Chapter 2 Constitutive Equations for Isotropic and Anisotropic Linear Viscoelastic Materials

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1 Chaper 2 Consiuive Equaions for Isoropic and Anisoropic Linear Viscoelasic Maerials Jacek J. Skrzypek and Arur W. Ganczarski Absrac In case of isoropic maerial symmery, he elasic-viscoelasic correspondence principle is well esablished o provide he soluion of linear viscoelasiciy from he coupled ficiious elasic problem by use of he inverse Laplace ransformaion (Alfrey Hoff s analogy). Aim of his chaper is o show useful enhancemen of he Alfrey Hoff s analogy o a broader class of maerial anisoropy for which separaion of he volumeric and he shape change effecs from oal viscoelasic deformaion does no occur. Such exension requires use of he vecor marix noaion o descripion of he general consiuive response of anisoropic linear viscoelasic maerial (see Pobiedria Izd. Mosk. Univ., (984) []). When implemened o he composie maerials which exhibi linear viscoelasic response, he classically used homogenizaion echniques for averaged elasic marix, can be implemened o viscoelasic work-regime for associaed ficiious elasic Represenaive Uni Cell of composie maerial. Nex, subsequen applicaion of he inverse Laplace ransformaion (cf. Haasemann and Ulbrich Technische Mechanik, 3( 3), (2)) is applied. In a similar fashion, he well-esablished upper and lower bounds for effecive elasic marices can also be exended o anisoropic linear viscoelasic composie maerials. The Laplace ransformaion is also a convenien ool for creep analysis of anisoropic composies ha requires, however, limiaion o he narrower class of linear viscoelasic maerials. In he space of ransformed variable s, insead of ime space, he classical homogenizaion rules for ficiious elasic composie maerials can be applied. For he above reasons in wha follows, we shall confine ourselves o he linear viscoelasic maerials, isoropic, or anisoropic. Keywords Linear viscoelasicy Anisoropic correspondence principle Anisoropic inegral equaions of linear viscoelasiciy Ficiious anisoropic elasic problem Homogenizaion of linear viscoelasic composies J.J. Skrzypek (B) A.W. Ganczarski Solid Mechanics Division, Insiue of Applied Mechanics, Cracow Universiy of Technology, al. Jana Pawła II 37, Kraków, Poland Jacek.Skrzypek@pk.edu.pl A.W. Ganczarski Arur.Ganczarski@pk.edu.pl Springer Inernaional Publishing Swizerland 25 J.J. Skrzypek and A.W. Ganczarski (eds.), Mechanics of Anisoropic Maerials, Engineering Maerials, DOI.7/ _2 57

2 58 J.J. Skrzypek and A.W. Ganczarski 2. Seleced Uniaxial Models of he Isoropic Linear Viscoelasic Maerials Creep phenomena a elevaed emperaure are usually reaed as nonlinear creep phenomenon problems. There exiss broad lieraure in he field of nonlinear creep, for example, creep anisoropy Findley e al. [4], survey on consiuive models of nonlinear creep Skrzypek [3], Been [2], ineracion creep and plasiciy Krempl [6], coupling of creep and damage Skrzypek [4], Skrzypek and Ganczarski [5], creep faigue damage Murakami [7], and nonconvenional creep models of anisoropic maerial Alenbach [] and ohers. A he beginning, we confine o he commonly used uniaxial isoropic linear viscoelasic models for which a general differenial equaion models may be wrien as p σ + p σ + p 2 σ + p a a σ a = q ε + q ε + p b b ε b (2.) where p, p,...,q, q,... denoe maerial consans, and consiuive equaion is a linear funcion of he sress σ, srainε, and heir ime derivaives σ, σ, ec., and ε, ε, ec. In such a case by he use of he Laplace ransformaion L { f ()} = f (s) = e s d, a linear viscoelasic problem can be reduced o associaed ficiious elasic problem in erms of he ransformed variable s, σ ij (x, s), hen he viscoelasic problem σ ij (x, ) is obained by he inverse Laplace ransformaion. Symbol {}sands here for funcion argumen of he Laplace ransformaion and should no be confused wih he Voig vecor noaion. 2.. Maxwell Model The uniaxial Maxwell model (M) consiss of a linear elasic spring ε H = σ/e and a linear viscous dashpo elemen ε η = σ/η conneced in a series, Fig. 2.. Differeniaion of he firs formula wih ime yields ε H = σ/e. When he addiive Fig. 2. Maxwell s maerial: a mechanical model, b creep curve under consan loading (a) (b)

3 2 Consiuive Equaions for Isoropic and Anisoropic Linear 59 decomposiion of he srain or he sain rae ε = ε H + ε η is used, we arrive a equaion of he Maxwell model, hence ε = σ E + σ η or σ + η σ = η ε (2.2) E When he inegraion of above equaion a consan sress σ = σ = cons ( σ = ) and iniial condiion ε() = σ /E is performed, we arrive a he creep funcion given as, see Fig. 2.b ( ε = σ E + ) η (2.3) or ε = σ J M (), J M () = E + η (2.4) The ime funcion J M () is he creep compliance funcion of he Maxwell model Voig Kelvin Model The Voig Kelvin model (V K) consiss of a linear spring elemen and a linear dashpo elemen which are conneced in parallel as shown in Fig. 2.2a. Adoping he addiive separaion of sress ino wo pars applied o he spring σ H = Eε and o he dashpo σ η = η ε wih ε = ε H = ε η, he differenial equaion of he V K model akes he form ε + E η ε = σ (2.5) η If a consan sress σ = σ = cons ( σ = ) is applied o he V K model, we arrive a nonhomogeneous differenial equaion ε + E η ε = σ η (2.6) Fig. 2.2 Voig Kelvin model: a mechanical scheme, b creep srain a consan sress inpu (a) (b)

4 6 J.J. Skrzypek and A.W. Ganczarski The homogeneous equaion of (2.6) is an equaion of separae variables ε ε = E η he general inegral of which is given by ε = C exp ( Eη ) (2.7) (2.8) When variaion of inegraion consan C() wih iniial condiion ε() = is done we arrive a he soluion of (2.6) ε = σ E [ exp ( Eη )] (2.9) or ε = σ J VK (), J VK () = E [ exp ( Eη )] (2.) Funcion J VK () is he creep compliance funcion of he V K model. Noe ha V K model does no accoun for insananeous elasiciy, hence J VK () =, see Fig.2.2b. When he more general case of a ime funcion σ() is applied and variaion of consan C() is done in (2.8) we arrive a he differenial equaion for C() Ċ() = η exp ( E η )σ() (2.) he general inegral of which is expressed in form C() = C + η ( ) E exp η ξ σ(ξ)dξ (2.2) Subsiuion of (2.2)o(2.8) wih he iniial condiion ε() = yields C =, such ha he following general soluion for ε() holds ε() = η exp ( E η ) = η exp ( E exp [ Eη ( ξ) ] σ(ξ)dξ ) η ξ σ(ξ)dξ (2.3)

