Fracture predictions in impact three-point bending test of European beech

Kubíke al. Jou nal o Wood Science (2024) 70:42 h ps://doi.o g/10.1186/s10086-024-02157-x ORIGINAL ARTICLE Open Access © The Au ho (s) 2024. Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his licence, isi h p:// c ea i eco mmons. o g/ licen ses/ by/4. 0/. Jou nal o Wood Science F ac u e p edic ions inimpac h ee-poin bending es o Eu opean beech Pe Kubík1, F an išek Šebek1* , Moj aba Hassan Vand2, Ma in B abec2 and Jan Tippne 2 Abs ac Ha dwood has become widesp ead in Eu opean o es s. The s onges ac o is clima e change and damage o coni- e s by he ba k bee le. The e o o s udy ha dwoods g ows wi h inc easing olume o applica ions. The e o e, Eu opean beech wood was in es iga ed unde wo impac loads in wo ma e ial di ec ions, esul ing in ou unique combina ions supplemen ed by he measu emen o he ic ion coe icien . Then, i was compu a ionally simu- la ed o ep oduce he c acking, while he ma e ial model e lec ed he o ho opic beha iou in elas ici y, plas ici y and ailu e. The model was coded using he use sub ou ine in Abaqus o ini ia e and p opaga e he c ack using he elemen dele ion. The esul ing eac ion o ces we e in good ag eemen wi h hose om he expe imen s. C ack- ing was nume ically simula ed in h ee o ou cases as expe imen ally obse ed, howe e , upon la ge de lec ions. The e o e, he model is applicable o u he in es iga ions. Keywo ds Aniso opy, Ba la , Dynamic, Explici , Flexu e, Rup u e In oduc ion Wood is a widesp ead na u al composi e, which has been used as a cons uc ion ma e ial o cen u ies because i is enewable, biologically deg adable, en i onmen ally iendly, s ong, ligh weigh , easy o manu ac u e, elec i- cally esis an , abso bing noise and aes he ic. The e o e, models capable o p edic ing de o ma ion and ailu e unde ope a ional o andom loading a e needed. Bending is one o he common es s o wood. Jansson [1] in es iga ed ailu e modes and s esses using s a ic and impac bending es s showing dec easing impac bending s eng h wi h dec easing ime o ailu e o he Sp uce Pine Fi (SPF). Yoshiha a e al. [2] used a s a ic h ee-poin bending es o de e mine he shea modulus wi h he help o a co ec ion unc ion and a modi ied Timoshenko beam o six wood species: Si ka sp uce, wes e n hemlock, akama su, yellow popla , shioji and balsa. Yoshiha a e al. [3] conduc ed a simila s udy o asymme ic ou -poin bending es s. Yoshiha a and Oka [4] ca ied ou he comp ession bending es o de e - mine he elas ic modulus, p opo ional limi and bending s eng h on specimens o Japanese i o a ious leng h- o- hickness a ios. Then hey compa ed he esul s wi h he con en ional bending es o con i m ha he co ec lexu al p ope ies can be ob ained om he comp ession bending es o a la ge leng h- o- hickness a io. Kubo- jima e al. [5] used he impac bending es o es ima e he elas ic modulus o Japanese ceda , hondo sp uce, hiba a bo i ae, Japanese ed pine, paulownia, Manchu- ian ash and Japanese e e g een oak. Polocoşe e al. [6] in es iga ed he e ec o low- eloci y impac on ailu e s esses and s i ness using he bending es consis ing o a pendulum and specimens ha ing a leng h o 650mm, wid h o 50mm and hicknesses o 20, 30 and 40mm o ind ha ailu e was signi ican ly di e en compa ed o quasi-s a ic es s o beech, la ch and pine. Polocoşe e  al. [7] ound highe ene gy abso p ion in pine and sp uce ein o ced wi h E-glass on he su ace loaded in *Co espondence: F an išek Šebek [email p o ec ed].cz 1 Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, Technická 2896/2, 616 69 B no, Czech Republic 2 Depa men o Wood Science and Technology, Facul y o Fo es y and Wood Technology, Mendel Uni e si y in B no, Zemědělská 810/3, 613 00 B no, Czech Republic