Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2022.34.0105 Acta Polytechnica CTU Proceedings 34:105–115, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague FINITE ELEMENT SIMULATION OF SHEAR AND COMPACT TENSION TESTS ON TIMBER Eliška Šmídová∗, Petr Kabele Czech Technical University in Prague, Thákurova 7/2077, 166 29 Prague 6, Czech Republic ∗ corresponding author: eliska.smidova@fsv.cvut.cz Abstract. The non-linear finite element simulation of the ASTM shear and pre-stressed compact tension tests was conducted. Tested specimens were cut from the glued laminated timber made of European spruce (Picea abies) with the lamina thickness of 10.5 mm and 45 mm. The 2D homogeneous orthotropic constitutive model of the tensile-shear fracture in timber, which has been proposed by the authors, was used. The model calibration was adopted from the authors’ recent experimental study. The numerical results show that the model can adequately reproduce both the experimental response and the crack pattern, which in certain cases comprises of cracking parallel or perpendicular to fibers. Furthermore, the results confirm that the value of the parameter fASTMxy , obtained from ASTM shear test as the maximum attained force divided by the shearing area, represents the averaged stress at the failure plane, while the extreme stress experienced by the material is much higher. Keywords: ASTM shear test, pre-stress, compact tension test, glued laminated (GL) timber, tensile- shear fracture model, non-linear finite element analysis. 1. Introduction The procedure for the determination of shear strength according to the standard ASTM D143-94 [1], see Figure 2, is widely accepted and used for timber and wood products. However, as shown by numerical simulations by Sajdlová and Kabele [2], the stress distribution along the failure plane in such shear-test configuration is non-uniform with strong concentra- tions at the edges of the loading platens. As reported in [3], the compact tension (CT) test on pre-stressed specimens with reduced ligament depth was conducted with the primary aim of obtaining the fracture behavior in the crack-normal direction for the fibers’ rupture concentrated around or at the cross-section ahead of the notch. Nevertheless, such failure type occurred only in one case out of five accom- panied by fiber bundles’ disintegration propagating along the grain. In the rest of the specimens, vari- ous failure types with fibers’ rupture or disintegration or their combinations were observed. The authors recommended only qualitative interpretation of the obtained results due to low precision of prescribed strain, presence of disintegration propagating along the grain that accompanied fibers’ rupture, role of lateral compression in fibers’ rupture, or insufficient number of specimens for each failure type. The aim of this study is to conduct a numerical simulation of (i) the ASTM shear test and (ii) the pre- stressed CT test on glued laminated timber (GLT) specimens and analyse the behavior of the tensile- shear crack parallel or perpendicular to the grain. To this end, the material model of the tensile-shear frac- ture in timber implemented in finite element method is used. 2. Constitutive model We briefly review the material model of tensile-shear fracture in timber that the authors’ team has re- cently developed and implemented in a FEM code [4]. Timber is considered as a 2D homogeneous con- tinuum. The model aims at capturing fracture under tension, shear, and their combination, while taking into account the phenomena of the elastic and inelas- tic behavior in a small deformation range, material orthotropy, both in linear and non-linear range, crack- ing across and along fibers, and the behavior under unloading/reloading. Non-linear response under com- bination of compression and shear is considered as a perfectly plastic. The model for the nonlinear behavior is composed of: • (i) failure criterion defining the stress condition for crack initiation, • (ii) crack-type criterion that distinguishes whether the crack occurs across or along the fibers, • (iii) cohesive (traction-separation) law defining the response of a crack. Failure (fracture) is assumed to occur when the following condition is satisfied: F (σx, σy, τxy) = 1 (1) Here σx, σy, and τxy are the stress components with respect to the axes of orthotropy. The failure function F is defined by the Tsai-Hill formulation [5] as: F = σ2 x f2 x + σ2 y f2 y + τ2 xy f2 xy − σxσy f2 x (2) 105 https://doi.org/10.14311/APP.2022.34.0105 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Eliška Šmídová, Petr Kabele Acta Polytechnica CTU Proceedings if σx ≥ 0 then fx = ftx else fx = fcx (3) if σy ≥ 0 then fy = fty else fy = fcy (4) where fxy is the absolute value of the shear strength. Symbols fx and fy are tensile or compressive strength along and across the fibers, respectively: for the pos- itive values of σx and σy, the strengths fx and fy correspond to the tensile strength values ftx and fty, respectively, otherwise the compressive strength values fcx and fcy are adopted. The crack-type criterion is based on the consid- eration that when timber is exposed to tension in the direction parallel or nearly parallel to the grain, rupture occurs across the grain and the crack is per- pendicular to the principal stress direction. When the angle θ, describing the deviation of the principal ten- sion from the grain, is larger than a certain threshold (θc), then the crack forms along the grain regardless of the principal stress direction. To distinguish these cases, the crack-type function FCT , Eq. 5, and criteria for cracking across and along the fibers, Eq. 6 and Eq. 7, are introduced: FCT = ∣∣∣∣ 2τxy σx − σy ∣∣∣∣− tan(2θc) (5) Crack across the grain: FCT < 0 (6) Crack along the grain: FCT ≥ 0 (7) In the 2D stress space the crack-type function FCT defines a plane, which divides the failure surface into two parts. The concept of the cohesive crack model [6] is used to model the material response after failure. The bridg- ing effect of incompletely ruptured or delaminated fibers is represented by a cohesive traction acting be- tween the crack faces. The traction-separation law defines the relation between the normal and tangent tractions tn and tm and the relative normal and tan- gent displacement between the crack surfaces δn and δm. The exponential form of the traction-separation law originally proposed by Hordijk [7] was adapted in the present model as: tn(δn, δm) =  tcin ( 1− √ δ2 n δ2 n,crit + δ2 m δ2 m,crit ) ( 1 + ( c1δn δn,crit )3 ) exp ( c2δn δn,crit ) − exp (−c2) ( 1 + c3 1 ) δn δn,crit for δn ∈ [0, δn,crit) and δm ∈ [0, δm,crit) 0 for δn > δn,crit or δm > δm,crit (8) (a). (b). Figure 1. Cohesive law in (A) crack-normal direction tn(δn, δm) with fracture energy Gf as an integral of tn(δn, δm) and (B) crack-tangent direction tm(δn, δm) [4]. tm(δn, δm) =  tcim ( 1− √ δ2 n δ2 n,crit + δ2 m δ2 m,crit ) 2 π arctan  δm lch 1− ( δn δn,crit )p( δn δn,crit )p  for δn ∈ [0, δn,crit) and δm ∈ [0, δm,crit) 0 for δn > δn,crit or δm > δm,crit (9) where tcin is the crack-normal traction at crack initi- ation, tcim is the crack-tangent traction at crack initi- ation, c1 and c2 are fitting parameters defining the slope of the tn function, δn,crit and δm,crit are the val- ues of normal and shear displacement jump at which the cohesive traction diminishes to zero, lch is the characteristic length of the smeared crack model (the width of the fracture process zone), and p is a material parameter that determines the slope of the function tm. The same form of the traction-separation relation is used both for cracks along and across the grain, although the respective values of the parameters may be different. The plot of traction-separation law is shown in Figure 1. 3. Model calibration The tensile-shear fracture model has already been calibrated within authors’ recent experimental study [3]. To this end, experimental results from off-axis tensile and compressive tests and compact tension test were used. The parameters applicable to the states before failure, such as the elastic moduli (Ex, 106 vol. 34/2022 Finite Element Simulation of Shear and Pre-stressed Compact Tension Tests on Timber Parameter Unit Avg Ex [MPa] 12418.0 fx [MPa] 77.6 Ey [MPa] 371.0 fy [MPa] 3.2 νxy [-] 0.37 νyx [-] 0.01 Gxy [MPa] 310.0 Ec,x [MPa] 16142.0 fc,x [MPa] 56.3 Ec,y [MPa] 310.0 fc,y [MPa] 3.3 fASTMxy [MPa] 3.7 Table 1. Off-axis tension and compression and shear test results for 10.5 mm GLT where Avg is the mean (values adopted from [3]). Ey, Gxy) and Poisson’s ratios (νxy, νyx), are listed in Table 1. In this section, we briefly review the parameters obtained by the calibration of the failure- related model components. The failure surface is defined by five parameters: the uniaxial tensile and compressive strengths along and perpendicular to the grain (fx, fcx, fy, fcy, resp.) and the shear strength (fxy). The values of tensile and compressive strengths were directly retrieved from the tensile and compressive tests. They are listed in Table 1. The parameter fxy was obtained by the least- square fitting of the failure criterion, expressed using an uniaxial failure function defining the relation be- tween the axial stress at failure σaxial and the off-axis angle θ, to the failure data from tensile and compres- sive tests. This procedure yielded fxy = 8.5 MPa [3]. With the crack-type criterion, the parameter of the critical load-grain angle θc is introduced in the model to distinguish the cases when failure occurs across or along the grain. Based on visual inspection of tensile specimens that failed due to fibers rupture, it was found that the actual grain deviation from the load direction is in the interval of [0°, 2°]. Opposed