Estimation Of Strength for Inhomogeneous Sample of the Porous Plate with Taking into Account of the Internal Heat Sources Power

At investigation of propagation of electromagnetic waves of microwave frequency into porous wetted media the method of local space averaging was used [1], according to it for inhomogeneous (nonmagnetic) multiphase material with dielectric losses and low conductor properties [2] under frequency dispersion of dielectric permeability of components we are using the information about the effective generalized complex dynamic dielectric permeability (E.G.C.D.D.P.) [3]. Such dielectric permeability into approach of effective macroscopic field takes the possibility to determine the lineal response of not interacting between vise verse mezoscopic averaging volumes (R.E.V. - Representative Elementary Volume [1]) on the base of cluster approach, if into the considering object the harmonic disturbance is occurs as a reaction on the external plate monochromatic harmonically electromagnetic wave of microwave frequency. For modeling of the processes into object we introduce the dielectric permeability ef [3], which take a possibility to review the porous humidified media, as a homogeneous isotropic dielectric with continuously changed into coordinate space and time electro-physical properties (this is the continuous media or the field approach). Taking into account, that mutual displacement of phases joined with hydrodynamic (mechanical) moving of liquid or gas into porous skeleton is unsignifical on the distances of еру electromagnetic wave length, which interacts with body. Also, according to the slow varying conditions of volume characteristics of porous material we are considering the not important (light) time dependence of E.G.C.D.D.P. during the period of oscillation. Relaying to this the effective complex dielectric permeability is reviewing as slow varying function of coordinate and time. The equation of field from which we get the expression for the power of microwave heat sources is resolved abstractedly from the heat and mass transfer equations into modelled porous (humidified) material.

We are considering the porous media as composite concrete ceramics which consist the large number of interconnected pores with the little size and porous skeleton (S is the solid phase) [5]. We assume that the media of the object (body) is unsaturated, in particular, into pores exist the substance in the liquid (L) and gas like phase (G). The last one is the two-component mixture of atmospheric dry air and water vapor [3]. The pores into the fist approach are modelled by the spherical inclusions with linear radius R: the micro pores (R < 2 nm), mezzo pores (2nm 50 nm) [6].

We are considering the solution of received into harmonically approach the equations [4] for electromagnetic field (E.M.F.), in which the such parameters are fulgurated: the is the complex amplitude of the E.M.F. into porous cell of considering media (here is the real quantity of the complex amplitude ) as well, as is the effective generalized complex dynamical dielectric permeability (E.G.C.D.D.P.) [3] (here the time t is into content of a parameter, by means of which the moving of phases according to deformation process of solid phase (skeleton) and flowing phases (liquid or gas) under influence of evaporation or condensation during heat and mass transfer are considered. The mentioned parameters we are considering the slowly (light) varying functions of coordinate. Then equations for the plate (transverse) polarized T.E.M. [7] monochromatic wave in the stretch of electric field into porous composite media [2] takes the form

Here is the effective generalized complex dynamical dielectric permittivity, is the complex refractive index; is the wave vector into the porous (inhomogeneous) media; (here is the velocity of light) is the wave vector of electromagnetic wave into vacuum, ω = 2π v is the corner frequency of E.M.F. (ν is the linear frequency), μ₀ and ε₀ are correspondingly the magnetic and electric constants into vacuum.

The bounds between the measured and modeled indexes we describe into approach of the effective field (E.M.A.) according to well known Bruggeman [3] formula

here is the static complex dielectric permittivity for σ = {S, L,G} phase under E.M.F. frequency ω for investigated porous composite media, where is takes formally according to ergodicity hypotheses.

As a porous sample during numerical experiment, we review the porous wetted (unlimited) porous plate. By the selection of the direction of distribution for electromagnetic wave along the axis 0X at the fixed polarization along 0Y (axis of ordinate Y) (Figure1) the wave equation (2) takes [2] the following view

where is the complex amplitude of electric field stretch along the axis 0Y.

We use the Wentzel-Kramers-Brillouin (W.K.B) approximation [9] and will look for a partial solution of equation (3) in the form , where A is the unknown complex constant and φ (x,t ) is the the phase of an electromagnetic wave varies depending on coordinates and time. After substituting the specified expression for into relation (3), we obtain the equation

Let us assume that the last term in (4) can be neglected due to the weak phase variability perpendicular to the wave propagation front. Then the condition of weak variation in the coordinate space (for the variable x it is the one-dimensional case) according to the relation (4) takes the following form:

Relation (4) under the condition of (5a) is fulfilled, as a result of the series arrangement, it is simplified and will have the following form

We seek a partial solution of the wave equation (3) in the following (6) form

The complex amplitude of the magnetic field strength is obtained according to the wave equations [2] by differentiation .

