Thermodynamic investigation of yield-stress models for amorphous polymers J Richeton, Said Ahzi, Loïc Daridon To cite this version: J Richeton, Said Ahzi, Loïc Daridon. Thermodynamic investigation of yield-stress models for amorphous polymers. Philosophical Magazine, Taylor & Francis, 2007, 87 (24), pp.3629-3643. 10.1080/14786430701381162. hal-00513838 HAL Id: hal-00513838 https://hal.archives-ouvertes.fr/hal-00513838 Submitted on 1 Sep 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Philosophical Magazine & Philosophical Magazine Letters For Peer Review Only Thermodynamic investigation of yield-stress models for amorphous polymers Journal: Philosophical Magazine & Philosophical Magazine Letters Manuscript ID: TPHM-06-Jun-0219 Journal Selection: Philosophical Magazine Date Submitted by the 21-Jun-2006 Author: Complete List of Authors: Richeton, J; Universite Louis Pasteur, IMFS Ahzi, Said; University Louis Pasteur, IMFS Daridon, L; Universite de Montpellier II, LMGC mechanical behaviour, polymers, thermodynamics, plastic Keywords: deformation Keywords (user supplied): yield stresse, amorphous polymers http://mc.manuscriptcentral.com/pm-pml Page 1 of 24 Philosophical Magazine & Philosophical Magazine Letters 1 2 3 Thermodynamic investigation of yield-stress models for amorphous polymers 4 5 6 7 8 J. Richeton, S. Ahzi * and L. Daridon + 9 10 11 12 13 Institut de Mécanique des Fluides et des Solides - UMR 7507 14 15 Université Louis Pasteur, 2 Rue Boussingault, 67000 Strasbourg, France 16 For Peer Review Only 17 18 19 20 21 22 Abstract 23 24 25 26 27 A thermodynamic study of the yield process in amorphous polymers is proposed to 28 29 investigate four yield theories: the Eyring model and its linearized form, the cooperative 30 31 32 model and the Argon model. For a poly(methyl methacrylate) (PMMA) and a polycarbonate 33 34 (PC), the corresponding apparent activation volumes and apparent activation energies are 35 36 calculated and compared for a wide range of temperatures and strain rates. In the case of the 37 38 39 cooperative model, we show that the secondary molecular relaxation is a key parameter in the 40 41 explanation of the specific mechanical behaviour of glassy polymers at low temperatures and 42 43 44 at high strain rates. For the three other models, thermodynamic inconstancies were found and 45 46 these are discussed. 47 48 49 50 51 52 53 * Corresponding author. Tel +33-3902-42952; fax +33-3886-14300; e-mail: [email protected] . 54 + 55 Present address: Laboratoire de Mécanique et de Génie Civil - UMR 5508, Université de Montpellier II 56 57 58 59 60 http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters Page 2 of 24 1 2 3 1. Introduction 4 5 6 7 8 The stress-strain response of amorphous polymers is known to be dependent on 9 10 temperature and on strain rate. In particular, the yield stress obeys thermally activated 11 12 13 processes. Many molecular theories have been proposed for the prediction of the yield stress 14 15 in the case of amorphous polymers [1-15]. Most of these models give an acceptable 16 For Peer Review Only 17 18 description of the yield stress, but they are either unable to account for the dramatic increase 19 20 of the yield stress at higher strain rates or are not valid through the glass-transition 21 22 temperature region [16]. Recently, a new formulation of the cooperative model [17,18] has 23 24 25 been shown to give excellent results for a wide range of strain rates and temperatures, 26 27 including the high strain rates and the glass-transition region. 28 29 The purpose of the present paper is to study and compare the thermodynamic 30 31 32 properties associated with four different yield-stress models: the two forms of Eyring’s 33 34 models, Argon’s model, and the cooperative models. This step will be helpful for an 35 36 understanding of the temperature and strain-rate limitations of the models. The 37 38 39 thermodynamic formalism of the yield stress for amorphous polymers is originally contained 40 41 in the work of Lefebvre, Escaig and co-workers [19-21] according to the concept of apparent 42 43 44 activation volume and apparent activation energy. This formalism could be applied to any 45 46 yield model. However, an analytical form for the apparent activation volume and energy 47 48 could be derived only for the four predicted models. Numerical study of the thermodynamic 49 50 51 formalism for other models such as