Introduction
The mechanical properties of wood are a crucial factor in the analysis and assessment of natural wood structures, such as trees. Wood in standing trees typically exceeds the fiber saturation point (FSP), with a moisture content (MC) that can significantly impact its mechanical behavior compared to construction lumber. To accurately model the mechanical response of trees and other green wood structures, having reliable data on the elasto-plastic properties of green hardwood is essential.
To address this need, researchers developed and validated an orthotropic elasto-plastic (E-P) material model for European beech (Fagus sylvatica L.) green wood. This model is capable of capturing both the non-linearity and orthotropic properties of the material, allowing for more realistic numerical simulations of wood structures.
The researchers selected 655 clear beech wood samples with the unique orthotropic structure of annual rings. All samples were prepared immediately after felling, with an average MC of 80%. The mechanical responses in normal directions and shear were represented by bi-linear stress-strain curves, from which the E-P model parameters were derived.
The E-P model was validated by comparing the force-deflection response of three-point bending experiments with finite element method (FEM) simulations. The average relative error was 4.6% for point-wise and 1.7% for integral-wise comparisons, demonstrating the model’s ability to accurately predict the non-linear response of green beech wood above the proportional limits.
This work provides a consistent set of material constants for the E-P material model that can be used for the structural analysis of beech wood with MC above the FSP, particularly green wood subjected to relatively high loads where plastic deformation occurs. The availability of these material parameters is a valuable resource for researchers and engineers working on the mechanical analysis of natural wood structures, such as trees, and the development of numerical models for green wood behavior.
Experimental Methodology
Sample Preparation and Mechanical Testing
Four European beech trees were selected from a natural forest stand and felled. One-meter-long logs were taken from a height of approximately 10 meters on each trunk to obtain the test samples. The trees had an average height of 32 meters, a breast height diameter of 45 cm, and an age of 95 years.
To maintain the high MC and prevent losses, the wood samples were processed within 4 days after felling and stored in sealed boxes with high relative humidity (99%) until testing. This ensured that the MC did not fall below 50% during the sample preparation.
Ten types of specially orthotropic samples were crafted for mechanical testing, including compression, tension, and shear in the longitudinal (L), radial (R), and tangential (T) directions, as well as three-point bending (Figure 1). The experimental setup and dimensions of the samples followed the British standard BS 373 and relevant Czech standards.
All samples were tested on a universal testing machine equipped with a 50 kN load cell. Digital image correlation (DIC) was used to measure the displacement and strain of the samples, except for the tension parallel to the grain, where mechanical extensometers were used.
The dimensions and weights of the samples were measured before the tests to determine the MC and density. In total, 655 samples were tested, with the number of samples per group varying due to the careful selection of defect-free, orthotropic specimens.
Bi-linear Material Model Identification
The stress-strain curves obtained from the experiments were fitted with bi-linear functions (piecewise linear) to identify the parameters for the E-P material model, including the yield stress (σ), elastic modulus (E), tangent modulus (Et), and shear modulus (G and Gt) for each loading mode and direction.
The shear characteristics in the radial-tangential (RT) plane were not experimentally determined but were calculated using empirical equations from the literature.
Poisson’s Ratio Determination
Poisson’s ratios were calculated from the active (compressive) and passive (tensile) strains measured by the DIC system, using the following equation:
νAP = -εP/εA
where νAP is the Poisson’s ratio in the XY plane, εA is the strain parallel to the loading (active), and εP is the strain perpendicular to the loading (passive).
Numerical Material Model Development
The experimental data from the uni-axial tests were used to establish the constants for the E-P material model, referred to as the bi-linear material model (B-MM). However, the B-MM could not be directly used in the FEM solver, as it needed to satisfy the restrictions of the generalized Hill’s plasticity theory.
To overcome the limitations of Hill’s theory, the researchers made the following adjustments:
- The tangent moduli (Et and Gt) were redefined to be close to zero, ensuring the validity of the consistency equations from Hill’s theory.
- The yield stresses (σ) were adjusted to satisfy the conditions of Hill’s theory, while maintaining realistic values for compression and tension.
This resulted in the computational material model (C-MM), which was then used in the FEM simulations.
Finally, a parameter study was conducted using the ANSYS Probabilistic Design System (PDS) to identify the most significant material parameters and further refine the C-MM. This produced the adjusted computational material model (AC-MM), which was used for the final FEM validation.
FEM Validation and Calibration
The FEM simulations were performed using ANSYS Mechanical APDL, with the wood modeled as a 3D orthotropic elasto-plastic material with non-linear isotropic hardening. The geometry of the FE model reflected the experimental three-point bending setup, including the specimen dimensions, support, and loading conditions.
The force-deflection curves from the FEM simulations were compared to the experimental results, and the mean relative error (MRE) was calculated to assess the accuracy of the material models. The MRE was calculated as the arithmetic average of the relative errors for the various material properties.
The initial C-MM showed good agreement with the experimental data, with an average MRE of 4.56%. Further calibration using the PDS analysis led to the final AC-MM, which reduced the MRE to 1.68% for the integral-wise comparison and 4.21% for the point-wise comparison.
The parameter study revealed that the most significant material parameters were the longitudinal elastic modulus (E_L) and the yield stress in compression parallel to the grain (σ_-L). Adjusting these parameters based on the PDS analysis resulted in the AC-MM, which showed excellent agreement with the experimental three-point bending results.
Discussion and Conclusions
The researchers successfully developed and validated an orthotropic elasto-plastic material model for green beech wood, which can be used for the structural analysis of wood structures with MC above the FSP, particularly trees and other green wood elements subjected to high loads.
The E-P material model was able to accurately capture the non-linear response of green beech wood, with the FEM simulations showing excellent agreement with the experimental data. The final AC-MM had a mean relative error of only 1.68% for the integral-wise comparison and 4.21% for the point-wise comparison, demonstrating the model’s reliability.
The availability of these material parameters is a valuable resource for researchers and engineers working on the mechanical analysis of natural wood structures, such as trees, and the development of numerical models for green wood behavior. The model can also be used in the processing and analysis of green wood in various applications.
The researchers note that further work could focus on calibrating the material model for other wood species, investigating the influence of defects (e.g., knots, drying cracks), and exploring the application of the model in more realistic, larger-scale scenarios involving the cylindrical orthotropy of wood.
Overall, this study provides a comprehensive and validated elasto-plastic material model for green beech wood, which can significantly improve the accuracy and reliability of numerical simulations in the field of wood biomechanics and engineering.
Acknowledgments
The authors gratefully acknowledge the support by the Ministry of Education Youth and Sports in the Czech Republic, and the authors would also like to thank the Mendel University in Brno for providing the trees from which the wood for the experiments was taken.