In this is the component of the magnetic field strength vector along the 0Z axis, and is the wave resistance (impedance) in vacuum. If the weak variation condition (5) is satisfied, we obtain

It should be noted that the choice of sign in expressions (6) and (7) has more illustrative than physical meaning.

The condition of weak variability of the field (5a) in the coordinate space is clearly formulated if we take into account that

where (here and is the electromagnetic wavelength and wave vector in vacuum) is the variable wavelength of the microwave EMF and the wave vector in a continuous porous (effective) medium. Then the corresponding relation (5) acquires a transparent physical meaning and the condition for it can be written:

where (here is the speed of light propagation in an effective medium) is the wave vector in a porous (wetted) medium modeled according to the effective dielectric characteristics [3].

The condition of weak variation of the dielectric properties of the material in the coordinate form (6) is usually ensured for most composite bodies with low electrical conductivity (2) due to the large value of the linear frequency v₀ of microwave radiation in vacuum. In the following, we will neglect the dispersion of the wave characteristics of the material under study at a constant value ω₀ = 2π v₀ of the angular frequency of microwave radiation in vacuum, considering the propagation of electromagnetic waves in the internal volume of a porous (wetted) flat plate, which takes place when taking into account scattering effects.

Thus, we arrive at a closed (complete) set of conditions for weak variability of dielectric (wave), bulk (phase) and wave properties of a three-phase (wetted) porous medium in the coordinate and time space of variables

Here is the wave vector and phase velocity of electromagnetic (TEM) wave propagation in the simulated medium, is the effective value of the refractive index, is the volume fraction σ = {S, L,G} phase ω₀ , - constant angular frequency of EMF in the microwave frequency range, l is the characteristic area size (REV) [1] for space averaging.

Relations (9)-(10) make it possible to apply the approximation of the effective macroscopic field [4] when studying (calculating) the effective electrophysical characteristics of a porous material within the framework of the approximation [1] of the theory of local spatial averaging.

Relations (9)-(10) make it possible to apply the approximation of the effective macroscopic field [4] when studying (calculating) the effective electrophysical characteristics of a porous material within the framework of the approximation [1] of the theory of local spatial averaging.

According to the properties of the solutions of the W.K.B model, when the conditions of weak variability (9) and (10) of the electrophysical characteristics of the porous (wetted) material are satisfied, the general solution of the wave equation (3) for a plane transverse (T.E.M.) electromagnetic wave can be written [14] in the form

where is the value of the refractive index that varies over time due to the sorption or desorption of liquid on the surfaces of the plate, A(t) and B(t ) are unknown functions of time, andexp[...] is notation of the exponent.

In the left (L) and right (R) half-space, external (Figure 1) with respect to the boundaries (edges) of the plate, which is modeled by vacuum, the electric field strength (under external microwave irradiation) will be considered in the form of expressions

where C(t) and D(t) are unknown functions, EL and ER are actual amplitudes of the external electromagnetic field (E.M.F.).

Let us write the boundary conditions for the components (Figure 1) of the electric field strength

and derivatives that correspond to the component strengths of the magnetic field

Such conditions (14) and (15) make it possible to take into account the dependence of the field amplitudes on the wetting at the boundaries (edges) of the plate, the value of which (volume fraction θ_L of liquid or saturation η_L of pores with liquid) changes dynamically depending on the power of the internal (thermal) heating sources and the implementation of heat and mass transfer conditions at the edges of the plate.

In accordance with the described expressions (11), (12) and (13) for the electric field strengths according to the continuity condition (14) of the components

and continuous (15) of derivatives

we obtain a system of linear equations (16) and (17) for determining the unknown (variable in the simulation time interval) coefficients that specify the amplitude of the E.M.F. at the edges of the plate, in the accepted notations

Such coefficients, taking into account the relation (18), will have the form

where (here α = {L, R} is the index corresponding to the left (L) and right (R) edges of the plate) are, respectively, the transmission and reflection coefficients of the T.E.M wave at the plate boundaries.

Thus, according to (11) and the expressions for the coefficients and (19) which are determined from the continuity conditions (14) and (15) of the field components, we obtain a partial solution of the wave equation (3) in the electric and magnetic field strengths

here determined according to relations (19) and normalized according to the values (where α = {L, R} - index of the plate edges) of the refractive index at the edges of the plate complex amplitudes of the electromagnetic wave, and characteristic resistance (impedance) in a vacuum.

It is known [7] that for the wave vector of a T.E.M. wave in a continuous (effective) medium the following relation holds: , where are absorption and propagation indicators of T.E.M waves.

Let us define the tangent of the dielectric loss angle in the simulated porous (wetted) material

where is the effective generalized complex dynamic permittivity (E.G.C.D.D.P), and і denoting the real and imaginary parts respectively.

