \end{equation}\)

where, according to [8], the molar energy could be expressed as follows:

\(\begin{equation} \bar{W}_L=\gamma(t) |\sigma(t)| \tag{11.1} \end{equation}\)

\(\begin{equation} \bar{W}_L=2W_\sigma(t)Sh(t). Where \: W_\sigma=\frac{\sigma^2}{2E} \tag{11.2} \end{equation}\)

From (11.2) it is apparent that the molar energy is invariant to the sign of stresses.

\(\begin{equation} \bar{W}_L=\frac{|\sigma(t)|}{E}Gr(t) \tag{11.3} \end{equation}\)

Generally an analytical physical energy requirement of macroscopic spontaneous brittle fracture is:

\(\begin{equation} U_o-\gamma(t)|\sigma(t)|=0,\: U_o-\bar{W}_L=0 \\ T=const \tag{12} \end{equation}\)

Physical criterion of fracture for the arbitrary limiting condition is:

\(\begin{equation} U_o-\bar{W}_L=A \tag{12.1} \end{equation}\)

where \(A\) is the normalized index of the molar energy of state of creepage, loosening, brittleness, etc.

To solve the equation (3.1) it is necessary to set the stress function \(\sigma(t)\), and the initial conditions: \(U_o=const\) - an activation energy of atomic bonds fracture, \(\gamma_o=\gamma(t=0)=Sh_o(E)\) - an initial value of the material structural parameter, \(\tau_o=const\) a physical parrameter.

For the arbitrary type of loading the solution of the equation (11) is not discovered.

At paper [12] examples of the numerical solution of the state equation for non-stationary loadings of the low-cycle fatigue are shown, the characteristic curves of effect of a loading frequency on the fatigue limit were gained, the calculations of aluminum alloys creeping at various stresses, temperature, etc. are fulfilled. The gained results well coincide with experimental data of independent studies. For a particular case of \(\sigma=const\) exemplary calculation is given below. 

5. Theses of the structure-energy model of rigid body strength, fracture and straining.

Resuming the experimental facts in the mechanics of SRB (the strained rigid body), the materials technology, the kinetic theory of crystals, the theory of ideal gas, etc., at paper [9] the kinetic structure-energy idealized model of a real rigid body composed of atoms and ideal base units was formulated. The model primarily takes into consideration the flaws of the rigid body monatomic structure. These flaws are expressed in the value of structure-energy parameter \(\overleftrightarrow{Gr}\), which is determined experimentally through the Zhurkov's formula parameter \(\gamma\) and further depends on the conditions and the continuance of the loading process. It is supposed that the boundaries of base units of SRB are microscopic volumes in which there are the "overstress" boundary fluxes of the molar energy. Boundaries of SU are intercrossed with the associated base atomic bonds (an energy flux of the kinetic thermo-mechanical motion), which periodically are broken irreversibly from fluctuations. The deforming of the rigid body macroscopic volume is considered in the theory as simultaneous process of elastic microstructural strains of SU, irreversible mutual microscopic displacement of idealized SU, as a result we obtain the macroscopic plasto-elastic volume deformation of the whole SU conglomeration. Irreversible macroscopic strains are considered as a result of accumulation in the macroscopic volume of the cumulative spatially oriented microscopic intermittent random elementary detrusions, displacements among elastic strained idealized ISFs (in the model they replace a lattice, a molecule segment, a cluster). The elementary detrusion is similar to the physical mechanism of the intermittent microstructural detrusion of Fraenkel – Eyring.