5 2 Consiuive Equaions for Isoropic and Anisoropic Linear 6 When inegraion by pars is applied o (2.3), we arrive a so-called inegral represenaion of he V K model ε() = σ() E E exp [ Eη ] ( ξ) σ(ξ)dξ (2.4) in which i is clearly seen ha he creep funcion J VK ()[ has wo erms: independen of ime J = /E and dependen on ime ϕ() = E exp E η ]. ( ξ) Analogous soluion may be reached by use of he Laplace ransform mehod (2.48). In order o do his he nonhomogeneous V K equaion (2.5) is muliplied boh-side by e s and inegraed wih respec o variable in range from o ε()e s d + E η ε()e s d = σ() η e s d (2.5) Consequenly, he algebraic equaion of he ransformed variable s is obained s ε(s) ε() + E σ(s) ε(s) = η η (2.6) When he iniial condiion ε() = is used, he soluion of (2.6) wih respec of ransformed variable ε(s) is given as he following ε(s) = s + E σ(s) (2.7) η Applying nex he inverse Laplace ransform and aking advanage of propery ha muliplicaion of wo ransforms in ficiious domain of variable s corresponds o he convoluion of wo funcions in real ime space, we arrive a he soluion of linear viscoelasic problem idenical o (2.3). ε() = η exp [ Eη ] ( ξ) σ(ξ)dξ (2.8)

6 62 J.J. Skrzypek and A.W. Ganczarski 2..3 Sandard Model The Maxwell and he Voig Kelvin wo-elemen uniaxial models described in he Secs. 2.. and 2..2 are very simple, alhough hey exhibi srong limiaions. The linear creep funcion a consan sress inpu corresponding o he Maxwell model does no confirm experimens, whereas he Voig Kelvin model is no capable o capure he insananeous elasic srain effec. Trying o overcome he above objecions, he commonly used hree-parameer sandard model is composed of wo pars, a spring elemen (E) and he V K uni (E, η) conneced in a series as shown in Fig. 2.3a. The differenial equaion of he sandard model can be derived in an analogous way as for he Maxwell and he Voig Kelvin simple models, such ha afer necessary rearrangemen used, he following is obained ηe E + E ε + E E E + E ε = σ + η σ (2.9) E + E The simple creep funcion, when he sandard model is subjeced o a sep funcion σ = σ = cons ( σ = ) and inegraed wih he iniial condiion ε() = σ /E used, akes one of wo equivalen forms ε = σ E ) [( + EE E ( exp E )] E η (2.2) or ε = σ J s (), J s () = E ) [( + EE E ( exp E )] E η (2.2) if he ime-dependen creep compliance funcion characerizing he sandard model J s () is used. Noe he horizonal asympoe of ε() curve as shown in Fig. 2.3b wih he new definiion used: /H = /E + /E. (a) (b) Fig. 2.3 The sandard model: a mechanical scheme, b creep a consan sress inpu wih insananeous elasic srain buil-in

7 2 Consiuive Equaions for Isoropic and Anisoropic Linear Burgers Model Alhough he sandard model is free from aforemenioned inconvenience of he Maxwell and he Voig Kelvin models, i sill exhibis a horizonal asympoe (srain sabilizaion) when, which usually is no rue; since according o experimenal findings, he creep srain shows raher he infinie increase wih ime. In order o conrol such behaviors, a more complex four-parameer Burgers model which consiss of wo simple unis, he Maxwell uni (E, η ), and he Voig Kelvin uni (E 2, η 2 ) coupled in a series can be used, as presened in Fig. 2.4a. The differenial consiuive equaion of Burgers model may be wrien in he forma η η 2 ε + η ε = η ( η 2 η σ + + η + η ) 2 σ + σ (2.22) E 2 E E 2 E E 2 E 2 Noe ha he above equaion is he second-order linear differenial equaion wih respec o srain and sress bu of consan coefficiens being funcions of four parameers E, E 2, η and η 2. I means ha all srain and sress funcions and heir ime derivaives are he linear funcions, whereas he coefficiens in Eq. (2.22) are consans: wo Young s modules E, E 2 and wo viscosiy parameers η, η 2. When he Burgers model is loaded by a sep sress inpu applied a = he inegraion of Eq. (2.22), wih wo iniial condiions ε() = σ /E, ε() = σ /η + σ /η 2 used, leads o one of equivalen relaionships ε = σ { + E + E [ ( exp E )]} 2 (2.23) E η E 2 η 2 or ε = σ J B (), J B () = { + E + E [ ( exp E )]} 2 E η E 2 η 2 (2.24) where he creep compliance funcion characerizing he Burgers model compliance funcion J B () is applied. (a) (b) Fig. 2.4 The Burgers model: a he mechanical scheme, b he simple creep curve a consan sress inpu applied a =

8 64 J.J. Skrzypek and A.W. Ganczarski Noe ha he creep curve described by he Burgers model exhibis a skewed asympoe which corresponds o he unlimied srain increase wih decreasing srain rae, which beer fis he experimenal findings, see Fig. 2.4b Creep Compliance and Relaxaion Behavior of he Seleced Linear Viscoelasic Models More complex models of linear viscoelasic maerials consising of one Hooke s elemen and n Voig Kelvin s unis coupled in a series or one Hooke s elemen and n Maxwell s unis coupled in parallel are analyzed by Been [2]. Consider now sress relaxaion of simple uniaxial linear viscoelasic models discussed in Secs , subjec o a consan srain ε a =, from he iniial sress level σ = Eε. In case of he Maxwell model, hesress relaxaion from he iniial level σ a = o is described as follows ( σ() = σ exp E ) (2.25) η Noe ha he rae of sress decrease changes from he iniial σ() = σ E/η o zero, σ( ) =. The so-called relaxaion ime r = η/e corresponds o he ficiious case if he sress decreases coninuously a he iniial rae and finally i would reach zero a = r. The Voig Kelvin model does no exhibi sress relaxaion effec. In his singular case applicaion of he consan srain inpu, ε = ε a = can be achieved only by an infinie iniial sress response σ(), such ha he following holds σ() = ηε δ() + Eε H() (2.26) where he erm conaining he Heaviside uni funcion H() describing he consan sress in he spring, followed by he infinie sress inpu in dashpo described by he δ-dirac funcion, appears. The sandard model is free from he above singulariy and if i is subjec o a consan srain a =, he sress coninuously decreases from he iniial level Eε ( = ) o he asympoically approached value Hε ( ) such ha sress relaxaion funcion of he sandard model is wrien as where he definiions hold: σ() = Eε [ H E + ( H E ) exp H = E + E and n = ( )] n η E + E. (2.27)

