**"Plasticity analysis of the strain in the tangential direction of solid wood subjected to compression load in the longitudinal direction,"**

*BioRes*. 9(1), 1097-1110.

#### Abstract

Uniaxial compression tests in the grain (longitudinal) direction of solid wood were conducted using specimens of Sitka spruce and Japanese birch. The nonlinear stress-strain behavior was analyzed using plasticity theory, which is typically applied to ductile materials such as metals. The relationship between the longitudinal and tangential directions obtained from the experimental results showed nonlinearity, as predicted based on plasticity theory. Nevertheless, it was more pronounced in the experimental results than in the plasticity analysis.

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**Plasticity Analysis of the Strain in the Tangential Direction of Solid Wood Subjected to Compression Load in the Longitudinal Direction**

Hiroshi Yoshihara

Uniaxial compression tests in the grain (longitudinal) direction of solid wood were conducted using specimens of Sitka spruce and Japanese birch. The nonlinear stress-strain behavior was analyzed using plasticity theory, which is typically applied to ductile materials such as metals. The relationship between the longitudinal and tangential directions obtained from the experimental results showed nonlinearity, as predicted based on plasticity theory. Nevertheless, it was more pronounced in the experimental results than in the plasticity analysis.

*Keywords: Compression test; Longitudinal strain; Material nonlinearity; Plasticity theory; Solid wood; Tangential strain*

*Contact information: Faculty of Science and Engineering, Shimane University, Nishikawazu-cho 1060, Matsue, Shimane 690-8504, Japan; *Corresponding author: yosihara@riko.shimane-u.ac.jp*

**INTRODUCTION**

When solid wood is subjected to compression load, its stress-strain diagram is initially linear, and it successively demonstrates nonlinearity similar to that of ductile materials such as metals. After the nonlinearity is induced, the stress-strain relationship of solid wood should be described using a theory other than anisotropic elasticity, such as plasticity. Plasticity theory, the details of which are described below, was originally proposed to describe the deformation properties of metals (Hill 1950; Yamada 1980) with consideration of the microscopic crystalline structure. Additionally, it has been developed to describe the nonlinear stress-strain behaviors of non-metallic materials, such as concrete (Chen and Chang 1978; Cheng and Suzuki 1980; Chen *et al.* 1980). There are several examples of the application of plasticity theory to describe the macroscopically nonlinear stress-strain behaviors of solid wood, even though its microstructure is different from that of metals (Norris 1962; Zakic 1975; Tujino 1975, 1976; Okusa 1977, 1978; Yoshihara and Ohta 1992, 1994, 1996; Moses and Prion 2002, 2004; Mackenzie-Helnwein *et al.* 2003, 2005; Hong *et al.*2011; Hering *et al.* 2012). Nevertheless, there have been few reports examining the validity of plasticity theory itself (Yoshihara and Ohta 1992, 1994; Mackenzie-Helnwein *et al.*2003, 2005; Hering *et al.* 2012). In particular, it is difficult to find any examples discussing the relationship between the longitudinal and transverse strains in solid wood under uniaxial compression loading condition after the nonlinearity is induced. Examples examining the relationship between the longitudinal and transverse strains are often restricted to the discussion of the elastic condition related to Poisson’s ratio (Taniguchi and Ando 2010a,b; Ando *et al.* 2013; Mascia and Vanalli 2012; Mascia and Nicolas 2013).

In this study, the longitudinal and tangential strains of Sitka spruce (*Picea* *sitchensis* Carr.) and Japanese birch (*Fagus**crenata* Endl.) were measured during uniaxial compression loading in the longitudinal direction until the compressive stress reached its maximum. The stress-strain relationships in the post-elastic region were analyzed based on plasticity theory. By comparing the stress-strain relationships obtained from the compression test and plasticity analysis, it was determined whether the plasticity theory (mathematical theory of plasticity in this study) is applicable for describing the nonlinear stress-strain behavior about the longitudinal-tangential plane of Sitka spruce and Japanese birch.

**STRESS-STRAIN RELATIONSHIP DERIVED FROM PLASTICITY THEORY**

The stress-strain behavior of ductile material is often analyzed based on the mathematical theory of plasticity (Hill 1950). The plasticity theory adopted in this study is based on the small deformation of material. In the post-peak stress region, large deformation can involve the buckling and cracking of cell walls, and such phenomena cannot be described by the plasticity theory described below (Easterling *et al.* 1982; Ashby *et al.*1985). From this point of view, the stress-strain behaviors in the post-peak stress region are not considered in this study.

