Dynamic Testing of Lime-Tree (Tilia Europoea) and Pine (Pinaceae) for Wood Model Identification

materials

Article Dynamic Testing of Lime-Tree (Tilia Europoea) and Pine (Pinaceae) for Wood Model Identiﬁcation

Anatoly Bragov 1, Leonid Igumnov 1,2,*, Francesco dell’Isola 1, Alexander Konstantinov 1,2, Andrey Lomunov 1 and Tatiana Iuzhina 1

1 Research Institute for Mechanics, National Research Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia; [email protected] (A.B.); [email protected] (F.d.); [email protected] (A.K.); [email protected] (A.L.); [email protected] (T.I.) 2 Research and Education Mathematical Center “Mathematics for Future Technologies”, 603950 Nizhny Novgorod, Russia * Correspondence: [email protected]; Tel.: +7-910-792-7826

Received: 22 October 2020; Accepted: 19 November 2020; Published: 20 November 2020

Abstract: The paper presents the results of dynamic testing of two wood species: lime-tree (Tilia europoea) and pine (Pinaceae). The dynamic compressive tests were carried out using the traditional Kolsky method in compression tests. The Kolsky method was modified for testing the specimen in a rigid limiting holder. In the first case, stress–strain diagrams for uniaxial stress state were obtained, while in the second, for uniaxial deformation. To create the load a gas gun was used. According to the results of the experiments, dynamic stress–strain diagrams were obtained. The limiting strength and deformation characteristics were determined. The fracture energy of lime and pine depending on the type of test was also obtained. The strain rates and stress growth rates were determined. The influence of the cutting angle of the specimens relative to the grain was noted. Based on the results obtained, the necessary parameters of the wood model were determined and their adequacy was assessed by using a special verification experiment.

Keywords: timber; natural composite; Kolsky method; deformation diagrams; wood species; energy absorption; wood model; veriﬁcation

1. Introduction Wood is a complex natural composite. It is widely used as a material for damping intensive dynamic loads of shock or explosive nature. In order to conduct reliable numerical analysis of the designs of containers for transporting hazardous substances, using wood as shock damping components, reliable mathematical models are required that take into account its complex multicomponent structure of wood. The actively developing models of wood deformation and destruction can be used in various design complexes to simulate the behavior of technically complex structures that incorporate wood elements. However, the complex nature of wood means that not a single model can be used for all purposes, thus different models must be used to solve various specific problems. In order to reliably describe the behavior of wood in a dynamically loaded structure, its model should include a sufficiently large set of parameters that take into account the deformation anisotropy and the dependence of the strength properties on the strain rate, density, temperature and moisture content. For example, in the manual for the LS-DYNA software package, the wood model No. 143 has 29 parameters: five modules for transversely isotropic constitutive equations, six tensile strengths for yield criteria, four hardening parameters to peak stresses, eight softening parameters after peak, and six parameters speed effect [1]. The LS-DYNA calculation complex library contains model parameters for only two wood species: southern yellow pine and Douglas fir. This necessitates detailed studies of various wood species

Materials 2020, 13, 5261; doi:10.3390/ma13225261 www.mdpi.com/journal/materials Materials 2020, 13, 5261 2 of 12 for various types of stress–strain states in a wide range of strain rates and temperatures, in order to equip mathematical models of wood with parameters (identification) that can adequately describe the behavior of the engineering structure containing wood components under shock wave loads. The first detailed review of using wood as an element of damping structures was given by Johnson [2]. In the report, he noted that the dynamic properties of wood are not well understood. In the past three decades, quite a lot of works have appeared that explored various aspects of the high-speed deformation and destruction of wood, including the use of wood as a damping material in containers for transportation of radioactive materials by air, road and rail transport [3]. A large amount of research into the dynamic properties of wood was performed by Reid et al. [4–6]. In these works, the dependences of destructive stress and energy absorption on the impact velocity were obtained. It was noted that dynamic destructive stresses are several times higher than static ones. The authors also postulated that failure stresses of specimens fabricated along the grain are an order of magnitude higher than that of the specimens cut transverse to the grain. The relationship between mechanical parameters of selected wood species (Carya sp., Fagus sylvatica L., Acer platanoides L., Fraxinus excelsior L., Ulmus minor Mill.) and the grain orientation angle concern loading direction was investigated in [7]. It was observed that as the angle between fibers and loading direction increases whereas the mechanical characteristics of all wood species decrease. In the works of Bragov et al., using the Kolsky method, dynamic diagrams of birch, aspen, 3 1 and sequoia were obtained at strain rates of ~10 s− with different direction of cutting specimens relative to the direction of timber fibers [8,9]. The deformation diagrams of specimens under loading along the grain are much higher than those under loading transverse to the grain. The limiting deformative characteristics have the opposite direction. In recent years, interest in studying the effect of moisture, density and cutting angle on the mechanical properties of wood has increased significantly. Adalian et al. [10] and Eisenacher et al. [11] 3 3 1 carried out a cycle of tests on wood in the range of strain rate from 10− to 10 s− . Wouts et al. [12], Ha et al. [13,14] and Hao et al. [15] considered the damping ability of various biological materials of plant origin from the point of view of improving artificial damping structures. Mach et al. [16] calculated the hysteresis losses (dissipated energy) as the area bounded by the loading and unloading curves, whereas stored energy (recoverable energy) is defined as the area under the unloading curve. Thus, the total energy is calculated by summing the area of the hysteresis loss and the stored energy. In the work of Novikov et al. [17], a large number of tests of sequoia, birch, aspen, and pine was carried out at different cutting angles, temperatures ranged from 30 C to +65 C, and at the moisture − ◦ ◦ content of 5%, 20% and 30%. As a result, the dependence of strength on moisture and cutting angle was determined. The mechanical properties of wood strongly depend on the place of growth, its age, the place of cutting and the type of stress–strain state. Therefore, the results obtained by different authors may differ significantly from each other. The purpose of this work is to conduct a detailed study of the effect of strain rate and type of stress–strain state on the mechanical properties of wood as well as to identify and verify the wood model.