to the recent numerical simulation analysis [8], we set θc = 1.6° to enable cracking both parallel and perpendicular to the grain. The traction-separation relationship tn vs. δn for the crack along the grain was obtained by the inverse analysis of the compact tension (CT) tests results [9]. The inverse analysis was applied on three load vs. crack mouth opening displacement curves, which corresponded to the minimum, intermediate, and max- imum measured response. Corresponding values of parameters in Table 2 are denoted accordingly. CT test Cohesive law Fit tcin δn,crit c1 c2 response [MPa] [mm] [−] [−] Minimum 3.6 0.52 8.8 21.5 Intermediate 3.8 0.18 1 3.7 Maximum 5.3 0.20 1.0 3.7 Table 2. Values of the cohesive law parameters for the crack along the grain calibrated to the results of inverse analysis for the minimum, intermediate, and maximum response of the CT test results (transcribed from [3]). 4. Shear test and pre-stressed compact tension tests 4.1. Material The specimen for the ASTM shear test [1] was a notched block saw-cut from larger pieces of glued laminated timber (GLT) made of European spruce (Picea abies) with the lamina thickness of 10.5 mm. The dimensions and configuration are depicted in Figure 2. The compact tension (CT) specimens were cut from larger pieces of European spruce GLT with the lamina thickness of 45 mm with the load-grain angle θ = 0°. The specimen size and test set-up are shown in Figure 5A. Both samples were fabricated to be representative elements of the GLT structure and free of defects such as knots or cracks. The tested specimens were condi- tioned in an indoor environment with a temperature of 21° and relative humidity of 55% for 2 months. During testing, the average moisture content of the samples was 8% and the density was 463 kg/m3. 4.2. Equipment and data acquisition The shear and compact tension tests were conducted in MTS Alliance RT/30 electromechanical testing ma- chine equipped with a 30 kN load cell. Force and crosshead displacement were recorded at 10 Hz. Im- age data were collected for a subsequent digital image correlation (DIC) analysis with snapshots taken ev- ery 5 sec using Canon EOS 70D 20 megapixel digital camera fitted with Canon EF 100 mm f/2.8 lens. The camera and data logger records were synchro- nized so as to link the photographs with the applied load. The Ncorr v. 1.2 open-source software [10] with the Ncorr_post in-house graphical interface [11] were involved to run the DIC analyses. 4.3. Shear test The shear test set-up was conducted according to the standard ASTM D143-94 [1], see Figure 2. The speci- mens were freely laid on the steel base plate without capping or gluing. No extensometer was attached and the vertical relative displacements ∆x,DIC depicted in Figure 2A were acquired from DIC measurements. 107 Eliška Šmídová, Petr Kabele Acta Polytechnica CTU Proceedings (a). (b). Figure 2. Shear-parallel-to-grain test according to the standard ASTM D143-94: (A) set-up and rela- tive vertical displacement ∆x,DIC calculated by DIC analysis, (B) a mounted specimen. [3] The load was applied with the crosshead displacement control at a rate of 0.6 mm/min. The valid specimens failed as the plane defined by the inner edge of the base plate and the inner corner of the notch sheared off in the direction parallel to the grain. The specimens with the fracture plane extended back onto the supporting surface e.g. due to material imperfections were ignored, since this failure is governed by the compressive resistance between the load and support plates, rather than by shear. We calculated the shear strength parameter fASTMxy according to the standard ASTM D143-94 [1] as: fASTMxy = Fmax A (10) where Fmax is the maximum attained force, and A is the shearing area. As seen in Figure 2 the design shearing area is 2500 mm2. The actual size of this area was calculated using the measured dimensions of each specimen. The Figure 3 shows the valid responses as load vs. relative vertical displacement. For the curve S1, the peak value can be clearly identified and it is followed by a decrease and hardening. Compared to it, the Figure 3. Results of the shear-parallel-to-grain test according to the standard ASTM D143-94 [1]: load vs. DIC relative vertical displacement ∆x,DIC . [3] curves S2-S4 exhibit plateau. Despite not being plot- ted in this figure, all the curves S1-S4 end with sudden drops, which are associated with an unstable fracture propagation. The Table 1 lists the obtained values of the shear strength fASTMxy . The specimens failed with the crack parallel to the grain more or less along the expected shearing area. The crack of the specimen S1 was slightly skewed head- ing from the inner edge of the notch towards the inner edge of the bottom support, see Figure 4A. Compared to it, the cracks of the specimens S2-S4 were almost aligned with the specimens’ longer edges