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Model-Based Framework Rheology for the Interpretation of Laos

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Model-Based Framework Rheology for the Interpretation of Laos ISSN 2176-5480 22nd International Congress of Mechanical Engineering (COBEM 2013) November 3-7, 2013, Ribeirão Preto, SP, Brazil Copyright c 2013 by ABCM MODEL-BASED FRAMEWORK RHEOLOGY FOR THE INTERPRETATION OF LAOS Roney Leon Thompson Universidade Federal Fluminense [email protected] Paulo Roberto de Souza Mendes Alexandra Araujo Alicke Ricardo Teixeira Leite Pontifífica Universidade Católica do Rio de Janeiro [email protected], [email protected], [email protected] Abstract. The more stablished method employed to generalize the linear viscoelastic oscillatory material functions for the interpretation of Large Amplitude Oscillatory Shear is based on the so-called “Fourier-transform rheology", where the nonlinear response of the material is decomposed into a Fourier series. In this approach, the harmonics associated to the frequencies that are higher than the imposed one are the measurers of the nonlinear response. This methodology has its merits, since it offers an objective rationale for the treatment of complex behavior. However, it has received a lot criticism due to lack of physical interpretation of the role played by the different higher harmonics. This drawback is founded on the very soul of these methodologies: the necessity of a basis of infinite functions in order to provide the full description. Towards the task of joining understanding and predictability of complex material behavior we propose a different philosophy for the interpretation of LAOS results. The methodology consists on choosing a constitutive model whose parameters are clearly and physically interpreted and using it as a framework to understand material behavior. We call this methodology Model-Based Framework Rheology (MBFR). A model can be roughly seen as a combination of basis functions and coefficients or parameters. The philosophy consists on taking the advantage of a our experience and knowledge with respect to a certain model parameter and generalize its concept to a more complex situation by relaxing the usual restriction this parameter has in the model where it was conceived and defined. The resulting analysis is born with a physical interpretation and is ready to be implemented in a different problem. The difficulty, and the strength, of this philosophy comes from finding a model framework with a reasonable degree of complexity. If the basis functions form a too simple set, then the parameters will carry all the complexity of the material. In this case it seems that is not reasonable to expect that the model will perform adequately in different conditions. On the other hand, if one defines an enormous number of functions, there are lots of coefficients to be determined and it is hard to attribute a physical interpretation to these coefficients, what is not desirable also. However, an appropriate choice of the model framework provides a rich interpretation of the results and a reasonable predictiveness of the material behavior in other conditions are obtained. We show that the usual approaches of SAOStrain and LAOStrain have a clear interpretation from the Kelvin-Voigt framework while the usual approaches employed in SAOStress and LAOStress are better interpreted from a Maxwell perspective. A more robust model, where elasticity, viscous response, plasticity and thixotropy are present, is offered as a framework for the analysis of the LAOS response of complex materials. Keywords: keyword 1, keyword 2, keyword 3. (up to 5 keywords) 1. INTRODUCTION 1.1 General aspects of LAOS After a long tradition of usage of oscillatory motion in the linear viscoelastic regime, where the material rheologi- cal functions G0 and G00 are the acclaimed measurable quantities, nowadays, LAOS is considered the most promising methodology to understand the behavior of complex material. LAOS experiments combine two important features: it probes the material in the nonlinear viscoelastic regime and it uses the advantages of the oscillatory motion. The former is a practical need, since many industrial processes subject complex material to complex behavior and because of that, understanding how this material behaves in the nonlinear regime is fundamental for optimization and design purposes. The latter concerns the strength that is attributed to oscillatory motions, namely its capacity of tuning amplitude and frequency independently (Pipkin, 1972). Since we can construct independent Weissenberg and Deborah dimensionless numbers from amplitude and frequency (Giacomin et al., 2011), a sweep of these two entities provides a large spectra of material behavior. One important and largely employed methodology to understand the complex behavior of these materials is to rep- 2813 ISSN 2176-5480 F. Author, S. Author and T. Author (update this heading accordingly) Paper Short Title (First Letters Uppercase, make sure it fits in one line) resent its response in the so-called Lissajous-Bowditch curves. The elastic form is obtained