9 2 Consiuive Equaions for Isoropic and Anisoropic Linear 65 Table 2. Properies of Maxwell, Voig Kelvin, sandard and Burgers viscoelasic unis, afer Skrzypek [3] Model Creep compliance funcion J() Maxwell V K Sandard Burgers Model Maxwell E ( + Eη ) ( ) e E η E E E ( + E E ) e E η E E [ + E + E ( )] e E 2 η 2 η E 2 Relaxaion modulus E() Ee E η V K Sandard Burgers E [ + η E δ()] ( E E E + E + E E + E e ) E +E η (q q 2 r )e r (q q 2 r 2 )e r2 A If he Burgers model is subjec o a consan srain ε = ε a =, a coninuous sress relaxaion is described by he combinaion of wo exponenial funcions exp( r ) and exp( r 2 ) (cf. Table 2.), where afer Findley e al. [4] he new definiions are used p = η + η + η 2 p 2 = η η 2 q = η E E 2 E 2 E E 2 q 2 = η η 2 E 2 r,2 = p A 2p 2 A = p 2 4p 2 (2.28) 2.2 The Uniaxial Bolzmann Superposiion Principle of he Isoropic Linear Viscoelasic Maerials 2.2. Bending of a Beam Subjec o Saionary Load Summarizing he resuls of previous subsecion, response of he arbirary linear viscoelasic maerial a sep sress inpu σ = σ = cons can be wrien as follows ε(x, ) = ε e (x)ej() (2.29)

10 66 J.J. Skrzypek and A.W. Ganczarski In oher words, srain a he arbirary maerial poin, being a funcion of he space x and ime variables, can be presened as a produc of he insananeous elasic srain ε e (x) depending on x only and he creep compliance funcion J() specifically chosen for given maerial model dependen on ime only. In he ligh of commens presened in Sec. 2..5, Eq.(2.29) does no apply o he Voig Kelvin model in a sraighforward manner. The V K model does no comprise he iniial elasic srain. In paricular, a deflecion of beam made of he linear viscoelasic maerial w ve (x, ) a consan loading can be presened as he produc of he elasic deflecion w e (x) and he dimensionless creep compliance funcion EJ() as follows w ve (x, ) = w e (x) EJ () (2.3) For creep compliance funcions shown in Table 2., wearrivea w M (x, ) = w e (x) ( + Eη ) [ w VK (x, ) = w e (x) exp ( Eη )] [ w s (x, ) = w e (x) + E E exp E E w B (x, ) = w e (x) { + E η + E E 2 ( E [ exp )] η ( E 2 η 2 )]} (2.3) The aforemenioned relaionships hold for all linear viscoelasic models discussed even hough V K model does no exhibi insananeous response. This commen holds for all linear viscoelasic models ha do no have free elasic spring Bending wih Tension of a Beam Subjec o Nonsaionary Load Consider a prismaic beam of doubly symmeric cross-secion subjec o he axial force and he bending momen being boh funcions of coordinae x and ime : N = N(x, ), M = M(x, ). Assume also ha boh exernal forces N and M can be expressed as producs of funcion dependen of x co-ordinae N = N(x) or M(x) and one common ime funcion f (): N(x, ) = N(x) f () and M(x, ) = M(x) f (). Supposing for simpliciy ha viscoelasic deformaion fulfils he Bernoulli hypohesis of sraigh and normal segmens ε(x, z, ) = λ(x, ) + zκ(x, ), iispossibleo separae Eq. (2.29) ino he viscoelasic axial elongaion λ ve and he viscoelasic curvaure κ ve

11 2 Consiuive Equaions for Isoropic and Anisoropic Linear 67 λ ve (x, ) = N (x) A κ ve (x, ) = M (x) I J( ξ) f (ξ) dξ ξ J( ξ) f (ξ) dξ ξ (2.32) In a paricular case, if boh generalized forces are applied insananeously a = : N(x, ) = N(x)H() and M(x, ) = M(x)H(), remembering ha δ-dirac funcion is defined as δ() = Ḣ() and applying (see Byron and Fuller [3]) J( ξ)δ(ξ)dξ = J(ξ)δ( ξ)dξ = J() (2.33) we finally obain equaions for viscoelasic elongaion λ ve (x, ) and curvaure κ ve (x, ) in a form λ ve (x, ) = N (x) EA J () κ ve (x, ) = w (x, ) = M (x) EI J () (2.34) Hence, he axial elongaion of he linear viscoelasic beam λ ve is a produc of insananeous (elasic) elongaion and he creep compliance funcion. Analogously, he curvaure of he linear viscoelasic beam κ ve is a produc of insananeous curvaure and creep compliance funcion J(). For simpliciy, he convenional beam heory is adoped here Inegral Represenaion of Creep and Relaxaion Funcions in Case of Arbirary Loading Hisory A general differenial equaion of uniaxial linear viscoelasic models can be wrien as follows: p σ + p σ + p 2 σ + =q ε + q ε + q 2 ε + (2.35) where he consan p i, q i are coefficiens of he linear arbirary order differenial consiuive equaion, see Eq. (2.). Order of Eq. (2.35) is equal o number of viscous elemens (dashpos) appearing in he mechanical model. A compac operaor forma can be used insead of Eq. (2.35) Pσ() = Qε() (2.36)

12 68 J.J. Skrzypek and A.W. Ganczarski where P and Q sand for linear differenial operaors wih respec o ime acing on sress σ() and srain ε(), respecively 2 P = p + p + p Q = q + q + q (2.37) I is clear ha he linear elasic (Hooke s) maerial is a paricular case of linear viscoelasic maerial in equaion of which all ime derivaives disappear, whereas q /p = E. Noe ha differenial operaors P and Q are linear wih respec o all derivaives, hence he operaor forma of Eq. (2.36) can formally be reaed as an algebraic equaion as follows σ() ε() = E ve() E ve () = Q() P() (2.38) The raional operaor E ve () used in Eq. (2.38) plays a role of he ime-dependen siffness operaor. As a consequence by conras o elasiciy a fracion σ()/ε() is no consan bu depends on ime. Hence, Eq. (2.38) should be read in a symbolic way as follows Q() P() σ() ε() (2.39) Class of he linear viscoelasic maerials is a subclass of he nonlinear viscoelasic maerials; however, he Bolzmann superposiion principle holds for he linear viscoelasic maerials only. The superposiion principle saes ha resulan response of he sysem ε() under he sum of causes is equal o he sum of responses corresponding o causes acing separaely. In paricular if sress σ is applied a ime ξ and, hen, sress σ 2 is applied a ime ξ 2, he resulan srain ε() a any ime > ξ 2 is represened as he sum of he srains resuling from boh sresses considered as hough each were acing separaely ε [σ ( ξ ) + σ 2 ( ξ 2 )] = ε [σ ( ξ )] + ε [σ 2 ( ξ 2 )] (2.4) In case of arbirary loading hisory, sress σ() can be approximaed by a sum of n sress inpus Δσ i, hence from he Bolzmann principle he srain oupu holds ε() = n ε i ( ξ i ) = i= n J( ξ i )Δσ i (2.4) i=