spreading from the inner edge of the notch, see Figure 4B. 4.4. Pre-stressed compact tension test The outer dimensions and loading of the CT spec- imen were based on the ASTM E1820-09 standard [12]. To induce the fiber rupture propagating along the ligament and obtain the fracture behavior in the crack-normal direction, the test arrangement was mod- ified with (i) pre-stress ∆σya between 2.7 MPa and 3.3 MPa (Figure 5A) and (ii) reduction of the ligament depth to 10 mm (Figure 5B). The load was applied with the crosshead displacement control at a rate of 0.5 mm/min. All five specimens failed with various failure types. Two specimens failed with the failure type F0 - disinte- gration parallel to the grain initiated at or around the notch tip where the crack propagated perpendicular to the ligament, see Figure 6A. The failure type F1, in which fibers rupture is concentrated around or at the cross-section ahead of the notch (Figure 6B), occurred once. Two specimens exhibited the failure type F2, in which fibers’ rupture concentrated around or at the cross-section ahead of the notch was either followed by disintegration along the grain propagating from the cross-section’s middle, denoted as F2-a (Figure 6C), or accompanied by disintegration propagating from both the notch tip and the cross-section’s middle, denoted as F2-b (Figure 6D). The recorded load vs. crosshead displacement re- sponses are presented in Figure 7. Their differences 108 vol. 34/2022 Finite Element Simulation of Shear and Pre-stressed Compact Tension Tests on Timber (a). (b). Figure 4. The final fracture of the shear-parallel-to- grain test according to the standard ASTM D143-94: (A) the specimen S1 and (B) the specimen S3. (a). (b). Figure 5. Pre-stressed compact tension (CT) test with the load-grain angle θ = 0°: configuration and dimensions. (A) Front view. (B) Side view. [3] in shape reflected various failure types that occurred. All five curves were linear at the beginning. Two curves with failure type F0 proceeded with harden- ing branch and a final drop. In one curve exhibiting failure type F1, softening branch with little drops and hardening portions followed. In the curve with failure type F2-a, the softening and then hardening branches followed. After the linear branch, the curve with failure type F2-b captured a decrease in slope, softening, and hardening. (a). (b). (c). (d). Figure 6. Failure of pre-stressed CT specimens with the load-grain angle of θ = 0°: (A) type F0 - disin- tegration parallel to the grain initiated at or around the notch tip, (B) type F1 - fibers’ rupture concen- trated around or at the cross-section ahead of the notch, (C) type F2-a - fibers’ rupture concentrated around or at the cross-section ahead of the notch followed by disintegration propagating from the cross- section’s middle, and (D) type F2-b - fibers’ rupture concentrated around or at the cross-section ahead of the notch accompanied by disintegration propagating from both the notch tip and the cross-section’s middle. [3] 5. Numerical simulations of shear test 5.1. Introduction The aim of the ASTM shear test simulations was to demonstrate the capability of the constitutive model outlined in Section 2 to reproduce the behavior of timber dominated by shear crack parallel to the grain. As the critical crack-sliding displacement δm,crit for 109 Eliška Šmídová, Petr Kabele Acta Polytechnica CTU Proceedings (a). Figure 7. Load vs. crosshead displacement for pre- stressed CT specimens with the load-grain angle of θ = 0° that failed in modes F0, F1, and F2. [3] the crack parallel to the grain is not known from the model calibration reviewed in Section 3, numerical simulations were run with different values from an estimated range. The mesh used for discretization of the timber shear blocks (Figure 8) was regular consist- ing of four-node quadrilateral plane-stress elements with the size of approx. 2.5 mm by 2.5 mm. Only the elements along the shearing line had the width of 2.0 mm, which was used as the characteristic length parameter lch for the crack parallel to the grain. As the material was orthotropic, the material axes were assigned to each element so as the grain direction was parallel to the specimen longitudinal direction. The vertical displacement Uy with the unit magnitude of U0 y = 0.1 mm was prescribed in half of the nodes along the horizontal notch edge, except for the inner corner node enabling shearing of the column of ele- ments. This unit load magnitude was scaled by an appropriate incremental loading factor, see Section 5.2. The vertical load plate that prevents the specimen from rotation was modelled as prescribed supports Ux = 0 mm along the vertical notch edge. Similarly, the base plate and the crossbar were considered as the supports Uy = 0 mm and Ux = 0 mm, respec- tively. The zero vertical displacement was prescribed only in the nodes placed further from the base plate - crossbar corner to minimize the effect of friction. The material-nonlinear calculations were performed using the Newton-Raphson method. 