Issue 48

M. Estrada et alii, Frattura ed Integrità Strutturale, 48 (2019) 348-356; DOI: 10.3221/IGF-ESIS.48.33 350 rectangular cross-section or panels of constant thickness. Particularly, the laminated bamboo derived from Guadua Angustifolia is called Laminated Guadua Bamboo (LGB). M ETHODS : N UMERICAL MODEL FOR TWO - DIMENSIONAL MEMBERS OF LGB he proposed numerical model for LGB represents the crack pattern and the mechanical response of two- dimensional specimens subjected to static loads, which produce tensile and shear stress [14]. This model assumes perfect adherence among slats and uniform distribution of the fibers into the matrix. The constitutive model of the composite material is derived from the mixing theory [15], the classical constitutive models of two-dimensional damage for matrix [16] and one-dimensional plasticity for the fiber bundle [17,18]. The fracture process is modeled by means of continuum strong discontinuity approach for composite materials reinforced with long fibers [19,20]. Constitutive model for a composite material LGB In this model, the behavior of the lignin matrix is represented by a two-dimensional damage constitutive model [16], in which the stress tensor ( ) m  and the constitutive tangent tensor ( ) m tg C are obtained from the strain tensor ( ) m  and the internal variables ( ) m r . In contrast, the behavior of the fiber bundle is described with the one-dimensional plasticity model [17], defined from the proposed progressive failure model [18]. In the fiber bundle constitutive model, the normal stress ( ) f ss  and the tangent modulus of elasticity ( ) f tg E are calculated from the longitudinal strain ( ) f ss  and the internal variables ( ) f r , while the strength and stiffness contribution in the transverse direction are neglected. The constitutive model of the composite material LGB is based on the classical mixing theory for parallel systems and with the negligible fiber diameter simplification [15]. In this model the stress and stiffness are obtained from the weighted sum of its components through its volumetric fractions, while the strain is common for the fibers and the matrix. The strain rate of the matrix ( ) m   is equal to that of the composite material, at the same time that the strain rate of the fibers group ( ) f   is equal to the strain component of the composite material along the longitudinal direction of fiber s , that is: ( ) ( ) , m f     s s         (1) The tensor stress rate of LGB is equal to the sum of the products between the volumetric participation coefficients,   m k , and   f k , and the stress rate, ( ) ( ) , m f     of the matrix (m) and of the fibers (f) , respectively, so that: ( ) ( ) ( ) ( ) ( ) m m f f ss k k     s s      (2) By substituting the constitutive equation of each component in the constitutive equation of LGB, the following expression is obtained for the constitutive tangent tensor of the composite: ( ) ( ) ( ) ( ) ( ) ( ) m m f f tg tg tg k k E      C C s s s s (3) Constitutive model for fibers bundle The high dispersion in the mechanical properties of the fibers comes, in part, from measurement errors in the laboratory, but above all, from the natural characteristics of the material. This has been found on different natural fibers, testing by other authors [21–24]. The strength ( ) f u  of the fibers is related to their volume. When the gauge length of the fiber in tensile tests is constant, it depends only on their cross-section area ( ) f a . Therefore, a modified Weibull probability distribution [23,25] is used to represent the fiber strength of LGB, thus: ( ) ( ) ( ) ( ) 0 ( , ) 1 exp f f f f u u a P a a                      (4) where 0 a is the average value of the cross-section area of all the fibers, and the scale parameter  is a measure of the characteristic strength of the fibers. The shape parameter  defines the variability of the data. From a progressive failure T