Introduction

Nowadays, computer-aided engineering (CAE) methods such as finite-element modeling (FEM) are used for designing and engineering of high-value products. Reliable FEM is based on a sound data basis of material properties, such as elastic constants, including the Poisson’s ratio (ν). For anisotropic materials like wood, the Poisson’s ratio for one orthogonal direction is the ratio of the transverse contraction (transverse strain (εq)), to the axial extension (axial strain (εl)). These parameters have mainly been investigated in the last 4–5 decades by mechanical or electrical measurement systems (e.g., strain gauges, mechanical extensometer systems, inductive strain measurement devices), because of the lack of availability and the high price of optical measurement systems (Davis 2004).

The first examinations of the Poisson’s ratio of spruce wood were conducted by Carrington (1921, 1922a, b). He deduced the Poisson’s ratio from flexure experiments by measuring the curvature in lateral direction (transverse strain (εq)) and longitudinal direction (axial strain (εl)) with a telescope. Hörig (1931) re-evaluated the data and adopted the ideas of Voigt (1882, 1887, 1966), about the orthotropic behavior of materials on wood. The model by Hörig (1935) is the basis for the orthotropic description of wood that is used nowadays. Further substantial studies on spruce wood were carried out by Wommelsdorff (1966) and Neuhaus (1981). They determined the six orthotropic Poisson’s ratios using inductive strain measurement devices and also strain gauges by means of tensile and flexure experiments. Furthermore, Niemz and Caduff (2008) and also Keunecke et al. (2008) investigated the Poisson’s ratio of spruce wood. Keunecke et al. (2008) have chosen digital image correlation (DIC) to measure the strain distribution. DIC is a non-contact optical surface deformation gauging technique (Chu et al. 1985; Zink et al. 1995; Pan et al. 2009; Valla et al. 2011).

In more recent studies, three-dimensional optical digital measurements (3D ODM) and also resonant ultrasound spectroscopy (RUS) methods were used to establish all elastic constants by using only one type of specimen (Forsberg et al. 2010; Majano-Majano et al. 2012; Vorobyev et al. 2016). A clear advantage is that all components of the stiffness tensor are established with the same specimen by employing a consistent method. Currently, though, not all elastic constants can be derived robustly (e.g., Poisson’s ratio)—as for example the viscoelastic damping of wood may cause an overlapping of resonant peaks (Longo et al. 2018) which eventually may lead to a wrong iterative deduction of the elastic constants in the inverse identification. This applies in particular to wood with low density. Further, Longo et al. (2018) pointed out that the free resonance frequencies are very insensitive to νRT and not sensitive at all to vLR and νLT. Three-dimensional ODM requires high-resolution cameras and a high level of expertise in material characterization, due to the need of the mathematical implementation of the procedure to establish consistent measurements (Majano-Majano et al. 2012). Moreover, the setup may bias the measurement results, which needs to be clarified before this technique can be considered as standard method for future wood material characterization. In summary, RUS and 3D ODM for establishing Poisson’s ratio are uncertain and may only be indicated when nondestructive testing is required (Bachtiar et al. 2017).

Optical gauging techniques that provide independent mechanical material properties in micro- or even nanoscale were found to be suitable for wood characterization (Xavier et al. 2007, 2013; Valla et al. 2011; Toussaint et al. 2016). Furthermore, these methods have the advantage to be contactless, which means avoiding any mechanical influences on the specimen. Therefore, in this study the optical gauging techniques “Electronic Speckle Pattern Interferometry (ESPI), laser extensometry and video extensometry” are compared head-to-head for establishing the Poisson’s ratio of (spruce) wood. Former studies prove that these methods are suitable for the mechanical characterization of the elastic properties of wood (Gingerl 1998; Eberhardsteiner 2002; Samarasinghe and Kulasiri 2004; Gindl et al. 2005; Müller et al. 2005; Konnerth et al. 2006; Gindl and Müller 2006; Dahl and Malo 2009; Valla et al. 2011; Bader et al. 2015; Crespo et al. 2017; Milch et al. 2017).

