FORMATION OF GEOMETRICAL CHARACTERISTICS OF A THIN LAYER OF POROUS AND ANISOTROPIC MATERIAL. Ireneusz Malujda

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1 METL , Hradec nad Moravicí FORMTION OF GEOMETRICL CHRCTERISTICS OF THIN LER OF POROUS ND NISOTROPIC MTERIL Ireneusz Malujda Poznań University of Technology Chair of Machine Design Fundamentals, Poznań, Piotrowo 3, bstract The article presents analysis of a stress distribution in a thin layer of wood being subject to a plasticization process under the influence of pressure and temperature. The problem was solved on the basis of the Huber-Mises-Hency volumetric strain critical energy hypothesis. This aim was achieved by solving a system of plasticity equations for the porous and orthotropic layer,, thus by arriving at a critical stress formula. s a result, an illustration of stress distribution in a layer of wood under a hot roller as a function of time with its surface temperature remaining constant was produced. The rolling process is executed using a rolling mill designed for a furniture manufacturer and implemented several times. The paper presents stress distribution in a thin layer of wood under a hot turning roller. Formulation of interrelations between the temperature, time and pressure is of paramount importance for determining effectiveness of wood refinement. The process is executed with a rolling mill designed for a furniture manufacturer and implemented several times [3]. The machine is intended for rolling of wood veneers and veneered furniture elements by applying the pressure of hot rollers with adjusted speed [5]. The analyzed process taes place in the layer close to wood surface at a high temperature (of about 30 0 C. In such conditions the lignin, maing up about 40% of wood s weight, undergoes the plasticization process. Research shows that with increasing the temperature to about 30 0 C the strength of wood in a direction perpendicular to its fibers decreases several times.. PLSTICIT CONDITION OF POROUS ND NISOTROPIC MTERIL The plasticity theory uses the term: plasticity condition in order to determine the conditions required at a given point of a body for a plastic deformation to occur. nalysis of plastic deformations in the thin layer of wood under consideration was based on the generalized model of an ideally plastic medium. The use of the said model for describing behaviour of wood under pressure and temperature provides a sufficient approximation of real-life situations []. In case of cold rolling of metal (Fig.a. the actual pressure diagram under the roller [9] consists of two parts: the lower one C D, where the pressure curve required to overcome the yield stress R e increases slightly due to metal strengthening, and the upper one C D E, which represents the pressure values required to overcome the horizontal components, being the result of metal flow resistance occurring during the rolling process. S e = S r + S p, ( where: S e - deformation resistance during the rolling process, S r - metal yield stress, S p - metal flow resistance during the rolling process. Wood is a porous material and unlie metal, during its rolling, no higher strengthening or flow resistance can be expected [4]. Exceeding yield point R e during the compression results in a deformation and compacting of the structure.

2 METL , Hradec nad Moravicí a b E D C C D Figure.Roller pressure distribution, i.e. deformation resistance K w along contact line s during cold rolling of: a steel, b wood In practice, it is often accepted for rolling process calculations that S r is constant throughout the length of arc s. Thus it is possible to assume, that the distribution of pressure applied by the roller on the layer of wood is constant during cold rolling (Fig.b. The analysis of a plastic flow in a thin layer of wood was based on a somewhat modified Huber Mises-Hency volumetric strain critical energy hypothesis [7], [8], [9]. It is assumed that owing to wood's porosity the measure of effort is this part of energy, which is attributed to the change of the form of the body as well as its deformation. This condition has to tae anisotropy into account as well []. The yield condition describes the interrelation between the constituents of a stress and temperature condition and material constants for yield point, at which the first plastic deformations occur at a given point of the object. For anisotropic and porous material subjected to the effect of an increased temperature the above relation will tae the form expressed by the following scalar function [6]: F( σ, f v, T = 0, ( where:σ - anisotropic stress condition, f v - scalar function of porosity, T - scalar function of temperature. The above formula expresses the so-called yield criterion. ssuming the framewor to be a perfectly plastic non-strengthened material, for which plastic deformations considerable prevail over flexible strains and by introducing the porosity function the above condition taes the following detailed form: tr S + ψ t r σ = ψ i, ( where: tr S - trace of stress deviator, ψ and ψ - porosity functions, tr σ - trace of stress tensor, i - yield point of a material tensioned in a specific principal direction. For an orthotropic case the yield conditions taes the following form: ( σ + σ + ( σ - σ = i, (3 where: σ and σ - main stresses on anatomic directions: longitudinal, tangent (Fig. and Fig. 3, and - porosity functions.

