New effective moduli of isotropic viscoelastic composites. Part I. Theoretical justification

Transcription

1 IOP Conference Series: Maerials Science and Engineering PAPE OPEN ACCESS New effecive moduli of isoropic viscoelasic composies. Par I. Theoreical jusificaion To cie his aricle: A A Sveashkov and A A akurov 16 IOP Conf. Ser.: Maer. Sci. Eng elaed conen - New effecive moduli of isoropic viscoelasic composies. Par II. Comparison of approximae calculaion wih he analyical soluion N A Kupriyanov F A Simankin and K K Manabaev iew he aricle online for updaes and enhancemens. This conen was downloaded from IP address on 3/1/18 a :1

2 IOP Conf. Series: Maerials Science and Engineering 14 (16) 199 doi:1.188/ x/14/1/199 New effecive moduli of isoropic viscoelasic composies. Par I. Theoreical jusificaion A A Sveashkov A A akurov Naional esearch Tomsk Polyechnic Universiy 3 enina ave. Tomsk 6345 ussia sveashkov@pu.ru Absrac. According o he approach based on he commonaliy of problems of deermining effecive moduli of composies and viscoelasic solids which properies are imeinhomogeneous i is assumed ha a viscoelasic solid is a wo-componen composie. One componen displays emporal properies defined by a pair of Casiglianian-ype effecive moduli and he oher is defined by a pair of agrangian-ype effecive moduli. The oig and euss averaging is performed for he obained wo-composie solid wih he inroducion of a ime funcion of volume fracion. In order o deermine closer esimaes a mehod of ieraive ransformaion of ime effecive moduli is applied o he viscoelasic oig euss model. The physical jusificaion of he mehod is provided. As a resul new ime effecive moduli of he viscoelasic solid are obained which give a closer esimae of emporal properies as compared o he known models. 1. Inroducion The problems of calculaing he sress-srain sae of viscoelasic solids are reduced o he soluion of a sysem of inegro-differenial equaions of equilibrium in he region occupied by he solid wih specified boundary loads and displacemens. A he same ime hese problems can be considered from he viewpoin of composie mechanics because composie properies are inhomogeneous wih respec o coordinaes and properies of viscoelasic solids which are ime-inhomogeneous. This makes he applicaion of effecive modulus heory o viscoelasiciy [1-4] possible. Earlier we derived expressions for ime effecive moduli of he agrangian and Casiglianian ype [5] and used hem o consruc ransformaions for he oig euss model [6]. However his model gives insufficienly close esimaes for approximae soluions as in he case of elasic composies. The presen paper is aimed a obainmen of ieraively ransformed ime effecive characerisics. Earlier we successfully applied his approach for composies [7]. The requiremens o ieraive ransformaions are he following: (i) classical heorems on minimum of srain energy and addiional work funcionals mus be valid and (ii) inequaliies for effecive moduli (oig euss bounds) on each ieraion sep mus be saisfied. The convergence of ieraive ransformaion of ime effecive characerisics is verified boh numerically and by analyical deerminaion of he limi of he ieraion sequence.. Derivaion of expressions for ieraed effecive moduli The consiuive equaions of a linearly viscoelasic solid is as follows: Conen from his work may be used under he erms of he Creaive Commons Aribuion 3. licence. Any furher disribuion of his work mus mainain aribuion o he auhor(s) and he ile of he work journal ciaion and DOI. Published under licence by d 1

3 IOP Conf. Series: Maerials Science and Engineering 14 (16) 199 doi:1.188/ x/14/1/199 ij τ ij τ ij s de G e σ K τ dθ τ K θ. Here s ij e ij are he deviaoric sress and srain componens and σ() = σ ii () θ() = ε ii () (i = 1 3) () K() are he shear and bulk relaxaion funcions. The relaions inverse o Equaions (.1) have he following form: 1 ij τ ij τ ij e ds G s 1 θ 1 τdστ K θ Π() Π 1 () are he shear and bulk creep funcions and G K G 1 K 1 are he shorhand noaions for direc and inverse inegral operaors. Approximae consiuive equaions can be represened as k k sij g eij eij gc sij (.3) σ θ θ C σ where he agrangian and Casiglianian-ype ime effecive moduli are deermined by relaions [5]: C 1 C 1 1 g G h g G h 1 k K h k K h h is he Heaviside funcion. Using Equaions (.4) we consruced he oig euss models of ime effecive characerisics [6]: γ 1 γ G γgc 1 γ g G gc g gc g gc g γ 1 α g C where α is a numerical parameer [6]. e us inroduce expressions for ieraion sequences: n n1 n1 G γg 1 γ G n I is possible o show ha sequences G G Sequences n K K n 1 n γ 1 γ G 1 1 n 1... n n G G n n 1 converge o he same limi a n : (.1) (.) (.4) (.5) (.6) G G G G. (.7) n have he same limi a n : n n K K K K. (.8) 3. Physical jusificaion e us consider he srain energy and addiional work funcionals of a homogeneous isoropic elasic solid wih ime-dependen elasic moduli:

