Issue 48

M. Estrada et alii, Frattura ed Integrità Strutturale, 48 (2019) 348-356; DOI: 10.3221/IGF-ESIS.48.33 351 model and the classic uniaxial plasticity model, the following plasticity model is proposed to represent exclusively the mechanical response of the fibers group of LGB. The axial equilibrium allows establishing that ( ) ( ) ( ) ( ) f f f f ef ef A A    , where ( ) f  and ( ) f ef  are nominal and effective stress, respectively, and ( ) f A and ( ) f ef A are nominal and effective area cross-section, respectively. During the failure process, the effective area of the fiber group is reduced from ( ) f A to 0. Consequently, the scalar relation ( ) ( ) ( ) ( ) / / f f f f ef ef A A    is a possible measure of the integrity of the material. The failure probability of the fibers group ( ) ( ) 0 ( , ) f f ef P a a   is equal to the percentage of fibers whose strength is equal to or less than the effective stress applied to the LGB specimen. That is, the percentage of broken fibers as a function of effective stress indicates the level of material degradation or capacity loss. The failure probability corresponds to: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 exp exp f f f f ef ef ef ef f f f f f f u ef ef f f A A P E A A                                                    (5) From the tangent constitutive equation ( ) ( ) ( ) f f f tg E      of LGB, the tangent elasticity modulus ( ) f tg E can be obtained as follows. ( ) ( ) ( ) ( ) ( ) ( ) exp 1 exp f f f ef ef f f tg f E E                                                (6) Figure 2 : Relationship between the normal stress and longitudinal strain for the fiber bundle. Substituting of the previous expression into the classical one-dimensional plasticity model, the behavior of the fiber bundle during the stages of inelastic loading can be represented by the softening modulus ( ) f H , which takes the following form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (1 ) , where and exp( ) 1 f f f f f f f tg f f E H E E E H E H                     (7) The Weibull shape and scale parameters,  and  , come from experimental tensile tests on single fibers. For LGB fibers, these values are 1.3236   and 678.56 MPa   . Fig. 2 shows a schematic view of this model. From the observation of the process of failure of traction and flexion tests in the LGB, it is assumed that the mechanical behavior of the lignin matrix can be represented by a model of continuous damage with differential thresholds of tensile strength and compression [17,26]. The continuous damage of the matrix is attributed to the breaking of the polymer chains during the loading process. The code of LGB material constitutive model was written in Fortran 90 and was included into a version of the open-source program COMET. This version allows the no-linear finite element analysis,