the Robertson model [3] and the Ree-Eyring formulation 52 53 [2,4-9] are undoubtedly of interest but this is out of the scope of the present paper. In addition, 54 55 we note that the pressure can be included in each of the analysed models, as was done by 56 57 58 Ward [22] for the Eyring models, by Boyce et al. [23] for the Argon model and by Richeton et 59 60 al. [18] for the cooperative model. However, for the sake of simplicity and since we are http://mc.manuscriptcentral.com/pm-pml Page 3 of 24 Philosophical Magazine & Philosophical Magazine Letters 1 2 3 interested in the temperature and strain-rate effects, we neglect the pressure effect. The yield- 4 5 6 stress models were also extrapolated through the glass-transition region where the yield stress 7 8 is expected to vanish. 9 10 11 12 13 14 15 2. Thermodynamic analysis of the models 16 For Peer Review Only 17 18 19 20 2.1. Background 21 22 23 24 25 Lefebvre, Escaig and co-workers [19-21] introduced a thermodynamic development 26 27 for the yield stress relating strain rate, applied stress and temperature at comparable structural 28 29 states. They focused their interest on the apparent (or operational) activation volume V and 30 0 31 32 on the apparent activation energy ∆H0 . Expressions for V0 at a fixed temperature and 33 34 35 structure and that for ∆H0 at fixed yield stress and temperature are given by: 36 37 38 39 40 ∂ ln ε& 41 V0 = kT 42 ∂σ y T, struct . 43 (1) 44 2 ∂ ln ε& 45 ∆H0 = kT ∂T struct 46 σ y , . 47 48 49 50 51 where k is the Boltzmann constant, T is the absolute temperature, ε& is the strain rate and σ y 52 53 is the yield stress. 54 55 56 57 58 2.2. Application to the Eyring model 59 60 http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters Page 4 of 24 1 2 3 Eyring's model derives from the formal development of the transition-state theory [1]. 4 5 6 The resulting strain rate is classically given by: 7 8 9 10 E E 11 E ∆H σ yV ε&= ε & 0 exp − sinh (2) 12 kT2 kT 13 14 15 16 For Peer Review Only 17 E E E 18 where ε&0 is a constant pre-exponential strain rate, ∆H is an activation energy and V is an 19 20 activation volume. 21 22 23 24 25 Using equations (1) and (2), the operational activation volume of the Eyring model is 26 27 derived as: 28 29 30 31 32 V E 1 V E = (3) 33 0 E 34 2 tanh()σ yV 2 kT 35 36 37 38 39 We propose to rewrite this equation as a function of temperature and strain rate by solving Eq. 40 41 −1 E −1 2 42 (2) for σ yV2 kT and given that for x > 0 , (tanh() sinh() 1x) = 1 + x . The activation 43 44 45 volume is then reduced to a handier form: 46 47 48 49 2 50 EE E E Vε&0 ∆ H 51 V0 =1 + exp − (4) 52 2 ε& kT 53 54 55 56 57 In a similar way, the operational activation energy is given by: 58 59 60 http://mc.manuscriptcentral.com/pm-pml Page 5 of 24 Philosophical Magazine & Philosophical Magazine Letters 1 2 3 σ V E 1 4 ∆HE =∆ H E − y (5) 5 0 E 2 tanh()σ yV 2 kT 6 7 8 9 10 According to Eq. (3), this can also be written as 11 12 13 14 15 ∆HE =∆ H E −σ ⋅ V E (6) 16 0Fory Peer 0 Review Only 17 18 19 20 Here, we notice that the operational activation energy is composed of two separate entities. 21 22 23 The first represents the activation energy in absence of stress and the second contributes to the 24 25 reduction of the total energy barrier arising from the presence of stress. 26 27 28 29 30 2.3. Application to the linearized Eyring model 31 32 33 34 Eyring’s equation is frequently presented in the following linearized form: 35 36 37 38 39 ∆HLE − σ V LE 40 ε= ε LE exp − y (7) & & 0 41 kT 42 43 44 45 46 where all the parameters have the same meaning as in Eq. (2). This form is simply obtained 47 48 49 by replacing the hyperbolic function of Eq. (2) by an exponential under the condition 50 51 E σ yV2 kT > 1 . As a matter of fact, the original Eyring parameters and the linearized ones are 52 53 54 linked as follow: 55 56 57 58 E E 59 LEε&0 LE E LE V ε&0 = ∆=∆H H V = (8) 60 2 2 http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters Page 6 of 24 1 2 3 4 5 6 For this model, the operational activation volume is found to be independent of the 7 8 temperature and the strain rate: 9 10 11 12 LE LE 13 V0 = V (9) 14 15 16 For Peer Review Only 17 18 For the operational activation energy, we obtain: 19 20 21 22 LE LE LE 23 ∆H0 =∆ H −σ y ⋅ V (10) 24 25 26 27 28 LE LE Since V0 = V , this equation is simply the classical form of the operational activation 29 30 31 energy as given in Eq.