2. Test Methods, Materials and Specimens For dynamic compressive tests of wood, the Kolsky method and split Hopkinson pressure bar (SHPB) technology were used [18]. Figure1 shows the installation diagram, as well as the main formulas for determining the parametric dependencies of deformation, stress and strain rate of the sample under compression. Two measuring bars were made of high-strength aluminum alloy and had a diameter of 20 mm and length of 1.5 m. During testing, a striker accelerated in the barrel of a gas gun impacts the SHPB and excites an elastic compression wave in the loading measuring bar, which, upon reaching the specimen, deforms it. The second (support) measuring bar acts as a dynamometer-waveguide and allows you to register the transmitted strain pulse εT(t) and then to determine the process of stress developing in the specimen. The pulse εR(t) reﬂected from the specimen in the loading bar makes it MaterialsMaterials 20202020,, 1313,, 52615261 33 ofof 1212

specimen, and its integration allows to determine the process of the specimen deformation possible to determine the process of strain rate change in the specimen, and its integration allows to development. Small-length foil strain gages were used for registration elastic strain pulses in the bars. determine the process of the specimen deformation development. Small-length foil strain gages were Based on these pulses, the parametric processes of stress, strain and strain rate over time are used for registration elastic strain pulses in the bars. Based on these pulses, the parametric processes of calculated using the formulae shown in the lower part of Figure 1. Then, excluding time as a stress, strain and strain rate over time are calculated using the formulae shown in the lower part of parameter, a dynamic stress–strain curve is constructed with the known law of variation of the strain Figure1. Then, excluding time as a parameter, a dynamic stress–strain curve is constructed with the rate [19]. known law of variation of the strain rate [19].

Figure 1. Installation scheme and basic dependencies. Figure 1. Installation scheme and basic dependencies. Dynamic tests were carried out with specimens of lime-tree (Tilia europoea) and pine (Pinaceae) with aDynamic diameter tests of 20 were mm andcarried a length out with of 10 specimens mm, cut from of lime-tree solid wood (Tilia at an europoea angle of) and 0◦ and pine 90 (◦Pinaceaerelative) towith the a axis diameter of the treeof 20 trunk. mm and The a ﬂat length ends of the10 mm, samples cut from were carefullysolid wood hand-grinded at an angle beforeof 0° and testing. 90° Therelative physical to the parameters axis of the oftree tested trunk. materials The flat areends shown of the in samples the Table were1. carefully hand-grinded before testing. The physical parameters of tested materials are shown in the Table 1. Table 1. The physical parameters of tested materials. Table 1. The physical parameters of tested materials. Wood Species Lime-Tree Pine WoodParameter Species Along the Grain Lime-Tree Transverse to the Grain Along the Grain PineTransverse to the Grain Parameter Along the Grain Transverse to the Grain Along the Grain Transverse to the Grain Density, g/cm3 0.505 0.02 0.512 0.015 0.435 0.01 0.484 0.015 Density, g/cm3 0.505 ± ±0.02 0.512 ±± 0.015 0.435 ± 0.01 0.484± ± 0.015 Moisture content, % 7.5 0.2 7.9 0.3 6.8 0.3 7.1 0.2 Moisture content, % 7.5 ± ±0.2 7.9 ±± 0.3 6.8 ± 0.3 7.1± ± 0.2

TheThe totaltotal amountamount of of tested tested samples samples of of each each wood wood species species amounted amounted 50 pieces: 50 pieces: 40 pieces 40 pieces were were used forused mechanical for mechanical testing testing and 10 and pieces 10 pieces were were used used for subsequent for subsequent wood wood model mode veriﬁcation.l verification. In each In each test modetest mode during during mechanical mechanical testing testing (strain (strain rate rate and and loading loading direction), direction), 4–5 experiments4–5 experiments were were carried carried out, theout, results the results of which of which were then were averaged. then averaged. All the experimentsAll the experiments were carried were out carried in laboratory out in conditionslaboratory atconditions room temperature at room temperature and 50% air and humidity. 50% air humidity.

3.3. Experimental Results UsingUsing the above above methods, methods, the the dynamic dynamic tests tests of pi ofne pine and and lime-tree lime-tree were were carried carried out. As out. a result, As a result,their dynamic their dynamic stress–strain stress–strain curves, curves, the theultimate ultimate strength strength and and deformative deformative characteristics characteristics of of thethe specimensspecimens cutcut alongalong and and transverse transverse to to the the grain grain were we obtained.re obtained. Dynamic Dynamic strain strain diagrams diagrams for pinefor pine and lime-treeand lime-tree under under uniaxial uniaxial stress stress state are state presented are presented in Figures in 2Figures and3. The2 and ﬁgures 3. The show figures the averaged show the stress–strainaveraged stress–strain curves and curves appropriate and appropriate strain rate-strain strain rate-strain curves. The curves. spread The ofthe spread obtained of the properties obtained wasproperties no more was than no 6%. more In than all ﬁgures 6%. In solid all figures lines show solid the lines dependences show the dependences of the true stress of the of thetrue specimen stress of

Materials 2020, 13, 5261 4 of 12 Materials 2020, 13, 5261 4 of 12 Materials 2020, 13, 5261 4 of 12 the specimen on its deformation σ~ε, and the dotted lines (in the lower part of the diagrams) on its deformation σ~ε, and the dotted lines (in the lower part of the diagrams) correspond to the thecorrespond specimen to onthe itshistory deformation of the strain σ~ε rate, and change the dotted έ~ε (the lines appropriate (in the lower axis is part on the of right).the diagrams) history of the strain rate change ε´~ε (the appropriate axis is on the right). correspond to the history of the strain rate change έ~ε (the appropriate axis is on the right).

Figure 2. Deformation diagrams of pine under loading along and across the grain. Figure 2. Deformation diagrams of pine under loading along and across the grain.