In the plane stress condition, the relationship between the stress increment and the strain increment in the post-elastic region is represented as follows (Hill 1950; Yamada 1980),

(1)

where [*D*^{e}] is the stiffness matrix in the elastic region and is the plastic strain increment. When Young’s moduli in the *x* and *y* directions are defined as *E _{x}* and

*E*, respectively, and Poisson’s ratio and the shear modulus in the

_{y}*xy*plane are defined as

*v*and

_{xy}*G*, respectively, [

_{xy}*D*

^{e}] is obtained as follows:

(2)

In this study, the *x* and *y* directions coincided with the longitudinal (grain) direction and tangential direction (direction parallel to the ring direction) of solid wood, respectively, and the compression load was applied in the *x* direction. According to the Prandtl-Reuss theory (Prandtl 1924; Reuss 1930), is derived as follows,

(3)

where *f* is the plastic potential, is the equivalent stress, is the equivalent plastic strain increment, and *g* is the inverse of the gradient of the relationship. By substituting Eq. (3) into Eq. (1), the following results:

(4)

According to plasticity theory, *f* and are similar to a yield criterion. In this study, the yield criterion proposed by Hill (1950), the applicability of which has been verified in several previous studies (Yoshihara and Ohta 1992, 1994, 1996), was used, and it is derived as follows:

(5)

where *X* and *Y* are the yield stresses in the *x* and *y* directions, respectively, *S* is the yield shear stress in the *xy* plane, and *Q* is a constant with the dimension of (stress)^{2}, such that the quantities *f* and also have the units of stress. In this study, *Q* is assumed to be equal to *X*^{2}, so *df* is obtained from Eqs. (1) and (4) as:

(6)

Therefore,

(7)

By substituting Eq. (7) into Eq. (4), the stress-strain relationship in the plastic region is derived as follows,

(8)

where [*D*^{p}] is the stiffness matrix in the plastic region. From Eq. (5), is derived as follows:

(9)

In this study, a uniaxial compression load was applied along the *x* direction, so

. Therefore,

(10)

The quantity [*D*^{p}] then is derived from Eqs. (8) through (10) as,

(11)

where

(12)

and

(13)

To determine the stiffness matrix [*D*^{p}], it is necessary to derive the value of *g*. The *g* value can be obtained from the stress-plastic strain relationship in the loading direction, The plastic strain is obtained by subtracting the elastic strain component, which can be derived as , from the total strain as follows:

(14)

From Eqs. (4) through (6), = and in the uniaxial condition ( ≠0, ). The relationship is derived by the following equation, which was proposed by Ludwik (Hill 1950):

(15)

where *n* and *K* are the material’s parameters obtained by the regression of relationship. The relationship is derived as , similarly to Eq. (15). Therefore, 1/*g* is obtained from Eqs. (5) and (19) as follows:

(16)

**EXPERIMENTAL**

**Specimens**

Sitka spruce (*Picea* *sitchensis* Carr.) and Japanese birch (*Fagus**crenata* Endl.) were investigated. The densities of these lumbers at 12% moisture content (MC) were 373 ± 2 and 587 ± 3 kg/m^{3}, respectively. These lumber samples were free of defects, such as knots or grain distortions, so specimens cut from them could be regarded as “small and clear”.

The samples were stored at a constant 20 ºC and 65% relative humidity (RH) before and during the test, so the specimens were confirmed to be in an air-dried condition. The moisture content of the specimens was 11.7± 0.2%. The room temperature and RH were maintained throughout the tests.

The lumber was sawn into several boards with thicknesses of 45 mm, and the specimens were then cut from these plates. The specimens were long-matched and they had dimensions of 60 (L) 30 (R) 30 (T) mm^{3}. The dimensions of the specimens were determined according to JIS Z2101-2009. Figure 1 shows the photograph of the cross-section of the specimens. As shown in the photograph, the annual ring orientation could be ignored and its orthotropic symmetry was confirmed. Ten specimens were used for each species.

**Fig. 1.** Photograph of the cross-section of the specimen used in the experiment. Left: Sitka spruce, Right: Japanese birch

**Compression Tests**

To measure the normal strain in the longitudinal and tangential directions of each specimen, and , a biaxial strain gage (Tokyo Sokki FCA-2-11, gage length = 2 mm) was bonded at both centers of the longitudinal-tangential plane. The and values were obtained by averaging the strain gage outputs detected from both planes. The specimen was set on a steel plate, and a compression load, *P*, was applied to the specimen with a cross-head speed of 0.5 mm/min until the load markedly decreased. The cross-head speed was determined such that the strain rate effect was not significant (Okuyama and Asano 1970).