Macroscopic deformations are considered as the total of projections of the elementary detrusions in the ISF in three orthogonal axes directions of the tensor of main true stresses. The elementary detrusion is a result of fracture of one idealized strong bond of atomic level in an idealized structural fragment. Fracture of a strength atomic bond changes the molar density of energy, causes (activates) waves and the divergence flux of the molar energy portions – the strength quasi-particles. This process in the macroscopic volume is characterized by the following parameters: the density of energy of elastic deformations, the molar energy and the molar local power, absolute and relative changes of a physical value of molar volume of strength quasi-particles, evaluated in three orthogonal directions of tensor of principal stresses (strains). Energy of quasi-particles \(\bar{W}_L\) is a value on which the level of energy barrier of fracture activation (the limiting molar density of energy) owing to heat-mechanical motion of atoms (a motion flux) of the strained rigid body in the fields of stress and temperature drops:

\(U_R=U_o-\bar{W}_L\)

The macroscopic forming in the offered model is considered like rheological fluxion of the viscous compressible fluid organized by idealized base units – a viscous conglomeration. Body base units interact among themselves by means of the collective associated bonds of monatomic level. Mechanical and physical properties of a rigid body (irreversible deformations, formation of flaws and free surface, a heat generation, etc.) are considered through the properties of functions, derived from the molar volume and the molar energy of the medium organized by strength quasi-particles. Fracture of ideal atomic bonds are characterized by energy of quasi-particles, power of processes of quasi-particles destruction and other properties of connections fracture (energy quasi-particles destruction) in a rigid body volume. These parameters uniquely determinate mechanical strength, plasticity of medium, etc. Local molar energy of quasi-particles of strength is the thermodynamic (isothermic – isobaric – isochronous) molar potential characterizing the dissipative (irreversible) losses of an elastic energy.

Thus, in this model of strength it is supposed that the volume of real solid medium is filled by the strength quasi-particles characterizing energy and power of process of fracture of base bonds of atoms in the ISF, being in the fields of mechanical stresses and temperature, and the state of atoms is not considered. In the structure-energy model, in the course of time of stresses impact, the volume of solid medium is continuously elastically and plastically changes its shape, these processes go concurrently (simultaneously). The total macroscopic strains of a rigid body volume are the sum of the elastic relative deformations and the irreversible relative deformations of the molar volume of quasi-particles of the strained rigid body. At destruction of quasi-particles (fracture of base atomic bonds) the elementary flaws and microscopic hollows are formed. In polycrystalline materials it is the elementary pointwise dislocations or their assemblage (linear, helical, etc.). Assemblage of such elementary flaws of a rigid body is a free surface, a microfracture, etc. With use of Hart – Gibbs equations, there was gained the dependency for determining energy of boundaries of a tridimensional phase which is related to the molar energy of quasi-particles. Thus, the formation energy of the free surface of a rigid body at accumulation of the strength quasi-particles destructions was estimated. The gained dependence allows to take account of a scale factor, the effect of the free surface on the strength of a material [12].

The irreversible microscopic forming (deforming) in the body volume happens from the first moment of the rigid medium solidification and is sped up (initiated) as a result of composition of fields of microscopic (local temperature) stresses and strains, macroscopic mechanical strains and stresses and a field of macroscopic temperature. Using general concepts of geometry, it is easy to show that the total relative elementary volume deformation (strain) of a real rigid body conglomeration   is the sum of the relative elementary deformations of geometrical space, relative deformations of an ideal elastic body   and relative deformations of the quasi-particles molar volume   (an idealized inelastic real body):

\(\varepsilon_{\Sigma}=\varepsilon+\varepsilon_r\)

where \(\varepsilon_r\) is the true irreversible strain, for the case of uniaxial deformation.

Properties of the ideal irreversibly destroyed structurally non-uniform rigid body model comply with the physical law that follows from the experimental dependences of the kinetic concept of strength and the wave theory [9, 10].

 

Variant 1. Analog of the Boyle – Mariotte law: 

\(pV=D, \: D=const, \: T=const\)

mass of gas is constant. All values have equal dimensionalities, the dependency is homothetic. The structure-energy law for a stationary state (8), at the constant energy of quasi-particles:

\(\sigma Sh=Gr, \: Gr=const\) 

\(Gr\) is the molar energy necessary for fracture of atomic bonds in the present structurally-physical condition of a material (solid substance). \(|\sigma|\) are the main true stresses, at \(T=const\), \(|\sigma|>0\). 