ESPI is a non-contact gauging technique based on the Michelson interferometer (Meschede 2015), which is used for planar strain measurement in the present study. The technique uses laser light (coherent light wave) together with a CCD camera to record displacements of the specimen surface. The surface is illuminated with a laser beam from two different planar directions, and the reflected light is registered by a CCD sensor. The ESPI system converts the light information into a speckled image, which describes the surface of the object. Deformation of the specimen results in a new speckle pattern. By subtracting the new speckle pattern from the reference pattern, an illustration with typical fringe pattern is obtained (Jones and Wykes 1989). In the next step, a phase-shift method is used to transform the fringe picture into a so-called 2π-modulo image, which is used to create a map of displacement (Eberhardsteiner 1995; An and Carlsson 2003; Müller et al. 2015). Additional material data (for example strain distribution) can be gained from the deformation map through post-processing. More comprehensive information about the ESPI technique is available in other studies (Gingerl 1998; Rastogi et al. 2001; Eberhardsteiner 2002; Müller et al. 2005).

The basic principles of the laser extensometry method are similar to the ESPI technique. A laser source radiates a beam, which is projected on the surface of the specimen. The reflected light beams are recorded on a camera sensor, which generates a speckle pattern on the basis of the intensity distribution (Messphysik—Materials Testing 2017). The mechanical load induces movements on the object surface. Those movements indicate displacements of the speckle pattern as well. The core of the technique is to identify pattern areas of the initial picture in the upcoming images (Zwick/Roell 2017a). Due to the unique gray value distribution of any defined pattern area, it is possible to find these speckle zones in any upcoming deformation image. After that, a complex algorithm runs to find the motion of the defined speckle zone between the initial picture and the following images. For the estimation of the strain in one direction, it is necessary to perform this procedure on two self-contained pattern zones at least. More comprehensive information about the laser extensometry technique can be found elsewhere (Choi et al. 1991; Kamegawa 1999; Anwander et al. 2000; Jin et al. 2013; Messphysik—Materials Testing 2017; Zwick/Roell 2017a).

The video extensometry method is based on capturing ongoing images of the specimen, for example, during a tensile test by using a digital video camera. To capture the lateral movements, the specimens need to be marked somehow (e.g., sticker and pen marker) at least on two different positions. By using this method, it is important to have high contrast between the object surface and the measurement points (markers) to ensure unaltered results. While the specimen is stressed, the pixel distance between these markers is tracked continuously. Image processing algorithms are used to track these motions in real time. Automatically, a direct strain measurement value can be obtained by mapping these motion measurements against the initial specimen image. For recording the transverse deformations of the specimens, no extra marking is required. In this case, special edge detection algorithms are applied (Zwick/Roell 2017b). The video extensometry technique provides non-contact real-time strain measurement in lateral and transverse direction independently from each direction. More specific information about the fundamentals of the technique can be found in Vial (2004), Wolverton et al. (2009), Bovik (2010) and Zwick/Roell (2017b).

The present study focuses on determining the Poisson’s ratio of spruce wood in all main orthogonal directions by means of the non-contact optical gauging techniques ESPI, laser extensometry and video extensometry, respectively. Therefore, a uniaxial tensile experiment was designed under real measuring conditions to generate comparable and truthful values. The main hypotheses of this study are:

  • ESPI, laser extensometry and video extensometry are suitable for the detection of the Poisson’s ratio of wood.

  • Poisson’s ratio gained by means of ESPI, laser extensometry and video extensometry will show no statistically significant differences.

Materials and methods

Material

Solid wood made of Norway spruce (Picea abies (L.) Karst.) specimens was used in the experiments. Sawn timber without noticeable defects like knots or cracks was meticulously selected. Semifinished elements in the different anatomical directions were cut out of the boards by means of a circular saw. For producing the samples out of these elements, a CNC and a conventional planning and a milling machine were used. After manufacturing the raw material to the desired shape, samples were conditioned at a temperature of 20 ± 2 °C and a relative humidity of 65 ± 5% (after ISO 554) to an average moisture content of ω = 12%. Under this condition, the average sample density (ρ) was 465 ± 30 kg/m3.

