Issue 49

Y. Saadallah et alii, Frattura ed Integrità Strutturale, 49 (2019) 666-675; DOI: 10.3221/IGF-ESIS.49.60 668 describe the viscoplastic behavior of wood [29, 30] and polymers [28, 31]. The proposed model, by its components, is extensible and therefore likely to simulate a wide variety of materials. Figure 1 : Viscoelastic-viscoplastic rheological model Mathematical formulation The viscoelastic response is represented by the rheological model composed of Kelvin-Voigt with instant elasticity. It follows that the total viscoelastic strain  is all of an instantaneous elastic part e  and a deferred part ve  . e ve      . (1) The stress  generated in the viscoelastic mechanism is expressed by: e ve ve ve E K          (2) Where E, K and ve  represent the viscoelastic parameters of the model (Fig. 1). The speed of instantaneous elastic strain e   being zero, replacing e  by its value, we obtain:   ve E K E K         (3) Where   . is the rate of total viscoelastic strain. The threshold of plasticity being reached, the response of the material takes a second component that is the viscoplastic strain vp  . The latter is described by the generalized model of Bingham with nonlinear hardening. e ve vp        (4) It follows that the equations governing the viscoplastic behavior of material are expressed by:   n e vp vp vp H          (5) Where H, n and vp  represent the viscoplastic parameters of the model (Fig. 1); e  is the elastic limit. This allows us to write: Kelvin-Voigt Bingham Elastic Viscoelastic Viscoplastic ௩௘ ௘ ௩௣