Figure 3. Deformation diagrams of lime-tree under loading along and across the grain. Figure 3. Deformation diagrams of lime-tree under loading along and across the grain. Two testFigure modes 3. Deformation on strain rate diagrams were chosen: of lime-tree with unde non-destructiver loading along and and destructive across the results.grain. From the obtainedTwo stress–strain test modes on diagrams, strain rate it followswere chosen: that for with both non-destructive wood species the and specimens destructive cut results. and loaded From alongthe obtainedTwo the graintest stress–strainmodes have theon largeststrain diagrams, rate modulus were it followschosen: of the load thwithat branch for non-destructive both in thewood diagram, species and dest as the wellructive specimens as ultimate results. cut stressFrom and andtheloaded obtained energy along absorption. stress–strainthe grain have diagrams, the largest it modulus follows thofat the for load both branch wood in speciesthe diagram, the specimens as well as ultimatecut and loadedstressWood and along energy endurance the grain absorption. have can be the indicated largest modulus in Figure of4 the, where load branch diagrams in the of diagram, ﬁvefold loadingas well as of ultimate a pine specimenstressWood and withenergy endurance its absorption. minor can damages be indicated (curves in 1–5) Figure and 4, a singlewhere independentdiagrams of loadingfivefold ofloading another of similar a pine specimenspecimenWood with with endurance its its complete minor can damages failurebe indicated (curve (curves 6)in are 1–5)Figure presented. and 4, a where single The diagrams independent areof fivefold locatedloading on loadingof the another deformation of similara pine axisspecimenspecimen conditionally withwith its its inminor ordercomplete damages to make failure (curves it easier (curve 1–5) to assess6) and are a the presented.single eﬀect independent of multipleThe diagrams loadingloading are on of theanotherlocated steepness similaron the of specimen with its complete failure (curve 6) are presented. The diagrams are located on the1 thedeformation load sections axis of conditionally the stress–strain in order curves. to Themake average it easier strain to assess rate at the repeated effect loadsof multiple was ~600–800 loading s −on, deformationthe steepness axis of the conditionally load sections in of order the stress–strain to make it easier curves. to Theassess averag the effecte strain of rate multiple at repeated loading loads on thewas steepness ~600–800 of s− 1the, and load in thesections case of of specimen the stress–strain failure, curves.the strain The rate averag was aboute strain 2200 rate s −at1. Onerepeated can clearly loads was ~600–800 s−1, and in the case of specimen failure, the strain rate was about 2200 s−1. One can clearly

Materials 2020, 13, 5261 5 of 12 Materials 2020, 13, 5261 5 of 12 Materials 2020, 13, 5261 5 of 12 see a decrease in the steepness of the load section of stress–strain curve1 during repeated loads by a and in the case of specimen failure, the strain rate was about 2200 s− . One can clearly see a decrease seefactor a decrease of 2–3, which in the issteepness associated of thewith load partial section destruction of stress–strain of the specimen curve during during repeated each loading loads by and a in the steepness of the load section of stress–strain curve during repeated loads by a factor of 2–3, factorviolation of 2–3, of the which flatness is associated of its ends. with partial destruction of the specimen during each loading and which is associated with partial destruction of the specimen during each loading and violation of the violation of the flatness of its ends. ﬂatness of its ends.

Figure 4. An example of multiply repeated loading of the specimen with its minor damage. Figure 4. An example of multiply repeated loading of the specimen with its minor damage. Figure 4. An example of multiply repeated loading of the specimen with its minor damage. As anan importantimportant characteristic characteristic of of the the timber timber damping damping capacity, capacity, the energythe energy absorption absorption of the of tested the woodtestedAs specieswood an important species was estimated was characteristic estimated underloading underof the loadingtimber along anddamping along transverse and capacity, transverse to the the grain to energy the by grain calculating absorption by calculating the of areathe testedunderthe area wood the under curve species theσ~ curveε was(Figure σestimated~ε5 ).(Figure under 5). loading along and transverse to the grain by calculating the area under the curve σ~ε (Figure 5).

a) b) a) b) Figure 5. Energy absorption of the tested wood species: (a) for pine, (b) for lime-tree. Figure 5. Energy absorption of the tested wood species: (a) for pine, (b) for lime-tree. For both wood species, there is a significant excess of energy absorption by the specimens cut and For both wood species, there is a significant excess of energy absorption by the specimens cut tested along the grain, compared to the specimens cut and tested transverse to the grain. and testedFor both along wood the species, grain, comp thereared is a tosignificant the specimens excess cut of andenergy tested absorption transverse by to the the specimens grain. cut It is interesting to compare the results on the damping ability of wood-based materials obtained and testedIt is interesting along the tograin, compare comp theared results to the on specimens the damping cut and ability tested of wood-based transverse to materials the grain. obtained by other researchers. In the work [12] the damping capacity of two types of deciduous and coniferous by otherIt is interestingresearchers. to In compare the work the [12] results the damping on the damping capacity abilityof two oftypes wood-based of deciduous materials and coniferous obtained wood: beech and spruce under loading in the longitudinal, tangential and radial directions were bywood: other beech researchers. and spruce In the under work [12]loading the damping in the longit capacityudinal, of twotangential types of and deciduous radial directions and coniferous were compared. The highest specific energy absorbed was noted for specimens under longitudinal loading, wood:compared. beech The and highest spruce specificunder loading energy inabsorbed the longit wasudinal, noted tangential for specimens and radial under directions longitudinal were and the smallest—under tangential loading. This is traditional tendency for wood materials. compared.loading, and The the highestsmallest—under specific tangentialenergy absorbed loading. was This notedis traditional for specimens tendency under for wood longitudinal materials. While in [14], the damping ability of the mesocarp layer in durian shell (tropical fruit) was loading,While and in the [14], smallest—under the damping tangentialability of loading.the mesocarp This is layer traditional in durian tendency shell for(tropical wood materials.fruit) was investigated and, as a result of the research, it was found the inverse effect: specific energy absorption investigatedWhile in and, [14], as the a result damping of the ability research, of itthe wa mesocarps found the layer inverse in durianeffect: specificshell (tropical energy absorptionfruit) was investigatedof the mesocarp and, layeras a result under of lateral the research, loading it iswa highers found than the thatinverse under effect: axial specific one. This energy may absorption be due to of the mesocarp layer under lateral loading is higher than that under axial one. This may be due to

