Issue 49

Y. Saadallah et alii, Frattura ed Integrità Strutturale, 49 (2019) 666-675; DOI: 10.3221/IGF-ESIS.49.60 667 Many models have been developed to represent the rheological behavior of thermoplastics [3-8]. Phenomenological models inspired by analog mechanisms such as spring, damper and pad combinations are well placed to represent viscoelastic and viscoplastic responses. The simplest mechanisms are the Maxwell and Kelvin-Voigt viscoelastic models and the Bingham viscoplastic model. The combination of these mechanisms in series or in parallel makes it possible to represent a wide variety of mechanical behaviors capable of representing linear or non-linear viscoelastic and viscoplastic rheological behaviors. The mechanical behavior characterization of thermoplastic polymers is always a complicated task. This behavior depends on many external factors such as temperature and strain rate, but also other internal factors including entanglements, crosslinking and crystallinity degree [2, 6]. This dependence is physically explained by their structure in macromolecular chains whose mobility, entanglement, branching and crosslinking are the main factors at the origin of the viscous aspect [9]. Taking into account these microscopic factors makes it possible to better understand the macroscopic response and contributes to the establishment of relevant behavior models [10]. On the other hand, it is very difficult to account the influence of all the physical mechanisms in the behavior laws, from which the application of the phenomenological models . Demonstration of time behavior dependence is usually done by means of creep, relaxation or tensile tests at different strain rates. Experimental data is used to identify the model parameters under consideration using different methods. Two major classes of methods are distinguished: analytical-graphical methods and inverse identification methods. Methods based on the formulation of inverse problems are the most used. They consist in the minimization of a function that establishes the gap between the experimental data and the results of the modeling. It is therefore a question of solving an optimization problem by setting up the appropriate calculation algorithm. In recent years, so-called metaheuristic optimization methods have been developed. As an example, genetic algorithms and ant colony algorithms are mentioned [refs]. Genetic algorithms based on evolutionary ideas of natural selection and genetics [11, 12]. They serve as powerful tools for solving very complicated optimization problems when the derivative of the objective function is very difficult to obtain or does not exist [13, 14]. So, they saw their use for determining material parameters for models with a large number of parameters. The authors of the reference works [14-16] used them for the estimation of the viscoelastic parameters while those of the references [6, 17, 18] used them for the determination of the viscoplastic parameters. Numerous studies relating to the sensitivity of polymer behavior to time and temperature are identified in the literature. The references [19-24] are cited for illustrative purposes. Since mechanical behavior models are controlled by parameters, it is natural that these parameters also depend on time and temperature. Several researchers have investigated the relationship of the sensitivity of elastic parameters, including Young's modulus and elastic limit, to strain rate and temperature [20, 21, 25, 26]. It is found that increasing the rate of strain increases the elastic parameters while the increase in temperature decreases them. On the other hand, although there is also much published work on the sensitivity of viscoplastic parameters, few of them have established dependence functions. The present work is an analysis of the sensitivity to strain rate of the parameters of a viscoelastic-viscoplastic behavior model applied to thermoplastic materials. The proposed model is an assembly of the Kelvin-Voigt viscoelastic mechanism in series with the Bingham generalized viscoplastic mechanism with nonlinear hardening. The material under study is a polyamide 6. Tensile tests at different strain rates were conducted to serve as data necessary for the identification of model parameters. The latter are identified by means of an inverse analysis based on the technique of genetic algorithms. As test- model results are confronted and good consistency is enforced, dependency functions are then derived from non-linear regression. R HEOLOGICAL MODELING Viscoelastic-viscoplastic model he proposed rheological model is illustrated in Fig. 1. It is a series assembly of two mechanisms, one representing the viscoelastic behavior and the other the viscoplastic behavior. The viscoelastic mechanism is represented by the instantaneous Kelvin-Voigt model, consisting of a spring representing instantaneous elasticity mounted in series with a parallel combination of another spring and another damper thus reproducing a viscoelasticity. This model is adopted to describe the viscoelastic behavior of polymers [27, 28]. The viscoplastic mechanism, which is activated only when the loading exceeds a critical value known by the plasticity threshold, is described by the generalized model of Bingham defined by the parallel association of a pad indicating the threshold of plasticity, a non-linear spring representing the work hardening of the material and a damper simulating the viscoplastic strain. The Bingham model is applied to T