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N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000
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Procedia Structural Integrity 28 (2020) 2168–2173 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete mechanical systems Nikita Kazarinov a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersburg, 190031, Russia 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete mechanical systems Nikita Kazarinov a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersburg, 190031, Russia N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete mechanical systems Nikita Kazarinov a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersburg, 190031, Russia N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete mechanical syste s Nikita Kazarinov a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersb rg, 190031, Russia N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete echanical systems Nikita Kazarin v a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersb rg, 190031, Russia Abstract Dynamic fracture effects such as fracture del y and dependen e of the material strength on the loading rate are known from multiple experiments on crack initiation and spallation. Clas ic linear oscillator (mass-spring system) is considered and critical spring stretch fracture condition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic loading. Moreover, the mass-spring model can be calibrated to simulate some known experimental results on dynamic crack initiation due to pulse loading. © 2020 he Auth rs. Pu lish by ELSEVIER B.V. This is an en ac ess article under the CC BY-NC-ND lic n e (h tps://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the Europ an Str ctural Int rity Society (E I ) xCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 1 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete echanical systems Nikita Kazarin v a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersb rg, 190031, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia Abstract Dynamic fracture effects such as fracture delay and dependen e of the material strength on the loading rate are known from multiple exp rim nts on crack initiation and spallation. Classic linear oscillator (mass-spring system) is considered and critical spring stretch fracture condition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic loading. Moreover, the mass-spring model can be calibrated to simulate some known experimental results on dynamic crack initiation due to pulse loading. © 2020 The Authors. Published by ELSEVIER B.V. This is an ope access article under th CC BY-NC-ND license (h tps://creativecommons. rg/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity Nikita K c P t Saint Petersburg, 199034, Russia ic ure e fects such a fract d lay and de c on a s a s r t n o D r l c ut r analyti l d i lo t m s simulate some known experimental results on dynamic crack initiation due a © A h t P K y ds: ture, fracture delay, rate sensitivity . azari t l./ tr t r l I t rit r i ( ) n discrete e nical syst N a b c a Emperor Alexander I St. t rs rg State Tr s ort U i rsit , aint Petersburg, , ssi b Inst robl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersbur t t i rsit , Saint Petersburg, 199034, Russia A tract Dynamic fracture effects such as fracture de y and de f t t ri l tr t t l i r t r fr lti l ri t on crack initiatio ll ion. la si lin r oscill t r ( -s ri y t m) i i r a iti al spring stretch fr t r c iti i . Due to relative simplicity of the model and availability of analytical solutions, th dynamic fracture effects can be studie analyti ll f r t f i l i . r r, t - ri l li r t to simulate some known experimental results on dynamic crack initiation due t l l ing. © 2020 T t . l I . . i i rti l r - - li ( tt :// r ti commo . r /li / - - / . ) r-r i r r i ilit f t r tr t r l I t rit i t ( I ) r s: iscrete models, dynamic fracture, fracture delay, rate sensitivity N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 1 1st Virtual European Conference on Fracture Dynamic fracture effects observed in discrete echanical systems Nikita Kazarinov a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersburg, 190031, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia Abstract Dynamic fracture effects such as fr cture de y and dependence of the material strength on the loading rate are known from multiple experim nts on crack initiation and spalla ion. Classic linear oscillator (m ss-spring system) is considered and c tical spring stretch fracture condition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic l adi g. Moreover, the mass-spring model can be calibra ed to simulate so e known experimental results on dynamic crack initiation due to pulse loading. © 2020 The Autho s. Publ shed by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity N. Kazarinov et al./ Structural Integrity Procedia 00 (2020) 000–000 1 1st Virtual Europ an Conference on Fracture Dynamic fracture effects observed in discrete echanical systems Nikita Kazarinov a , Yuri Petrov b , Alexander Smirnov c a Emperor Alexander I St. Petersburg State Transport University, Saint Petersburg, 190031, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia Abstract Dynamic fracture effects such as fracture del y and dependence of the material strength on the loading rate are known from multiple experim nts on crack initiation and spalla ion. Clas ic linear oscill tor (m ss-spring system) is considered and c itical spring stretch fracture condition is added. Due to rel tive simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic l adi g. Moreover, the mass-spring model can be calibra ed to simulate some known exp rimental results on dynamic crack initiation due to pulse loadi g. © 2020 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity 2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo 10.1016/j.prostr.2020.11.044 In (1) critical spring deformation fracture condition is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: s: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator model. Moreover, appropriate choice f the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) 0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring deformation fracture condition is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: More complicated discrete systems such as chains of oscillators have b en thoroughly studied in many works: infinite oscillator chains were used by Slepyan and Troyankina (1984), oscillator chains are used to investigate heat transfer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this paper a si gle linear oscillator system will be studied for a case of various dynamic