Page 2 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 ension on a pendulum compa ed o specimens wi h- ou ein o cemen . The ac o was 1.4 o pine and 2.5 o sp uce. Jacques e al. [8] ca ied ou ull-scale es ing on indi idual ligh ame lumbe s made om SPF. Then, he ou -poin bending es s wi h s ain a es o 6 × 10–6 o 4 × 10–1 s–1 se ed o p opose he s ess–s ain ela- ionship applicable o he modelling o blas loadings. Olmedo e al. [9] ollowed he dynamic esponse o esh s ems unde impac loading using he Mou on–Cha py pendulum. Then hey analysed he load-bea ing capac- i y o wooden cons uc ions made o elled ees se ing as p o ec ion agains alling ocks. B anche iau e al. [10] p oposed an analy ical ela ionship, which con e s he elas ic modulus es ima ed om he h ee-poin bend- ing es o he elas ic modulus appea ing in he ou - poin bending es . The esul s we e expe imen ally alida ed on specimens om six wood species wi h di - e en densi ies. Di e en elas ic moduli o sp uce and oak in h ee- and ou -poin bending es s we e also analysed by Babiak e al. [11]. Ga e al. [12] examined he e ec o he mal modi ica ion o Eu opean oak and No way sp uce using he Cha py pendulum o ind ha changes in basic chemical componen s and impac bend- ing s eng h we e less a ec ed o oak han o sp uce. The e ec o he mal modi ica ion on impac bend- ing s eng h was also in es iga ed by Hassan Vand and Tippne [13] o ind ha he mal modi ica ion caused a dec ease in de lec ion and maximum longi udinal s ain up o app oxima ely 50% acco ding o digi al image co - ela ion (DIC) o h ee-poin bending es s o i e wood species: ash, beech, la ch, oak and sp uce. Hassan Vand e al. [14] also es ed he e ec o mois u e con en on he beha iou o h ee wood species o beech, oak and sp uce. They e alua ed he wo k equi ed o ini ia e he c ack and b eak he specimens and ound, using DIC, ha he maximum de lec ion and longi udinal ensile s ain inc eased wi h inc easing mois u e con en . The minimum o he li e a u e deals wi h he compu- a ional modelling o wood unde impac . The e o e, his wo k ocuses on he dynamic beha iou o Eu opean beech (Fagus syl a ica L.) unde he h ee-poin bend- ing es . All es s we e ca ied ou using a d op-weigh impac es ing machine wi h wo hamme s o achie e a ious impac eloci ies. Then, nume ical simula ions ollowed o de elop a ma e ial model wi h good ac- u e p edic abili y compa ed o expe imen s. I should be no ed ha many wo ks [15, 16] use he o ho opic yield c i e ion acco ding o Hill [17] o he desc ip ion o plas ic beha iou . Howe e , some ma e ial pa ame e s ( h ee ou o six) o he equi alen s ess can be nega i e due o signi ican ly dis inc yield s esses in espec i e wood di ec ions [18–20]. The nega i e ma e ial pa ame- e s can hen cause he equi alen s ess o be unde ined. The e o e, he o ho opic yield c i e ion acco ding o Ba la e al. [21] will be used o o e come his issue. Expe imen s Ma e ial The ma e ial s udied was Eu opean beech belonging o ha dwoods. The specimens we e made om ee g own nea B no (Czech Republic). The wood was conside ed o ho opic wi h a di ec ion pa allel wi h h ee p ima y (heigh ) g ow h called longi udinal (L), a di ec ion pe - pendicula o he ee p ima y g ow h passing h ough he pi h o he ee called adial (R) and a di ec ion pe - pendicula o ee g ow h and simul aneously as a an- gen o annual ings called angen ial (T) as depic ed in Fig.1. All specimens we e s o ed in a clima ic chambe a a empe a u e o 20°C and a ela i e humidi y o 65% o p oduce a uni o m equilib ium mois u e con en o 12%. Th ee‑poin impac bending es All specimens we e manu ac u ed wi h ib es o ien ed along he leng h o 300mm ha ing a squa e c oss sec- ion wi h an edge o 20mm. Be o e es ing, specimens we e weighed