Relation between the quasi-particles molar volume and Zhurkov's structural parameter:

\(Sh_o(\sigma=E)=\gamma_o\)

where \(Sh_o\) is the minimum theoretical molar volume of base quasi-particles at a stress of \(\sigma=E\).

Variant 2. In the theory of elasticity there exists a model of an ideal elastic indestructible rigid body (reversible processes). The state of such body follows the Hooke law: 

\(\frac{\sigma}{E}=\varepsilon\)

By analogy to the Hooke law, the structure-energy law for a steady state could be put down as follows:

\(\frac{\sigma}{Gr}=Sr\)

where \(Sr=\frac{1}{Sh}, \: Gr=const\) 

Analogy of various relationships:

\(\varepsilon=\frac{\Delta V}{V}\) - is the relative volume;

\(Sr = \frac{n}{V}\) - is the relative number of quasi-particles in the volume;

\(n\) - is the number of quasi-particles.

For non-stationary and complicated stressed state at paper [10] the universal differential form of the structure-energy law and the combined equations of state were formulated respectively. 

6. Definitions. Dependences of properties of the strained rigid body, gained with use of physical molar characteristics of the body state with strength quasi-particles.

Physical molar properties of the strained rigid bodies are explicitly related to the common parameters of deforming and fracture in the mechanics and strength theory, objectively represent the irreversible physical microstructural processes of formation of microscopic flaws (in metals it is dislocations), of macroscopic free surface, generation of heat. Thus, in the structure-energy theory it is possible to present the processes of deforming and destruction by means of physical dependences. It is analytically possible to determine the microscopic and macroscopic parameters of the strained rigid body in the course of time through the changes of molar volume, molar energy, molar power, etc. [12].

Short definitions of the basic molar quantities.

\(Sh(t), \: m^3/mole\), is the molar volume of the elementary quasi-particles of the kinetic microscopic energy of atomic bonds fracture.

\(\bar{W}_L, \: J/mole\), is the molar energy of strength quasi-particles.

\(\gamma(t,\gamma_o,\sigma,T), \: m^3/mole\), is a structural function that characterizes an irreversible change in the course of time of a material structural condition. 

\(\gamma', \: m^3/mole/s\), is the rate of an irreversible change of the base quasi-particles volume. 

\(r(t,\sigma,T), \: mole/m^3\), is the structural density or amount of base quasi-particles in the unit of rigid body volume, \(r=1/\gamma\). 

Let's itemize the characteristics of the strained rigid body that follows from the equation of state and gained dependences of the theory.

Characteristic1. 

Time under load before origination of state of the macroscopic brittle fracture of a rigid body (durability) \(\tau_*\) at any rate of constant strains. The generalized Zhurkov's formula of durability: 

\(\begin{equation} \tau_*=\tau_o exp\frac{U_o-\bar{W}_{Lo}}{RT}, \: \bar{W}_{Lo}=\gamma_o\sigma=W_\sigma Sh_o \tag{13} \end{equation}\)

Свойство 2.

Isochronous state of a thermodynamic system is an energy requirement for retention of unvarying time before destruction:

\(\bar{W}_L=U_o-RT ln \frac{\tau_*}{\tau_o}=const\)

Characteristic 3. 

(experimentally gained characteristic of materials [4]).

Principle of summation of continuances of remaining the materials under the influence of equal stresses till the durability estimated value.

\(\tau_*=\tau_1+\tau_2+t_{*i}, \: \sigma=const\)

where \(\tau_i\) - is a continuance of deforming with unload (repose), \(t_{*i}\) - is the last period before destruction.

Characteristic 4.  

Irreversible destructions, changes of local energy in the course of time, density and specific molar volume of base quasi-particles \(\sigma=const\), \(T=const\). Diagrams of the dependences are presented on the fig. 1.