loads: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator model. Moreover, appropriate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) 0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring deformation fracture condition is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , wher and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: Mass – elastic spring models is a rather simple mechanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheles , the oscillator model can b a useful tool to describe complicat d mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element method to investigate acoustic emission and in McNelley and Gates (1978) mass-spring models are used to address inverse rate sensitivity of materials. More complicated discrete systems such as chains of oscillators have been thoroughly studied in many works: infinite oscillat r chains were used by Slepyan and Troyankina (1984), oscillator chains are used to investigate heat transfer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this paper a single linear oscillator system will be studied for a case of various dynamic ads: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the line r oscillator model. Moreover, appropr ate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) 0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring deformation fracture condition is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , wher and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: M ss – elastic spring models is a rather simple mechanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheless, the oscillator model can be a useful tool to describe complicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element me d to inve igate acoustic emission and in McNelley and Gates (1978) mass-spri g models re used to address inverse rate sensitivity of materials. More complicated discrete systems such as chains of oscillators have been thoroughly studied in many works: infinite oscillator chai s were us d by Slepyan nd Troyankina (1984), oscillator chains are used to inve tigate heat tr n fer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this paper a single linear oscillator system will be studied for a case of various dynamic loads: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator mod l. Mor ov r, appropriate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) 0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring deformation fracture condition is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: Mass – elastic sp ing models i simp e scillat s N illator- ike model is coupled with finite lement method to inve tigate aco G e i t y c ns were used b 84), oscill c investigat h an ki m c cr o di in work Gorbushin and Mishuris (2019). a m w ied of io s ic l a short pulse load rl r d e o d ar present for the l o d . ate choice of the o cillator model parameters o da k u c a t’s consider mass on an elast ( ) i ̈ = ( ) 0 = ̇( 0 ) ! 1), I 1) critical spring deformation fracture condition is also listed. Prob r pulse load ng case, when = , whe tion. s i n oblem ( l ti i l i t i l i al system adm tt n li it l ti ti ti i r l . t l , t ill t l l t l t i li t i al phenomena. For example, in ll t t ill t li l i l it i it l t t t i i t ti i i i ll t i l t i t nsitivit t i l . li t i t stems such i illators have n t ly tudie i s: infinite oscillator i l i , ill t i t i ti t t t t i i t . , i tion phenomenon is studied using lattice mo els in work Gorbushin and Mishuris (2019). t i i l li ill t t ill e studied for a case of various dynamic lo : t l l li l ri in l . D t i ti t a t t l t t t ence n th l in r t t t li os ill t l. , i t i t ill t l t ll ws one to describe some known experim tal ta n cr i iti ti i l in c ditio s. . l t’ onsi ass on l ti sp i it ti li t t . l ti i i ti , i ti i t e following balance equation: ̈ = ( ) ( ) = ̇( ( ) ⇔ (1), iti l i ti t iti i l li t . l il l t l l loading , ( , t l lit ti i l i i i te nction. T e l ti bl i t ll i : Mass – elastic spring models is a rather simple m chanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheless, the oscill tor model can be a useful tool to describe co plicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element met d to inve igate acoustic emission and n McNelley and Gates (1978) mass-spring models are used to address inverse rate sensitivity of materials. More complicated discrete systems such as chains of oscillators have been thoroughly studied in many works: infinite oscillator chai s were us d by Slepyan and Troy nkina (1984), oscillator chains are used to inve tigate heat tr n fer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this pap r a single linear oscillat r system will be studied for a case of various dynamic loads: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator mod l. Mor ov r, appropriate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) (0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring def rmation fracture conditi n is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration orrespondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: 1. Introduction Mass – elastic spring models is a rather simple m chanical system adm tting explicit solutions of equation of motion for a wide range of loads. Nevertheless, the oscill tor model can be a useful tool to describe complicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element met d to inve igate acoustic emission and n McNelley and Gat s (1978) mass-spri g mod ls are used to address inverse rate sensitivi y of material . More c mplicated discrete systems such as chains of scillators have been thoroughly studied in many works: infinite oscillator chai s were us d by Slepya and Troy nkina (1984), oscillator chains are used to i ve tigate heat tran fer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenom non is stu ied using lattice mo els in w rk Gor ushin and Mishuris (2019). In this pap r a single linear oscill or system will be studied for a case of va ious dynamic loads: a short pulse load and linearly rising load. Due to nertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator model. Mor over, appropriate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) (0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring def rmation fracture conditi n is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Abstract Dynamic fracture effects such as fracture delay and dependence of the material strength on the loading rate are known from multiple experiments on crack initiation and spallation. Classic linear oscillator (mass-spring system) is considered and critical spring stretch fracture condition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic loading. Moreover, the mass-spring model can be calibrated to simulate some known experimental results on dynamic crack initiation due to pulse loading. © 2020 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity Abstract Dynamic fracture effects such as fracture delay and dependence of the material strength on the loading rate are known from multiple experiments on crack initiation and spallation. Classic linear oscillator (mass-spring system) is considered and critical spring stretch fracture condition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic loading. Moreover, the mass-spring model can be calibrated to simulate some known experimental results on dynamic crack initiation due to pulse loading. © 2020 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity Abstract Dynamic fracture effects such as fracture delay and dependence of the material strength on the loading rate are known from multiple experiments on crack initiation and spallation. Classic linear oscillator (mass-spring system) is considered and critical spring stretch fracture condition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic loading. Moreover, the mass-spring model can be calibrated to simulate some known experimental results on dynamic crack initiation due to pulse loading. © 2020 The Authors. Published by ELSEVIER B.V. This is an open acce s article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of th European Structural Integri y So iety (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity Abstract Dynamic fracture effects such as fracture del y and dependen e of the material strength on the loading rate are known from multiple exp riments on crack initiation and spallation. Clas ic linear oscillator (mass-spring system) is considered and critical spring stretch fracture ondition is added. Due to relative simplicity of the model and availability of analytical solutions, the dynamic fracture effects can be studied analytically for the case of dynamic loading. Moreover, the mass-spring model can be calibrated to simulate some known experimental results on dynamic crack initiation due to pulse loading. © 2020 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND lic e (h tps://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: discrete models, dynamic fracture, fracture delay, rate sensitivity b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia b RAS Inst Probl Mech Engng, Saint Petersburg, 199178, Russia c Saint Petersburg State University, Saint Petersburg, 199034, Russia 1. Introduction 1. Introduction 1. Introduction 1. Introduction 1. Introduction 1. Introduction 1. Introductio 1. Introduction Mass – elastic spring models is a rather simple mechanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheless, the oscillator model can be a useful tool to describe complicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element method to investigate acoustic emission and in McNelley and Gates (1978) mass-spring models are used to address inverse rate sensitivity of materials. More complicated discrete systems such as chains of oscillators have been thoroughly studied in many works: infinite oscillator chains were used by Slepyan and Troyankina (1984), oscillator chains are used to investigate heat transfer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this paper a single linear oscillator system will be studied for a case of various dynamic loads: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator model. Moreover, appropriate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) (0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), In (1) critical spring deformation fracture condition is also listed. Problem (1) can be easily solved for a rectangular pulse loading case, when ( ) = [ ( ) − ( − )] , where and are the pulse amplitude and duration correspondingly and ( ) is Heaviside step function. The solution of problem (1) is the following: Mass – elastic spring models is a rather simple mechanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheless, the oscillator model can be a useful tool to describe complicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element method to investigate acoustic emission and in McNelley and Gates (1978) mass-spring models are used to address inverse rate sensitivity of materials. More complicated discrete systems such as chains of oscillators have been thoroughly studied in many works: infinite oscillator chains were used by Slepyan and Troyankina (1984), oscillator chains are used to investigate heat transfer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this paper a single linear oscillator system will be studied for a case of various dynamic loads: a short pulse load and linearly rising load. Due to inertia of the mass the fracture delay effect and strength dependence on the loading rate are present for the linear oscillator model. Moreover, appropriate choice of the oscillator model parameters allows one to describe some known experimental data on crack initiation under dynamic loading conditions. 2. Fracture delay effect Let’s consider mass on an elastic spring with stiffness and force ( ) applied to the mass. The mass deflection is described by function ( ) , which satisfies the following balance equation: ̈ + = ( ) (0) = ̇(0) = 0 ( ) ≥ ! ⇔ (1), Mass – elastic spring models is a rather simple mechanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheless, the oscillator model can b a useful tool to describ c mplicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite element method to investigate acoustic emission and in McNelley and Gates (1978) mass-spring models are used to address inverse rate sensitivity of materials. More complicated discrete systems such as chains of oscillators have been thoroughly studied in many works: infinite oscillator chains were used by Slepyan and Troyankina (1984), oscillator chains are used to investigate heat transfer effects (Kuzkin and Krivtsov (2018)). Moreover, dynamic crack propagation phenomenon is studied using lattice models in work Gorbushin and Mishuris (2019). In this paper a single linear oscillator system will be studied for a case of various dynamic l Mass – elastic spring models i a r ther simple mechanical system admitting explicit solutions of equations of motion for a wide range of loads. Nevertheles , the scillator model can be a useful tool to describe complicated mechanical phenomena. For example, in Ngo and Pellet (2020) the oscillator-like model is coupled with finite el ment method to investigate acoustic emission and in McNelley and Gates (1978) mass-spring models are used to address inverse rate sensitivity of materials. . i
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