4 IOP Conf. Series: Maerials Science and Engineering 14 (16) 199 doi:1.188/ x/14/1/199 W e dv ρ f u d u ds ij i i i i S i σ dv u ds 1 W εij K G θ G εijε ij 3 σij σijσ ij ii ii ij i s Su 1 1 3K G G 6K G σ θ ε i j 13. i Here K() G() are he elasic bulk compression and shear moduli i u s are he sresses and displacemens specified on porions of boundary S = S US u σ ij ε ij u i are he sress ensor srain ensor and displacemen vecor componens and f i i are he bulk and surface force vecor componens. Funcionals Π Ψ assume minimum values on admissible displacemen and sress fields ha obey equilibrium and compaibiliy equaions. Now le us consider oig euss bounds G G G K K K. (3.) Here G K are he moduli of he homogeneous elasic medium whose specific poenial energy is equal o he corresponding poenial of a wo-componen composie wih moduli (.4). Thus he oig and euss inequaliies give he upper and lower bounds for moduli G K. However using Equaions (3.) i is impossible o calculae G and K as well as specific srain energy W and addiional work Λ. e us now demonsrae ha ieraion sequences (.6) can be used o derive expressions for G K and W Λ. Since we consider an elasic medium wih consiuive equaions (.4) funcionals Π Ψ and specific poenials W Λ depend on ime. The same is rue of sough moduli G K. We assume ha minimum values Π = Π min Ψ = Ψ min correspond o moduli G K for each paricular value of. We 1 n 1 n denoe he specific poenials corresponding o G G... G and G G... G by W W 1 W n and Λ Λ 1 Λ n. Then owing o inequaliies 1 n 1 n G G... G G G... G (3.3) he following inequaliies are fulfilled: W W1... W n 1... n. (3.4) n n Since moduli G and G for any n saisfy inequaliy n n G G G (3.5) inequaliies Wn W n (3.6) are also saisfied for he specific poenials. In he limi a n we have: Wn W n. (3.7) n n The oig euss bounds shrink o a single poin in which moduli G G become equal o he limi (.7). Thus he sough limi values of G K really correspond o specific poenials W (ε ij ) Λ (σ ij ) equal o he specific poenial energy and specific addiional work of he wo-componen elasic solid wih consiuive equaions (.3). (3.1) 3

5 IOP Conf. Series: Maerials Science and Engineering 14 (16) 199 doi:1.188/ x/14/1/ Discussion of resuls Expressions for new effecive characerisics of viscoelasic solids were derived using he following mahemaical procedures: 1. Approximae represenaion of consiuive equaions of viscoelasiciy hrough relaions of elasic Hooke's law wih ime-dependen moduli of agrangian and Casiglianian ype.. epresenaion of he elasic medium as a wo-componen one and formulaion of he oig euss model for he averaging of properies based on his represenaion. 3. Consrucion of ieraion sequences for he oig euss moduli. 4. Applicaion of minimum variaional principles for he funcionals of poenial srain energy and addiional work. I has been shown ha he sequence of ieraively ransformed oig and euss moduli allows he conracion of he oig euss bounds. In he limi we obain effecive characerisics of a homogeneous isoropic elasic medium wih specific srain and sress poenials equal o he corresponding poenials of a wo-componen medium. These effecive characerisics can be called energeically equivalen. The repored heoreical findings can be applied o calculae he sress-srain sae of srucures made of viscoelasic maerials which will be discussed in he second par of his paper []. 5. Conclusion Conracion mappings for he oig euss bounds were obained for a wo-componen homogeneous isoropic elasic medium wih ime-dependen moduli. I was shown using heorems on minimum poenial srain energy and addiional work ha he limi of he sequence of ieraively ransformed oig euss moduli is he effecive characerisics of energeically equivalen specific srain and sress poenials of he homogenous medium of comparison. Expressions for hese characerisics have been derived. eferences [1] Chrisensen M 1979 Mechanics of Composie Maerials (New York: Academic Press) [] Sveashkov A A Mechanical of composie maerials 36(1) 37 [3] Sendeckyj G P 1974 Composie Maerials Academic Press 45 [4] Pobedrya B E 1984 Mechanics of Composies (Moscow : Izd-vo MGU) [5] Sveashkov A A 1 Applied Problems of he Mechanics of iscoelasic Maerials (Tomsk : Izd-vo TPU) [6] Sveashkov A A Kupriyanov N A and Manabaev K K 1 Comp. Con. Mech. 9 [7] Sveashkov A A Kupriyanov N A and Manabaev K K 13 Izv. UZov. Fizika 6 4