Local energy:

\(\begin{equation} \bar{W}_L(t)=U_o-RT \cdot ln \frac{\tau_{*o}-t}{\tau_o}, \: J/mole \tag{14} \end{equation}\)

Density of base quasi-particles:

\(Sr(t)=\frac{W_\sigma}{U_o-RT \cdot ln \frac{\tau_{*o}-t}{\tau_o}}, \: mole/m^3\) 

Specific molar volume of base quasi-particles:

\(\begin{equation} Sh(t)=\frac{1}{W_\sigma} \left [ U_o-RT \cdot ln \frac{\tau_{*o}-t}{\tau_o} \right ], \: m^3/mole \tag{15} \end{equation}\)

Structure-sensitive function: 

\(\begin{equation} \gamma_r(t)=\frac{1}{\sigma} \left [ U_o-RT \cdot ln \frac{\tau_{*o}-t}{\tau_o} \right ], \: m^3/mole \tag{16} \end{equation}\)

Characteristic 5.

Local molar power of destruction of base quasi-particles (the dissipative power in gram-molecules):

\(\begin{equation} q_L=\frac{\mathrm{d} \bar{W}_L}{\mathrm{d} t}=\frac{RT}{\tau_o} exp\frac{\bar{W}_L-U_o}{RT}, \: J/mole \cdot s \tag{17} \end{equation}\)

Characteristic 6.

Absolute rate of irreversible destruction of base quasi-particles, rate of dislocations generation, of atomic bonds fractures:

\(\begin{equation} I_R=r'(t)=\frac{RT}{\tau_*(W_L)W_L(t)\gamma_r(t)}, \: mole/m^3 \cdot s \tag{18} \end{equation}\)

Characteristic 7.

The relative rate of irreversible fractures of base quasi-particles \(I_r=I_R \cdot r^{-1}\).

\(\begin{equation} I_r=\frac{r'}{r}=\frac{RT}{\tau_*(W_L)W_L(t)}, \: 1/s \tag{19} \end{equation}\)

Characteristic 8

The absolute amount of irreversibly destroyed base quasi-particles. The same, of monatomic level flaws, of conventional pointwise dislocations, in the body volume unit, for a time span \(t\), in the direction of an axis of the stress tensor \(\sigma_1\).

\(\begin{equation} r_{n1}=N_A RT \int_{0}^{t} \frac{1}{\tau_*(W_L)W_L(t)\gamma_r(t)}\mathrm{d}t, \: unit/m^3  \end{equation}\)

Characteristic 9.

Formation of microcavities (microcracks) and the free surface of a deformed rigid body at irreversible destruction of base quasi-particles: 

\(\mathrm{d}A_s=\frac{1}{\delta_s}\mathrm{d}W_L, \: m^2/mole\)

where \(\mathrm{d}A_s, \: m^2/mole\), is the elementary specific area of the formed free surface in a deformable rigid body; \(\delta_s, J/m^2\), is the rheological coefficient of surface tension, it is spotted from experiments on destruction of samples different in sizes. Dependence is gained on the ground of Hart's equations [15], and the theory of phase boundaries of Gibbs.

Characteristic 10.

Macroscopic irreversible deformations of rigid bodies. Rate of the true irreversible relative deformations:

\(\begin{equation} \bar{\varepsilon}(t)=\int_{0}^{t} \frac{1}{\tau_*(W_L)W_L(t)}\mathrm{d}t=\int_{0}^{t}I_r \mathrm{d}t, \: \sigma=const \tag{20} \end{equation}\)

\(\bar{\varepsilon}\) is the integral true irreversible deformations, for the uniaxial deforming.

\(\begin{equation} \dot{\bar{\varepsilon}}_1(t)=\frac{RT}{\tau_*(W_L)W_L(t)}, \: 1/s \tag{21} \end{equation}\)

\(\dot{\bar{\varepsilon}}_1\) is the rate of the integral true irreversible deformations, for the uniaxial deforming.