13 2 Consiuive Equaions for Isoropic and Anisoropic Linear 69 If he ime sep ends o zero, we arrive a he inegral form of uniaxial creep srain for he linear viscoelasic maerial ε() = J( ξ) σ(ξ) ξ dξ = J( ξ) σ(ξ)dξ (2.42) If he analogous reasoning is applied o he arbirary srain hisory (kinemaic conrol), we arrive a he inegral form of he uniaxial sress relaxaion for he linear viscoelasic maerial σ() = E( ξ) ε(ξ) ξ dξ = E( ξ) ε(ξ)dξ (2.43) In above inegral equaions, J( ξ) and E( ξ) denoe he creep funcion and he relaxaion funcion of he maerial considered, respecively. In pracical applicaions, he alernaive forms o (2.42) or(2.43) are more convenien ε() = J σ() + ϕ( ξ) σ(ξ)dξ (2.44) or σ() = E ε() ψ( ξ) ε(ξ)dξ (2.45) where separaion of he insananeous and ime-dependen oupus are disinguished. The general inegral forms (2.42) or(2.43) do no comprise explicily iniial condiions, whereas in he forms (2.44) or(2.45) J or E denoe iniial value of creep or relaxaion funcions (a = ) whereas ime funcions ϕ( ξ) or ψ( ξ) denoe ime-dependen pars of creep or relaxaion funcions. Noe ha in Eqs. (2.44) and (2.45) symbol ξ denoes ime when he sress or he srain inpus are imposed, whereas denoes he observaion ime when srain response ε() or sress response σ() are measured. This approach can be idenified as he linear herediary model where kernel funcion depends on ime inerval ξ by conras ohenonlinear herediary models where kernel funcions depend on, ξ separaely, cf. Rabonov [].

14 7 J.J. Skrzypek and A.W. Ganczarski 2.3 Muliaxial Isoropic Linear Viscoelasic Maerials In wha follows, we briefly summarize he fundamenals of he linear viscoelasiciy in case of maerial isoropy. This will serve as he saring poin for furher exension of isoropic o anisoropic linear viscoelasic equaions. Such exension will furher be used as convenien ool for analysis of anisoropic viscoelasic composies Differenial Represenaion Under a Muliaxial Sress Sae In case of isoropic linear viscoelasic maerials under he muliaxial saes of sress and srain, i is convenien o separae volumeric effec from he shape change effec. Similar o elasiciy, such separaion is possible only in case of maerial isoropy (see Sec..4.8). Direc exension of linear isoropic viscoelasic consiuive equaions (2.36) and (2.37) ohemuliaxial saes akes he form P s ij () = Q e ij () P 2 σ kk () = Q 2 ε kk () (2.46) where P, Q, P 2, and Q 2 are he linear differenial operaors applicable o he separable shape change and he volume change effecs. In he explici forma, equaions (2.46) can be rewrien as ( p + p + 2 p a ) p a a s ij () = ( q + q + 2 b ) q q b b e ij () ( p 2 a ) (2.47) σ kk () = + p ( q + p p 2 a a + 2 b q q b b + q ) ε kk () For he purpose of furher consideraion, i is convenien o ransform differenial equaions (2.47) expressed in erms of physical ime, f () o he equivalen algebraic equaions expressed in erms of ransformed variables s, f (s) according o he Laplace inegral ransform L { f ()} = f (s) = e s f () d (2.48)

15 2 Consiuive Equaions for Isoropic and Anisoropic Linear 7 The use of above ransformaion allows o replace he real iniial differenial problem of he linear viscoelasic maerial (differenial equaion and appropriae iniial condiions) by he equivalen elasic algebraic equaion of a ficiious elasic maerial. In he nex sep, when he ficiious algebraic problem is solved in elemenary way, applicaion of he inverse Laplace ransform allows o reurn o he original viscoelasic problem. Described procedure leads o he soluion faser han he sraighforward inegraion of a differenial equaion due o he Laplace ransform pairs known from lieraure. Basic properies of he Laplace ransform commonly used in heory of viscoelasiciy can be found among ohers in, e.g., Nowacki [8], Pipkin [9], Findley e al. [4]. Exemplary Laplace ransforms for seleced elemenary funcions f () are shown in Table 2.2. By use of he Laplace ransformaion equaions of ransformed isoropic linear viscoelasiciy (2.46) can be expressed in erms of he ransformed variable s as follows P ŝ ij (s) = Q ê ij (s) P 2 σ kk (s) = Q 2 ε kk (s) (2.49) Table 2.2 Laplace ransforms of frequenly used funcions f () f (s) f () f (s) f () s f (s) f () H() s s a f (ξ)dξ H( a) f (s) a s s e as δ() = Ḣ() δ( a) e as e a s + a e a e b b a (s + a)(s + b) e a a s(s + a) s 2 n n! s n+ n e a n! (s + a) n+ ae a be b a a 2 ( e a ) s (a b)s (s + a)(s + b) s 2 (s + a)

16 72 J.J. Skrzypek and A.W. Ganczarski or ( p + p s + p 2 s2 + + p a sa) ŝ ij (s) = ( q + q s + q 2 s2 + +q b sb) ê ij (s) ( p + p s + p 2 s2 + + p a sa) σ kk (s) = ( q + q s + q 2 s2 + +q b sb) ε kk (s) (2.5) Based on he analogy beween Eq. (2.49) ha describe ransformed viscoelasic problem and linear isoropic elasic equaions s ij = 2Ge ij, σ kk = 3K ε kk (2.5) i is possible o find ou he generalized modules of viscoelasiciy G ve () and K ve () as G ve () = Q 2P K ve () = Q 2 3P 2 (2.52) which are ime-dependen funcions of. Addiionally, if he definiions of Young s modulus E and Poisson s raio ν known from he elasiciy are used E = 9KG 3K + G, 3K 2G ν = 6K + 2G (2.53) subsiuion of (2.52) o(2.53) furnishes he generalized Young s modulus E ve () commonly called relaxaion modulus and generalized Poisson s raio ν ve () of isoropic linear viscoelasiciy ha can be expressed in erms of ime-dependen operaors (2.46) E ve () = 3 Q Q 2 P P 2 3Q Q 2 Q P + 2 Q = 2 P P 2 Q + 2P Q 2 2 ν ve () = Q 2 P 2 Q P 2 Q 2 P 2 + Q = P Q 2 P 2 Q P P 2 Q + 2P Q 2 (2.54) By conras o elasiciy, he above modules are ime-dependen differenial operaors bu no maerial consans. The deviaoric P, Q and he volumeric P 2, Q 2 differenial operaors and he corresponding ransformed operaors P, Q and P 2, Q 2 for seleced isoropic linear viscoelasic models are given in Table 2.3. When he addiional assumpion of hydrosaic pressure independence of he elasic response is used i is necessary o consequenly apply P 2 =, Q 2 = 3K. Noe ha he above Eqs. (2.52) and (2.54) refer o isoropic linear viscoelasic maerial for which number of independen generalized modules equals 2, namely G ve () and K ve () or equivalenly E ve () and ν ve (). In paricular case of isoropic elasiciy, he above creep modules reduce o wo consans G ve () = G and K ve () = K or equivalenly E ve () = E and ν ve () = ν.

