Научная статья на тему 'Mathematical modeling of timber elastic-viscous-plastic deformation in the wood drying process'

Mathematical modeling of timber elastic-viscous-plastic deformation in the wood drying process Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

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МАТЕМАТИЧЕСКАЯ МОДЕЛЬ / MATHEMATICAL MODEL / МЕТОД КОНЕЧНЫХ ЭЛЕМЕНТОВ / METHOD OF FINITE ELEMENTS / ТЕПЛОМАСО-ПЕРЕНОС / УПРУГОВЯЗКОПЛАСТИЧЕСКОЕ СОСТОЯНИЕ / ПРОЦЕСС СУШКИ / DRYING PROCESS / ДРЕВЕСИНА / TIMBER / HEAT AND MASS TRANSFER / ELASTIC-VISCOUS-PLASTIC STATE

Аннотация научной статьи по электротехнике, электронной технике, информационным технологиям, автор научной работы — Prusak Yuriy, Kroshnyy Igor, Sokolovskiy Yaroslav

In the paper there is described solving a current scientific problem of mathematical modeling of heat-mass transfer processes and elastic-viscous-plastic deformation taking into account the mechanics and sorption creep in hygroscopic capillary-porous materials with variable anisotropic heat and mechanics characteristics what is of importance for the rational choice and substantiation of energy conservative technologies of timber drying under the conditions of necessary qualitative production providing.

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Текст научной работы на тему «Mathematical modeling of timber elastic-viscous-plastic deformation in the wood drying process»

Деревопереработка. Химические технологии

ритные размеры, низкую энергоемкость технологического процесса.

Разработанные конструкции малогабаритных ленточнопильных станков могут обеспечить повышение качества пилопродукции от рубок ухода и реконструкции лесных насаждений.

Библиографический список

1. Свиридов, Л. Т. Ленточнопильное оборудование для лесоматериалов: теория,

конструкция, расчет [Текст] : монография / Л. Т. Свиридов, А. И. Макси-менков; ВГЛТА. - Воронеж, 2007. - 325 с.

2. Попиков, П. И. О динамике режущего инструмента в ленточнопильном станке [Текст] / П. И. Попиков, С. А. Чепе-лев, К. А. Чернышков // Математическое моделирование, компьютерная оптимизация технологий, параметров оборудования и систем управления : межвуз. сб. науч. тр. / ГОУ ВПО «ВГЛТА». - Воронеж, 2010. -Вып. 15. - 175 с.

DOI: 10.12737/4522 УДК 004.94-674.047

MATHEMATICAL MODELING OF TIMBER ELASTIC-VISCOUS-PLASTIC DEFORMATION IN THE DRYING PROCESS

head of the department of computer science and modelling of technological processes, doctor of Sc., professor Ya. I. Sokolowskyy

philosophy doctor, teacher of the department of computer science and modelling of technological

processes I. N. Kroshnyy postgraduate Yu. V. Prusak

National University of Forestry and Wood-Technology of Ukraine sokolowskvv@ukr.net, kroshny.igor@gmail.com, prsk@ukr.net

Actuality of theme. During drying of hygroscopic capillary-porous bodies irreversible physical processes arise up and cooperate between itself, there are phase transitions, which predetermine changes in physical mechanical properties of materials in general, initial form of body, formation of cracks in it and its possible destruction. These phenomena which take place in the conditions of high changeability of structural and physical properties of hygroscopic materials are basic restrictive factors for development of effective methods of management the capillary-porous materials

drying processes. The improvement of existing and creation of new resources saving technologies of drying is closely related to the detailed study deformation-relaxation and heat-and-mass transfer processes in capillary-porous materials. Applying only experimental methods of the deformation run and heat-and-mass transfer research in such systems is connected with considerable financial expenses and technical difficulties. Therefore there is an objective necessity of development of twodimensional mathematical models and effective programmatic-algorithmic facilities of their

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realization, which would bind the technological parameters of drying process to development of the elastic-viscous-plastic state of wood taking into account the anisotropy of heat-mechanic properties and peculiarities of wood deformation mechanisms. There has been elaborated the applied software which consists of the documented classes and provides the possibility to automate the finite-elemental analysis of timber intensedeforming state during the convective drying process.