5.2. Numerical solution aspects The element size and loading increment magnitude were determined by the numerical analyses of two mod- els with meshes with typical element size of 2.5 mm and 1.0 mm. The mesh with 2.5 mm element size was used to compare the incremental loading factor of 0.15 and 0.30, that is the actual load increment of 0.015 mm and 0.030 mm, respectively. The results in Figure 9 show that the load-relative displacement curves are similar for both models up to the relative displacement of ∆x,DIC = 0.175 mm, where the finer Figure 8. 2D FE mesh composed of approx. 2.5 mm by 2.5 mm quadrilaterals aligned with the specimen longitudinal direction and lamina. Figure 9. The effect of element size and load step size on calculated response in the shear test simulation. mesh calculation was ended due to exceeded conver- gence criteria. The obtained fracture patterns were also similar: the crack parallel to the grain spread from the notch edge downwards along the shearing line towards the compression-shear zone with non-linear deformation around the edge of the bottom support. A conclusion was drawn to be used for all the subsequent analyses, that the mesh with the 2.5 mm element size provided a converged solution. Comparing the calcula- tion results with the incremental loading factor of 0.15 and 0.30, we can see a good agreement in responses and crack patterns. 5.3. Results First, in Figure 10, we compare the experimental results with the numerical responses obtained with the intermediate parameters in Table 2 for different values of critical crack-sliding displacement δm,crit = {0.03, 0.055, 0.100, 0.600} mm. We can see, that the model reproduces well the initial slope, the post-peak branch, and the lowest, intermediate, and the highest level of the load capacity observed in tests. We can no- tice, that the results with δm,crit = {0.100, 0.600} mm yield almost the same response and the parameter δm,crit ≥ 0.100 mm does not influence the responses. Second, in Figure 11, we compare the experimental 110 vol. 34/2022 Finite Element Simulation of Shear and Pre-stressed Compact Tension Tests on Timber Figure 10. Load vs. DIC relative vertical displace- ment ∆x,DIC for the results of ASTM shear test ex- periments and simulations calculated with the inter- mediate parameters from Table 2 and different values of δm,crit. Figure 11. Load vs. DIC relative vertical displace- ment ∆x,DIC for the results of ASTM shear test exper- iments and simulations calculated with the minimum, intermediate, and maximum parameters from Table 2 and δm,crit = 0.055 mm. results with the numerical responses obtained with δm,crit = 0.055 mm and the minimum, intermediate, and maximum parameters from Table 2. The calcula- tions with the minimum and the intermediate parame- ters reproduce well the S1 experiment, as for the peak value followed by a decrease, hardening and a final drop. They differ only in the final drops that occur beyond ∆x,DIC = 0.2 mm and ∆x,DIC = 0.26 mm, respectively. Compared to it, the calculation with the maximum parameters reproduce well the S2 and S4 experiment exhibiting the plateau followed by the final drop. As expected, a similar crack pattern occurred in the numerical simulations: the crack parallel to the grain initiated under tensile-shear stress states at the upper edge of the shearing area propagating downwards. A zone, in which the non-linear deformation was devel- oping under compression and shear, spread upward from the lower support along the shear plane until it reached the downward-growing crack, see Figure 12A. The final strain states εxx and γxy are shown in Fig- ure 12B and Figure 12C, respectively. Comparing the strain states with the image of cracking at the maximum relative vertical displacement in Figure 4, we can see, that the model captures the crack pattern adequately. (a). (b). (c). (d). Figure 12. ASTM shear test simulation result at (A- C) ∆x,DIC = 0.3 mm and (D) ∆x,DIC = 0.16 mm for intermediate parameters from Table 2 and δm,crit = 0.055 mm: (A) crack parallel to the grain (red) and non-linear deformation zone (green) developed under compression and shear stress states, (B) normal strain εxx, (C) shear strain γxy, and (D) shear stress τxy. As overviewed in Section 4.3, the shear strength according to the ASTM standard fASTMxy is calculated by dividing the maximum attained force Fmax and the shearing area A. Therefore, it corresponds to the average shear stress on the area at the specimen failure. The Figure 12D shows the shear stress along the shearing area around the peak load at ∆x,DIC = 0.16 mm. 6. Numerical simulations of pre-stressed CT test 6.1. Introduction The main objectives of the pre-stressed CT test simu- lations were