Dog-bone-shaped specimens corresponding to DIN 52188:1979-05 (1979) were used for testing in longitudinal (L) direction. Due to the high stiffness of wood in longitudinal direction, high forces and thus also high clamping forces had to be applied. To ensure no material failure and no slipping in the clamping area, those dog-boned shaped specimens were required. For samples, where the load was applied in tangential (T) or radial (R) direction, simplified strip-shaped specimens with uniform cross section were manufactured (20 × 6 × 120 mm3 = width × height × length). Because of the low stiffness in the transverse direction of wood, only low clamping forces act and therefore no slippage was guaranteed. In accordance with the theory by Hörig (1935), six different samples (Fig. 1) were necessary to measure all main orthogonal Poisson’s ratios: νLR, νLT, νRL, νRT, νTL and νTR independently. In this case, the first character of the subscript stands for the direction of the longitudinal extension [longitudinal strain (εl)] and the second character of the subscript describes the direction of the transverse contraction [transverse strain (εq)]. In total, 60 specimens were tested, i.e., 10 in every direction.

Fig. 1
figure 1

3D illustrations of the specimens; a dimensions corresponding to DIN EN 52188:1979-05 (1979); b, c 20 × 6 × 120 mm3 (width × height × length)

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Designation of Poisson’s ratio

In general, the phenomenon that axial extension (εl) of a rod results in transverse contraction (εq) was discovered 1760 by Poisson. The ratio of the passive deformation (normal to the direction of the applied force) to the active deformation (in direction of the applied force) is defined as Poisson’s ratio:

$$\nu = - \frac{{\varepsilon_{\text{q}} }}{{\varepsilon_{\text{l}} }}.$$
(1)

Thus, for specimen RT as an example (Fig. 1b left), the Poisson’s ratio is defined as:

$$\nu_{\text{RT}} = - \frac{{\varepsilon_{\text{T}} }}{{\varepsilon_{\text{R}} }}\left( {\nu_{ij} = - \frac{{\varepsilon_{j} }}{{\varepsilon_{i} }}\,i,j \in {\text{R}},{\text{L}},{\text{T}}\;{\text{and}}\;i \ne j} \right).$$
(2)

Calculation of Poisson’s ratio as compliance coefficient

Hörig’s (1935) idealization of wood as a crystalline material with linear elastic and orthotropic mechanical material properties leads to the following notation of Hooke’s law, when the material axes longitudinal (L), radial (R) and tangential (T) are orthogonal to each other:

$$\left[ \varepsilon \right] = \left[ S \right] \cdot \left[ \sigma \right] \Leftrightarrow \left( {\begin{array}{*{20}c} {\varepsilon_{\text{L}} } \\ {\varepsilon_{\text{R}} } \\ {\varepsilon_{\text{T}} } \\ {\gamma_{\text{RT}} } \\ {\gamma_{\text{TL}} } \\ {\gamma_{\text{LR}} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{1 \mathord{\left/ {\vphantom {1 {E_{\text{L}} }}} \right. \kern-0pt} {E_{\text{L}} }}} & { - {{\nu_{\text{RL}} } \mathord{\left/ {\vphantom {{\nu_{\text{RL}} } {E_{\text{R}} }}} \right. \kern-0pt} {E_{\text{R}} }}} & { - {{\nu_{\text{TL}} } \mathord{\left/ {\vphantom {{\nu_{\text{TL}} } {E_{\text{T}} }}} \right. \kern-0pt} {E_{\text{T}} }}} & 0 & 0 & 0 \\ { - {{\nu_{\text{LR}} } \mathord{\left/ {\vphantom {{\nu_{\text{LR}} } {E_{\text{L}} }}} \right. \kern-0pt} {E_{\text{L}} }}} & {{1 \mathord{\left/ {\vphantom {1 {E_{\text{R}} }}} \right. \kern-0pt} {E_{\text{R}} }}} & { - {{\nu_{\text{TR}} } \mathord{\left/ {\vphantom {{\nu_{\text{TR}} } {E_{\text{T}} }}} \right. \kern-0pt} {E_{\text{T}} }}} & 0 & 0 & 0 \\ { - {{\nu_{\text{LT}} } \mathord{\left/ {\vphantom {{\nu_{\text{LT}} } {E_{\text{L}} }}} \right. \kern-0pt} {E_{\text{L}} }}} & { - {{\nu_{\text{RT}} } \mathord{\left/ {\vphantom {{\nu_{\text{RT}} } {E_{\text{R}} }}} \right. \kern-0pt} {E_{\text{R}} }}} & {{1 \mathord{\left/ {\vphantom {1 {E_{\text{T}} }}} \right. \kern-0pt} {E_{\text{T}} }}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {{1 \mathord{\left/ {\vphantom {1 {G_{\text{RT}} }}} \right. \kern-0pt} {G_{\text{RT}} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{1 \mathord{\left/ {\vphantom {1 {G_{\text{TL}} }}} \right. \kern-0pt} {G_{\text{TL}} }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{1 \mathord{\left/ {\vphantom {1 {G_{\text{LR}} }}} \right. \kern-0pt} {G_{\text{LR}} }}} \\ \end{array} } \right) \cdot \left( {\begin{array}{*{20}c} {\sigma_{\text{L}} } \\ {\sigma_{\text{R}} } \\ {\sigma_{\text{T}} } \\ {\tau_{\text{RT}} } \\ {\tau_{\text{TL}} } \\ {\tau_{\text{LR}} } \\ \end{array} } \right)$$
(3)