Such models will enable to offer more effective, energy saving technologies of capillary-porous materials drying, especially wood.

Analysis of literary sources. The conducted analysis of mathematical models deformation-relaxation processes and methods of their calculation during drying of capillary-porous materials, especially woods, showed that the resilient and viscoelastic area of deformation is mainly investigated in unidimensional and two-dimensional cases for permanent mechanical descriptions, independent of temperature and humidity change [1, 2, 3, 4, 5]. Construction of mathematical models of wood deformation during drying taking into account resilient, viscoelastic, plastic deformation and features of their regeneration in a wide turndown physical mechanical properties of wood is a complicated and not fully decided problem. Various models of idealizing rheological bodies and proper to them differential and integral equalizations were offered and applied for the design of wood standard in the limited intervals of temperature and humidity change indexes.

Taking into account difficult connected with each other processes of deformation and mass transfer during the design of the

convective drying of hygroscopic capillary-porous materials substantially complicates mathematical models and requires the improvement of numeral methods for their realization and software development.

The comparative analysis of mathematical models of stress-strain state of wood during drying showed that existent models did not take into account all complex of elastic-viscous-plastic deformation taking into account the anisotropy of heat-mechanic descriptions of material in the conditions of non-isothermal humidity transfer, features of deformation which consider the speed of mass transfer change and the mechanism of deformations regeneration.

Synthesis of mathematical model. According to the basic laws of thermodynamics of irreversible processes, mechanics of the inherited environments, contraction of hygroscopic materials, the mathematical model of two-dimensional viscoelastic deformation is formed and heat-and-mass transfer in the process of drying of capillary-porous materials taking into account the anisotropy of heat-mechanical descriptions of material, resilient, viscoelastic, plastic deformations and deformations, caused by a mechanism of mechanical and sorption creep.

System of model equalizations for determination component of deformation vector £ = (sn, £22, si2)T, strains o= (on, 022, o12)T, temperature T(X,t) and humidity content U(X,t) during drying wooden bar during time re[0,Tdry] in the area of cross-cut section Q= {Х= (xi,X2); xig[0, li], X2 e[0, h]} the center of which is combined with beginning of co-ordinates, and the axes of anisotropy coincide with co-ordinate axes, it is built so.

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The components of tensions vector are satisfied by equalizations of equilibrium with boundary conditions which take into account absence of external efforts:

da,, ^ da

dx1 dx.

'12 = 0: da ; —12 + da 22 =

•2 dXj dX2

а г, = 0.

lj Ixj =l 1, x2 =l2

(1)

Modeling of connection between the components of tensions and deformations during wood drying is based on Boltzmann-Volterra integral equalizations and laws of contraction of hygroscopic materials [1, 6]. For an account mechanical and sorption deformations, predefined speed of change of humidity during drying, the functions of rheological conduct of materials are complemented correlations mechanical and sorption creep. Therefore connection between tensions and deformations taking into account the anisotropy of mechanical properties in material looks like:

a11 (t) = Cn (T,U)[s„ (t)-em] -Cu (T,U) •

fRn (t - s,T,U) x [sn (t)- SU1 ] ds + C12 (T,U)•

0

{S22 (T)- SU2 ]- C12 (T,U)• т

• f R12 (T- s, T ,U ) • [S22 (T) - SU 2 ] ds;

0

CT22 (T) = C21 (T, U) [s11 (T) - SU1 ] - C21 (T ,U) •

t (2)

•f R21 (T - ^ T,U )[S11 (T)-SU1 ] ds +

+C22 (T,U)(s22 (t)-6u2)-C22 (T,U)•

т

f R22 (t -s,T,U)\s21 (t) -Su2]ds;