from the trajectory of the response in the stress-strain plane while in the viscous form the response is represented in the stress-strain rate plane. Besides the obvious results expected from a Newtonian and a Hookean materials, the linear response is also predictable in a Lissajous-Bowditch representation, where an elliptical orbit is the projected curve in each plane. Hence, the degree of nonlinearity can be inferred by a “degree of non-ellipticity" of the curve form. This way of interpreting the data can give useful insights and for sure increases the comprehension of material behavior. However, mathematically defined and physically soundable quantities are sine qua non conditions for a description of the material behavior. Therefore, the challenge faced by the rheology community is to present measurable and meaningful rheological quantities that can be consider the nonlinear counterparts of G0 and G00. Until now, there is no consensual rationale for such generalization. The first and more stablished method employed to generalize the linear viscoelastic oscillatory material functions is based on the so-called “Fourier-transform rheology" (Wilhelm, 2002), where the nonlinear response of the material is decomposed into a Fourier series. In this approach, the harmonics associated to the frequencies that are higher than the imposed one are the measurers of the nonlinear response. An alternative representation form is to use a different-from- sinusoidal set of orthogonal basis. The most representative approach of this form is the Chebyshev polynomials of the first kind (Ewoldt et al., 2008). These two methodologies have their merits, since they offer an objective rationale for the treatment of complex behavior. However, they have received a lot criticism (Cho et al., 2005; Rogers and Lettinga, 2012; Rogers, 2012) due to lack of physical interpretation of the role played by the different higher harmonics. This drawback is founded on the very soul of these methodologies, i.e. on the necessity of a basis of infinite functions in order to provide the full description. There are few options available in the literature that can be used as alternatives to the Fourier-Chebyshev approach. Two of them are worthy of mentioning here. The first one is the Stress Decomposition (SD) of Cho et al. (2005) and the second is the Sequence of Physical Processes (SPP) of Rogers (2012). Although the two of them have different rationales, they offer approaches to analyze the nonlinear material response by considering two “material" functions (instead of infinite functions of Fourier and Chebyshev analysis) that can be interpreted as generalized dynamic moduli. The two approaches present alternative rheological functions that recover the linear viscoelastic moduli in the limit of the linear regime. The SD approach is based on a decomposition of the stress response, of a LAOStrain input of the form 0 00 0 00 γ = γo sin wt, into two additive parts: σ and σ . Where σ is a function of the strain, γ, and σ is a function of the strain rate, γ_ = !γ. By symmetry arguments, requiring that σ0 is an odd function of γ and an even function of γ_ and that σ00 is an even function of γ and an odd function of γ_ , they come to this unique decomposition and stated that σ0 is the elastic part of the stress and σ00 is the viscous part of the stress. Writing σ0 = Γ0γ and σ00 = Γ00γ_ , they called Γ0 and Γ00 the generalized dynamic moduli. The SPP approach was developed in Rogers et al. (2011) and Rogers and Lettinga (2012), where the sequence of physical processes was identified in trajectory through the 3D space defined by stress-strain-strain rate. A quantitative oriented form of SPP was presented by Rogers (2012). In this approach, the projection of the binomial vector on the strain-strain rate plane is assumed to be the generalized complex modulus. The projection of this vector into strain and strain rate directions, R0 and R00=!, are considered the generalized dynamic moduli. Although Γ0 6= R0 and Γ00 6= R00=!, in the linear viscoelastic regime, both tend to G0 and η0. It is important to notice that G0 and G00 moduli are considered the representatives of elastic and viscous behavior, respectively, in the linear viscoelastic regime irrespectively of the constitutive model of the material. Hence, discovering which are the “true" representatives counterparts of elastic and viscous behavior seems to be the natural path to follow as far as the nonlinear regime is considered. Therefore, the Fourier-Chebyshev analyses generalize G0 and G00 by a series 0 00 of coefficients, Gn and Gn in the case of the Fourier series, and en and vn in the case of the Chebyshev functions. In these cases, the first coefficient reduces to linear viscoelastic counterpart and the coefficients associated to functions of higher order vanish in the linear viscoelastic regime. On the other hand the SD-SPP analyses generalize each modulus as a single generic version, and hence, there are two new entities, Γ0 and Γ00 in the case of SD, and R0 and R00 in the case of SPP, that reduce to their linear viscoelastic counterparts.
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