where [ε] is the strain vector described by the elongations (εL, εR, εT) and shear strains (γRT, γTL, γLR). [S] is the compliance matrix which includes 12 compliance coefficients (1/EL, − νRL/ER, 1/GRT, etc.) which are a function of the Young’s moduli (EL, ER, ET), shear moduli (GRT, GTL, GLR) and Poisson’s ratios (νLR, νLT, νRL, νRT, νTL, νTR). [σ] is the stress vector which contains the tensile stress (σL, σR, σT) and shear stress (τRT, τTL, τLR) components. In elasticity theory, the constants of the compliance matrix are shown to satisfy symmetry condition as elastic deformation shall be non-dissipative. Due to the symmetry of the compliance matrix [S], two of the Poisson’s ratios are related to each other by means of below equations:

$$- \frac{{\nu_{\text{RL}} }}{{E_{\text{R}} }} = - \frac{{\nu_{\text{LR}} }}{{E_{\text{L}} }}$$
(4)
$$- \frac{{\nu_{\text{TL}} }}{{E_{\text{T}} }} = - \frac{{\nu_{\text{LT}} }}{{E_{\text{L}} }}$$
(5)
$$- \frac{{\nu_{\text{TR}} }}{{E_{\text{T}} }} = - \frac{{\nu_{\text{RT}} }}{{E_{\text{R}} }}.$$
(6)

It means that the linear elastic mechanical behavior can be described by three moduli of elasticity, three shear moduli and three Poisson’s ratios, whereas only three of the six main orthogonal Poisson’s ratios are independent material constants. Other bounds on the moduli of orthotropic materials are caused by the requirement that the compliance matrix must be positive definite. More fundamentals about the calculation and estimation of the Poisson’s ratio of wood and wood-based materials can be found in Kollmann and Côté (1968), Bodig and Jayne (1982) and Niemz and Sonderegger (2017).

Experiments

The material characterizations with ESPI, laser extensometry, video extensometry and mechanical extensometer were performed by applying uniaxial tensile tests. All experiments were performed on Zwick/Roell universal testing machines (Ulm, Germany), equipped with the control software Zwick/Roell testXpert 2 V3.5 (Ulm, Germany). The characteristics of the gauging techniques ESPI, laser extensometry and video extensometry are summarized in Table 1.

Table 1 Summarized comparison of the characteristics of the gauging techniques ESPI, laser extensometry and video extensometry with respect to the conducted measurements
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ESPI measurement

In the first step, uniaxial tests were performed and elongation and contraction of the specimens were measured by means of the ESPI technique. For this, two ESPI Q300 Dantec-Ettemeyer (Ulm, Germany) devices, with a maximal measurement resolution of 0.03 µm (Dantec Dynamics A/S 2017), were mounted on the machine to allow measuring deformation on the surface of the specimens from both sides at the same time (Fig. 2a). Müller et al. (2015) showed that in shear experiments on spruce wood, deformation on the front and back side of a specimen can vary significantly. ESPI is highly sensitive, which can be affected by different factors, such as vibrations. Comparing test results from the front and the back side indicated immediately biased data. Additionally, higher precision of the ESPI results could be achieved by using the mean value of the deformation measured on the front and back of the specimens.

