where is the uniform static deformation

where ɛ is the uniform static deformation. Evidently, the functional dependence can be represented as
Respectively, the following equation can be written for the stresses σ:
We should note that σ is connected to , i.e., there are induced stresses connected to the reorganization of the internal structure that generates the stressses . The complex relationship of the stress and the deformation transform the first equation of the system (5) into the equation
Staying within the static approach, we can rewrite this equation as
Here
The second equation in the system (6) describes the dependence of stress pattern induced by bond restructuring, for the case when the principal term is known
We can write the following relations for the second continuum: where is the diffusely mobile hydrogen density. The Eq. (8) are converted to a form
The last equation is similar to Darcy\'s law where the polo-like kinase 1 coefficient depends on the strain field ɛ. Therefore, we obtain the following expression for the hydrogen particle speeds (the second continuum)
The above-listed equations should be complemented with the the balance equations for the bound and the diffusely mobile hydrogen particles:
Using Eq. (9), we obtain from these equations a new equation for the proprotion of the bound hydrogen particles in the material:
Eq. (10) is a mixed-type equation, as it contains terms typical for hyperbolic equations: , , as well as terms typical for parabolic equations: , . This means that a comprehensive analysis of a non-stationary task for a finite initial distrubance must reveal a characteristic moving front of an increase (or a decrease) in hydrogen particle concentration. In order to analyze this equation, we shall use the Fourier method of variable separation. For this end, we are going to assume that
Then
Hence, we obtain an ordinary differential equation for : and also an ordinary differential equation for :
Let us solve this problem with the following initial conditions: where the parameter λ is determined by the microstructural parameters of the material in question. First let us construct the equations for the constant term of the series based on the initial conditions (11). In this case and the equation for the time multiplier has the form:
The solutions of this equation will be the functions of the form where the constants T0 and T1 are determined by the initial conditions. For the second term of the series we have the value, and the equation for has the form
Let us introduce the notation
Its solution is a function where
Substitution into the initial conditions produces, after integration over a full harmonic period, the following relations:
These expressions allow to write the final form of the solution for the relative proportion of bound hydrogen particles:
Predictably, the diffusely mobile hydrogen that is uniformly distributed over the volume increases its binding energy independent of diffusion, while the non-uniformly distributed hydrogen diffuses, with the diffusion rate determined by the function F(ɛ) that is a conventional size of the flow section of the hydrogen diffusion channels depending on the strain ɛ. A decrease in the value of F(ɛ) leads to an increase in G(ɛ). If F(ɛ) → 0, then ξ1 will tend to 0, while the multiplier in Eq. (13) will tend to . These difference between the exponents determines the non-uniformity in hydrogen concentration distribution along the spatial coordinate. Therefore, a redistribution of hydrogen concentrations due to diffusion will decelerate. In the limiting case we will obtain the following formula:
The equations of the system (7) can be used to model the effect of hydrogen on the σ(ɛ) dependence, which is easily measured. We should note that the effects connected to the changes in temperature, and also non-linear phenomena caused by the changes in the bound hydrogen contents due to material deformation can be described by the suggested model.