Materials 2020, 13, 5261 6 of 12 of the mesocarp layer under lateral loading is higher than that under axial one. This may be due to the fact that the durian shell does not have the same clearly pronounced ﬁber structure as in wood, therefore the authors’ accentuation on the lower strength and damping ability of the material in the axial direction of load application compared to lateral loading refers to the radial and tangential strength of shell of this fruit.

4. Identification and Verification of the Wood Model To describe the behavior of wood under dynamic loads in the library of the LS-DYNA calculation complex, there is model No. 143 MAT_WOOD. This is a model of a transversely isotropic material for solid elements. It is possible to set material properties or use a predefined set of constants, but only for two species of pine growing in the USA: southern yellow pine and Douglas fir. The primary features of the model are: Transverse isotropy for the elastic constitutive equations (different properties are modeled parallel • and perpendicular to the grain). Yielding with associated plastic flow formulated with separate yield (failure) surfaces for the • parallel- and perpendicular-to-the-grain modes. Hardening in compression formulated with translating yield surfaces. • Post-peak softening formulated with separate damage models for the parallel and • perpendicular-to-the-grain modes. Strength enhancement at high strain rates. • Model 143 for pine contains a predefined set of 29 parameters depending on moisture content and temperature. According to the results of the pine tests, some parameters of this model were adjusted for a particular batch of samples under study. These parameters are shown in Table2.

Table 2. Main mechanical properties of pine for model identiﬁcation.

Designation Value Density 450 kg/m3 Modulus of elasticity in the direction of the grain 10,500 MPa Flow stress during compression in the direction of the grain 80 MPa The modulus of elasticity in the direction perpendicular to the grain 246 MPa Flow stress during compression in the direction perpendicular to the grain 10 MPa

It should be noted that the elastic moduli of pine under loading along and across the grain were obtained as a result of quasi-static tests, since the Kolsky method, in principle, does not allow constructing a stress–strain curve in which the slope dσ/dε of the elastic loading section would be equal to the static modulus of elasticity. Usually the steepness of this section is several times less than the static Young’s modulus. The reasons for this are as follows: - The difference in the cross-sectional areas of the measuring bars and the sample (some part of the incident wave is reflected from the free surface around the sample), causing an apparent deformation of the sample, - The presence of gaps between the ends of the measuring bars and the sample (due to the non-parallelism of its ends and their roughness or insufficient “pressing” of the sample during the preparation of the test), - Dispersion during the propagation of waves in the bars (decrease in the steepness of the transmitted pulse during the propagation time to the recording strain gauge). In order to confirm the adequacy of the model parameters determined from experiments, it is necessary to verify it, but in experiments other than those in which these parameters were Materials 2020, 13, 5261 7 of 12 obtained.Materials 2020 For, 13, this 5261purpose, we used a model experiment in full-scale and numerical implementation.7 of 12 The simulation of the indentation process was performed using the finite element method. The explicit timeperformedMaterials integration 2020 using, 13, 5261 the scheme Dynamics-2 was used® software for solving package the [20]. equations The nume of motionrical experiment in time. Modelingwas simulated7 was of 12 performedin an axisymmetric using the formulation, Dynamics-2® andsoftware its design package corresponded [20]. The numericalto a similar experiment natural test. was simulated in performed using the Dynamics-2® software package [20]. The numerical experiment was simulated an axisymmetricTo verify the formulation, model of andmateri itsal design behavior, corresponded we used to an a similarexperimental natural test.scheme for dynamic in an axisymmetric formulation, and its design corresponded to a similar natural test. indentationTo verify by the using model the of system material ofbehavior, a split Hopkinson we used an bar. experimental The indicated scheme procedure for dynamic is schematically indentation To verify the model of material behavior, we used an experimental scheme for dynamic byshown using in the Figure system 6. ofThe a splitsample Hopkinson 5 and a bar.removable The indicated indenter procedure 4 with a is hemispherical schematically shownhead are in indentation by using the system of a split Hopkinson bar. The indicated procedure is schematically Figuresandwiched6. The between sample 5 the and measuring a removable bars indenter 2 and 6. 4 Upon with a impact hemispherical loading headby the are striker sandwiched 1, a compression between shown in Figure 6. The sample 5 and a removable indenter 4 with a hemispherical head are theimpulse measuring is generated bars 2 andin the 6. Uponbar 2 the impact duration loading of which by the strikerdepends 1, aon compression the length of impulse the striker is generated and the sandwiched between the measuring bars 2 and 6. Upon impact loading by the striker 1, a compression inamplitude the bar 2 on the the duration speed ofof whichthe striker. depends In the on thecase length of hemispherical of the striker indenter, and the amplitudethe contact on area the speedat the impulse is generated in the bar 2 the duration of which depends on the length of the striker and the ofinitial the striker.moment In of the indentation case of hemispherical is very small, indenter, therefore, the contact the amplitude area at the of initial the reflected moment ofpulse indentation is very amplitude on the speed of the striker. In the case of hemispherical indenter, the contact area at the issignificant, very small, and therefore, so the sample the amplitude is loaded of several the reflected times. pulse For reliable is very significant,recording of and additional so the sample loading is initial moment of indentation is very small, therefore, the amplitude of the reflected pulse is very loadedcycles, it several is necessary times. Forto increase reliable recordingthe length of of additional the supporting loading bar cycles, in comparison it is necessary with to the increase length the of significant, and so the sample is loaded several times. For reliable recording of additional loading lengththe loading of the bar supporting as much as bar required in comparison to register with loading the length cycles of the[21]. loading bar as much as required to cycles, it is necessary to increase the length of the supporting bar in comparison with the length of register loading cycles [21]. the loading bar as much as required to register loading cycles [21].