Then the absorption α (x,t ) and propagation β (x,t ) indices of the T.E.M wave are described by the expressions

subject to performance normalizing ratio.

Considering the above ratio , where are the above-mentioned absorption and propagation parameters of a plane electromagnetic wave, we obtain the effective values of the wavelength and the depth of wave penetration

where is the dielectric loss tangent.

The power of the E.M.F energy flow in a diffuse wave process can be calculated according to the definition of the Umov-Pointing vector [4] by the relation

where is the complex conjugate value of the averaged amplitude of the magnetic field strength

Analyzing expression (18), for the model parameter we obtain

here are integral values of the absorption and propagation coefficients of an electromagnetic wave through the thickness L of plate correspondingly, and is the amplitude multiplier, which is determined by the damping coefficient that varies with time (due to the movement of fluid phases in a porous material).

Then, according to the definition (25) and the expressions for the electric (20) and magnetic (21) field strengths for the microwave radiation flux power in the accepted (26) notations in the large thickness approximation L of plate (26) and symmetrical edges (if E₀ - amplitude value of the external E.M.F intensity), the relationship for the power of the energy flow in a porous wetted material with effective wave properties will be

Were,

is the notation; is the power of external microwave radiation; and is the the value of the refractive index and transmission coefficient of the electromagnetic T.E.M. wave at the edges of the plate; is the attenuation coefficient of the T.E.M. wave.

Let us assume that the total amount of E.M.F energy supplied to the material under study at microwave intermediate resolution is completely converted into heat due to scattering (diation) effects. Thus, we neglect the processes of performing work by the studied thermodynamic system due to internal (ponderomotor) forces acting on the porous frame (including thermal expansion of the plate).

Then the expression for calculating the power of electromagnetic (dielectric or microwave) heating sources has the form

where is the value of the Umov-Pointing vector [4] at the edges and in the internal volume of the plate.

For the power of internal heat sources under symmetric external (microwave) irradiation, which is determined by the expression for the E.M.F energy flow in the material (25), when approximating large thicknesses of the plate according to (26), we obtain

In relation (29), the notations are equivalent to the physical quantities described according to (28) and

As before, the Bruggeman formula [3] was used in the numerical simulation of the heat source power (28) in the EMA approximation.

and the expression (31) obtained in [15] for finding the uniform wetting of a porous material

here ϕ - averaged porosity, - critical and residual pore saturation with liquid, amb P - pressure value in the surrounding environment, k_in - inherent permeability of the frame, σ - surface tension coefficient, m - model parameter.

For the material under study, namely historical ceramic brick (ϕ = 0, 46 - averaged porosity) and two modification (ϕ = 0, 4 - modA and ϕ = 0,52 - modB), with known values of dielectric (static) permittivities of phases,

Table 1: Dielectric constants of components for the studied material (historical ceramic brick) according to the spherical inclusion model [ ε_S - solid phase, ε_L - liquid phase (water), ε_G - gas phase ( dry air )].

which are given above in Table 1, by relation (27) we obtain the corresponding graphical dependencies for the real (Figure 2) and imaginary (Figure 3) part of effective generalized complex dynamic permittivity (EUCDDP) porous moistened material at a typical (standardized) frequency of external microwave irradiation from the saturation η_L by liquid.

Below is also graphically depicted (Figure 4) the dependence of the dielectric loss tangent angle at different humidity values η_L of the investigated porous material at a specified frequency of microwave irradiation.

For such a material, according to the moisture dependence L η from the effective pore radius is established [15], that under porosity ϕ = 0, 46 from the effective pore radius and critical pore fluid saturation, as well as the value of the model parameter m = 0,67 in the semi-empirical model of moisture release or retention by van Genuchten [16]. Then, according to relation (31), the macroscopic (homogeneous or uniform) value of wetting of such (porous) material under normal conditions is (when , where - external pressure relative to the moistened macroscopic porous material) corresponds to value , which coincides with the maximum of the pore size distribution function over the effective radius.

Under the described conditions of uniform (homogeneous) wetting of the porous material, the absorption indices (23) and propagation (24) of the TEM wave, as well as the transmission and reflection coefficients in expressions (1.16) for the amplitudes of the electromagnetic wave on the boundary surfaces of the porous plate (Figure 1) do not depend on time due to the constant value of liquid saturation. In this case, the distribution of EMF power in the material under symmetric microwave irradiation according to (27) is determined by the thickness L of plate, equilibrium value of pore saturation with liquid and dielectric (relaxation) properties of the phases. This is due to the fact that, that ЕУКДДП moistened porous material (ϕ = 0, 46 )is defined according to the EMA approximation [3] (30) (Figure 2 and Figure 3), as well as the dielectric loss tangent tg δ_ω (Figure 4), are determined by the equilibrium constant value saturation of pores with liquid.

