Sofia Holovata

Department of Information Technologies

National Forestry and Wood-Technology University of Ukraine

Lviv, Ukraine

e-mail: [email protected], [email protected]

Bohdan Pobereyko

Department of Automation and Computer-Integrated Technologies

National Forestry and Wood-Technology University of Ukraine

Lviv, Ukraine

e-mail: [email protected]

*Abstract *— a new mathematical model was synthesized to determine the strength of anisotropic materials under biaxial stressed state and its approbation was held using broad – and needle – leaved trees. In particular, the short – term strength curved lines were made for wood of pine and oak. On the basis of their analysis it revealed that the offered model reflects the features of the limiting stressed state of the investigated timber species in radial and tangential plane of structural symmetry satisfactorily.

Keywords — mathematical model, strength, strain, warp, volumetric strain, potential energy density.

# I. Introduction

The increase of product quality indicators and reducing energy costs for its production is one of the main conditions for the successful development of woodworking enterprises. The decisive role in solving this problem belongs to the study of the short-term strength of wood with a complex stress state, because strength is one of those factors that significantly limit the intensification of lumber dehydration processes.

The duration of these processes cannot be arbitrarily small, its value must be such that at all points of the dried material the values of the components of the stress tensor do not exceed the limit values. Otherwise, there will be residual stresses, which are the main factors in reducing the quality of the material. For determination of the ultimate stresses in wood with a uniaxial stress state, experimental research methods are used, and in wood with a complex stress state, the method of mathematical modeling is used.

The method of mathematical modeling is implemented through mathematical models, which are usually called strength criteria. They include the criteria of R. Mises, Mises-Hill, E.K. Ashkenazi, K.V. Zakharov, O.K. Malmeister, Goldenblatt-Kopnov, etc. However, their use for modeling wood strength curves and surfaces with biaxial, flat, and three-dimensional stress states is not sufficiently substantiated. None of the mentioned models has been adapted and tested on softwood and hardwood. The difficulty of solving this problem lies in the fact that the input data of these models (criteria) are strength constants (for example, shear limits of an orthotropic material along the diagonal planes of structural symmetry), which are currently not subject to experimental determination.

Nowadays, there are no technical solutions for determining and controlling stresses in dried lumber, and the existing mechanical theories of strength do not fully describe the strength of wood under complex stress conditions, so the task of modeling the strength of wood is urgent. The lack of empirical data on the determination of pure shear strength limits along the main and diagonal planes of structural symmetry for wood of different breeds does not allow to confirm or deny the validity of the currently known strength criteria for the studied material. In particular, no model has been adapted or tested on softwood and hardwood.

**II. Mathematical model for strength determination of anisotropic materials**

In order to solve this problem, a new mathematical model was built to determine the short-term strength of anisotropic materials under conditions of a biaxial stress state. The components of this model are:

- strength condition [1]:

- dependence of the modulus of elasticity of the material on the method of loading [2]:

The developed model, in contrast to the known ones, makes it possible to calculate the ultimate stress states of anisotropic materials with both weak and strong asymmetry of strength limits in the main directions.

**III. Adaptation of mathematical to wood of deciduous and coniferous specie**s

It is shown that the solutions of the system of the equations (6) are the following formulas for determination of the constants *a* and *b* of the mathematical model of the short-term strength of anisotropic materials (1) – (3):

The value of the *k* constant of the mathematical model is determined by the formula given in [1]

**IV. Results of the mathematical model testing on hardwood and softwood**

Using this algorithm, the ultimate stress states of pine and oak wood were determined, the values of the physical and mechanical characteristics of which are given in Table 1. Namely, the strength curves for these materials under conditions of a biaxial stress state in the radial-tangential planes of structural symmetry were constructed (Fig. 1). The analysis of the coordinates of such points of the constructed curves, which characterize such a limit stress state of the material, the characteristics of which are empirically determined, was carried out. These points are the points of intersection of these curves with the axes of coordinates А_{С}, С_{С}, В_{С}, D_{C} and А_{Д}, С_{Д}, В_{Д}, Д_{Д }(A_{D}, C_{D}, B_{D}, D_{D}).