Figure 6. High speed indentation experiment scheme. Figure 6. High speed indentation experiment scheme. Using strain gauges 3,Figure elastic 6. strainHigh speed pulses indentation are recorded experiment in the scheme.measuring bars (Figure 7). The followingUsing pulses strain are gauges indicated 3, elastic by numbers: strain 1—incident pulses are recorded(loading) strain in the pulse measuring εI(t), 2—reflected bars (Figure pulse7). The followingUsing strain pulses gauges areε indicated R3, elastic by strain numbers: pulses 1—incident are recorded (loading) in theε I measuring strain pulse barsεI(t ),(Figure 2—reflected 7). The in the first loading cycle 1 (t) , 3—incident (loading) strain pulse 2 (t) in the second loading cycle, pulsefollowing in the pulses first loading are indicated cycle εbyR( numbers:t), 3—incident 1—incident (loading) (loading) strain strain pulse pulseεI (t) inεI(t the), 2—reflected second loading pulse 1 ε R 2 εT 4—reflected strain pulse inε theR second loading cycle 2 (t) , 5—transmittedεR( ) ε I pulse (first cycle) 1 (t) cycle,in the 4—reflected first loading straincycle pulse1 (t) , in 3—incident the second (loading) loading strain cycle pulse2 t , 5—transmitted2 (t) in the second pulse loading (first cycle) cycle, T εT T ε, 6—transmitted(t), 6—transmitted pulse pulse (second (second cycle) cycle) 2 (tε) .( t). ε R εT 14—reflected strain pulse in the second loading2 cycle 2 (t) , 5—transmitted pulse (first cycle) 1 (t) εT , 6—transmitted pulse (second cycle) 2 (t) .

Figure 7. Typical waveform obtained in the experiment for high-speed indentation, taking into account Figure 7. Typical waveform obtained in the experiment for high-speed indentation, taking into additional cycles: upper beam—data from bar 2; lower beam—data from bar 6. account additional cycles: upper beam—data from bar 2; lower beam—data from bar 6. Figure 7. Typical waveform obtained in the experiment for high-speed indentation, taking into The loading of the SHPB system in the numerical experiment (Figure8) is performed in the same Theaccount loading additional of the SHPBcycles: systemupper beam in the—data numerical from bar experiment 2; lower beam (Figure—data 8) from is performed bar 6. in the same way as in the natural experiment—with the help of a striker having an initial velocity V0. In the way as in the natural experiment—with the help of a striker having an initial velocity V0. In the calculationThe loading process, of the the incident SHPB system and reflected in the numerical strain pulses experiment are calculated (Figure in 8) element is performed 2 and inthe the past— same inway element as in the6. Thenatural time experiment—with dependences obtained the help during of a striker the simulation having an areinitial compared velocity withV0. In the the correspondingcalculation process, values the recorded incident during and reflected the experiment. strain pulses are calculated in element 2 and the past— in element 6. The time dependences obtained during the simulation are compared with the corresponding values recorded during the experiment.