3 METL , Hradec nad Moravicí f f where: for f v 0 we obtain 0 i, for f v we obtain, 0. v =, = - f v, (4 v For othotropic cases the number of modules, functions or material constants is reduced to 6 (from in a spatial stress condition: L, M, N, P, Q, R, where : L, M, N are material functions describing the yield points during tensioning as a function of temperature, P, Q, R - are material functions describing the yield points during shearing, L = +, M = +, N = +, ( P =, Q =, R =. (6 3 Having introduced the orthotropic stress coefficients defined as follows: α = (, β = (, (7 where: longitudinal yield point, - transverse yield point, tangent direction, 3 transverse yield point, diagonal direction. The yield criterion for plane stress and orthotropic case may be expressed as follows: 3 3 σ ( + +α ( + σ + ( α - βσ σ =. (8 The process of rolling has been analysed for plane strain condition with the following input criteria: dε = 0, l : h > 0, 3 the effect of stress σ has been ignored due to insignificant friction in the direction of rolling. y differentiating the equation (8 we obtain: α ( + σ + ( ( + α - β σ = 0, (9 and then the stress σ in direction (Fig. 3 is described by the following formula: ( α β σ = σ. (0 α 3

4 METL , Hradec nad Moravicí Fig.. natomic directions of wood Fig. 3. Principal stress directions for a rolled layer of wood where:,, 3 - principal directions, P - rolling force, s - contact length between roller and rolled wood layer, h - initial thicness of wood layer, h - thicness of rolled wood layer, l - width of wood layer. y way of introducing the relation (0 to equation (8 we obtain the following form of yield criterion: σ ( α β [( + - ] =, ( σ = ( α β. ( For isotropic tensioning the yield criterion (8 is described by the following formula: P + αp + α - β P = F, (3 where: P critical force exerted on the surface of the plasticized material. 4

5 METL , Hradec nad Moravicí This formula is used to obtain the critical force value in the following form: P = (3+ 3α β F. (4 In the end, the plasticity condition for the flat deformation state during the rolling process (after subtracting ( from equation (4 shall have the following form: P ( α β (3+ 3α β = [ ] F. (5 Equation (5 maes it possible to determine the critical value of the force in the layer of wood under consideration, which must be exceeded for a plastic deformation to occur. In the case under consideration the plasticization process occurs in line with the increasing temperature over time, thus the plasticization yield point is defined by the function, which taes the said relation into consideration. 3. CONCLUSIONS. The analysis of plastic deformations in the process of rolling a layer of wood has been based on a generalized model of perfectly plastic material. The yield criterion has been formulated using the hypothesis of ultimate energy of non-dilatational strain, which has been modified to account for porosity and anisotropy of wood.. The obtained formula for the critical force in a layer of porous and anisotropic material can be used to rationalize the process of shaping its geometrical characteristics and designing equipment used for this purpose. 3. The model developed here serves as a tool, which can assist in the design of geometrical, material and functional parameters. 4. s the strength criterion in the process of rolling a layer of wood there may be used the longitudinal tensioning yield point and that stress should be temperature-dependent. ILIOGRPH. ednarsi, T. Mechania plastycznego płynięcia w zarysie. Warszawa: PWN, Dudzia, M., Malujda, I., Mielniczu,J., Mosalewsi, K. nalysis of plastic deformations in a thin layer of porous and anisotropic material during hot rolling (Non classicalmaterial models in engineering design. Poznań: Zeszyty Nauowe Politechnii Poznańsiej, nr 75, Dudzia,M., Malujda, I. 003, Machine for rolling wooden elements. Lviv: 6-th InternationalSymposium of Urainian Mechanical Engineers, Lviv, Krzysi, F. Naua o drewnie. Warszawa: PWN, Malujda, I. Walcowanie na gorąco zaoleinowanych elementów meblowych. Poznań: Wydawnictwo Politechnii Poznańsiej, Malujda, I., Mielniczu, J. Mathematical Model of wood hot rolling process. Dresden: GMM, 75-th nnual Scientific Conference, March 7, Szczepińsi, W. Mechania plastycznego płynięcia. Warszawa: PWN, Walcza, J. Wytrzymałość materiałów oraz podstawy teorii sprężystości i plastyczności. Warszawa Kraów: PWN, Wusatowsi, Z. Podstawy procesu walcowania. Katowice: PWT, 95. 5