to demonstrate the capability of the con- stitutive model outlined in Section 2 to reproduce the behavior of timber dominated by tensile crack parallel or perpendicular to the grain or combination of both, i.e., the failure modes F0-F2. Based on the results from Section 5.3, the parameter δm,crit = 0.055 mm for the crack along the grain was used in pre-stressed CT test simulations. As the cohesive law parameters for the crack across the grain are not known from the model calibration reviewed in Section 3, numerical sim- ulations were run with assumed values of critical crack- opening and sliding displacements δacrossn,crit = 1.1 mm and δacrossm,crit = 1.1 mm, and parameters cacross1 = 3.0 and cacross2 = 6.7. The pre-stressed CT specimens were discretized with a regular mesh consisting of four-node quadri- lateral plane-stress elements with the size of approx. 111 Eliška Šmídová, Petr Kabele Acta Polytechnica CTU Proceedings Figure 13. 2D FE mesh composed of approx. 2.0 mm by 2.0 mm quadrilaterals aligned with the specimen edges. 2.0 mm by 2.0 mm (to model the specimen and con- finement plattens) and four external cable elements (to induce the confinement), seeFigure 13. The axes of the orthotropic timber material were assigned to each element so as the grain direction was perpendicular to the direction of the specimen’s notch. The loading was applied as the vertical displacement Uy with the unit magnitude of U0 y = 0.01 mm to the upper loading hub as shown in Figure 13. This unit load magnitude was scaled by an incremental loading factor specified in Section 6.2. The loading point was placed at a small vertical offset of 4 mm from the loading hub’s centroid to enhance the non-symmetric failure as observed in the experiments. The bottom hub was fixed at its center. The material-nonlinear cal- culations were performed using the Newton-Raphson method. 6.2. Numerical solution aspects Using two models with meshes with typical element size of 2.0 mm and 1.0 mm, the appropriate element size and loading increment magnitude were deter- mined. The mesh with 2.0 mm elements was used to run calculations with the incremental loading factor of 50 and 35, that is the actual load increment of 0.50 mm and 0.35 mm, respectively. As seen in Fig- ure 14, the load-crosshead displacement curves were similar for both models up to the displacement of 3.0 mm, where the calculation with the finer mesh ended due to exceeded convergence criteria. The ob- tained fracture patterns were similar, both of the type F1 - fibers’ rupture concentrated around or at the cross-section ahead of the notch. It was concluded that the mesh with 2.0 mm element size provided a converged solution and, thus, can be used for all subse- quent calculations. Comparing the calculation results with the incremental loading factor of 50 and 35, we can see a good agreement in responses in Figure 14 and the same can be said about the crack patterns. The slight difference in slope of the softening branches is attributed to the fact that the disintegration of the fibers’ bundles, that accompanied fibers’ rupture along the ligament within the failure type F1, occured at different locations and had a different extent in Figure 14. The effect of element size and load step size on calculated response in the CT test simulation with the pre-stress of 3.3 MPa, θc = 1.6°, and failure type F1. Figure 15. The effect of element size and load step size on calculated response in the CT test simulation with the pre-stress of 3.3 MPa, θc = 0.4°, and failure type F0. each calculation. To confirm that the element size of 2.0 mm and the loading increment of 0.50 mm are also suitable for other failure types, numerical analysis similar to the previous one (Figure 14) was conducted with the critical load-grain angle of θc = 0.4° which hindered propagation of the main crack across the grain in the notch direction and encouraged failure due to vertical cracking along the grain (type F0) - see also the discussion in Section 6.4. The results in Figure 15 show that both models yielded similar response up to the displacement of 2.5 mm, where the finer mesh calculation ended due to exceeded convergence criteria. The obtained fracture patterns were very similar, both categorized as failure type F0. The comparison of the simulations with the incremental loading factor of 50 and 35 showed a very good agreement in responses and crack patterns. 6.3. Results Results of the calculations with pre-stress levels of 3.3, 3.0, 2.7, and 2.1 or 1.0 MPa are summarized in Figure 16 and Figure 17. Figure 17 displays the normal strains εxx and εyy. Concentrations of these strains occur due to opening of horizontal (εyy) or vertical (εxx) cracks; therefore their contour plots indicate the crack patterns, which can be compared with those observed in experiments and shown in Figure 6. 