Issue 50

M.A. Khiat et alii, Frattura ed Integrità Strutturale, 50 (2019) 595-601; DOI: 10.3221/IGF-ESIS.50.50 597 fiber breaks and the corresponding stress concentrations in intact neighboring fibers. Ineffective length  is generally defined as the length along a fiber from the site of fiber break, necessary to regain the capability to carry full load. In the case of Lin fibers reinforced polymers, the fiber strength and modulus are relatively insensitive to temperature and moisture concentration variation, but epoxy matrix is influenced by theses environmental conditions. Consequently, the interface fiber/matrix is con cerned by changes in matrix properties. However, because interfacial properties are difficult to measure, the interfacial bonding strength is assumed to be directly related to the yield stress of the matrix at interface, τ0. Interfacial failure is said to occur when the shear stress of the interface reaches τ0 [12]. As illustrated in figure 1, the model considers a central core of i broken fibers which are flanked by unbroken fibers which are subject to stress concentrations due to broken fibers. The unbroken fibers are, in turn, flanked by a homogenous effective material that is considered to be strained uniformly. Although the fiber bundle is not arrayed neatly, hexagonal array of fibers is assumed. It is also assumed that the influence of broken fiber is limited to a critical length, and that the stress concentration occurs to only neighboring fibers according to the arrangement. It is further assumed that broken core can be approximated by a homogeneous material with circular cross-section whose Young’s modulus may be obtained by the rule-of-mixtures:       2 2 0 0 2 0 f f m m i A E i A r r d E E r d               (1) A and E are the area and modulus of hexagonal arrangement of i broken fibers flanked by unbroken fibers, respectively. In the present model, we have included the local damage by introducing a region of debonding and local plasticity where the shear strength of the matrix τ0 is multiplied by a constant η. In the region 0 x a   , the equilibrium equations are given by: 2 0 0 0 2 2 0 f f d U iA E r dx      (2) 2 1 0 2 1 0 0 2 2 ( 2 2 ) ( ) 2 0 2 m f f f G d U iA E r d r U U r d dx          (3) where β is given as a function of geometry and fiber and matrix modulus and a is the half length of the region of debonding and plasticity. U0, U1 and U2 are tensile displacements in the three regions (figure 1). The solutions of the differential Eqns. (2) and (3) are given below: 2 0 0 0 0 1 ( ) U x x C r E    (4)     1 1 1 0 0 1 1 2 2 2 2 0 1 2 1 ( ) C 1 ( ) x x x C C r U x e e e x E R r E               (5) Similarly, for the region a x    where no interfacial yielding occurs, the equilibrium equations are: 2 2 0 0 0 1 0 2 ( ) 2 ( ) 0 2 m d U G E r d r U U d dx       (6) 2 1 0 2 1 0 1 0 2 2 ( 2 2 ) ( ) 2 ( ) 0 2 2 m i f f f G d U Gm n A E r d r U U r U U d d dx          (7)