17 2 Consiuive Equaions for Isoropic and Anisoropic Linear 73 Table 2.3 Differenial operaors for secleced simple linear viscoelasic models (afer Findley e al. [4]) Model Hooke Maxwell Voig Kelvin Sandard Burgers Differen equaion σ = Eε σ + η E σ = η ε σ = Eε + η ε σ + p σ = qε + q ε σ + p σ + p2 σ = q ε + q2 ε Deviaoric operaors P + η E η E + η +E + p + p Q 2G η E + η Volumeric operaors P2 + η E + η E E E E + η E +E E +E q + q E + p + p Q2 3K η E + η Transformed deviaoric operaors E E E + η E +E E +E q + q P + η E s + η E s + p +E s + p 2 s2 Q 2G η s E + η s E E E +E + η E E +E s q s + q 2 s2 Transformed volumeric operaors P2 + η E s + η E +E s + p s + p 2 s2 Q2 3K η s E + η s E E E + η E +E E s q +E s + q 2 s2

18 74 J.J. Skrzypek and A.W. Ganczarski Inegral Represenaion Under a Muliaxial Sress Sae As menioned above, in case of isoropic viscoelasic maerials number of he maerial ime funcions is equal o wo. Hence, i is possible o separae effecs of shape change from he volume change. In such a way, we arrive a he inegral form of he consiuive equaions of isoropic linear viscoelasic maerial ha generalize he analogous uniaxial equaion (2.43) formuliaxial saes s ij () = 2 σ kk () = 3 G ve ( ξ) e ij (ξ) dξ ξ K ve ( ξ) ε kk (ξ) dξ ξ (2.55) In a paricular case, if he volume change is pure elasic,eq.(2.55) ake he simplified form s ij () = 2 G ve ( ξ) e ij (ξ) dξ ξ (2.56) σ kk = 3K ε kk 2.4 Elasic-Viscoelasic Correspondence Principle for he Case of Isoropic Maerials Consider a he beginning, a paricular case of isoropic linear viscoelasic behavior for which separaion of he volume change from he shape change holds in a similar fashion as in case of isoropic elasic behavior, see Sec Remember however ha in a more general case of he anisoropic behavior, linear elasic, and linear viscoelasic, his separaion is no possible, see Sec..8. Analogy beween he ransformed equaions of linear isoropic viscoelasic maerials (2.49) and convenional equaions of isoropic elasiciy (2.5) leads o he searching of he soluions of viscoelasiciy on basis of a priori known coupled elasic problems. This analogy is known as he elasic-viscoelasic correspondence principle, see Findley e al. [4]. Le us summarize a complee se of equaions of linear isoropic viscoelasiciy, see Skrzypek [3] he equilibrium equaions σ ij (x, ) x i + b j () = (2.57) he consiuive equaions formed eiher in he forma of differenial operaor represenaion (2.46)

19 2 Consiuive Equaions for Isoropic and Anisoropic Linear 75 P s ij (x, ) = Q e ij (x, ) P 2 σ kk (x, ) = Q 2 ε kk (x, ) (2.58) or in he inegral represenaion (2.55) s ij (x, ) = 2 σ kk (x, ) = 3 he linearized geomeric equaions ε ij (x, ) = 2 G ev ( ξ) ė ij (x, ξ) dξ K ev ( ξ) ε kk (x, ξ) dξ [ ui (x, ) + u ] j (x, ) x j x i (2.59) (2.6) he boundary condiions under assumpion ha boundary beween domains of force Γ P and displacemen Γ U remain unchanged P i (x, ) = σ ij (x, ) n j na Γ P U i (x, ) = u i (x, ) na Γ U (2.6) For simpliciy independence of he viscoelasic modules, G ve,k ve from he spaial coordinaes holds. In oher words, maerial homogeneiy is assumed. In a paricular case of composie maerials alhough ha maerial is inhomogeneous a microlevel a homogenizaion echnique allows o reduce such problem o homogeneous a he RUC level of he represenaive uni cell, see Chap. 3. When he Laplace ransformaion of he above se of Eqs. (2.57) (2.6) is done we arrive a he ficiious coupled elasic problem σ ij (x, s) x i + b j (x, s) = ŝ ij (x, s) = 2sĜê ij (x, s) = Q P ê ij (x, s) σ kk (x, s) = 3s K ε kk (s) = Q 2 ε kk (x, s) P 2 ε ij (x, s) = [ ûi (x, s) + û ] j (x, s) 2 x j x i (2.62) P i (x, s) = σ ji (x, s) n j a Γ P Û i (x, s) = û i (x, s) a Γ U in which he body forces b j (x, s), exernal forces P i (x, s), displacemens Û i (x, s) as well as ficiious elasic consans Ĝ (s) and K (s) are funcions of he ransformed variable s.

20 76 J.J. Skrzypek and A.W. Ganczarski Finally, he analogy beween viscoelasic and coupled elasic problems can be formulaed. Namely if a soluion of coupled ficiious elasic problem is known (2.62) σ ij (x, s) and û i (x, s), he soluion of corresponding linear viscoelasic problem (2.57) (2.6) can be obained on he way of he inverse Laplace ransformaions σ ij (x, ) and u i (x, ). Simulaneously, following relaions mus hold Ĝ ve = Q (s) 2s P (s) K ve = Q 2 (s) 3s P 2 (s) (2.63) The correspondence principle can be applied only o he boundary problems where he inerface beween he boundary Γ P (where he exernal forces are prescribed) and he boundary Γ U (where he surface displacemens are given) is independen of ime, see Findley e al. [4]. The above limiaion does no hold in case of some maerial forming processes, for insance rolling, where he inerface beween boundaries Γ P and Γ U varies wih ime. An example of elasic-viscoelasic correspondence principle applied o muliaxial sress and srain saes he hick walled ube made of he isoropic sandard maerial subjec o inernal pressure p() = ph() applied insananeously a = is considered afer Findley e al. [4]. Taking advanage of he correspondence principle and recalling Lamé s soluion for coupled elasic problem u e = ( pa2 + ν b 2 b 2 a 2 E r + ν ) + ν r subsiuion of (2.54) fore, ν in (2.64) gives (2.64) + ν E P Q + ν ν 2 P 2 Q + P Q 2 3 P Q 2 In his way, a ficiious coupled elasic problem in erm of s û(s) = p(s)a2 b 2 a 2 P Q ( b 2 r + 2 P 2 Q + P Q 2 3 P Q 2 ) r (2.65) (2.66) is find ou. Applying ransformed operaors P, P 2, Q, and Q 2 of sandard model (see Table 2.3) under addiional assumpion ha shape change creep deformaion is accompanied by he elasic hydrosaic deformaion P 2 =, Q 2 = 3K, we find a soluion of he ficiious coupled elasic problem û(s) = pa2 b 2 a 2 A E η + B (2.67) s

