ON LINEAR VISCOELASTICITY WITHIN GENERAL FRACTIONAL DERIVATIVES WITHOUT SINGULAR KERNEL

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1 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 S335 ON LINEAR VISCOELASTICITY WITHIN GENERAL FRACTIONAL ERIVATIVES WITHOUT SINGULAR KERNEL by Feng GAO ab Xiao-Jun YANG ab* and Syed Tauseef MOHYU-IN c a Sae Key Laboraory for Geomechanics and eep Underground Engineering China Universiy of Mining and Technology Xuzhou China b School of Mechanics and Civil Engineering China Universiy of Mining and Technology Xuzhou China c eparmen of Mahemaics Faculy of Sciences HITEC Universiy Taxila Can Pakisan Original scienific paper hps://doi.org/.98/tsci73897g The Riemann-Liouville and Capuo-Liouville fracional derivaives wihou singular kernel are proposed as mahemaical ools o describe he mahemaical models in line viscoelasiciy in he presen aricle. The fracional mechanical models conaining he Maxwell and Kelvin-Voig elemens are graphically discussed wih he Laplace ransform. The resuls are accurae and efficien o reveal he complex behaviors of he real maerials. Key words: fracional derivaives wihou singular kernel Laplace ransform Maxwell elemen Kelvin-Voig elemen linear viscoelasiciy Inroducion Fracional derivaives (F) of he Riemann-Liouville () and Liouville-Capuo () ypes [-5] due o he power-law funcions has been successfully uilized o invesigae he imporan models in he fields of solid mechanics especially in linear viscoelasiciy (see e. g. [6-] and he relaed references herein). The general F involving he kernels of he exponenial and Miag-Leffler funcions were proposed in he sense of he and ypes see [-]. For example a new fracional-order hea-diffusion equaion (FHE) was proposed in []. The fracional-order groundwaer flow (FGF) was repored in [3]. The fracional-order C circui (FCC) was suggesed in [4]. The hea-flows (HF) of fracional-order were invesigaed in [5-7]. The anomalous diffusion [8] and relaxaion [9] models wihin general F were proposed. The rheological models in he differen operaors were repored in [-]. Recenly he general F of Capuo-Fabrizio ype was used o model he fracional-order Maxwell model (see e. g. [3]). The main aim of he previous aricle is o invesigae he fracional-order Maxwell and Kelvin-Voig models in he general F wih he aid of he Laplace ransform. Mahemaical ools In his secion we presen he definiions of he general F wihou singular kernel in he sense of he and ypes used in his paper. * Corresponding auhor dyangxiaojun@63.com

2 S336 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 The general F wihou singular kernel of he funcion ψ () in he sense proposed by Capuo and Fabrizio is defined as see [ -8]: p( ) /( ) and Μ( ) represens (see [3] and he cied references herein): ( ) [ p( )( τ)] () ψ Μ e ψ τ dτ () where Λ( ) is a normalizaion parameer wih respec o ( (]) and p( ) /( ) and Μ( ) represens (see [3] and he cied references herein). ( ) Λ( ) Μ ( ) () ( ) The general F wihou singular kernel of he funcion ψ () in he sense of he ype proposed by Yang Srivasava and Machado is defined as [5 8]: ( ) [ p( )( τ)] ψ Μ e ψ τ dτ (3) The relaionship of eqs. () and (3) is presened as [3]: ( ) ( ) ( ) ψ ψ e ψ Μ (4) The general F wihou singular kernel of he funcion ψ () in he sense is defined as see [8]: [ ( τ )] () e ψ e ψ τ dτ (5) and he general F wihou singular kernel of he funcion ψ () in he sense as [8]: [ ( τ )] e ψ e ψ τ dτ (6) The relaionship of eqs. (5) and (6) is as follows [8]: e ψ ψ e ψ (7) e The general F wihou singular kernel of he funcion ψ () in he sense is defined as see [8]: ( τ ) () e ψ e ψ τ dτ + (8) and he general F wihou singular kernel of he funcion ψ () in he sense by [8]: ( τ ) e + ψ e ψ τ dτ (9) The relaionship of eqs. (8) and (9) is as follows [8]: e ψ ψ eψ () e + + The general F wihou singular kernel of he funcion ψ () in he sense is defined:

3 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 S337 ψ () ψ ( τ) dτ () τ and he general F wihou singular kernel of he funcion ψ () in he sense as: ψτ ψ dτ () τ The relaionship of eqs. () and () is as: ψ ψ ψ (3) We noice ha eqs. () and (3) are he special cases where he orders of he and CL derivaives is equal o. The Laplace ransform (LT) of he funcion ψ () is defined as [8]: s ψ ψ s : e ψ d (4) The LT of he general F operaors [8] are lised in ab.. Fracional mechanical models In his secion we discuss he laws of deformaion for he real maerial and general mechanical models. Laws of deformaion for he real maerial According o he equaion of he power-law sress relaxaions [] we easily wrie he equaions of he sress relaxaion: () ε( τ) () ε() (5abcd) () ()e ε () e ε() where is he sress and ε is he srain. We noice ha eq. (5a) is he classical case [7] and do no discuss i here. Fracional mechanical models In order o presen he general fracional-order Maxwell and Kelvin-Voig models we consider he following general fracional mechanical elemens conaining spring and dashpo models: ( τ ) ε ( τ ) (6) Table. The LT of he differen general F operaors General F LT operaors () Μ( ) sψ( s) ψ() s+ p( ) ψ [ ] d ε () τ (7) d respecively where is he Young s modulus is he viscosiy is he maerial-dependan parameer and / represens he general F operaors menioned above. Expressions ( ) ψ () Μ( ) s ψ () s s+ p( ) s + s s ψ () e [ ψ ψ() ] e( ) ψ () sψ () s s + s s s ψ () e + [ ψ ψ() ] e( + ) ψ () sψ () s s ψ ss [ ψ( s) ψ() ] () ψ () sψ () s