Then the power of internal heat sources (microwave heating) according to expression (28) for the EMF energy flow in the plate material can be calculated numerically and displayed graphically in Figure 5 and Figure 6.

Graphical results of numerical modeling (Figure 5 and Figure 6) of the power of internal (volumetric) heat sources in a porous (unconfined) moistened plate at different thicknesses L of plate (where L = nΔL / L(n = ΔL / L) - dimensionless plate thickness, and n - number of breakpoints), which are multiples of the wavelength in the environment, obtained for the following values of microwave irradiation parameters and properties (parameters or characteristics) of the investigated porous material: 1. External microwave irradiation: power - , frequency - ; 2. Porous humidified medium: ϕ = 0, 46 - averaged porosity, - intrinsic permeability of the skeleton, - equilibrium saturation of pores with liquid, T = 298,15[K] - thermodynamic temperature, - pressure value in the gas phase, - the length of an electromagnetic wave in a medium.

The strength criterion for a moistened porous plate, the dielectric properties of which are described by relations (1)-(31), is formulated as [18]:

where P(J) - function of the boundary surface in the deviatoric plane; σ₀ =I₁ /3 - average tension; τ₀ =(2J2/3)^0,5 -octahedral shear stress; I1 the first invariant of the stress tensor; J₂, J₃ - second and third invariant of the stress deviator; - alternative invariants of stress deviators J₂, J₃; α, β, C₀, C₁, C₂ – material constants that are established experimentally based on the methodology [18].

There is a transition layer with sufficient adhesion between the liquid in the pore and the solid phase. Adhesive bond energy γad and it changes Δγ_ad depend on the energy characteristics of the coating and the porous carcase [19]. Let us formulate the strength criterion [19] for the transition layer at the pore boundary in the form

where Δσ_m – change in interfacial tension; Δγ_m – change in interfacial energy; Δ – symbol of deviation (change) of a parameter or energy characteristic of the surface (interfacial) layer; ΔA_ad – change in adhesion work; Δσ_m#, Δγ_m#, Δ A_ad#, Δγ_ad#– empirical constants.

Relations (31), (32) together constitute a generalized version of the strength criterion for a porous material (such as concrete). The parameters of expression (31) are determined on the basis of experiment, and the parameters of relations (32) are estimated on the basis of a computational experiment [20].

If adhesion is broken, the pore size increases, the pores merge and can form a crack (a microcrack to begin with).

The criteria for ultimate plasticity ΘT and strength ΘS for the tip of a defect (microcrack) can be formulated as dimensionless relations.

where ε_i - intensity of deformations; ε₀ - volumetric deformation; ϕ_r - volumetric deformation, рад; ε_ic - intensity of destructive deformations; ε_0c - destructive volumetric deformation; ε_iu – ultimate strain intensity; ε_0u -limiting volumetric deformation.

Transition from deformation components ε_jk (34) and (35) to mechanical stresses σ_jk We carry out the calculations based on the model relations described in the work [19] (j, k =1, 2, 3). In particular, an equation of the Hooke’s law type is used.

Due to the influence of internal gas and liquid pressure in the pore, a zone of plastic deformation appears in the vicinity of the crack tip, to which the pressure corresponds pcr1(σT), and in the second stage – a critical state p_cr2 (σ_S) [σ_T – plastic limit; σ_S – strength limit (destructive stress of a solid body, i.e. a skeleton)]

Parameters p_cr1 and p_cr2 are important for assessing the strength, reliability, and service life of a porous material under the influence of internal (volumetric) heat sources.

Let us supplement the relation (32)-(35) with the expression of the limitation on the power of internal heat sources , which is determined from the relation (29) taking into account (30) and (31):

where – the maximum value of the power of internal heat sources that lead to the destruction of the porous body.

Relations (29)–(36) constitute a new version of the strength criterion for a porous body (compared to (32)–(35)) taking into account the power of internal heat sources that lead to the destruction of the porous body (i.e., to the formation of a micro crack, which over time can turn into a main crack).

From the considered graphical dependencies it follows that at plate thicknesses the exponential law of power attenuation of Beer-Lambert microwave heating sources is realized [16]. If wave a periodical effects occur (see Figure 5 (when and Figure 6 (when and , which can be localized at plate thicknesses that are commensurate with the minimum (Fig. 6,when and ) symmetric with respect to the center of the plate, the anharmonicity value. It is clear that the wave (aperiodic) properties of heat sources will change depending on the value of equilibrium humidification porous plate material, which is determined by the structure parameters: ϕ - porosity and k_in - the inherent permeability of the skeleton. Criterion ratios have been formulated for assessing the strength of a porous material, taking into account the power of internal heat sources that lead to the destruction of a porous body.

The author is grateful to the parents for their patience in conducting this study.

No conflict of interest.

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