Table 1. Physical – mechanical characteristics of pine and oak wood with temperature T = 20 °C and relative humidity W = 12% [3]: in the number – tensile value and in the denominator – compression value

Wood species | Modulus of elasticity, hPa | Strength limits, MPa | ||||

E_{a} | E_{r} | E_{t} | σ_{r} | σ_{t} | ||

Pine | 11,9 11,9 | 0,54 0,67 | 0,47 0,55 | 3,23 5,10 | 2,63 7,50 | |

Oak | 14,2 14,2 | 1,18 1,40 | 0,91 1,01 | 8,0 – | 6,5 – | |

Poisson’s ratios | ||||||

μ_{ar} | μ_{at} | μ_{rt} | μ_{ra} | μ_{ta} | μ_{tr} | |

Pine | 0,03 | 0,037 | 0,38 | 0,49 | 0,41 | 0,79 |

Oak | 0,07 | 0,09 | 0,34 | 0,43 | 0,41 | 0,83 |

Because, according to the Fig. 1, the abscissa values of the A_{C} and C_{C} and A_{D} and C_{D} points are the values of the tensile and compressive strength limits of pine and oak wood, respectively, in the radial direction of orthotropy, and the ordinates of the В_{С} and D_{C} and В_{D} and D_{D} points are the values of the limits tensile and compressive strengths of the specified wood species in the tangential direction, which are currently known and given in the reference literature [2].

The analysis of the coordinates of these points showed that the mathematical model (1) – (4) satisfactorily reflects the anisotropy and asymmetry of the strength of the studied wood species along the main directions of symmetry. Indeed, since the abscissa of point AC is greater than the ordinate of point BC, and the absolute value of the abscissa of point CC is less than the absolute value of the ordinate of point DS, the limit of short-term tensile strength of pine wood in the radial direction is greater than the limit of tensile strength in the tangential direction, and in in the case of compression, on the contrary: the absolute value of the strength limit in the radial direction is smaller than the absolute value of the strength limit in the tangential direction, which is confirmed by the results of experimental studies [2].

The found ratio of tensile strength limits in the radial and tangential directions of anisotropy for pine wood also holds for oak wood.

Therefore, according to the results of currently known experimental studies of the short-term strength of wood with uniaxial stress states in the directions of anisotropy, the mathematical model (1) – (12) adequately describes the ultimate stress state of pine and oak wood under conditions of biaxial tension-tension, compression-compression and tension-compression.

In addition, it should be noted that the characteristics of the constructed curves (Fig. 1) satisfy the main heuristic requirements for the construction of known phenomenological mechanical theories of the strength of solid bodies [4, 5]: 1) the curves of the ultimate stress state (curves of short-term strength) should be smooth and convex; 2) curves of short-term strength of the material must cover the origin of the Cartesian coordinate system.

Fig. 1. Curves of short-term strength of wood with temperature T=20 °С, relative humidity W = 12% and biaxial tension state in the radial-tangential plane of structural symmetry: 1 – pine, 2 – oak

**Conclusions**

It is shown that the calculation curves of the short-term strength of pine and oak wood with a biaxial stress-strain state in the radial-tangential plane of structural symmetry satisfactorily describe the results of experimental tests of the material for uniaxial tensile and compressive strength in the radial and tangential directions of anisotropy.

In particular, it was confirmed: a) the short-term tensile strength limit of pine wood in the radial direction is greater than the tensile strength limit in the tangential direction, and vice versa in the case of compression: the absolute value of the strength limit in the radial direction is less than the absolute value of the strength limit in the tangential direction; b) the absolute values of the tensile strength limits in the radial and tangential directions of anisotropy of pine wood are smaller than the corresponding absolute values of the compressive strength limits.

##### References

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