21 2 Consiuive Equaions for Isoropic and Anisoropic Linear 77 where A and B are funcions of radial coordinae r exclusively A = E ( b 2 r + r 3 ), B = E + ( E b 2 E E r + r ) + 2r 3 9K (2.68) Finally, he soluion of he real linear viscoelasic problem is achieved by use of he inverse Laplace ransform of (2.67) in he following forma { ( u() = pa2 b 2 b 2 a 2 E r + r ) + 2r 3 9K + ( b 2 E r + r ) [ ( )]} 3 (2.69) exp E η 2.5 Inegral Represenaion of he Linear Viscoelasic Equaions of Anisoropic Maerials The elasic-viscoelasic correspondence principle applied for isoropic maerial presened in previous Sec. 2.4, was based on mahemaically convenien separaion of he volumeric and he shape change effecs from oal viscoelasic deformaion. However, in case of any class of maerial anisoropy such separaion does no occur. Hence for sake of generaliy, we change formulaion of he correspondence principle o he uniform fashion ha does no employ he above separaion. For convenience, he vecor-marix noaion will be used. In a general case of anisoropic linear viscoelasic maerial, heinegral form of consiuive equaions is furnished as (see Shu and Ona [2]) or ε ij () = σ ij () = ve J ijkl ( ξ) σ kl (ξ) dξ (2.7) ve E ijkl ( ξ) ε kl (ξ)dξ (2.7) where ve J ijkl ( ξ) defines he fourh-rank ensor of creep funcions; whereas, ve E ijkl ( ξ) is he fourh-rank ensor of relaxaion funcions which characerize viscoelasic properies of anisoropic maerial. Assuming he symmery condiions: ve J ijkl = ve J klij = ve J jikl = ve J ijlk,or ve E ijkl = ve E klij = ve E jikl = ve E ijlk, boh ensors of viscoelasic anisoropy have 2 independen funcions. Boh consiuive ensor funcions ve J ijkl or ve E ijkl depend on curren ime (inegraion limi).

22 78 J.J. Skrzypek and A.W. Ganczarski When he vecor-marix noaion is used, he general consiuive equaion of anisoropic linear viscoelasic maerial defined by Eqs. (2.7) and (2.7) akes equivalen inegral form {ε()} = [ ve J( ξ)] {σ(ξ)} dξ (2.72) ξ or {σ()} = [ ve E( ξ)] {ε(ξ)} dξ (2.73) ξ When nonabbreviaed noaion is used inroducing marix of creep compliance funcions ve J ij ( ξ), we arrive a he following consiuive inegral equaions of anisoropic linear maerial ε xx () ε yy () ε zz () = γ yz () γ zx () γ xy () ve J ve J ve 2 J ve 3 J ve 4 J ve 5 J 6 σ xx (ξ) ve J ve 2 J ve 22 J ve 23 J ve 24 J ve 25 J 26 ve J ve 3 J ve 32 J ve 33 J ve 34 J ve σ yy (ξ) 35 J 36 ve J ve 4 J ve 42 J ve 43 J ve 44 J ve σ zz (ξ) 45 J 46 dξ (2.74) ve J ve 5 J ve 52 J ve 53 J ve 54 J ve τ yz (ξ) 55 J 56 τ zx (ξ) ve J ve 6 J ve 62 J ve 63 J ve 64 J ve 65 J 66 τ xy (ξ) where [ ve J] ij =[J( ξ)] ij is he creep compliance marix. For Eq. (2.73) he inverse relaion holds σ xx () ve E ve E ve 2 E ve 3 E ve 4 E ve 5 E 6 ε xx (ξ) σ yy () σ zz () ve E ve 2 E ve 22 E ve 23 E ve 24 E ve 25 E 26 = ve E ve 3 E ve 32 E ve 33 E ve 34 E ve ε yy (ξ) 35 E 36 τ yz () ve E ve 4 E ve 42 E ve 43 E ve 44 E ve ε zz (ξ) 45 E 46 dξ (2.75) τ zx () ve E ve 5 E ve 52 E ve 53 E ve 54 E ve γ yz (ξ) 55 E 56 γ zx (ξ) τ xy () ve E ve 6 E ve 62 E ve 63 E ve 64 E ve 65 E 66 γ xy (ξ) In paricular case of orhoropic linear viscoelasic maerial, Eqs.(2.74) and (2.75) reduce o narrower forms ε () ve J ve J ve 2 J 3 σ (ξ) ε 22 () ε 33 () ve J ve 2 J ve 22 J 23 = ve J ve 3 J ve σ 22 (ξ) 32 J 33 σ 33 (ξ) γ 23 () ve J 44 dξ (2.76) γ 3 () ve τ 23 (ξ) J 55 τ 3 (ξ) γ 2 () ve J 66 τ 2 (ξ)