0

a (t) = 2C33 (T ,U )S12 (t)-2C33 (T ,U )•

t

• f R33 (t- s, T ,U )s^(s)ds,

0

where - su = (su 1,su2,su3)t - a vector

component of deformations, predefined the change of wood temperature and humidity content;

Cij - components of tensor of resiliency woods, which depend on the temperature and humidity. Values sUi, C, are determined with the help of formulas:

SU1 a1AT + ДАи

SU 2 = a2AT + P2AU

0 0

C

Ei 1 -

ЦЕ2

1 - Ц^2 0

ЦЕ2 1 - Ц^2

е2

1 - Ц^2 0

0

0 ,

ц

(3)

where - AT, AU - accordingly increase of temperature and humidity content,

Ei(T, U) - Young’s elastic modulus, vi(T,U) - Poisson’s ratio,

[л(Т, U) - modulus of shear.

For the modeling of mechanical and sorption deformations, predefined by humidity speed change in wood, such equalizations are used [7]:

ds.

дт

= m (a - EmSM (t))

dU

дт

(4)

where - m - is a tensor of mechanical and sorption deformations, which depend on the temperature, in radial and tangential directions of wood anisotropy, the coefficients of which are determined with the help of experimental data.

For the modeling of plastic deformations in wood the Prandtl-Reuss equations of plastic flow is used. The relation between differentials of tensions and deformations for the flat tense state looks like:

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da = ф) I

dsn +—djftej _

у i_2v у у у

S

2-2 s=-а 3

r1+2М '

3E

(5)

у

where - s,j - deviators of deformation,

Sjj - Kronecker’s delta.

The functions of rheological conduct of wood during drying taking into account the mechanism of piling up of irreversible deformations choose in a kind:

R (t,u , t ) = M ao _Z ai exp (_biT) ■ 2,-

_ i=1 _ Sn

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a0 _

■ h(t)h(t0 _t)_

M

Ea exp (_p (t _to ))

(6)

h (t _to ) ,

where - h(r) - Heaviside step function, and

unknown coefficients a,, b

a,

ei are

described by the system of differential equalizations of non-isothermal humidity transfer:

ST S

cp— = — St Sx,

f

л dT , Sx1.

\

+-

S_

Sxn

f

SU S

St Sx l Sx

a -

SU

+-

+-

S_

Sx1

ST

1 Sx1

+-

S_

Sxn

S

ST

2 д,,

SX2 У

f

+pr

SU; St ’

SU

a

V

Sx

+

(7)

f

Sx

2 У Л

ST a2o —

2 Sx V UX2 у

Boundary conditions look like:

+ Po (1 _s)P, (U^-Up) =

=a h _ TL, ==,);

P (u _ Ul,=,) ■

SU _ST

a.----h a,o —

, Sn Sn

(8)

determined by the method of the least squares on the basis of approximation of experimental information of creep of wood samples on-loading and after unloading. They are the functions of temperature and humidity.

A synthesized mathematical model (1)-(6) enables to define stress-strain state of wood during drying taking into account resilient, viscoelastic, plastic deformations and deformations, caused by the mechanism of sorption creep, taking into account savings of irreversible deformations. For its realization needed to determine the temperature and humidity during wood drying, which are in formulas (2), in the tensors of resilient descriptions of C and mechanical and sorption deformations of m.

During the convective drying of wood the two-dimensional mathematical model of heat-and-mass transfer taking into account the anisotropy of thermo-physical descriptions is

Лесотехнический журнал 2/2014

SU ST

a----ha0 —

Sn Sn

= 0;

ST

Sn

= 0;

u = U • T = T

^1г=0 U0’2 |t=0 0>

i = 1,2.

x, =0

Denotations used here: T0( X), U0(X) -

primary apportionment of the temperature and humidity content in the material;

U (Tc ,ф) - equilibrial humidity;

c(T, U) - heat capacity; p(U) - density;

2 (T, U) - coefficients of heat-

conducting in the directions of anisotropy; s - coefficient of phase transition; p0 - basic density; r - specific heat of vaporization;

0 (T, U) - thermal-gradient coefficient;

ai (T ,U)- coefficients of hydraulic conductivity in the directions of anisotropy; a (Tc, v) - coefficient of heat exchange;

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xi =0

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Д (Tc ,p, v)- coefficient of humidity exchange;

Tc - environment temperature; p(r) and v(r) - relative humidity and rate of movement of drying agent;

n - vector of external normal of area limit

A

т - current time. During modeling of drying process in the period of the irregular mode the primary apportionment of humidity content in wood is accepted permanent, and in the period of the regular mode initial humidity content changes after a parabolic law.