Performing simple transformations (21), we gain the usual form of the final creep equation, the rate of the relative irreversible true deformation.

\(\begin{equation} \dot{\bar{\varepsilon}}_1(t)=\varepsilon_{ro} exp\frac{\gamma_r\sigma-U_o}{RT}, \: 1/s, \: \sigma=const \tag{21.1} \end{equation}\)

where \(\varepsilon_{ro}=\frac{RT}{\gamma_r\sigma}\frac{1}{\tau_o}, \: \gamma_r=\gamma_r(t)\).

Characteristic 11.

Heat generation, power and work of the heat generation process, at plastic deforming and destruction of base quasi-particles.

The power of heat generation as a result of irreversible destructions:

\(\begin{equation} q_1(t)=U_o \frac{RT}{\tau_*(W_L)W_L(t)\gamma_r(t)}, \: J/m^3/ s  \tag{22} \end{equation}\)

Work of destructions, energy of the heat generation in the volume unit:

\(\begin{equation} Q_1(t)=U_o \int_{0}^{t} \frac{RT}{\tau_*(W_L)W_L(t)\gamma_r(t)}\mathrm{d}t, \: J/m^3 \tag{22.1}  \end{equation}\)

where \(Q_1, \: J/m^3\), is the thermal energy generated in the volume unit of a rigid body, from effect of stresses \(\sigma_1\). \(\sigma=const\) during time \(t\), in the conditions of \(T=const\) the thermal energy is discharged which could be determined by computing the current definite integral.

New generic universal physical parameters of material durability and strength [12]..

\(I_r=\frac{r'}{r}, \: 1/s\), is the relative rate of irreversible destructions of base quasi-particles (Characteristic 7, or the relative rate of an irreversible change of the molar density of base quasi-particles). The quantitative physical measure of a deformable solid material ability to accumulate irreversible destructions (fractures) of atomic bonds at the time unit. New generic physical parameters of absolute \(I_{RG}\) and relative (dimensionless) \(I_{rG}\) accumulation rates of the irreversible changes of the strained rigid body, or the physical measure of damageability, it is spotted for conditions of  \(\sigma=const\), \(T=const\).

\(I_{rg}\) is the relative modulus of initial rate of the strength activation, of the material destruction, which is determined for the stresses value of  , during the initial moment of load application. It is the relative in re  parameter of the material.

\(\sigma_G\) is the stresses of the uniaxial tension at which durability (time period till the sample fracture) is equal to \(\tau_G=31.536 \cdot 10^6 s, \: (1 year)\).

\(I_{rG}=\frac{RT}{\sigma_{_G}\tau_*(t)\gamma_r(t)}\)

\(\sigma=const, \: T=const, \: T=293^{\circ}K, \: (T=20^{\circ}C)\)

\(I_R=r'(t), \: mole/m^3 \cdot s\), is an absolute rate of the irreversible destruction of base quasi-particles (Characteristic 6). It determines the forming absolute rate, accumulation of microscopic destructions, damageability, heat generation that are caused by the irreversible plastic changes in the strained rigid body.

\(I_{RG}\) is an absolute in re parameter, destruction rate of the material base strength quasi-particles, an absolute modulus of initial rate of the strength activation, of the material destruction. \(I_{RG}\) is the fracture rate of atomic bonds of the material, it is calculated for the uniaxial constant tension stress \(\sigma_G\), at the initial moment of load application.

\(I_{RG}=\frac{RT}{\sigma_{_G}\tau_{_G}\gamma_o^2},\: mole/m^3/s\)

Here \(\sigma_{_G}\) are the stresses, at which durability is equal to \(\tau_{_G}=31.536 \cdot 10^6 \: s \: (1 year),\: \sigma=const,\: T=const=293K \: (20^{\circ}C)\).

At the paper [12] values of \(I_{RG},\: I_{rG}\) for some materials are given.