23 2 Consiuive Equaions for Isoropic and Anisoropic Linear 79 or σ () σ 22 () σ 33 () = τ 23 () τ 3 () τ 2 () ve E ve E ve 2 E 3 ε (ξ) ve E ve 2 E ve 22 E 23 ve E ve 3 E ve ε 22 (ξ) 32 E 33 ε 33 (ξ) ve E 44 dξ (2.77) ve γ 23 (ξ) E 55 γ 3 (ξ) ve E 66 γ 2 (ξ) being exension of equaions of orhoropic linear elasiciy (.3). Noe ha boh sresses and srains are funcions of ime ve σ ij = ve σ ij (), ve ε ij = ve ε ij (), in similar fashion as elemens of creep compliance ve J ij = ve J ij ( ξ) and relaxaion ve E ij = ve E ij ( ξ) marices. Applying he Laplace ransform o Eqs. (2.74) and (2.75), we arrive a he associaed ficiious elasic consiuive equaions in he ransformed domain ε xx (s) Ĵ Ĵ 2 Ĵ 3 Ĵ 4 Ĵ 5 Ĵ 6 σ xx (s) ε yy (s) Ĵ 2 Ĵ 22 Ĵ 23 Ĵ 24 Ĵ 25 Ĵ 26 σ yy (s) ε zz (s) Ĵ = s 3 Ĵ 32 Ĵ 33 Ĵ 34 Ĵ 35 Ĵ 36 σ zz (s) (2.78) γ yz (s) Ĵ 4 Ĵ 42 Ĵ 43 Ĵ 44 Ĵ 45 Ĵ 46 τ yz (s) γ zx (s) Ĵ 5 Ĵ 52 Ĵ 53 Ĵ 54 Ĵ 55 Ĵ 56 τ zx (s) γ xy (s) Ĵ 6 Ĵ 62 Ĵ 63 Ĵ 64 Ĵ 65 Ĵ τ 66 xy (s) or σ xx (s) Ê Ê 2 Ê 3 Ê 4 Ê 5 Ê 6 ε xx (s) σ yy (s) Ê 2 Ê 22 Ê 23 Ê 24 Ê 25 Ê 26 ε yy (s) σ zz (s) Ê = s 3 Ê 32 Ê 33 Ê 34 Ê 35 Ê 36 ε zz (s) τ yz (s) Ê 4 Ê 42 Ê 43 Ê 44 Ê 45 Ê 46 γ yz (s) τ zx (s) Ê 5 Ê 52 Ê 53 Ê 54 Ê 55 Ê 56 γ zx (s) τ xy (s) Ê 6 Ê 62 Ê 63 Ê 64 Ê 65 Ê γ 66 xy (s) (2.79) 2.6 Applicaion of he Anisoropic Correspondence Principle o he Case of Orhoropic Composie Maerials For a purpose of engineering applicaion, for insance o some composie maerials in which a leas one of phases exhibis viscoelasic behavior, i is sufficien o assume he narrower case of orhoropic linear viscoelasic equaions (2.76) and (2.77). When he Laplace ransformaion is applied o he inegral consiuive equaions of he orhoropic linear viscoelasic maerial (2.76) and (2.77), we arrive a he associaed ficiious orhoropic elasic equaions in erms of s

24 8 J.J. Skrzypek and A.W. Ganczarski { σ(s)} = s[ê(s)] { ε(s)} (2.8) or { ε(s)} = s[ĵ(s)] { σ(s)} (2.8) When he vecor-marix noaion is applied he corresponding formulas can be wrien in an expanded fashion σ (s) Ê Ê 2 Ê 3 ε (s) σ 22 (s) Ê 2 Ê 22 Ê 23 ε 22 (s) σ 33 (s) Ê = s 3 Ê 32 Ê 33 ε 33 (s) (2.82) τ 23 (s) Ê τ 3 (s) 44 γ 23 (s) Ê 55 γ 3 (s) τ 2 (s) Ê γ 2 (s) 66 or ε (s) Ĵ Ĵ 2 Ĵ 3 σ (s) ε 22 (s) Ĵ 2 Ĵ 22 Ĵ 23 σ 22 (s) ε 33 (s) Ĵ = s 3 Ĵ 32 Ĵ 33 σ 33 (s) γ 23 (s) Ĵ γ 3 (s) 44 τ 23 (s) Ĵ 55 τ 3 (s) γ 2 (s) Ĵ τ 2 (s) 66 (2.83) The above equaions generalize consiuive equaions of isoropic viscoelasic maerial (2.62) 2,3 o he case of maerial orhoropy, where Ê ij = Ê ij (s) and Ĵ ij = Ĵ ij (s) sand for he orhoropic relaxaion funcion marix and he creep compliance marix, respecively. Soluion of he problem of viscoelasic orhoropy can be obained on he way of he Laplace inverse ransform applied o he ransformed variables σ ij (s), ε ij (s), which are reransformed o physical variables σ ij (), ε ij (). I should be emphasized ha in case of anisoropic linear viscoelasic maerials, similar o anisoropic elasic maerials, i is no possible o separae he consiuive equaions ino he volumeric change and he shape change uncoupled equaions since anisoropy resuls in full coupling beween he volume and he shape viscoelasic deformaion. In case of composie maerials, he properies of which exhibi he linear viscoelasic feaures, a generalizaion of commonly used homogenizaion echniques (see Chap. 3) o viscoelasic work-regime can be proposed (cf. Haasemann and Ulbrich [5]). The frequenly applied concep of he represenaive uni cell (RUC) originally developed for elasic composies can also be adaped o nonelasic behavior of composies (marix and/or fiber). To his end, he class of linear viscoelasic maerial occurs o be very convenien when use of he correspondence principle which enables o ransform he real viscoelasic (ime-dependen) problem o a ficiious elasic (ime-independen) problem in he domain of new variable s (see Table 2.4).

25 2 Consiuive Equaions for Isoropic and Anisoropic Linear 8 Consiuive equaion of viscoelasic maerial can be formulaed wice: a he level of subcell (βγ) where componens are described by differen viscoelasic equaions (marix, fibers, paricles) and a he level of RUC where differen properies of maerial are homogenized in a paricular way o yield mean or effecive viscoelasic response. The above described sages of descripion are briefly presened as follows: a level of subcell he local consiuive equaions of linear viscoelasiciy hold ve ε (βγ) ij () = ve J (βγ) ijkl ( ξ)ve σ (βγ) kl (ξ)dξ (2.84) or ve σ (βγ) ij () = ve E (βγ) ijkl ( ξ)ve ε (βγ) kl (ξ)dξ (2.85) where he local variables, microsress, and microsrain are combined by local consiuive ime-dependen fourh-rank ensors (he creep compliance ensor or he relaxaion ensor) for he homogeneous consiuen maerial. In a formal fashion when a homogenizaion inside he RUC is used, we arrive a ve ε ij () = ve J ijkl ( ξ) ve σ kl (ξ)dξ (2.86) or ve σ ij () = ve E ijkl ( ξ) ve ε kl (ξ)dξ (2.87) Mean or effecive fourh-rank ensors of creep compliance ve J ijkl or relaxaion ve E ijkl are defined a he level of RUC in erms of he corresponding local ensors ve J (βγ) ijkl or ve E (βγ) ijkl a he level of subcell by he use of a homogenizaion procedure in an analogous way as for he elasic composie (3.58). Homogenizaion of he viscoelasic properies of he composie maerial is no a rivial problem and is seldom me in lieraure, cf. e.g., Haasemann and Ulbrich [5]. Applicaion of correspondence principle occurs o be very useful since i makes possible o conver ime-dependen heerogeneous viscoelasic problem o associaed ime-independen elasic problem for which homogenizaion ools can direcly be applied. In paricular when elasic-viscoelasic analogy is applied for anisoropic composies a he level of RUC, he applicaion of he Laplace ransform allows o reduce inegral equaion of real maerial (2.86)or(2.87) o coupled se of equaions of a ficiious elasic problem in space of he ransformed variable s