For numeral realization of mathematical models of connected with each other processes of heat-mass transfer during wood drying (7)-(8) the Finite element method (FEM) is used [8]. Equivalent variation formulation of model is for this purpose got with assumption, that the change of humidity content in time is possible to show as a sum of constituents, related to the stream of mass transfer by the gradient of humidity content and temperature. Eventual system of equalizations for realization of mathematical model (7)-(8) after FEM looks like:

C + K ]U }+{F } = 0,

C

a{T)

dr

+ K ]T }+F)

0,

where:

(9)

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[C] = J Po [N]T [N] dV;

V

[К] = J[B]T [D* ] [B] dV + J PoPw [N]T [N] dS; -

VS

{F) = J[B]T [H][B][T]dV -JPoPUp [N]T dS

VS

according to the matrix of thermo-physical

properties of material, damping and loading,

{N} - matrix of form functions.

Analogical matrixes are those Ic ]. 1K ].{F 1 related with the coefficient of heat conductivity and heat exchange.

For the values of change of temperature {Т} and humidity {U} in time the Finite difference method is used. Then numeral realization of mathematical model (7)-(8) is taken to the decision of the kind of system equalizations:

MU L, = (R); [at UL, = {R }• (io)

As thermo-physical descriptions of wood depend on a temperature and humidity, and equalization of model (7)-(8) are related with each other, the iteration process of equalizations realization (10) is carried out on every sentinel step taking into account additional iteration procedure, which specifies influence of humidity on apportionment of temperature in material and vice versa. Completing of iterations for equalizations (10) foresees implementation of conditions:

{Un} - {Uni}<W-4 i {Тп} - (Тп-1}<10-4.

For numeral realization of mathematical model (1)-(6) elastic-viscous-plastic deformation of wood during drying developed FEM for the research of wood deformation taking into account mechanical-sorption and plastic deformations and mechanism of regeneration of deformations. For this purpose on the basis of a minimum of complete potential energy equivalent variation formulation of tasks was received. Lagrange’s equation, minimum of which coincides with the decision of mathematical model (1)-(6) is written down like:

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Q=-

1

2j((U}T [ B [С][ «+2MT [ B C

•йп « <*ччг [ в [с]

0

dU ^

•I («}АТ+(^}Ди+[^]

dr

(ii)

J+2W [B [C

dU \ ^ dr

dr ' )

j[%r)][£](U}| (о}АГ+(^}Аи+[п\

Finite difference approximation of vectors of moving {u(t)} and deformation {s(t)} and function of rheological conduct of wood R(t, t') in time were used to receive the basic correlations of FEM. Especially for {s (t)} and the core of relaxation are got:

(s(r)} = (s(r)} + (g(r+1)} S(r)}(r - r) ;(12)

Г+1 -r

R* =A2r R* (r ) + Аг£ R* (r )+АГ R* (tm ) .(13)

2 j=1 2

From the condition of a minimum of functional SQ = 0 system of equalizations of algebra is received for finding of the unknown

moving on every sentinel step At (i = 1, M, М - an amount of sentinel intervals):

N

Z

n=1

f

K

(n)

(U } = Z(F'n >}-Z

K

(n)

dU

dr

(14)

(a} AT + (p}AU + [ m]

4

where, integrals [k(ndefine the matrix of key inflexibility of material, which is defined by resilient or plastic characteristics of wood and geometrical sizes of elements of laying out. In case of resilient deformation it is taken that [k (n ^]=[k (n)]. For viscous-plastic

deformation Oy/^121 + ct222 - aua22 + to2 > apl) matrix of inflexibility consists of two matrices [K(n)] i [K(n) pl ], and [C(n) pl ] is calculated with

the help of (5). The matrix of loading (f (n)} is determined by the rheological conduct of wood, and also temperature and humidity descriptions of material. Vector of component to be found {U} on the i-step after laying out on time is unknown in relation to calculations {U} on previous i-1 steps depending on apportionment of temperature and humidity, which are determined on those steps using the algorithm from a previous paragraph.