The data of the load and strains were simultaneously recorded using a data logger (Tokyo Sokki TDS-303) at an interval of 2 sec. The total testing time was approximately 5 min. To prevent the bending moment induced at the end of the specimen, the compression plate was equipped with a spherical coupling, which allows the free rotation of the plate (Yoshihara and Yamamoto 2001) as shown in Fig. 2. Solid wood cannot be a perfectly homogeneous material, so inhomogeneous loading is easily induced based on various factors such as the density profile and distorted cutting in the specimen.

**Fig. 2.** Setup of the uniaxial-compression test

When considering these phenomena, it is difficult to compute the compressive stress while considering the inhomogeneity in the specimen. In this study, the load was assumed to be applied homogeneously to the specimen, and the compressive stress was obtained by dividing *P* by the cross sectional area *A*, whereas the longitudinal and transverse strains, and , respectively, were obtained by averaging the strain data obtained from both planes.

Figure 3(a) shows the definitions of Young’s modulus, Poisson’s ratio, and compressive strength,* E*_{L}, *v*_{LT}, and , respectively, whereas Fig. 3(b) shows the definitions of secant modulus and secant Poisson’s ratio, *E*_{sec} and *v*_{sec}, respectively, the details of which are described below. The Young’s modulus *E*_{L} and Poisson’s ratio *v*_{LT} were obtained from the initial slope of the and relationships, respectively. The compressive strength was determined from the maximum stress. The proportional limit stress was obtained from the stress at the onset of nonlinearity in the relationship. There are several methods to determine the proportional limit stress (Davies *et al.*2001). In this study, it was determined from the stress where the half-thickness of the plotter trace deviated from the straight line drawn in the elastic region of the relationship, as shown in Fig. 4 (Davies *et al.* 2001).

**Fig. 3.** (a) The definitions of Young’s modulus, Poisson’s ratio, and compressive strength, *E*_{L}, *v*_{LT}, and , respectively, and (b) the definitions of secant modulus and secant Poisson’s ratio in the nonlinear region, *E*_{sec} and *v*_{sec}, respectively

**Fig. 4. **Method of** **determination of the proportional limit stress_{pl}

Using Eq. (14), the plastic strain _{L}^{p} was obtained, and the parameters *n* and *K* were calculated by regressing the _{L}– _{L}^{p} relationship into Eq. (15). Note that _{L}, _{L}, *E*_{L}, _{LT}, and_{pl} obtained from the actual compression test correspond to* _{x}*,

*,*

_{x}*E*,

_{x}*, and*

_{xy}*X*in the plasticity analysis detailed in the previous section.

**Plasticity Analysis**

The stress-strain relationships in the nonlinear region were analyzed based on the plasticity theory by applying the following procedure:

(i) The stress and strain components and stiffness matrix at the stage *k*, { * _{ij}*}

*, {*

_{k}*}*

_{ij}*, and [*

_{k}*D*

^{p}

*], were assumed to be known.*

_{k}(ii) In this study, a compression load was applied in the longitudinal direction alone. Therefore, Eq. (1) was simplified to

(17)

In this study, the value of *d* * _{x}* was determined to be a constant value,

*i.e*., 0.1 MPa. From Eq. (8), the component of the strain increment at the stage (

*k*+ 1), {

*d*

*}*

_{ij}

_{k }_{+ 1}, was obtained as follows:

(18)

(iii) The stress and strain components at the stage (*k* + 1) were obtained as follows:

(19)

(iv) The process from (i) to (iii) was repeated until the * _{x}*value reached

_{max}. The

*–*

_{x}*and*

_{x}*–*

_{y}*relationships obtained from this procedure were compared with the*

_{x}_{L}–

_{L}and

_{T}–

_{L}relationships obtained from the actual compression test.

**RESULTS AND DISCUSSION**

Figure 5 shows the compressive stress-longitudinal strain and transverse strain-longitudinal strain relationships, _{L}– _{L} and_{T}– _{L}, respectively. Figure 6 shows the photograph of the specimens obtained after the compression loading. As shown in Fig. 6, macroscopic fractures could not be found in the specimen until the compressive stress reached its maximum.