4 S338 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 J () ε()/ () and G () ()/ ε () are he creep compliance funcion (CCF) and relaxaion modulus funcion (RMF) where () is he iniial sress and ε () is he iniial srain. Wih he help of he Bolzmann superposiion principle and under he assumpion of causal hisories [7 3] we have he following consiuive equaions of sress and srain of he Volerra ype: and d τ ε() () J () + J ( τ) dτ dτ (8) d ετ () ε() G () + G ( τ) dτ dτ respecively. Taking he Laplace ransform for eqs. (8) and (9) yields: which leads o: where (9) ε() s J() s Ε() s () s () s G() s Ε()() s ε s () J() sgs () () Ε () s d ετ d τ dτ dτ Ε () s [ ε() ] [ () ] represens he parameer relaed o he corresponding general F operaors i. e. Ε () s s Ε ( s) s/( s ) Ε( s) Μ ( ) s/[ s+ p( )] and Ε ( s) s/( s ). The creep and relaxaion processes for he general fracional-order Maxwell models A general fracional-order Maxwell model (GFMM) is ha he branch of he spring and dashpo is in series [7]. The consiuive equaions (CE) for he GFMM are given as: d () () d ε() d d Subsiuing eqs. () (3) (5) (6) (8) (9) () and () ino eq. () yields: ( ) ( ) ε ε ε ε e e e e ε ε e + e + e + e + () (3abc def)

5 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 S339 ε ε (3gh) If he maerial is subjeced o he one-sep sress hisory () φ() () where φ () is he uni sep funcion [3] he creep differenial equaions (CE) for he GFMM are wrien as. Wih he use of eq. () we have: ( ) ( p( ) ) ( ) φ Μ φ ε e ε φ e φ ε ε e e φ e φ ε ε e + e + φ φ ε ε which deduce he corresponding CCF as: J ( ) ( ) p p + + J + + Μ ( ) Μ Μ ( ) Μ J + + J + + J + J + J + J + Γ ( + ) Γ + (4ab) (4cd) (4ef) (4gh) If he maerial is subjeced o he one-sep srain hisory ε() φ() ε () he relaxaion differenial equaions (RE) for he GFMM are given as: ( ) [ p( ) ] Μ( ) ε( ) e ε( ) e e e ε( ) e e + e + ( ) ε which leads o he corresponding RMF: G p( ) p( ) Μ( ) + Μ( ) + Μ Μ e G e Μ + Μ + (5abcd efgh) (6abcd efgh) (7ab)

6 S34 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S e e + + G G + + e e + + G G G cos G cos (7cde fgh) The creep and relaxaion processes for he general fracional-order Kelvin-Voig models A general fracional-order Kelvin-Voig model (GFKVM) is ha he branch of he spring and dashpo is in parallel [7]. The CE for he GFMM are given as: d ε () + ε () () (8) d Thus we reduce o he CE wihin he GF as follows: e( ) e( ) e( + ) e + ( ) ( ) ε ε ε ε ε + ε ε + ε ε + ε ε + ε ε + ε ε + ε If he maerial is subjeced o he one-sep sress hisory () φ() () he CE for he GFKVM are presened as: e( ) e( ) e( + ) e + ( ) ( ) ε ε φ ε ε φ ε + ε φ ε + ε φ ε + ε φ ε + ε φ ε + ε φ ε + ε φ which yields he corresponding CCF as follows: J p( ) p( ) p( ) Μ ( ) + Μ ( ) + Μ Μ ( ) + e + e + e Μ ( ) + Μ ( ) + [ ] (9abcd efgh) (3abcd efgh) J p( ) p( ) Μ ( ) + Μ ( ) + e + e Μ ( ) + (3abc) e + e + e + ( + ) J

7 Gao F. e al.: On Linear Viscoelasiciy wihin General Fracional erivaives... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S335-S34 S34 J J e + e e + e + e J e + e J cos + e + e J e e + If he maerial is subjeced o he one-sep srain hisory ε() φ() ε () he RE for he GFKVM are as follows: Μ [ p( ) ] + φ ε( ) φ ε( ) e + φ ε( ) φ ε( ) φ ε( ) e + φ ε( ) φ ε ( ) φ ε( ) + φ ε ( ) φ ε φ ε e φ ε which resuls in he corresponding RMF as follows: [ p( ) ] G G Μ e + G G e + G G e + G G + Conclusions and remarks From mahemaics and physics view poin we considered he general Riemann-Liouville and Capuo-Liouville F wihou singular kernel o characerize he rheological properies of he real maerials. The fracional-order Maxwell and Kelvin-Voig models were discussed wih he aid of he Laplace ransform. The proposed approaches are efficien for analyzing and modelling he creep and relaxaion behaviors in he linear viscoelasic maerials. Acknowledgmen This work is suppored by he Sae Key Research evelopmen Program of he People s Republic of China (Gran No. 6YFC675) he Naural Science Foundaion of China (Gran No.5334) and he Prioriy Academic Program evelopmen of Jiangsu Higher Educaion Insiuions (PAP4). Nomenclaure ime [s] Greek symbols ε() srain [ ] fracional dimension [ ] () sress [Pa] (3de fgh) (3abcd efgh) (33abcd efgh)

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