112 vol. 34/2022 Finite Element Simulation of Shear and Pre-stressed Compact Tension Tests on Timber Figure 16. Load vs. crosshead displacement for the results of CT test experiments and simulations calcu- lated with θc = 1.6° and different pre-stress levels. In the simulation with pre-stress of 3.3 MPa we observed propagation of a major crack across fibers in the direction of the notch (Figure 17A top) with multiple short branches along the grain (Figure 17A bottom). This pattern corresponds well with the failure type F1 seen in Figure 6B. The respective load- displacement curve (Figure 16) exhibits the highest peak and post-peak softening. In the simulation with 3.0 MPa, crack across fibers propagated in the di- rection of the notch up to approx. one third of the ligament’s length (Figure 17B bottom) from where crack along the grain started to propagate (Figure 17B top). Such a cracking corresponds to failure type F2-a shown in Figure 6C. The calculated load-displacement response (Figure 16) reaches the second highest peak, out of the curves for different pre-stress levels, fol- lowed by post-peak softening, hardening, and then a decrease at the end. In the simulation with 2.7 MPa, a major crack across fibers (Figure 17C bottom), that propagated in the direction of the notch, was accom- panied by the first branch along fibers (initiated at the notch tip) and followed by the second branch along fibers (initiated at approx. one half of the lig- ament’s length), see Figure 17C top. This cracking resembled failure type F2-b shown in Figure 6D. The obtained load-displacement curve (Figure 16) has the third highest peak followed by post-peak softening, hardening, and a decrease at the end. Crack along the grain initiated at or around the notch tip domi- nated in the calculations with 2.1 MPa and 1.0 MPa (Figure 17D top). Nevertheless, it was accompanied by crack across the grain in a few finite elements at the ligament’s beginning (Figure 17D bottom) with crack-normal strain (i.e. εyy) much lower than that of crack along the grain (i.e. εxx). Such failure was considered comparable to failure type F0 presented in Figure 6A. The obtained load-displacement curves are shown in Figure 16 with the fourth and fifth highest calculated peak. For the calculation with 2.1 MPa, the peak is followed by post-peak hardening and a decrease at the end. For the simulation with 1.0 MPa, the post-peak branch is softening. It is obvious that the numerical calculations were able to reproduce the different failure patterns (F0- F2) observed in the experiments. The models also (a). (b). (c). (d). Figure 17. Distributions of normal strains εxx and εyy obtained from the CT test simulations with differ- ent pre-stress levels: (A) pre-stress 3.3 MPa (state at Uy = 7.0 mm), (B) pre-stress 3.0 MPa (state at Uy = 9.0 mm), (C) pre-stress 2.7 MPa (state at Uy = 7.0 mm), and (D) pre-stress 1.0 MPa (state at 1.5 mm). captured well the initial slope, the peak, and the post-peak trends of the load-displacement records. 6.4. Effect of threshold angle θc The value of the threshold angle θc, which determines if a crack forms across the grain (splinter failure) or along grain, could not be exactly determined from uniaxial tension tests [3]. The tests only indicated that it falls within the range between 0° and 2°. Therefore, to investigate how sensitive is the model of the CT test to this parameter, two more sets of calculations were run with different values of θc = 0.4° and θc = 1.4°. Results are summarized in Figure 18 and Figure 19. In both sets of simulations with pre-stress of 1.0 MPa - 3.3 MPa, we observed a major crack along the grain initiated at or around the notch tip (Fig- ure 19 top), that was accompanied by a short branch, which propagated from the notch tip in the direction of the notch, either as crack along the grain for θc = 0.4° 113 Eliška Šmídová, Petr Kabele Acta Polytechnica CTU Proceedings (a). (b). Figure 18. Load vs. crosshead displacement for the results of CT test experiments and simulations calculated for different pre-stress levels with (A) θc = 0.4° and (B) θc = 1.4° leading to failure type F0. (a). (b). Figure 19. CT test simulation result of normal strains εxx and εyy at crosshead displacement of (A) 4.0 mm and (B) 5.0 mm with the pre-stress level of 3.3 MPa and critical load-grain angle of (A) θc = 0.4° and (B) θc = 1.4°. (Figure 19A bottom) or crack across the grain for θc = 1.4° (Figure 19B bottom). This crack pattern is considered to correspond well with the failure type F0 shown in Figure 6A. All the load-displacement curves (Figure 18) reach the peak with respect to the pre-stress level. The post-peak behavior exhibits hardening, except for the calculation with 1.0 MPa in which softening occurs. To summarize the CT test simulations with regard to critical load-grain angle θc and pre-stress level, different failures were observed: • (i) type F0 occurred with θc = 1.6° and the pre- stress level of 1.0 MPa and 2.1 MPa or with θc = 0.4° and 1.4° and all the pre-stress levels of 1.0 MPa to 3.3 MPa, • (ii) type F1 with θc = 1.6° and 3.3 MPa, • (iii) type F2-a with θc = 1.6° and 3.0 MPa, • (iv) type F2-b with θc = 1.6° and 2.7 MPa. The simulations with the pre-stress level of 2.7 MPa to 3.3 MPa and θc = 1.6° reproduced well the peak load, post-peak behavior, and respective failure types F1 - F2 (Figure 16), that were observed in three ex- perimental results out of five. Compared to it, the simulations with the same pre-stress level range and θc = 0.4° or θc = 1.4° reproduced adequately the load capacity, post-peak hardening, and respective failure type F0 (Figure 18), that were observed in two experiments out of five. Based on experimental and numerical results, the following recommendation was drawn up: to use θc = 1.6° and θc = 1.4° or less in 60% and 40% of pre-stressed CT test simulations, respec- tively, for a pre-stress level from the range between 2.7 MPa and 3.3 MPa. Treshold angle θc, determined as the interval be- tween 0° and 2° by visual assessment of off-axis tension specimens [3], may be regarded as one of timber mate- rial characteristics, which are typical for a significant variability. Thus, statistically significant results of not only off-axis tension tests but also CT experiments with regard to failure type are needed together with an appropriate statistical approach to numerical models. 7. Conclusion The finite element method with an anisotropic con- stitutive model were used for numerical simulation of ASTM shear test and pre-stressed compact tension test. The numerical results reproduced adequately both the experimental responses and the crack pat- terns. It was shown in shear test simulation that the value of the parameter fASTMxy , obtained from fASTMxy = Fmax A , represents the averaged stress at the failure plane, while the extreme stress locally experienced by the material is much higher. It is concluded that, if a finite element model of a structural member has a finer resolution of the stress field variation than is the size of the tested specimen, using the result of the ASTM shear test as the material strength may lead to underestimation of the member’s load capacity. The pre-stressed CT test simulation demonstrated the capability of the constitutive model to reproduce complex tensile-shear behavior of timber governed by tensile crack parallel or perpendicular to the grain or combination of both. The influence of the tresh- old principal load-grain angle θc to the results was analysed. 114 vol. 34/2022 Finite Element Simulation of Shear and Pre-stressed Compact Tension Tests on Timber List of symbols A Shearing area [mm] c1, c2 Parameters of the tn function [–] δn, δm Relative normal and tangent displacement [mm] δm,crit Shear displacement jump at which the cohesive traction diminishes to zero [mm] δn,crit Normal displacement jump at which the cohesive traction diminishes to zero [mm] ∆x,DIC Relative vertical displacement by DIC [mm] Ex Modulus of elasticity along fibers [MPa] Ey Modulus of elasticity across fibers [MPa] fcx Compressive strength along fibers [MPa] fcy Compressive strength across fibers [MPa] ftx Tensile strength along fibers [MPa] fty Tensile strength across fibers [MPa] fxy Shear strength [MPa] fAST M xy Shear strength according to standard ASTM [MPa] Gxy Shear modulus [MPa] lch Characteristic length of crack band model [mm] νxy, νyx Poisson’s ratios [–] p Material parameter of the slope of the tm function [–] Fmax Maximum attained force [N] σaxial Axial stress at failure for the off-axis test [MPa] σx, σy Normal stress along and across fibers [MPa] tn, tm Normal and tangent tractions [MPa] tci m Crack-tangent traction at crack initiation [MPa] tci n Crack-normal traction at crack initiation [MPa] τxy Shear stress [MPa] θ Principal load - grain angle [°] θc Treshold principal load - grain angle [°] Acknowledgements This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS20/155/OHK1/3T/11. 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ASTM International, West Conshohocken, 2010. 115 https://doi.org/10.4028/www.scientific.net/AMM.784.137 https://doi.org/10.4028/www.scientific.net/AMM.784.137 https://doi.org/10.1016/j.cam.2021.113676 https://doi.org/10.1016/j.compstruc.2018.10.005 Acta Polytechnica CTU Proceedings 34:105–115, 2022 1 Introduction 2 Constitutive model 3 Model calibration 4 Shear test and pre-stressed compact tension tests 4.1 Material 4.2 Equipment and data acquisition 4.3 Shear test 4.4 Pre-stressed compact tension test 5 Numerical simulations of shear test 5.1 Introduction 5.2 Numerical solution aspects 5.3 Results 6 Numerical simulations of pre-stressed CT test 6.1 Introduction 6.2 Numerical solution aspects 6.3 Results 6.4 Effect of threshold angle c 7 Conclusion List of symbols Acknowledgements References