Table 1 lists the mechanical properties of Sitka spruce and Japanese birch obtained from the compression tests. In the plasticity analysis, however, the test data listed in Table 2, which were obtained from a single specimen, were used instead of the properties shown in Table 1. The * _{x}*–

*and*

_{x}*–*

_{y}*relationships obtained from the plasticity analysis were compared with the actual*

_{x}_{L}–

_{L}and

_{T}–

_{L}relationships. The

*E*,

_{x}*,*

_{xy}*X*,

_{max},

_{max},

*n*, and

*K*values were obtained from the actual compression tests conducted in this experiment, whereas the

*E*values of Sitka spruce and Japanese birch were obtained from Hearmon (1948) and Forestry and Forest Products Research Institute, Japan (2004), respectively. Although the

_{y}*E*value was required for the analysis, it did not influence the

_{y}*–*

_{x}*and*

_{x}*–*

_{y}*relationships in this loading condition.*

_{x}**Fig. 5.** Compressive stress-longitudinal strain and transverse strain-longitudinal strain relationships, _{L}– _{L} and _{T}– _{L}, respectively

**Fig. 6. **Specimens obtained after compression loading. Left: Sitka spruce. Right: Japanese birch

**Table 1. **Mechanical Properties Obtained from the Compression Tests

Results are averages (SD). Parameters* n* and *K* are obtained by regressing the _{L}*–* _{L}^{p} relationship into Eq. (15). Ten specimens were used for each species.

**Table 2. **Mechanical Properties Used for the Plasticity Analysis Shown in Fig. 7

The secant modulus *E*_{sec} and secant Poisson’s ratio _{sec} in the nonlinear region are defined using the temporary stress and strains as _{L}/ _{L} and – _{T}/ _{L}, respectively, as shown in Fig. 3(b). Figure 7 shows typical examples of the *E*_{sec}– _{L} and_{sec}– _{L} relationships obtained from the actual compression test and plasticity analysis conducted using the data listed in Table 2. As shown in Fig. 5, the nonlinear region in the _{L}– _{L} relationship for Japanese birch is significantly larger than that of Sitka spruce. Nonlinearity clearly exists in the _{T}– _{L}relationship, although it is less significant than that in the _{L}–_{L} relationship. When the _{L}– _{L} and _{T}– _{L}relationships are absolutely linear, the *E*_{sec} and _{sec} values coincide with the *E*_{L} and _{LT} values, respectively, throughout the _{L} range. In Fig. 7, however, it is clear that nonlinearity is significant in the _{L}– _{L} and _{T}– _{L} relationships obtained from the experimental results and the plasticity analysis for both species. Therefore, plasticity theory may be effective for predicting the tangential strain in the nonlinear region in this loading condition, even if the deformation mechanism is essentially different from that of metals. For both species, however, the nonlinearity in the _{T}– _{L} relationship obtained from the experimental result is more pronounced than that predicted from the plasticity analysis. Therefore, the _{sec} value at the maximum compressive stress _{max} experimentally obtained is smaller than that obtained from the plasticity analysis, as shown in Table 3. According to several previous studies, transverse cracking and splitting reduces the increasing rate of transverse strain (Surgeon *et al.* 1999; Kashtalyan and Soutis 2000; Pidaparti and Vogt 2002; Amara *et al.* 2005; Yoshihara and Tsunematsu 2007a, b). In addition to the reduction of transverse strain predicted from plasticity theory, the damage to the cell wall may enhance the reduction of the transverse strain. Microscopic observations during the compression loading may effectively reveal the influence of the plastic deformation and damage propagation. Although microscopic observations have been conducted in several previous studies (Easterling *et al.* 1982; Ashby *et al.* 1985), it should be conducted more carefully, particularly paying attention to the deformation in the transverse direction. Additionally, the compression plate was equipped with a hinge to prevent the bending moment induced at the end of the specimen. Nevertheless, it was difficult to reduce the distortion and frictional force at the end of the specimen entirely, so the load may not have been applied perfectly along the loading axis. This issue might enhance the nonlinearity in the transverse strain-longitudinal strain relationship obtained from the experimental result. Further research should also be conducted on the influence of the loading eccentricity on the transverse strain.

**Fig. 7.** The *E*_{sec}– _{L} and _{sec}– _{L} relationships obtained from the compression test and plasticity analysis

**Table 3. ***E*_{sec }and** **_{sec} Values of the Maximum Compressive Stress _{max} Obtained from the Compression Test and Plasticity Analysis

Results are averages SD.

**CONCLUSIONS**

- Similar to the compressive stress-longitudinal strain relationship, the tangential strain-longitudinal strain relationship showed nonlinearity, which was more pronounced in the experimental result than in the plasticity analysis.
- Implicit damage might enhance the nonlinearity in the tangential strain-longitudinal strain relationship in addition to that predicted by plasticity theory.
- Microscopic observation during compression loading may effectively reveal the source of nonlinearity in more detail.

**ACKNOWLEDGMENTS**

This work was supported in part by a Grant-in-Aid for Scientific Research (C) (No. 24580246) of the Japan Society for the Promotion of Science.

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Article submitted: October 11, 2013; Peer review completed: December 14, 2013; Revised version received and accepted: December 20, 2013; Published: January 6, 2014.