Materials 2020, 13, 5261 8 of 12

Materials 2020, 13, 5261 8 of 12 calculation process, the incident and reflected strain pulses are calculated in element 2 and the past—in element 6. The time dependences obtained during the simulation are compared with the corresponding Figure 8. High speed indentation experiment simulation scheme. valuesMaterials recorded 2020, 13, 5261 during the experiment. 8 of 12 The indenter is made of high-strength tungsten–cobalt alloy and is modeled by a rigid non- deformable material. To assess the processes occurring in the sample, data obtained from measuring bars is used. In accordance with the Kolsky formulas, one can calculate the law of change of the indenter penetration rate into the sample V(t) and the penetration resistance force F(t): Figure 8. High speed indentationI experimentR simulationT scheme. Figure 8. High speedV(t) = indentation C1·(ε (t) + ε experiment(t)) − C2·ε simulation(t) scheme. (1) The indenter is made of high-strengthF(t) = tungsten–cobaltE2∙S2 εT(t). alloy and is modeled by a rigid non-deformableThe indenter material. is made of high-strength tungsten–cobalt alloy and is modeled by a rigid non- here εI(t), εR(t) and εT(t) denote the incident, reflected, and transmitted strain pulses in the measuring deformableTo assess material. the processes occurring in the sample, data obtained from measuring bars is used. bars, respectively, E is the elastic modulus, and C is the bar velocity of sound. The subscripts 1 and 2 In accordanceTo assess with the theprocesses Kolsky occurring formulas, in one the can sample, calculate da theta obtained law of change from ofmeasuring the indenter bars penetration is used. In refer to the first (loading) and second (supporting) measuring bars. rateaccordance into the with sample theV Kolsky(t) and formulas, the penetration one can resistance calculate forcethe lawF(t ):of change of the indenter penetration rate Wheninto the modeling sample V the(t) andprocess the penetrationof dynamic resistanceindentation force into F wood(t): samples, the following scheme was used: a sample and an indenterV(t) =wereC (εconsiderI(t) + εRed(t)) (FigureC εT (9).t) The spatial discretization of the V(t) = 1C1·(εI(t) + εR(t)) − C2·εT(t) indenter and the sample was done by using· a solidT three-dimensional− · finite element with one(1) F(t) = E2 S2 ε (t). (1) integration point. Due to the presence of symmetry,F(t) = E2· ∙aS 2quarter εT(t). of the geometric model was considered. On theεI planesεR of symmetry,εT the corresponding boundary conditions were specified: at the nodes herehere εI((tt),), εR(t) and εT((tt)) denote denote the the incident, incident, reflected, reflected, and transmittedtransmitted strainstrain pulsespulses inin thethe measuringmeasuring lying on the plane with the normal in the direction of the oY axis, zero velocities in the oY direction bars,bars, respectively,respectively,E Eis is the the elastic elastic modulus, modulus, and andC Cis is the the bar bar velocity velocity of of sound. sound. TheThe subscriptssubscripts 11 andand 22 and zero velocities of rotation about the oX and oZ axes were set; at the nodes lying on the plane with referrefer toto thethe firstfirst (loading)(loading) andand secondsecond (supporting)(supporting) measuring measuring bars. bars. the normal in the direction of the oZ axis, zero velocities in the oZ direction and zero velocities of WhenWhen modelingmodeling the the process process of of dynamic dynamic indentation indentation into into wood wood samples, samples, the followingthe following scheme scheme was rotation about the oX and oY axes were set. used:was used: a sample a sample and an and indenter an indenter were considered were consider (Figureed 9(Figure). The spatial 9). The discretization spatial discretization of the indenter of the Zero velocities in the direction of the oX axis were set at the nodes belonging to the surface of andindenter the sample and the was sample done by was using done a solid by three-dimensionalusing a solid three-dimensional finite element with finite one element integration with point. one the sample, which in full-scale experiments rested against the transmitted measuring bar. Dueintegration to the presence point. Due of symmetry,to the presence a quarter of symmetry, of the geometric a quarter model of the wasgeometric considered. model On was the considered. planes of The indenter was modeled by an absolutely rigid undeformable body. For the indenter, the law symmetry,On the planes the correspondingof symmetry, the boundary corresponding conditions boundary were specified: conditions at thewere nodes specified: lying onat the planenodes of the change in the speed of its movement in the axial direction was set. The time dependence of withlying the onnormal the plane in thewith direction the normal of the in oY the axis, direction zero velocities of the oY in axis, the oYzero direction velocities and in zero the oY velocities direction of indenter’s axial velocity for a particular experiment was calculated using the Equation (1) based on rotationand zero about velocities the oXof rotation and oZ axesabout were the oX set; and at theoZ axes nodes were lying set; on at thethe planenodes withlying the on normalthe plane in with the the signals recorded in the measuring bars. directionthe normal of thein the oZ axis,direction zero of velocities the oZ inaxis, the zero oZdirection velocities and in the zero oZ velocities direction of and rotation zero about velocities the oX of The “surface–surface” contact interaction was set between the indenter and the sample. androtation oY axes about were the set. oX and oY axes were set. Zero velocities in the direction of the oX axis were set at the nodes belonging to the surface of the sample, which in full-scale experiments rested against the transmitted measuring bar. The indenter was modeled by an absolutely rigid undeformable body. For the indenter, the law of the change in the speed of its movement in the axial direction was set. The time dependence of indenter’s axial velocity for a particular experiment was calculated using the Equation (1) based on the signals recorded in the measuring bars. The “surface–surface” contact interaction was set between the indenter and the sample.

a) b)

Figure 9. 9. GeometricGeometric statement statement of the of prob thelem problem of numerical of numerical simulation: simulation: (a) 3D configuration, (a) 3D conﬁguration, (b) plane configuration.(b) plane conﬁguration.

Zero velocities in the direction of the oX axis were set at the nodes belonging to the surface of the sample, which in full-scale experiments rested against the transmitted measuring bar. The indenter was modeled by an absolutely rigid undeformable body. For the indenter, the law ofa) the change in the speed of its movement in the axial b) direction was set. The time dependence of

Figure 9. Geometric statement of the problem of numerical simulation: (a) 3D configuration, (b) plane configuration.

Materials 2020, 13, 5261 9 of 12 indenter’s axial velocity for a particular experiment was calculated using the Equation (1) based on the signals recorded in the measuring bars. Materials 2020, 13, 5261 9 of 12 The “surface–surface” contact interaction was set between the indenter and the sample. InIn natural natural experiments experiments onon thethe indentationindentation the the samp samplesles of of pine pine were were used used in inthe the form form of oftablets tablets withwith a lengtha length of of 10 10 mm mm and and a diametera diameter of of 20 20 mm. mm. Some Some of of the the samples samples were were cut cut in in the the direction direction ofof the woodthe wood grain, grain, while while another another part part of the of the samples samples was was cut cut in in the the perpendicular perpendicular to to the the grain grain direction.direction. AsAs mentioned mentioned above, above, inin thethe processprocess of dynamic indentation, indentation, the the sample sample is issubjected subjected to tomultiple multiple loadingloading in in the the experiment experiment (see (see FigureFigure7 7),), thusthus undergoingundergoing a a certain certain deformation deformation in in each each loading loading cycle. cycle. High-speedHigh-speed ﬁlm film recording recording of theof the indentation indentation process process makes makes it possible it possible to estimate to estimate a number a number of loading of cycleloading and cycle indentation and indentation depth at depth which at which the destruction the destruction of the of material the material occurs. occurs. Figure Figure 10 10 shows shows the framesthe frames of such of such registration, registration, made made using using a high-speed a high-speed camera camera HSFC HSFC Pro. Pro.

FigureFigure 10.10. High-speedHigh-speed registration registration of of dynamic dynamic indentation. indentation.