In this way the mathematical modeling of heat-mass transfer is carried out and deformation-relaxation processes during drying of capillary-porous materials taking into account the anisotropy of heat mechanic descriptions, mechanical and sorption creep, plastic deformations and piling up of irreversible deformations.

Programmatic realization of mathematical models. We described the application software for numeral realization of mathematical models of heat-mass transfer (7)-(8) and elastic-viscous-plastic deformation (1)-(6) of wood during drying developed within the limits of the object-oriented approach [9]. Programmatic complex, developed with the help of programming language Java, contains an informative model and interface of the programmatic system, which looks like packages of classes and relations between them with the use of graphic diagrams of UML, components of programmatic code, calculative charts of FEM realization. Developed classes represent the essence of the object-oriented realization of Finite element method. It creates possibility of integration of the developed programs to the existent systems of the automated modeling with the purpose of expansion of their functional possibilities.

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There are separate packages for the classes, which realize: geometrical and physical and mechanical descriptions of researches object; laying out of area on finite elements with the help of mesh of knots and elements; determination of basic functions within the limits of eventual elements; calculable classes (squaring for numeral integration); interpolation functions; decision of the system of linear equations; classes, which are oriented on the concrete calculations of matrix and vectorial algebra; classes of saving loaded and received data; user interface.

Especially, calculating of the material temperature and humidity, which takes place in two basic classes - WFETriangulator (humidity) and TFETriangulator (temperature). These classes are streams, which are fulfilled parallel and cooperate after every iteration. All operations are executed in the method - run(), which is foreseen in the interface - Runnable. As global members of class the matrices of C, K and F are taken. A resulting matrix can be received, causing the method - getResultMatrix(), which get the array of results. As classes execute plenty of operations with matrices, most widespread of them were taken in the class-utility MatrixUtils, which is used during calculations. Loaded data for calculations streams take in the class InputDataHolder.

A separate stream (Thread) is a class for triangulation - laying out on triangles -Triangulator. It is inherited from the system class Thread and realizes the interface Runnable. To access the initial data the InputDataHolder class is used, which keeps all the loaded data of the program. For notification of other objects in the program about completing the process of laying out the

OnTriangulationListener interface is used. The method OnTriangulationFinished is caused in it, after the completing the process.

The class EquationSet is used for the decision of the system of equalizations, received by combining the results of triangulation and entry parameters. A basic method here is solve(), which redirect request to the system function decision, and then a result returns. Method solve() approach to the method simplify(), which simplifies the system at first, and then executes basic recursion calculations.

Thus realization of mathematical models, in which object-essence programed as a separate class software is developed. All classes can be repeatedly used for the computer modeling of other mathematical models with the use of Finite element method.

The order of convergence of close decision is analyzed using FEM and errors assessment estimate is shown. A mathematical model of heat-mass transfer verified by the comparisons of calculating values of temperature and humidity with the known experimental data.

For calculations permanent temperature conditions are chosen: (1st process -

ts = 70 °C, ф = 50 %, v = 2 m/s; 2nd process -ts = 80 °C, ф = 60 %, v = 1 m/s). Geometrical sizes of the wood sample of pine are: 90*50*25 mm.

Verification of adequacy of mathematical model of elastic-viscous-plastic deformation of wood taking into account the mechanism of sorption creep during drying took place by comparing of results of numeral modeling with the known researches without taking into account the sorption creep deformations (m = 0, Ems = 0) and viscous-

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plastic deformation (R(r-s) = 0) with the known resilient models. Also the results of two-dimensional modeling were compared with the results of modeling the elastic-viscous-plastic state of unidimensional case. The analysis of the conducted test calculations of the deformation fields and tensions testifies the satisfactory convergence of numeral and experimental definitions of the probed values.