Fig. 2
figure 2

Scheme of the uniaxial tensile experiments; a setup with two ESPI devices; b setup with the laser extensometry and video extensometry system

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The ESPI Q300 Dantec-Ettemeyer devices were mounted on the testing machine in such a way that the optical axis of both devices coincided and the specimen was clamped exactly in the center point in between both devices (Fig. 2a). Any vibrations of the devices were minimized by additional supporting frames. A free clamping length of 350 mm (for the dog-bone-shaped specimen Fig. 1a) and 70 mm (for the strip-shaped specimen Fig. 1b, c) was chosen. In both cases, a field of view (FoV) of 26 × 13 mm2 was used, to observe deformation on the specimen. To prevent biasing of the data due to vibration artifacts, a pre-force was applied prior to starting the test procedure to stabilize specimens. For longitudinal dog-bone-shaped samples (Fig. 1a), a pre-force of 100 N was applied, and for strip-shaped samples, a pre-force of 20 N (Fig. 1b) and 10 N (Fig. 1c) was applied. The total deformation had to be established by accumulating the deformation determined in several load steps, because of the high sensitivity of the ESPI technique. It was assumed that two to three fringes in the y-axis picture would give reliable results per load step. The load step had to be adjusted to the stiffness of the material. Stiff material would lead to large load steps, which could be selected. A load step of 100 N and 10 N was found to be appropriate for testing wood in longitudinal (Fig. 1a) and transverse (Fig. 1b, c) direction, respectively. At every load step, the crosshead of the testing machine was stopped for 5 s to capture the generated speckle images in x- and y-direction. For all samples, total deformation was divided into six load steps, which resulted in a maximum load of 700 N tested in longitudinal, 80 N (Fig. 1b) and 70 N (Fig. 1c) in transverse direction. Therefore, samples were stressed to σ = 5.83 MPa, σ = 0.67 MPa and σ = 0.58 MPa in longitudinal, radial and tangential direction, which corresponded to less than 20% of the breaking strength in the different directions. All ESPI images were recorded and analyzed with the post-processing software ISTRA 2001 Dantec-Ettemeyer (Ulm, Germany) to determine the Poisson’s ratio, afterward. For this, the mean values of axial and transverse strains were measured within the FoV and Poisson’s ratio was calculated corresponding to Eq. (1).

Laser and video extensometry measurement

The laser and video extensometry measurements were carried out on a universal testing machine Zwick/Roell Z020 (Ulm, Germany), equipped with an optical extensometer system including both gauging techniques. Therefore, the extensometer system contains a gauging sensor, a digital camera and a laser light source. An absolute measurement accuracy of 0.11 µm of the laser extensometry system, laserXtens (Zwick/Roell, Ulm, Germany), is specified by the manufacturer. With the laser extensometry system used, axial and transverse strain measurements can be performed simultaneously (Zwick/Roell 2017a). The video extensometry device, videoXtens (Zwick/Roell, Ulm, Germany), is a camera which is enclosed in a metal housing, with a measurement accuracy dependent on the field of view (FoV) (Zwick/Roell 2017b), for FoVs smaller than 200 mm with a measurement accuracy meeting the requirements specified in DIN EN ISO 9513:2013-05 (2013). In this study, the FoV was selected such as to ensure an accuracy of 0.2 µm. As illustrated in Fig. 2b, the video extensometry device was positioned in the center, which was flanked by two laser sources of the laser extensometry device. The measuring points of the laser extensometry device were positioned on the upper and lower side of the specimen. For the dog-bone shaped samples, the distance of the measuring points was 70 mm, whereas for the strip-shaped samples a distance of 40 mm was chosen. Contraction of the samples was measured in the center of the specimens. For this, the video extensometry device used the contrast of the edges of the specimen. Increased contrast of the edges was achieved by illuminating the specimen from the back. This setup was chosen because it is not possible to measure the axial strain (εl) and the transverse strain (εq) with videoXtens simultaneously.