TheThe left left part part of of the the figure figure corresponds corresponds to theto the experiment experiment with with the the sample sample cut cut in thein the direction direction along thealong grain, the whereas grain, whereas the right the part right corresponds part corresponds to the experimentto the experiment with the with sample the sample cut in thecut directionin the transversedirection totransverse the grain. to Thethe grain. numbers The on numbers the left on of thethe high-speedleft of the high-sp registrationeed registration images indicate images the numberindicate of the loading number cycles of loading (running cycles number (running of impulse number in of the impulse loading in measuring the loading bar) measuring whichone bar) can seewhich in Figure one 7can. It see can in be Figure seen in 7. Figure It can be10 thatseen thein Fi samplesgure 10 cuttingthat the o samplesff parallel cutting to the off grain parallel remain to the intact throughoutgrain remain the intact experiment, throughout which the is experiment, confirmed bywhich the finalis confirmed state of suchby the samples. final state Samples of such obtainedsamples. by cuttingSamples in theobtained direction by cutting perpendicular in the direction to the grainperpendicular retain their to the integrity grain retain during their the integrity first loading during cycle, however,the first cracksloading and cycle, gaps however, between cracks the layersand gaps of woodbetween appear the layers in the of second wood appear cycle, which in the growsecond and progresscycle, which in subsequent grow and loadingprogress cycles, in subsequent leading loading to the separation cycles, leading of the to samplethe separation into parts. of the sample intoIt parts. can be noted that for the same load amplitude, the samples obtained by cutting in the direction of the fiberIt can remain be noted intact, that for while the thesame samples load amplitude, cut perpendicular the samples to theobtained fiber exhibitby cutting failure in the already direction inthe firstof loadingthe fiber cycle.remain intact, while the samples cut perpendicular to the fiber exhibit failure already in the first loading cycle. A comparison of the shape of the imprint on the pine sample after the experiment on the dynamic indentation of a hemispherical indenter is shown in Figure 11.

Materials 2020, 13, 5261 10 of 12

A comparison of the shape of the imprint on the pine sample after the experiment on the dynamic Materialsindentation 20202020,, 1313 of,, 52615261 a hemispherical indenter is shown in Figure 11. 1010 ofof 1212

a) b)

FigureFigure 11. 11. TheThe shape shape of of the the imprints imprints obta obtainedinedined byby by numericalnumerical numerical modelingmodeling modeling ((aa (a)) )andand and inin in thethe the naturalnatural natural testtest test ((b (b).).).

FigureFigure 1212 showsshows a a comparison comparison of of the the time time dependences dependences of of the the indentation indentation resistance resistance of of pine pine samplesamplesample recordedrecordedrecorded in inin the thethe experiments experimentsexperiments (solid (solid(solid colored colorecolore lines)d lines) with thewith data the obtained data obtained by numerical by numerical modeling modeling(black dashed (black lines) dashed during lines) dynamic during indentation dynamic indentation of hemispherical of hemispherical indenter into indenter specimen into both specimen along bothand acrossalong and the grain.across the grain.

Figure 12. Comparison of experimental and calculated pulses in the supporting bar. Figure 12. Comparison of experimental and calculatedated pulsespulses inin thethe supportingsupporting bar.bar. It can be seen that the model used allows us to accurately reproduce the features of the deformation ItIt cancan bebe seenseen thatthat thethe modelmodel usedused allowsallows usus toto accuratelyaccurately reproducereproduce thethe featuresfeatures ofof thethe of real material, namely, the difference in its deformation behavior in different directions with respect deformation of real material, namely, the difference inin itsits deformationdeformation behaviorbehavior inin differentdifferent directionsdirections to grain orientation. with respect to grain orientation. The results obtained are in qualitative agreement with the data from previous studies of high The results obtained are in qualitativeitative agreementagreement withwith thethe datadata fromfrom previousprevious studiesstudies ofof highhigh strain rate behavior of wood [8,17]. Correction of some parameters of the MAT_WOOD model made it strainstrain raterate behaviorbehavior ofof woodwood [8,17].[8,17]. CorrectionCorrection of some parameters of the MAT_WOOD model made possible to describe the deformation behavior of the studied wood species with sufficient accuracy. itit possiblepossible toto describedescribe thethe deformationdeformation behaviorbehavior ofof thethe studiedstudied woodwood speciesspecies withwith sufficientsufficient accuracy.accuracy. A further increase in the quality of the mathematical model is possible by taking into account the effect A further increase in the quality of the mathematical model is possible by taking into account the of the strain rate in a wide range of its change, as well as taking into account the fracture of the material. effect of the strain rate in a wide range of its change, as well as taking into account the fracture of the material. In addition, it is necessary to conduct more complex model experiments to verify the fracturefracture criteria,criteria, forfor example,example, high-speedhigh-speed penetration and perforation of wood plates.

Materials 2020, 13, 5261 11 of 12

In addition, it is necessary to conduct more complex model experiments to verify the fracture criteria, for example, high-speed penetration and perforation of wood plates.

5. Conclusions Dynamic tests of lime-tree and pine were conducted. There is a strong anisotropy of the properties of the tested materials: the specimens exhibit the greatest strength under load applied along the grain, while the lowest strength is observed under loading transverse to the grain. The load branch module is non-linear and, as a rule, smaller than the unloading branch module (while maintaining the specimen integrity). At the specimen cutting angle of 90◦ with respect to the direction of the fibers, the stress–strain diagram after reaching a certain threshold stress value (about 8 MPa for lime-tree and about 10 MPa for pine) is close to the ideal plastic diagram. At 0◦ cutting angle, the initial section of the diagrams is close to linear. That is, an elastic deformation occurs. However, after reaching an ultimate stress value (yield strength), the diagram becomes nonlinear with stress relaxation. Especially such behavior is observed in the experiments in which the failure of specimens occurs. Using a modification of the SHPB method, experiments were performed on the dynamic indentation of indenters with a hemispherical head into the samples parallel and perpendicular to the direction of the fibers. Dependences of resistance to penetration on time are constructed. In the direction of the fiber (along the grain), the resistance force was more than three-times greater than in the transverse direction. The results of the natural experiment were compared with the results of numerical modeling, in which the wood behavior was described by the MAT_WOOD model from the LS-DYNA software, in which some of the parameters were obtained experimentally. A good coincidence was observed.