Analysis of modeling results and their discussion. Application of the developed mathematical models and applied programmatic facilities is shown for the research of processes of heat-mass transfer of the elastic-viscous-plastic state of wood during drying. For numeral experiments some thermo-physical descriptions of wood are specified. Especially, on the basis of working of experimental data dependence of coefficient of wood hydraulic conductivity as functions from a temperature and humidity is used: aml(T,U) = am(T) amil(U), aml / am2 = 1,25. For determination of coefficient of humidity exchange we use the dependence:

a = 0,95(Т/фе exp(-2oVp /rTR)) 10"9, where Vp, о

- molar volume and superficial strain of liquid, ф

- relative humidity of environment. Value r = r(U) have been received on the basis of wood structure modeling by the system of inconstant capillaries of radius r, that depends on humidity.

To research the elastic-viscous-plastic state taking into account the mechanism of sorption creep resilient and mechanical-sorption descriptions were chosen according to the temperature and humidity change on the basis of approximation of known experimental data that look like:

E,= E„ (1 + E, (T« - T)) + E„ (Up - U);

mi = mi,(l + mt (T0 - T)). * ^

where Eio, EiT, EtU, mio, mt - coefficient, determined on the basis of approximation of known experimental data.

For determination of the module of wood plasticity used experimental data, according to which the determination of the modules of plasticity have been received.

The analysis of apportionment of temperature and humidity dynamics affirms that mathematical models enable to take into account interconnectivity of processes, their physical non-linearity, predefined by dependence of thermo-physical properties of material temperature and humidity. Curves of the temperature (T - T0) / T0 and humidity content (U - U0) / U0 shown on the Fig. 1 and Fig. 2, testify that the presence of internal resistance of heat and humidity transference predetermines the unevenness of apportionment of the temperature-moisture fields and their interconnectivity. A process of humidity diffusion in wood is less intensive during the humidity change of material surface with an environment. Therefore the value of

1,2 1

0.8

o.e

0.4 0.2 о

0 2 4 6 8 10

Fig. 1. Dependence (U - U0)/ U0 of different determinations of time

(U - Uo)/ Uo

2 h

4 h

8 h

\

25 h ■ ■ ^ \ ‘' A

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Fig. 2. Dependence (T - T0)/ T of different determinations of time

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humidity on-the-spot diminishes and goes to balanced, and the maximal gradients of humidity content appears inside the material.

In a high-quality plan received results coordinate with the results of temperature and humidity fields modeling of drying wood in partial cases. The analysis of graphic dependences testifies the sharp intensity of heat and humidity transfer growth, which is related not only with the fact described higher about less influence of diffusion in comparison to external humidity content, but with the effect of thermohydraulic conductivity. The process of thermodiffusion influences on thermo humidity transfer. Especially, in case of only diffusive transference the curves of humidity transfer have monotonous character and evenly diminish from a maximal value in a central layer to the minimum value on-the-spot of wood sample. The presence of humidity transfer taking into account a thermo-diffusion, a value of humidity content in wood can be greater than an initial. It predetermines appearance of maximal values of humidity content not only “in a middle” layer

but also in other “near a middle” layers of wood sample. Diminishing of humidity exchange coefficient and increase of diffusion coefficient predetermines the increase of thermal-diffusion stream influence. Diminishing of gradient of temperature predetermines the decline of thermal-diffusion stream, and diminishing of humidity on the surface Us slows humidity exchange with an environment. If a diffusive stream is greater than a total stream, namely thermal-diffusion from the surface and humidity exchange with the agent of drying, humidity content of superficial layers increase. This growth is observed to the moment of smoothing of the described streams.