First, laser extensometry was applied to measure strain in axial and transverse direction. In transverse direction, results showed a high variability. Hence, a hybrid approach was selected, using laser extensometry for measuring axial extension (εl) and video extensometry for transverse contraction (εq). Thereafter, the Poisson’s ratio was calculated by means of the software testXpert 2 V3.5 (Ulm, Germany) automatically. The clamping and measurement length for the dog-bone shaped samples (LR and LT) amounted to 350 mm and 80 mm, respectively. For the strip-shaped samples (RL, RT, TL and TR), the distances amounted to 70 mm and 40 mm, respectively. The specimens LR and LT (Fig. 1a) were stressed at maximum with a force of 4000 N and a pre-force of 100 N, and the specimens RL, RT, TL and TR (Fig. 1b, c) with a pre-force of 10 N and a maximum load of 200 N. Accordingly, the maximum tensile stress ε amounted to 34.17 MPa (specimens LR and LT) and 1.75 MPa (specimens RL, RT, TL and TR), which is far lower than the yield stress of spruce wood.

Mechanical extensometer measurement

For evaluating optical gauging techniques, additional measurements were performed by using the same set of strip-shaped specimens with a mechanical extensometer, makroXtens (Zwick/Roell, Ulm, Germany). The axial elongation (εl) of the specimens for a load step of ∆F = 200 N was calculated corresponding to Eq. (7) (Bodig and Jayne 1982):

$$\varepsilon_{\text{l}} = \frac{\Delta F}{{A*{\text{MOE}}}},$$
(7)

where A is the cross section of the specimens and MOE represents the modulus of elasticity (i.e., slope of the stress–strain curve).

MakroXtens is a conventional clip-on mechanical extensometer with a measurement resolution of 0.5 µm, which meets the requirements specified in DIN EN ISO 9513:2012 class 0.5 (Zwick/Roell 2017c). More comprehensive information about mechanical extensometers can be found elsewhere in the literature (Figliola and Beasley 2001; Zwick/Roell 2001, 2017c; Davis 2004; Pan and Wang 2016).

Statistical evaluation

All statistical tests were carried out using the software package IBM SPSS Statistics 21. Initially, the Shapiro–Wilk test was applied to verify whether the measured data follow a normal distribution. Because the null hypothesis was rejected, which meant that the data did not follow a normal distribution, the Wilcoxon-matched pair test was used to determine the statistical equivalence of two data sets. To perform a statistical comparison with more than two data sets, the Friedman’s test was employed.

Results and discussion

Table 2 gives an overview of the sample’s moisture content (ω), density (ρ) in dependence of the orthogonal directions (ρL, ρR and ρT) and also the summarized test results of the non-contact optical gauging techniques in comparison with the literature references: Hörig (1935), Wommelsdorff (1966), Neuhaus (1981), Niemz and Caduff (2008) and Keunecke et al. (2008). The optical gauging techniques, ESPI and a combination of laser extensometry (for εl) and video extensometry (for εq) returned consistent results in terms of the Poisson’s ratio: νLR, νLT, νRL, νRT, νTL and νTR, and in terms of the modulus of elasticity (MOE) for all three orthogonal directions: EL (mean of ELR and ELT), ER (mean of ERL and ERT) and ET (mean of ETR and ETL). The results are presented in the form of the mean values (e.g., x̅[νLR]) and the coefficient of variation (e.g., CoV[ER]) of the Poisson’s ratio and the moduli of elasticity. The quantified moduli of elasticity and densities are in the expected magnitude range of spruce wood. The region from which the trees were removed, but also the number of trees from which the sample material has been obtained, must be taken into account since they could have an impact on the material properties. The shape and size of the specimens could also have an impact on the material properties. When using small-sized samples, the annual ring width may bias the measurement results (Niemz and Caduff 2008), because it correlates directly with the material density and stiffness. However, these effects cannot explain the significant differences between the presented Poisson’s ratios.

Table 2 Experimentally determined material data of spruce wood in comparison with the literature references: moisture content (ω), mean values (x̅) and coefficient of variation (COV) of the density: ρL, ρR, ρT, Poisson’s ratio: νLR, νLT, νRL, νRT, νTL and νTR, and the modulus of elasticity: EL, ER and ET
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Almost all Poisson’s ratios established in this study violate the symmetry condition of the linear elastic and orthotropic compliance matrix [S]. This phenomenon has also been mentioned in previous studies (Neuhaus 1981; Bodig and Jayne 1982; Garab et al. 2010; Hering et al. 2012; Bachtiar et al. 2017). The stiffness or compliance matrix is symmetric due to Betti’s reciprocity theorem [the deflection d (in direction A) due to a unit force p (in direction B) is equal to the deflection d (in direction B) due to a unit force p (in direction A)]. It becomes asymmetric as soon as this theorem is violated, i.e., when deformation is dissipative (friction, damage) or the deformation is non-local. In order to still obtain a symmetric compliance matrix [S], as required for time-efficient FEM (and for parameterizing orthotropic material models), the calculation of the average value from each corresponding off-diagonal term, i.e., Eqs. (4–6), followed by a backward calculation to re-obtain the elastic material parameters was pursued as proposed by Bachtiar et al. (2017).