in analy ical balance wi h 1mg eadabil- i y, which esul ed in an a i hme ic mean o densi y o 727kg/m3 wi h a s anda d de ia ion o 33kg/m3. The symme ic h ee-poin bending was ca ied ou in wo ma e ial di ec ions R and T (Fig. 1), espec i ely. The span- o-dep h a io was 12, esul ing in a suppo span o 240mm. Bo h he suppo s and he hamme had a adius o 15mm and a su ace oughness Ra o 0.2μm. The es s we e pe o med on he d op-weigh impac es ing machine DPFes 400 om Labo ech ( igh Fig.2) a oom empe a u e. Two di e en hamme s we e d opped om wo ini ial heigh s o ob ain wo impac eloci ies as gi en in Table1 o p o ide a di e se ma e ial o nume ical simula ions. The es ing machine au oma - ically measu ed he ue impac eloci ies (Table1). Fo each o hese wo condi ions, 6 specimens we e es ed in he R and T di ec ions, esul ing in 4 da ase s. The ham- me displacemen was measu ed wi h 0.01 mm p eci- sion, while he o ce was measu ed using a piezoelec ic o ce ansduce CFT + 50 kN om HBM a ached o he hamme . The es s we e eco ded wi h Fas cam SA-X2 om Pho on. The high-speed came a had a cell size o 20μm and was equipped wi h a Nikon Mic o-Nikko G lens wi h a ocal leng h o 105mm and a Nikon Z TC-2.0 × . I was placed app oxima ely 0.9m om he la e al speci- men su ace, which was pa allel o he came a senso . Addi ional ligh ing was ensu ed by wo Mul iLed QT s andalone lamps (le Fig.2). The ield o iew was i - ed o a cen al pa o la e al specimen’s su ace wi h a hamme . Images wi h a esolu ion o 1024 × 672px we e Page 3 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 cap u ed wi h a ame a e o 20,000 ps. The image pos - p ocessing was pe o med in Vic-2D 2010 so wa e om Co ela ed Solu ions. In o de o de e mine he con- e sion ac o , a simple calib a ion was done using he one-dimensional scale de ined by he known dis ance. The displacemen ield was calcula ed om he ield o 3 × 3p . The Lag ange s ain ield was de e mined wi h he lowes possible s ain il e size o 5 × 5p so ha he maximum possible spa ial esolu ion was achie ed. In 20 T LR L R T 20 240 300 R15 R15 R15 Fig. 1 Loading in he R ( op) and T di ec ions (bo om) wi h all ma e ial di ec ions du ing impac h ee-poin bending es s (all dimensions in mm) Impac es i ng mac hi n e H igh -spee d came a L ED ligh s D a a ac q u i s i i on s y s e m Fig. 2 High-speed came a wi h accesso ies (le ) and d op-weigh impac es ing machine ( igh ) Page 4 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 addi ion, he op ical measu emen se ed o he analysis o he ac u e (i any). Measu emen o  ic ion coe icien T ibological expe imen s we e pe o med o collec inpu o nume ical modelling. The es s we e ca ied ou using he Uni e sal Mechanical Tes e (UMT) T iboLab om B uke (Fig.3). The pin o 5mm in diame e was made o AISI 52100 alloy s eel ha ing a su ace oughness Ra o 0.2μm, he same as he suppo s and he hamme in he h ee-poin impac bending es . I was o ced agains wooden specimens by 196 N, which was measu ed by a dual o ce senso DFH-100 wi h a sampling equency o 100 Hz and co esponded o a con ac p essu e o 10MPa. This con ac p essu e was chosen as i eached ens o MPa in he ollowing nume ical simula ions. The pin oscilla ed wi h a equency o 2Hz and ampli ude o 5mm o 120s. Six specimens wi h dimensions o 40 × 20 × 10mm we e es ed. These specimens we e man- u ac u ed om he same ee as hose o he h ee-poin impac bending es and s o ed in he clima ic chambe o each he same uni o m equilib ium mois u e con en o 12%. The pin was o ced agains h ee LR and h ee LT su aces o 40 × 20mm mo ing in he R and L di ec ions, espec i ely. The e o e, he measu emen was wi hin one annual ing o he L di ec ion and app oxima ely 5 annual ings o he R di ec ion wi h an a e age annual