To research the influence of anisotropy of structural properties of wood the numeral experiment of determination the temperature-humidity fields in radial and tangential samples have been made. For the exposure of influence exactly of anisotropy of thermo-physical properties conditions of external environment were set such, for which withering coefficients would be the smallest. The analysis of researches testifies that humidity of superficial area of standard substantially does not influence on apportionment of humidity under the thickness of tangential board. During the long period of time humidity in central part of board doesn’t change. In initial moments of time the maximal values of humidity are observed in “near surface” layers. Diminishing of humidity in central layers occurs in a regular mode period. For a radial board humidity diminishes in all points of wood sample. To graphic dependences the humidity change is not peculiar inflection points for initial moments of wood drying, as in the case of tangential boards. The speed of humidity change in a radial board is more

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intensive, than in tangential. The increase of duration of drainage slows the change of humidity of radial board in comparison to the change of humidity of tangential board.

On the basis of the developed mathematical models we investigated influence of anisotropy of viscous-plastic descriptions of wood, descriptions mechanical and sorption creep, withering coefficients on apportionment of two-dimensional stress-strain state of wood during drying taking into account an isothermal process. For this purpose during numeral modeling the probed index changed to the half of its value and the change of normal and tangential tensions on-the-spot and on the “near surface” layers of wood was probed.

Especially, on the Fig. 3 graphic dependences of normal tension ox on-the-spot tangential saw-timbers for different values component of tensor of resiliency are shown. Continuous curve 1 answers a numeral calculation without the change of resilient descriptions. The biggest influence on the tension apportionment ox has the change of Young’s elastic modulus in tangential direction, thus a curve 2 (for which E2(2) = 0,5 E2 and other coefficient didn’t change during numeral modeling). Especially, the increase of deviation ox, for the case ox of Е2(2) = 0,5 Е2 without change E2 is more considerable with the drying time increase. A change of the module of resiliency in radial direction Ei(3) = 0,5 Ei doesn’t influence the apportionment a lot ox on the surface (a curve ox in this case almost coincides with ox without change of

characteristics). The diminution of the Poisson’s ratio v12(4) = 0,5 v12 to the change of component ox is characterized by a curve 4, and the diminution of the shear module G12(5) = 0,5 G12

on ax describes the curve 5. The analysis of graphic dependences testifies that growth of change of wood anisotropic mechanical descriptions, dependent upon humidity on apportionment of normal tensions ox on the surface during the time of wood drying.

Fig. 3. The influence of anisotropy of tensor resiliency component on the apportionment of normal tension ox on the surface

On the Fig. 4 influence of anisotropy coefficients of mechanical and sorption creep on tension in the “near surface” layer of wood sample is shown. Curve 1 meets a set value for calculation. The analysis of graphic dependences testifies that diminishing of coefficients mr (mr(2) = 0,5 mr, curve 2), цп W3 = 0,5 ^t,

curve 3) doesn’t substantially influence on the value of normal tensions ox. The change of mrt (mrt(4) = 0,5 mrt, curve 4) increases the value of normal tensions from 0,5 to 0,72 MPa after the initial stage of drying process. The most important influence has the change of mt (mt(5) = 0,5 mt, curve 5) on incremental tensions in a “near surface” area. On the initial stage of drying process maximal value increases from 0,75 MPa to 1,24 MPa. The increase of tension ox is observed with the duration of drying

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Fig. 4. Influence of mechanical and sorption creep on apportionment of normal tension in a “near surface” layer

process. Diminishing mt moves the change of sign of normal tensions on time axis.

Thus, received mathematical models and developed programmed complex give the opportunity to determine the changes of technological process parameters in wood drying to the apportionment of heat-and-mass transfer and elastic-viscous-plastic state of wood taking into account the mechanical-sorption creep and anisotropy of heat mechanic characteristics of wood.

Conclusions

1. We formulated a mathematical model of elastic-viscous-plastic deformation of wood during drying process, which takes into consideration plastic deformations, caused by mechanical-sorption creep and anisotropy of mechanical characteristics of the material and gives the opportunity to determine twodimensional stress-strain state in the conditions of non-isothermal humidity transfer.