Lekhnitskii et al. (1964) showed that all bodies can be divided into homogenous (physical properties remain invariant in all directions and all points) and non-homogenous bodies, as well as in isotropic and anisotropic. Perkins (1967) noted that wood is inhomogeneous at macro- and microscopic scale. Qing and Mishnaevsky (2010) investigated the effect of annual ring structure, microfibril angle and cell shape angle on the elastic constants in a numerical study employing a 3D micromechanical computational model of softwood, considering the wood’s structure at four scales from microfibrils to annual rings. They showed that vLR increases with increasing microfibril angle and decreases with wood density. The vLT increases with the increasing microfibril angle and cell shape angle. Hearmon (1948) showed that Poisson’s ratios can even gain negative values at certain microfibril angles. Consequently, it means that in future experimental testing, more parameters must be recorded (annual ring radius, content of early and latewood, microfibril angle, etc.).

In general, the Poisson’s ratios determined in this study have a higher mean value compared to the literature references. Particularly, the mean value x̅[νRL] is about 6.7 times higher than the lowest value found in the literature (Table 2). Even the values found in the literature are not consistent with each other and show wide dispersions. The differences to the literature references are open to speculation, because on the one hand, the scattering of the literature data could suggest similar median values compared to the own measurements. On the other hand, diverse gauging techniques and apparatuses with different measurement resolutions were used by each researcher. Nevertheless, in this study different gauging techniques were directly compared to each other with the same sample set.

To ensure the statistical reliability of the own measurements, a validation of the used gauging techniques was pursued. Related to the described approach, the summarized results of the statistical evaluation are presented in Tables 3 and 4. The test on normality distribution (Table 3) does not show a normal distribution for all investigated data (for example p = .013 for νRL), which is why the Wilcoxon test was applied (Table 4). This test confirmed the statistical equality between the measurements gained due to ESPI compared to laser extensometry (for εl) in combination with video extensometry (for εq). For example, the highest differences of means could be found for the MOE of ER, which shows a statistical value of Z (N = 22) = − 1.607, p = .108. Compared to that, the Poisson’s ratio of νRL illustrates a good accordance of the means with values of Z (N = 11) = − .533 and p = .594.

Table 3 Summarized results of the test of normality distribution (Shapiro–Wilk test) applied to the Poisson’s ratio: νLR, νLT, νRL, νRT, νTL and νTR, and to the modulus of elasticity: EL, ER and ET
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Table 4 Summarized results of the test on statistical equivalence (Wilcoxon test) of the data gained by means of ESPI compared to laser extensometry (for εl) in combination with video extensometry (for εq)
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In order to guarantee the accuracy of the own measurements by means of the chosen non-contact optical gauging techniques, a statistical validation with a mechanical extensometer was done additionally. For this validation, the axial extension (εl) of the specimens RL and RT has been investigated with the same sample set, because the means of the Poisson’s ratio x̅[νRL] had the biggest discrepancies to the lowest literature reference (Table 2). The axial extension (εl) of the tested specimens at a load level of ∆F = 200 N is illustrated as box plots in Fig. 3. Even the qualitative comparison of the gauging techniques does not show any discrepancies at all (e.g., x̅[ε ESPIl ] = 0.153%, x̅[ε laser extensometryl ] = 0.151%, x̅[ε video extensometryl ] = 0.147% and x̅[ε mechanical extensometerl ] = 0.155%). To confirm that there are no significant differences, the results of Friedman’s test and Wilcoxon test are presented in Table 5. The nonparametric Friedman test for repeated measurements shows a Chi-square of χ2 = 1.421 which was not significant (p = .701). As expected, the Wilcoxon test also indicates for all comparisons no statistically significant differences (all p ≥ .05). Based on this investigation, it is certain that the transverse strain (εq) would show a similar accuracy among the gauging techniques, because the physical principle for the measurements is exactly the same. Unfortunately, the mechanical extensometer does not allow measurements normal to the direction of the applied force, which would be needed for this kind of validation.