ing wid h o 2.5mm. The pin ajec o y was 10mm, bu he esul s we e p o- cessed in MATLAB R2024a o a 6-mm-long po ion o he specimen in o de o omi he dead ends whe e he pin decele a ed and accele a ed. Then, he conside ed ic ion was kine ic and a ea o e alua ion app oxima ely 50 mm2. Also, he e alua ion was no conduc ed o he beginnings o he es s whe e a ploughing was p esen . Nume ical simula ions The calcula ions we e pe o med in Abaqus/Explici comme cial code based on he explici o mula ion o he ini e elemen me hod. The ma e ial model was imple- men ed using he VUMAT use sub ou ine, as i is no a s anda d one. The ailu e was modelled by dele ing ele- men s ha eached c i ical damage, which is a simple echnique ha does no equi e emeshing, con a y o he node sepa a ion me hod. Model o ma e ial The hamme and suppo s we e made o s eel, which is much s i e han he es ed wood specimens. The e o e, he s eel pa s we e modelled as igid, sa ing some com- pu a ional ime. On he con a y, wood was modelled as a homogeneous o ho opic con inuum. The he e o- genei y was neglec ed o sa e some compu a ional ime again h ough ma e ial model simpli ica ion, including i s calib a ion. The o ho opic elas ici y was desc ibed by he gene al- ised Hooke’s law as whe e εi , Ei and σi a e he no mal s ain, elas ic modulus and no mal s ess, espec i ely, εij , νij , Gij and σij a e he enso ial shea s ain, Poisson’s a io, shea modulus and shea s ess o i,j=L,R,T , espec i ely. All elas ic con- s an s (Table2) we e aken om [20], whe e he disin e- g a ion o he same wood was ca ied ou unde a ious s ain a es in a ious ma e ial di ec ions. The o ho opic plas ici y was desc ibed by he model o Ba la e al. [21], who p oposed he yield condi ion as (1)        εL εR εT εLR εRT εTL        =           1 EL − νLR EL − νLT EL 000 − νRL ER 1 ER − νRT ER 000 − νTL ET − νTR ET 1 ET 000 0001 2GLR 00 0 0 001 2GRT 0 0 0 0001 2G TL                  σL σR σT σLR σRT σTL        (2) σB−σy=0, Table 1 Con igu a ion o he impac h ee-poin bending es s Hamme weigh (kg) Ini ial heigh (mm) A e age ue impac eloci y (m/s) Numbe o specimens (–) Ma e ial di ec ion (–) 9.05 458.87 2.70 6 R 9.05 458.87 2.70 6 T 4.55 815.77 3.25 6 R 4.55 815.77 3.25 6 T D i in g mechanism Sp ec i men h o ld e Sp ecimen Pin F o ce se n so L oa di n g mec h an i sm Fig. 3 UMT T iboLab used o measu emen o ic ion coe icien Page 5 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 whe e σy is he yield s ess, which is dependen on he equi alen plas ic s ain (Fig.4), while σB is he equi a- len s ess acco ding o Ba la e al. [21] as (3) σ B =1 m √2 m | K1 − K2 | m +| K2 − K3 | m +| K3 − K2 | m , whe e m is he exponen (in luencing he shape o he yield su ace), K1,K2 and K3 a e he p incipal alues o he linea ly ans o med de ia o ic s ess enso whe e a , b , c , , g , and h a e he ma e ial pa ame e s, which we e i ed (Table3; Fig.5) so ha he yield su ace co esponded o ha in [20], whe e he model o Hill [17] was used wi h a nega i e ma e ial pa ame e (H<0) . A change in he yield c i e ion was sough due o he elimi- na ion o a nega i e ma e ial pa ame e esul ing om a signi ican di e ence in he yield s esses in he L di ec- ion compa ed o o he s (R and T in Fig.5), which can be p oblema ic in some cases. The yield c i e ion was aken om [20], whe e i was no necessa y o model di - e en yield s esses in ension and comp ession in ini e elemen s. Then, he yield c i e ion was ecalib a ed o Eu opean beech, s ill neglec ing he s eng h di e en ial e ec . Howe e , mo e sophis ica ed yield c i e ia should be conside ed when signi ican di e ences in ensile and comp essi e yield s esses we e obse ed. (4) K =    c (σL − σR) −b (σT − σL) 3hσLR gσTL hσLR a(σR−σT)−c(σL−σR) 3 σRT gσTL σRT