Author Contributions: Conceptualization, A.B. and L.I.; data curation, T.I.; formal analysis, A.K.; funding acquisition, L.I.; investigation, A.L.; methodology, A.K. and A.L.; project administration, A.B.; resources, L.I.; software, A.K. and T.I.; supervision, F.d.; validation, F.d. and A.K.; visualization, A.K.; writing—original draft, A.L.; writing—review and editing, A.B. All authors have read and agreed to the published version of the manuscript. Funding: Experimental investigations of wood were performed with the financial support of the Ministry of Science and Higher Education of the Russian Federation (task 0729-2020-0054). Numerical simulation of wood behavior was supported by the grant of the Government of the Russian Federation (contract No.14.Y26.31.0031). Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References

1. Murray, Y.D. Manual for LS-DYNA Wood Material Model 143; Publication No. FHWA-HRT-04-097; Federal Highway Administration: Washington, DC, USA, 2007. 2. Johnson, W. Historical and present-day references concerning impact on wood. Int. J. Impact Eng. 1986, 4, 161–174. [CrossRef] 3. Neumann, M. Investigation of the Behavior of Shock-Absorbing Structural Parts of Transport Casks Holding Radioactive Substances in Terms of Design Testing and Risk Analysis. Ph.D. Thesis, Bergische Universität Wuppertal, Wuppertal, Germany, 2009. 4. Reid, S.R.; Reddy, T.Y.; Peng, C. Dynamic compression of cellular structures and materials. In Structural Crashworthiness and Failure; Jones, N., Wierzbicki, T., Eds.; Taylor & Francis Publication: London, UK; New York, NY, USA, 1993; pp. 257–294. 5. Reid, S.R.; Peng, C. Dynamic Uniaxial Crushing of Wood. Int. J. Impact Eng. 1997, 19, 531–570. [CrossRef] 6. Harrigan, J.J.; Reid, S.R.; Tan, P.J.; Reddy, T.Y. High rate crushing of wood along the grain. Int. J. Mech. Sci. 2005, 47, 521–544. [CrossRef] 7. Mania, P.; Siuda, F.; Roszyk, E. Eﬀect of Slope Grain on Mechanical Properties of Diﬀerent Wood Species. Materials 2020, 13, 1503. [CrossRef][PubMed] 8. Bragov, A.M.; Lomunov, A.K. Dynamic properties of some wood species. J. Phys. IV 1997, 7, 487–492. [CrossRef] Materials 2020, 13, 5261 12 of 12

9. Bragov, A.M.; Lomunov, A.K.; Sergeichev, I.V.; Gray, I.I.I.G.T. Dynamic behaviour of birch and sequoia at high strain rates. “Shock Compression of Condensed Matter”. In Proceedings of the Conference of the American Physical Society Topical Group “APS-845”, Baltimore, MD, USA, 31 July–5 August 2005; American Institute of Physics: Melville, NY, USA, 2006; pp. 1511–1514. 10. Adalian, C.; Morlier, P. Modeling the behaviour of wood during the crash of a cask impact limiter. In Proceedings of the PATRAM’98 Conference Proceedings, Paris, France, 10–15 May 1998; Volume 1. 11. Eisenacher, G.; Scheidemann, R.; Neumann, M.; Wille, F.; Droste, B. Crushing characteristics of spruce wood used in impact limiters of type B packages. Packaging and transportation of radioactive materials. In Proceedings of the PATRAM 2013, San Francisco, CA, USA, 18–23 August 2013. 12. Wouts, J.; Haugou, G.; Oudjene, M.; Coutellier, D.; Morvan, H. Strain rate effects on the compressive response of wood and energy absorption capabilities. Part A: Experimental investigation. Compos. Struct. 2016, 149, 315–328. [CrossRef] 13. Ha, N.S.; Lu, G. A review of recent research on bio-inspired structures and materials for energy absorption applications. Composites 2020, 181, 107496. [CrossRef] 14. Ha, N.S.; Lu, G.; Shu, D.W.; Yu, T.X. Mechanical properties and energy absorption characteristics of tropical fruit durian (Durio zibethinus). J. Mech. Behav. Biomed. Mater. 2020, 104, 103603. [CrossRef][PubMed] 15. Hao, J.; Wu, X.; Oporto, G.; Wang, J.; Dahle, G.; Nan, N. Deformation and Failure Behavior of Wooden Sandwich Composites with Taiji Honeycomb Core under a Three-Point Bending Test. Materials 2018, 11, 2325. [CrossRef] [PubMed] 16. Mach, K.J.; Hale, B.B.; Denny, M.W.; Nelson, D.V. Death by small forces: A fracture and fatigue analysis of wave-swept macroalgae. J. Exp. Biol. 2007, 210, 2231–2243. [CrossRef][PubMed] 17. Bol’Shakov, A.P.; Balakshina, M.A.; Gerdyukov, N.N.; Zotov, E.V.; Muzyrya, A.K.; Plotnikov, A.F.; Novikov, S.A.; Sinitsyn, V.A.; Shestakov, D.I.; Shcherbak, Y.I. Damping properties of sequoia, birch, pine, and aspen under shock loading. J. Appl. Mech. Tech. Phys. 2001, 42, 202–210. [CrossRef] 18. Kolsky, H. An investigation of the mechanical properties of materials at very high rates of loading. Proc. Phys. Soc. Lond. 1949, 62, 676–700. [CrossRef] 19. Bragov, A.M.; Lomunov, A.K. Methodological aspects of studying dynamic material properties using the Kolsky method. Int. J. Impact Eng. 1995, 16, 321–330. [CrossRef] 20. Bazhenov, V.G.; Zeﬁrov, S.V.; Kochetkov, A.V.; Krylov, S.V.; Feldgun, V.R. The Dynamica-2 software package for analyzing plane and axisymmetric nonlinear problems of nonstationary interaction of structures with compressible media. Mat. Modelirovanie 2000, 12, 67–72. 21. Bragov, A.M.; Lomunov, A.K.; Sergeichev, I.V. Modiﬁcation of the Kolsky method for studying properties of low-density materials under high-velocity cyclic strain. J. Appl. Mech. Tech. Phys. 2001, 42, 1090–1094. [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).