2. On the basis of the created mathematical models and methods of analysis application software is developed for the modeling of elastic-viscous-plastic state of

capillary-porous materials during the convective drying, which consists of the above classes and gives the opportunity to automatize finite element analysis stress-strain state of wood during convective drying.

3. As a result of calculable experiments, conducted with the use of developed applied programmatic facilities regularity of anisotropy influence of thermo-physical and mechanical descriptions of wood were set, it initial humidity on changing of two-dimensional temperature-humidity and elastic-viscous-plastic state of wood during the convective drying. Satisfactory coordination of the results of numeral modeling with known experimental researches for partial cases was discovered.

References

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2002. - 340 с.

2. Соколовский, Я. И. Моделирование деформационно-релаксационных процессов в древесине во время сушки [Текст] / Я. И. Соколовский, М. В. Дендюк, Б. П. Поберейко // Лесной журнал : изв. ВУЗов России. -Архангельск, 2007. - № 1. - С. 75-83.

3. Дорняк, О. Р. Моделирование реологического поведения древесины в процессах прессования [Текст] / О. Р. Дорняк // Инженерно-физический журнал. - 2003. -Т. 76. - № 3. - С. 150-155.

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ний в древесине при сушке [Текст] / П. В. Акулич, К. Е. Милитцер // ИФЖ. -1998. - Т. 71. - № 3. - С. 404-411.

5. Perre, P. A physical and mechanical model able to predict the stress field in wood over a wide range of drying conditions [Text] /

P. Perre, J. Passard // Drying Technology Journal. - 2004. - Vol. 22. - N. 1-2. - pp. 27-44.

6. Кристенсен, Р. Введение в теорию вязкоупругости [Текст] / Р. Кристенсен. -М. : Мир, 1974. - 338 с.

7. Rodic, J. Mechanics of wood and composites [Text] / J. Rodic, A. Jayne // Van No-

straind Reinhold. New York. - 1982. - 712 p.

8. Сегерлинд, Л. Применение метода конечных элементов [Текст] / Л. Сегерлинд. - М. : Мир. - 1979. - 378 с.

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програмне забезпечення системи моделюван-ня та анашзу процесу сушшня капшярно-пористих матерiалiв [Текст] / Я. I. Соколовський, I. М. Крошний // Вюник Нацюнального ушверситету «Львiвська полггехшка»:

Комп’ютерш науки та шформацшш

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DOI: 10.12737/4523 УДК 630*812: 666.974

ВЛИЯНИЕ ВОДОПОГЛОЩЕНИЯ НА СВОЙСТВА ДРЕВЕСИНЫ В ПОЛИМЕРЦЕМЕНТНОМ КОМПОЗИЦИОННОМ МАТЕРИАЛЕ

доктор технических наук, доцент, профессор кафедры промышленного транспорта, строительства и геодезии Т. Н. Стородубцева аспирант кафедры промышленного транспорта, строительства и геодезии А. И. Томилин ФГБОУ ВПО «Воронежская государственная лесотехническая академия»

tamara-tns@yandex.ru

Весьма актуальным в настоящее время является использование в композиционных материалах отходов лесного комплекса, промышленности, сельского хозяйства и т. д.

Характеристики разрабатываемых композитов получены с применением формул науки о сопротивлении материалов, т.е. в пределах справедливости закона Р. Гука, на который эта наука опирается.

Исследованы зависимости основных механических характеристик полимерцементной матрицы композиционного материала от содержания в ней структурообразующих компонентов древесного компози-

ционного материала - модифицирующих наполнителей (графитовая мука, мука из пиритовых огарков), замедлителя реакции кристаллизации (глицерин) бензолсульфокислоты (БСК) и армирующих заполнителей (стеклосетка и кусковые отходы переработки древесины - щепа), которые вводили в ее состав вначале порознь, а затем одновременно. В подавляющем большинстве случаев эти зависимости были представлены графоаналитическими моделями в виде полиномов третьей степени и построенными с их использованием кривыми, что подтверждалось минимальными значениями сумм квадратов

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