b(σT−σL)−a(σR−σT) 3   , Table 2 Elas ic cons an s o Eu opean beech wood [20] Ma e ial di ec ion L R T LR RT TL Elas ic modulus (MPa) 13,000 3500 3000 – – – Poisson’s a io (–) – – – 0.302 0.362 0.318 Shea modulus (MPa) – – – 1608 460 1059 00.2 0.4 0.6 0.8 1.0 Equi alen plas ic s ain [–] 0 10 20 30 40 50 Y i e ld s ess [MPa] Fig. 4 Flow cu e o he Eu opean beech wood [20] Table 3 Plas ici y- ela ed ma e ial pa ame e s o Eu opean beech m (–) a (–) b (–) c (–) (–) g (–) h (–) 2 2.357 0.862 0.009 0.456 0.642 0.525 15 30 37.5 22.5 7.5 –7.5 –22.5 –37.5 –75 –45 –15 15 45 75 –75 –45 –15 15 45 75 R[MPa] 2 3 ( ( 0,5 T[MPa] 2 3 ( ( 0,5 R[MPa] L[MPa] 2 3 ( ( 0,5 L [MPa ] LR [MPa] Fig. 5 Yield locus in he Haigh–Wes e gaa d space (le ) and he space o wo no mal and one shea s esses ( igh ) o he Eu opean beech wood Page 6 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 The o ho opic cumula i e damage was desc ibed by he damage pa ame e as: whe e ε i , εD i and εp i a e he ac u e s ain, plas ic s ain o a gi en loading pa h and plas ic s ain, espec i ely. As men ioned abo e, he elemen is emo ed when he c i ical alue is eached, speci ically when max (Di)=1 . Fu he mo e, ac u e s ains a e dependen on s ess iaxiali y (apa om he a e dependence in oduced la e ) o inco po a e he ension/comp ession ailu e asymme y (which is simple han he ac u e models in [15, 22–24]) whe e σm is he mean s ess and σ is he equi alen s ess acco ding o on Mises The dependence o ac u e s ains on s ess iaxial- i y was based on [20] and was u he e ined using he ial and e o me hod using ens o nume ical simula- ions so ha he esul s ma ched expe imen al obse a- ion in impac h ee-poin bending es s. F ac u e s ains dependen on he s ess iaxiali y, ε i (η ) , a e shown in Fig.6 o he e e ence equi alen plas ic s ain a e o 1 s–1. Finally, ac u e s ains we e addi ionally depend- en on he equi alen plas ic s ain a e as expe imen s we e conduc ed a a ious impac eloci ies. Con a y o he linea a e dependence in [20] o signi ican ly g ea e s ain a es, an exponen ial dependence on he equi alen plas ic s ain a e based on he equa ion p o- posed by Johnson and Cook [25] was ecalib a ed o i he expe imen s as ollows (using ε i (η ) ha has al eady been calib a ed p e iously o he e e ence equi alen plas ic s ain a e o 1 s–1) whe e C is he ma e ial pa ame e , calib a ed as 0.1 a e se e al addi ional nume ical simula ions o e e ence (5) D i= ε D i  0 1 ε i  dεp i , (6) η= σm σ , (7) σ m = σ L +σ R +σ T 3, (8) σ= 1 √2 (σL − σR)2 + (σR − σT)2 + (σT − σL)2 + 6σ2 LR + 6σ2 RT + 6σ2 TL . (9) ε i=ε i(η)  1+Cln  ˙ ε p ˙ ε0 , equi alen plas ic s ain a e ˙ ε 0=1s −1 . Finally, ˙ εp is he equi alen plas ic s ain a e whe e ˙εp is he plas ic s ain a e enso . Model o geome y andbounda y condi ions As men ioned abo e, he ad an age o he elemen dele- ion echnique is i s simple implemen a ion. Howe e , i has i s d awbacks as i depends on he size o he ele- men , which is usually kep as small as possible o eal- is ically p opaga e he c ack using one o wo elemen s. The e o e, he elemen s had a size o 0.1mm in a eas o wood ailu e as well as in con ac egions (Fig.7). O he a eas we e meshed wi h elemen s o a size o 1.82mm. Only a 0.1-mm-wid h specimen was modelled as a plane s ain o ep esen he inne laye o wood ins ead o using 20mm o sa e conside able compu a ional ime. Geome y was meshed wi h 8-node linea b ick ini e elemen s wi h educed in eg a ion and hou glass con- ol (labelled C3D8R in Abaqus). The mesh consis ed o 128,774 nodes and 63,850 elemen s o which 4 we e addi- ionally 6-node linea wedge ini e elemen s wi h educed in eg a ion and hou glass con ol (labelled C3D6R in Abaqus) nea he con ac egions whe e he ine mesh changed in o coa se