Interaction of whole fiber and destroyed neighbour fiber in reinforced plastic

Composite materials, both on the basis of polymeric binder, and other materials, for example, concrete are widely distributed. The paper considers the case of longitudinal location of elements. The paper deals with the question the intensity of the adhesive interaction, of course, with the true strength of the adhesive bond. To study the interaction of two adjacent fibers, whole and broken in work, it is proposed to use the concept of a contact layer [1 – 4]. To study the interaction two adjacent fibers, whole and torn, we propose use the concept of contact layer. This anisotropic layer has a certain thickness, equal to the length of the bonds, the shear modulus in the contact plane, and the Young’s modulus along the rod-bonds. In fact, it is an anisotropic continuous medium. When solving one-dimensional problems, the shear modulus and thickness of the contact layer are combined into one parameter, whose influence is decisive on the distribution pattern and stress intensity near the edge. This parameter can be determined quite easily from a macro experiment.


Introduction
Quite a number of works by various authors [2 -7] are devoted to the edge effect near the rupture (end) of a single fiber in the composite. But the role of adhesive interaction was not practically studied in them, since adhesive interaction in these works was not characterized in any way, and by default it was assumed absolute. The latter means that only the fundamental requirements of the continuity of displacement and stress vectors, i.e. the requirements of maintaining continuity and the implementation of Newton's third law. In this case, inevitably, solutions are obtained in which the tangential stresses at the corner points (at the ends of the fiber) tend to infinity, i.e. the so-called singularity is manifested. The use of physically justified strength criteria in these conditions becomes almost impossible.
Works on the contact layer method in adhesive mechanics (initially called the boundary layer) began in the Soviet Union in the early seventies (for example, [8][9][10][11][12][13][14][15]) and continue now [1][2][3]. In the West, the first works on the characterization of adhesive contact appeared only in the nineties [16 -22]. Basically, it is a shear spring model like the Winkler base. Figure 1 shows the studied here scheme, when only one fiber is broken, while the surrounding matrix and the neighbouring fiber remain solid. IOP Conf. Series: Journal of Physics: Conf. Series 1425 (2020) 012188 IOP Publishing doi:10.1088/1742-6596/1425/1/012188 2 We assume that there is a contact layer between the fiber and the matrix [1][2][3][4]. The contact layer is represented as a set of elastic short rods -bonds. These bonds are perpendicular to the surfaces of the adhesive and the substrate. They are rare compared to the density of atoms on the surfaces being joined (approximately n = 10141 / cm2). That is they do not touch each other and therefore the stresses perpendicular to the rods are zero. This contact layer is characterized by two parameters: shear modulus , and thickness equal to the length of the bonds . As a result, the contact can be characterized by one parameter -the ratio. It characterizes the contact of the adhesive with the substrate and is called the intensity of the adhesive interaction or the stiffness of the contact layer [3]. Using the contact layer method, the following flat problem was solved in the article (Figure 1). Nearby are one broken fiber and whole fibers ( Figure 2). The main question of the problem -how are distributed in this situation at the specified loading normal stresses in fibers & matrix and shear stresses at the matrix-fiber boundaries.

Figure 2.
Flat one-dimensional model for the calculation of the stressstrain state. 0 -broken fiber, to the right (x = l) there is no load; matrix (1) and fiber (2) remain solid; contact layers are shaded.
We introduce the notation: 0 For reduced but dimensional [1/m] shear stresses we have: The results of calculations of the distribution of all reduced stresses by formulas (1)

Calculation results and discussion
We write the boundary conditions for the desired reduced functions. The boundary conditions for normal stresses in the model are: Let us estimate the boundaries of change of the functions (6) and (7) when the stiffness parameters change.
1. The stiffness of the matrix is much less than the stiffness of the fibers  . Calculations on the basic parameters of the normal stress given are shown in Figure 3. There they are represented by solid curves 0, 1, 2. Curve 2 -reflects the change in stresses in the whole fiber, i.e. their decrease with distance from the point of broken fiber "0". The solid curve 0 represents how effectively the broken fiber "0" is included in the full-fledged work. It can be seen from the figure that this process of transferring forces from whole fiber to broken fiber can be considered practically complete at a distance of 0.00025m (20-25h 0thickness of the rods or nominal diameters D f = h 0 of fibers). The solid curve 1 at the bottom of the graph corresponds to the unloading of the matrix. Its unloading, as can be seen, is completed almost within five to seven diameters. which here represents the intensity of adhesive interaction of the adhesive material (i.e. matrix) with substrate material (fiber). And here it can be seen that the equality of the stresses in the whole and broken fibers is not achieved even at a distance of 0.0011m from the edge of the broken fibers. This means that "equality" comes much later than in the basic version. According to the calculations it is achieved at a distance of 250 fiber diameters from place of rupture fiber 0. As a result, whole fiber is congested at a much greater length compared to the base case. . Figure 4 shows the curves of the reduced but dimensional (1/m) shear stresses on the distance from the point of fiber break "0". Solid curves 1 and 2 correspond to the desired tangential stresses of the base case. Curves numbers correspond: 1 are the tangential stresses   1 between the broken fiber 0 and the matrix 1; 2 are the tangential stresses   2 between the solid fiber 2 and the whole matrix 1. The curves indicate a high concentration of tangential stresses in the model.
The width of the zone of concentration of tangential stresses, as can be seen from the comparison, is substantially greater than the width of the zone of concentration of normal stresses in fibers 0 and 2 (Fig. 4, solid curves 0 and 2). The narrow zone of concentration is compensated by high values of the tangential stresses at the maximum. And then the question arises about the strength of the adhesive bond of the fiber with the matrix, and after that, the question of the compromise between the strength of the fiber, the intensity of the adhesive interaction and the strength of the adhesive bond. Indeed, it is known that the strength of the fiber depends on its length -the shorter the fiber, the higher its tensile strength. This means, for example, that the lower the intensity of adhesive interaction, the longer the whole fiber next to the broken one will be overloaded compared to the rest of the working long-haul fibers. There is a danger that this overloaded state will coincide with the strength of the fiber at a certain length. Then the destruction of the fiber into short sections may begin [1,2].
The tangential stress distribution curves look completely different when the intensity of the adhesive interaction G*/h* (dashed curves) is substantially less than the baseline -by an order of magnitude. The maximum stresses are almost an order of magnitude smaller than the maximum stresses of the base case. And the width of the zone of concentration is more than an order of magnitude greater than that in the base case. But it should be noted that the shear stresses  If in the formulas (8) to maximum shear stress at the boundary of the fiber-matrix to require that the maximum shear stresses are less or equal to the strength of the adhesive bond in shear ( Relation (10) can be simplified for different ratios of composite parameters. For example, for today's high-strength fibers and their contents in the composite, a simple ratio is obtained: As a result, the approximate formula lacks the rigidity and strength of the matrix, which is possibly a consequence of that in the approximation, the rigidity of the matrix is small, and its strength in the model was not taken into account.

Conclusion
Based on the results obtained in the work, it can be stated that the contact (adhesive) interaction of the adhesive with the substrate (here -matrices with fiber) must be characterized. To characterize the interaction, the concept of a contact layer is used. Its characteristics are described at the beginning of the article. As a result of mathematical modeling of the force interaction of a torn fiber with a single adjacent fiber, a formula is obtained that reflects the conditions of monolithicity for a reinforced material.