26 82 J.J. Skrzypek and A.W. Ganczarski e ε ij (s) = s Ê ijkl (s)e σ kl (s) (2.88) or e σ ij (s) = s Êijkl(s) e ε kl (s) (2.89) Finally, soluion of he anisoropic linear viscoelasic problem can be obained by he use of inverse Laplace ransformaion from he ransformed domain e ε ij (s), e σ ij (s) o he physical domain ve ε ij (), ve σ ij (). When absolue noaion is used we ge (see Haasemann and Ulbrich [5]) ve σ() = ve E() : ve ε( = ) + ve E( ξ) : ve ε(ξ)dξ = ve E() : ve ε( = ) +[ ve E : ve ε]() (2.9) Recall definiion of he Laplace ransform of he funcion f () ( > ) ino he funcion of ransformed variable f (s) L { f ()} = f (s) def = and definiion of he convoluion of wo funcions f () def = f ()e s d (2.9) f ( ξ) f 2 (ξ)dξ f () f 2 () (2.92) Applying he Laplace ransform (2.9) o he convoluion inegral (2.92), we arrive a he convoluion heorem (see Findley e al. [4]) L f ( ξ) f 2 (ξ)dξ = L { f () f 2 ()} = f (s) f 2 (s) (2.93) Taking nex he Laplace ransform of he inegral form of consiuive equaion of he anisoropic linear viscoelasiciy (2.9), we arrive a he equivalen ransformed algebraic equaion of anisoropic linear elasiciy according o scheme ve σ() = ve E() : ve ε( = ) + ve E() : ve ε() L σ(s) = sê(s) : ε(s) (2.94) defined by funcion of he ransformed variable s. Theransformed marix of anisoropic ficiious elasiciy sê(s) is buil a he level of RUC of considered

27 2 Consiuive Equaions for Isoropic and Anisoropic Linear 83 composie. The above marix is obained by he homogenizaion of he ransformed isoropic local marices sê(βγ) (s) a he level of composie microsrucure (subcells). The procedure described above is skeched by he following scheme shown in Table 2.4. The homogenizaion procedures for he anisoropic elasic composies are well recognized (for deails see nex chaper). By conras, here is no unique and direc homogenizaion procedure o yield he effecive creep compliance ve J (βγ) ijkl () and relaxaion ve E (βγ) ijkl () ensors (e.g., Haasemann and Ulbrich [5]). Hence o overcome his deficiency, he suggesed scheme is as follows: firs apply he Laplace ransform a he level of subcell in order o eliminae physical ime (lef pah in Table 2.4), second use a homogenizaion mehod in order o reach he RUC level for ficiious elasic RUC of composie maerial and finally apply he inverse Laplace ransform o arrive a he physical viscoelasic RUC level (righ back pah in Table 2.4). Usually for sake of simpliciy of furher applicaions, he ransversely isoropic effecive relaxaion marix sê(s) a he level of RUC is sufficien, whereas a he microlevel (subcell) he isoropic marices for he consiuens (f) fiber and (m) composie marix sê(f) (s) and sê(m) (s) are usually acceped (see also (2.82)). Elasic-viscoelasic correspondence principle as applied o orhoropic viscoelasic Table 2.4 Elasic-viscoelasic homogenizaion mehod based on he represenaive uni cell (RUC) applied o he associaed elasic maerial by correspondence principle ε (βγ) = σ (βγ) = ve J (βγ) σ (βγ) dξ linear viscoelasic {ε} = ============= ve E (βγ) ε (βγ) dξ homogenizaion? {σ} = L ve J σ dξ ve E ε dξ L ε (βγ) = s J (βγ) σ (βγ) associaed elasic ε = s J σ ============= σ (βγ) = s E (βγ) ε (βγ) homogenizaion σ = s E ε

28 84 J.J. Skrzypek and A.W. Ganczarski u mm e 3 e 2 e E m,ν m s 8 s u () E f,ν f Fig. 2.5 Beam model and RUC of unidirecional composie according o Haasemann and Ulbrich [5] Table 2.5 Viscoelasic properies of fiber and marix maerial following Haasemann and Ulbrich [5] Relaxaion funcion Fiber Marix e./s e /s Poisson s raio.2.35 and viscoplasic maerials is applied by Haasemann and Ulbrich [5]. In case of unidirecionally reinforced composie considered by Haasemann and Ulbrich [5] (see Fig. 2.5), boh fiber and marix maerials were described as isoropic linear viscoelasic (see Table 2.5); whereas a macroscale, he sane composie maerial obeyed he ransverse isoropy symmery. The Laplace Carson ransform L C { f ()} = f C (s) = s e s f () d (2.95) was used in order o ransform equaions of ransversely isoropic linear viscoelasic maerial in domain ino corresponding equaions of ficiious linear elasic maerial in domain of ransformed variable s, in which convenional homogenizaion echniques of elasic composies were applicable (for deails see Chap. 3). References. Alenbach, H.: Classical and non-classical creep models. In: Alenbach, H., Skrzypek, J.J. (eds.) Creep and Damage in Maerials and Srucures. CISM Courses and Lecures No. 399, pp Springer, Wien (999) 2. Been, J.: Creep Mechanics, 3rd edn. Springer, Berlin (28) 3. Byron, F.W., Fuller, M.D.: Mahemaics of ellipic inegrals for engineers and physiciss. Springer, Berlin (975) 4. Findley, W.N., Lai, J.S., Onaran, K.: Creep and Relaxaion of Nonlinear Viscoplasic Maerials. Norh-Holland, New York (976)

29 2 Consiuive Equaions for Isoropic and Anisoropic Linear Haasemann, G., Ulbrich, V.: Numerical evaluaion of he viscoelasic and viscoplasic behavior of composies. Technische Mechanik 3( 3), (2) 6. Krempl, E.: Creep-plasic ineracion. In: Alenbach, H., Skrzypek, J.J. (eds.) Creep and Damage in Maerials and Srucures. CISM Courses and Lecures No. 399, pp Springer, Wien (999) 7. Murakami, S.: Coninuum Damage Mechanics. Springer, Berlin (22) 8. Nowacki, W.: Teoria pełzania. Arkady, Warszawa (963) 9. Pipkin, A.C.: Lecures on Viscoelasiciy Theory. Springer, Berlin (972). Pobedrya, B.E.: Mehanika kompozicionnyh maerialov. Izd. Mosk. Univ. (984). Rabonov, Ju.N.: Creep Problems of he Theory of Creep. Norh-Holland, Amserdam (969) 2. Shu, L.S., Ona, E.T.: On anisoropic linear viscoelasic solids. In: Proceedings of he Fourh Symposium on Naval Srucures Mechanics, p. 23. Pergamon Press, London (967) 3. Skrzypek, J.J.: In Henarski, R.B., (ed.) Plasiciy and Creep, Theory, Examples, and Problems. Begell House/CRC Press, Boca Raon (993) 4. Skrzypek, J.: Maerial models for creep failure analysis and design of srucures. In: Alenbach, H., Skrzypek, J.J. (eds.) Creep and Damage in Maerials and Srucures. CISM Courses and Lecures No. 399, pp Springer, Wien (999) 5. Skrzypek, J., Ganczarski, A.: Modeling of Maerial Damage and Failure of Srucures. Springer, Berlin (999)

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