mesh (highligh ed in ed in Fig.7). As men ioned ea lie , he hamme and suppo s we e modelled as igid, he e o e, as su aces which come in o con ac wi h he specimen. The wid h o he igid (10) ˙ ε p = 2 3˙ εp :˙ εp , S ess iaxiali y [–] –0.5 00.5 1.0 0 0.5 1.0 1.5 2.0 F ac u e s a i n [–] 0 L R T Fig. 6 F ac u e s ain o all ma e ial di ec ions and he e e ence equi alen plas ic s ain a e o 1 s–1 Page 7 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 bodies, 0.3mm, was g ea e han he wid h o he spec- imen, 0.1mm, and we e cen ed wi h each o he . The hamme and suppo s we e meshed wi h 4-node bilin- ea quad ila e al igid ini e elemen s (labelled R3D4 in Abaqus) ha ing a size o 0.1mm, he e o e, 3 ele- men s pe wid h. One suppo , modelled he same as a hamme , had 1892 nodes and 1416 elemen s. Mo eo- e , an elemen was added o he e e ence poin o he hamme (highligh ed by a blue pen ag am in Fig.7) in o de o inco po a e he olume load (labelled MASS in Abaqus), equi alen o he hamme weigh . Displacemen s and o a ions o e e ence poin s o he suppo s we e es ic ed. The same applies o he hamme excep o he e ical displacemen . The ini ial eloci y was p esc ibed in his di ec ion, co - esponding o he a e age ue impac eloci y o he expe imen (Table1). The specimen had cons ained displacemen s only in he wid h di ec ion o simula e he plane s ain condi ion ( he geome y o he speci- men was disc e ised by 1 elemen along he wid h). The accele a ion due o g a i y o 9.807 m/s2 was applied on he whole geome y in he di ec ion o impac . In addi ion o he de ini ion o he con ac be ween he specimen, hamme and suppo s, sel -con ac was applied o all elemen edges in he a ea o expec ed ailu e (whe e he elemen s wi h 0.1mm edges we e). I ensu ed he con ac o new ee su aces, which we e no ini ially p esen in he geome y, eme ging a e he elemen dele ion ha simula ed he c acking. The no mal beha iou o he con ac s was se as ‘ha d’ o allow any p essu e when he su aces a e in con ac , while he ic ion coe icien o 0.15 ob ained om he expe imen s in he p e ious sec ion was se in he an- gen ial di ec ion. Resul s anddiscussion Figu es8 and 9 summa ise all he o ce esponses agains he hamme displacemen s ob ained om he impac es ing machine o hamme s ha weighed 9.05 and 4.55kg, espec i ely. The DIC did no se e o ob ain he displacemen s. The do ed lines highligh he es s in which he specimens emained in ac . The co espond- ing specimens bended o wa d and back ( he hamme e u ned app oxima ely o he e e ence posi ion co e- sponding o he displacemen o 0mm—i is ze o de lec- ion). The dashed-do ed lines highligh he es s in which he specimens pa ially b oke (did no b eak h ough he whole heigh ). The co esponding specimens exhibi ed signi ican sp ingback, he e o e, he hamme e u ned app oxima ely o he e e ence posi ion. The solid lines highligh he es s in which he specimens ailed (b oke h ough he en i e heigh ). The co esponding specimens emained ben a e he es (all in Fig.8). Mo eo e , he momen o c ack ini ia ion is highligh ed by a ci cle. The a e age alue o he hamme displacemen co espond- ing o he momen o c ack ini ia ion is highligh ed by a hick e ical dashed g ey line. This line is no plo ed in le Fig.9, whe e only 2 ou o 6 specimens pa ially b oke and none ailed o he hamme weighing 4.55kg loading in he R di ec ion. Only 1 specimen ailed o he ham- me ha weighed 4.55kg in he T di ec ion ( igh Fig.9). O he specimens pa ially b oke, while he specimen co - esponding o Tes 6 in igh Fig.9 was c acked along almos he whole heigh unde hese condi ions, he e- o e, he hamme s a ed e u ning la e . The a i hme ic mean ic ion coe icien o 0.15 wi h a s anda d de ia ion o 0.005 was measu ed i espec- i e o he ma e ial di ec ion. Then, he