These in re parameters represent the physical process of accumulation of defects, destructions and they allow to make a comparative estimate of the ability (rate) of the given material in the given structural condition to expend (to lose) the physical strength during the normalized interval of time. It is possible to set the necessary normalized level of stress and temperature, and to make a comparative estimate of rate of the material destruction process as the strength property, etc.

in re – actually, in fact (Lat.).

7. Examples of calculations of parameters of strength, fatigue, etc. Calculation of molar characteristics of Д16Т alloy.

On fig. 1 there are given the dependences diagrams of the molar quantities for the alloy Д16Т.

Fig.1

Character of analytically gained time dependences of some structure-energy parameters of alloy Д16Т, from the moment of load application to the state of macroscopic destruction \(\tau_*\), under conditions \(\sigma=const\), \(T=const\).

\(I_r\) is the relative rate of irreversible destructions (forming rate) of activated volume of base boundary atoms (19). 

\(\gamma_r(t)\) is a structure-sensitive function of the strained rigid body state (16).

\(\bar{W}_L(t)\) is the local energy of the activated volume (14). 

\(\bar{\varepsilon}\) is the integral true relative irreversible deformations of the uniaxial tension in the loading direction   (20). Initial parameters of material Д16Т, conditions of loading, and the experimental value of durability are taken from [14].

The time interval before macroscopic destruction \(\tau_G=31.536 \cdot 10^6 s, \: (1 year)\) is calculated by formula (13).

References.

1. Zhurkov S.N. The kinetic concept of the solids strength. Bulletin of the USSR Academy of Sciences, No.3. 1968, pp.46-52.
2. Regel V.R., Slutsker A.I., Tomashevsky E.G. The kinetic nature of the solids strength. Moscow, Nauka (Science). 1974, 560 p.
3. Fraenkel Y.I. The kinetic theory of fluids. Leningrad, Nauka (Science). 1979, 592p.
4. Zhurkov S.N., Sanfirova T.P. The strength temperature-time dependency of pure metals. Reports of the USSR Academy of Sciences. 1955, 2.101. pp.237-240.
5. Shtyryov N.A. An approach to estimation of time interval till the material disruption at arbitrary conditions of loading. Transactions, Theses of the international symposium reports. Strength of materials and elements of constructions at audio and ultrasonic frequencies of loading. Kiev, Naukova dumka. 1984, 35p.
6. Shtyryov N.A. Determination of physical conditions of the polycrystalline bodies destruction at non-stationary cyclic tension. The collection of proceedings. Vessel structural mechanics. Nikolaev, Nikolaev Shipbuilding Institute (НКИ). 1987, pp.74-84.
7. Shtyryov N.A. Molar energy and local molar power – the physical parameters of state of the kinetic microscopic motion of idealized gas corpuscles. Part 2, 2013.
8. Shtyryov N.A. A structural parameter function of the strained rigid body of the strength kinetic concept. Part 1, 2013.
9. Shtyryov. N.A. The monatomic-structural kinetic model, molar energy and power of destruction of a deformable rigid body conglomeration. Part 3. 2013.
10. Shtyryov. N.A. State equation and the structure-energy kinetic law of the strained rigid body. Part 4, 2013.
11. Yavorsky B.M., Detlaff A.A. Handbook on physics. Nauka, Moscow. 1979, 942p.
12. Shtyryov. Physical parameters and properties of the strained rigid body in the structure-energy kinetic theory of strength. Instances of solving problems of strength and fatigue. Part 5, 2013.
13. 100 years from the birth of S.N. Zhurkov. Solid State physics (ФТТ), vol. 47, issue No.5, 2005.
14. Petrov M.G., Ravikovich A.I. About deforming and destruction of aluminum alloys from positions of the kinetic concept of strength. Journal of Applied Mechanics and Technical Physics (ПМТФ), 2004. Vol.45. No.1, pp.151-161.
15. Hart E. Phase changes on the boundaries of grains. Solid State Physics News (НФТТ), No.8, Mir, 1978.

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