Fig. 3
figure 3

Axial strain (εl) in dependence of the gauging techniques ESPI, laser extensometry, video extensometry and mechanical extensometer for the specimens RL and RT

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Table 5 Summarized results of the test on statistical equivalence (Friedman test and Wilcoxon test) for the axial strain (εL) measurement by means of ESPI versus laser extensometry versus video extensometry versus mechanical extensometer
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However, as the specimens and the testing conditions were identical for the own measurements, it is possible to say that there are no statistical differences between the measurement techniques ESPI, laser extensometry and video extensometry. Moreover, the results obtained confirm the first hypothesis of this study, i.e., the non-contact optical gauging techniques ESPI, laser extensometry and video extensometry are suitable for the detection of the Poisson’s ratio of wood.

Nonetheless, in dependence of the chosen measurement setup, it was not possible to gain reproducible Poisson’s ratio for all specimens. In Fig. 4, the failed measurements in dependence of the gauging technique and specimen type are illustrated. The ESPI measurements with the specimens LR and LT failed, because at a certain load level the specimens started to creep, while capturing the image. This instability led to noises, which caused the post-processing process not to be executable. Moreover, it was impossible to measure the transverse strain (εq) of any specimens via laser extensometry. This could be explained by the very small transverse contractions of the specimens that led to very small displacements of the speckle zones, which were not exceeding the measurement resolution of 0.11 µm (measurement resolution of the laser extensometry device). To confirm this hypothesis, further studies are needed to be carried out.

Fig. 4
figure 4

Illustration of the failed measurements in dependence of the chosen gauging technique and the specimen type (a LR and LT; b RL, RT, TL and TR)

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Conclusion

In this study, three optical gauging techniques (electronic speckle pattern interferometry (ESPI), laser and video extensometry) and one mechanical gauging technique were used to establish the six Poisson’s ratio of spruce wood (Picea abies (L.) Karst.) in uniaxial tensile experiments.

All techniques were found to be suitable for establishing the Poisson’s ratios and returned statistically equivalent results. However, there are limitations in terms of the setup and specimen type. For example, with the “dog-bone-shaped specimen” it was not possible to establish ESPI measurements, because at a certain load level the specimens started to creep while capturing the image. Furthermore, the measurement of the transverse strain of any specimens via laser extensometry was not possible to establish due to the very small transverse contractions of the specimens that led to very small displacements of the measurement zones, which were not exceeding the measurement resolution of the device.

In engineering, wood is usually assumed to behave orthotropic. This model implies that the material behaves load-symmetric elastic (tension/compression), is homogenous and features three mutually orthogonal elastic symmetry planes. Due to Betti’s theorem, the compliance tensor is generally assumed symmetric (about its diagonal), which is also advantageous for the time- and resource (memory)-efficient FEM calculations. The Poisson’s ratios established in this study, though, are violating the symmetry conditions of elastic orthotropic materials, which might be caused by, for example, non-local deformations. The authors recommend to follow the procedure outlined by Bachtiar et al. (2017) (calculating the average value from each corresponding off-diagonal term, followed by a backward calculation to re-obtain the elastic material parameters), to warrant efficient FEM calculation and the use of pre-implemented material models.

While the Poisson’s ratios established are consistent within the study, they were found considerably different to some of the values found in the literature. Various researches have shown that the elastic constants, including the Poisson’s ratios, are sensitive to the annual ring structure, microfibril angle and cell shape angle as well as density. The wide spread of values published in the literature clearly shows that in experimental testing of wood specimens more parameters (other than the density and moisture content) must be recorded: annual ring radius, content of early and latewood, and microfibril angle. Future testing may also distinguish between Poisson’s ratios in each lamina (early/latewood) of the wood specimen—something which might only be possible with full-field optical gauging techniques.

In summary, the study shows that optical gauging techniques are suitable for determining the Poisson’s ratios of (spruce) wood. The discrepancies with the values found in other studies, though, clearly show the need to characterize and record the morphology of each specimen. Optical gauging techniques may further offer the possibility to establish the Poisson’s ratio lamina-wise.