expe imen ally and compu a ionally ob ained esul s a e compa ed in Figs.10 and 11. The p edic ed o ces we e in good Fig. 7 Fini e elemen mesh wi h de ails Page 8 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 ag eemen wi h he expe imen s be o e he maximum de lec ion. Fo ces oscilla ed mo e in nume ical simula- ions han in impac h ee-poin bending es s. Howe e , his is an in insic ea u e o explici dynamics. C ack- ing was p edic ed la e in nume ical simula ion han in expe imen s o a 9.05kg hamme in Fig.10. No c acking 0246810 12 14 Displacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] Tes 1 Tes 2 Tes 3 Tes 4 Tes 5 Tes 6 0246810 12 14 Displacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] Tes 1 Tes 2 Tes 3 Tes 4 Tes 5 Tes 6 R, 9.05 kg T, 9.05 kg Fig. 8 Fo ce–displacemen esponses o a 9.05 kg hamme in he R (le ) and T di ec ions ( igh ), espec i ely ( o in e p e a ion o he e e ences o colou in his igu e legend, he eade is e e ed o he web e sion o his a icle) 0246810 12 14 Dis p lacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] T, 4.55 kg Tes 1 Tes 2 Tes 3 Tes 4 Tes 5 Tes 6 0246810 12 14 Dis p lacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] R, 4.55 kg Tes 1 Tes 2 Tes 3 Tes 4 Tes 5 Tes 6 Fig. 9 Fo ce–displacemen esponses o a 4.55 kg hamme in he R (le ) and T di ec ions ( igh ), espec i ely ( o in e p e a ion o he e e ences o colou in his igu e legend, he eade is e e ed o he web e sion o his a icle) 0246810 12 14 Dis p lacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] R, 9.05 kg Expe imen s Simula ion 0246810 12 14 Dis p lacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] T, 9.05 kg Expe imen s Simula ion Fig. 10 Compa ison o o ce–displacemen esponses om nume ical simula ions and expe imen s o a 9.05 kg hamme in he R (le ) and T ( igh ) di ec ions, espec i ely Page 9 o 12 Kubíke al. Jou nal o Wood Science (2024) 70:42 was achie ed o he R di ec ion wi h a 4.55kg hamme (le Fig. 9) compu a ionally o ep oduce he expe i- men s (le Fig.11). Howe e , pa ial c acking was no achie ed compu a ionally o he T di ec ion wi h a 4.55kg hamme as in he expe imen s ( igh Fig.9). The elas ic s ain ene gy was so high in nume ical simula ions o ha case ha led o he p edic ion o c ack p opa- ga ion along he whole heigh , con a y o he impac h ee-poin bending es s whe e specimens b oke jus pa ially ( igh Fig. 9). The homogenei y o he model also con ibu ed o he lack o p edic ion o c ack a es , which occu ed in expe imen s a e some delamina ion because, among o he s, he wood is s ongly inhomoge- neous. This should be aken in o accoun by modelling he geome y based on da a om compu ed omog aphy speci ically o each specimen, which migh no be p ac- ical o indus ial applica ions. The p edic ed con ou s o he damage pa ame e a e displayed in Figs.12 and 13 a e he impac h ee-poin bending es s compa ed o expe imen s app oxima ely in scale. Howe e , i should be no ed ha he p edic ed de lec ions a e g ea e (app oxima ely 1.5mm), as is e i- den om Figs.10 and 11. The damage pa ame e s DR and DT eached hei c i ical alue in espec i e R and T 0246810 12 14 Dis p lacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] R, 4.55 kg Expe imen s Simula ion 0246810 12 14 Dis p lacemen [mm] 0 1000 2000 3000 4000 Fo ce [N] T, 4.55 kg Expe imen s Simula ion Fig. 11 Compa ison o o ce–displacemen esponses om nume ical simula ions and expe imen s o a 4.55 kg hamme in he R (le ) and T ( igh ) di ec ions, espec i ely R, 9.05 kg T, 9.05 kg 38.029.000.185.076.057.033.024.005.080.071.052.00.00 Fig. 12 Expe imen s ( op) compa ed o p edic ed con ou s o he damage pa ame e (bo om) o he 9.05 kg hamme and he R (le ) and T ( igh ) di ec ions (app oxima ely in scale)