INTERFACE DESIGN OF CORD-RUBBER COMPOSITES Z. Xie*, H. Du, Y. Weng, - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS INTERFACE DESIGN OF CORD-RUBBER COMPOSITES Z. Xie*, H. Du, Y. Weng, X. Li National Key Laboratory of Science and Technology on Advanced Composites in Special Environment, Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, P.R. China * Corresponding author(xiezhm@hit.edu.cn) Keywords : Interface design, Cord-rubber, Neutral inclusion, Non-circular cross-section 2.1 A G eneral N eutrality C ondition 1 Introduction Cord-reinforced rubber composites are widely used According to the concept of the neutral inclusion, in tires, hoses, belts and various attenuation the embedded inclusion does not affect the original constructions. The interfacial properties play a stress field. When the given stresses are applied to leading role in the performance and duration of the the inclusion, the displacement field can be cord-rubber products. Therefore, optimal design of calculated in the region of the inclusion. Based upon the interfacial properties is of great importance for the transmission conditions along the interface, one improving and enhancing the product qualities. may get the neutrality condition. Ru [3] has studied More attention has been paid to two ways, i.e., the a general neutrality condition in this way for the addition of bonding agents to the rubber compounds neutral elastic inclusion. and the adhesive treatment of the cords, to improve Consider a single cord with shear modulus μ 2 the interface strength[1]. However, there has been a embedded in the rubber with shear modulus μ 1 in little work concerning the mechanical design of the anti-plane shear, where the subscripts 1 and 2 refer interface. Carman et al. [2] proposed an optimal to the rubber and cord, respectively. For the sake of method for the interfacial modulus by minimizing simplicity, the rubber and cord are assumed to be the the maximum principal stress and the strain energy linear isotropic material. In terms of the Ru’s density in the composite materials. A zero-thickness w x y ( , ) analysis[3], the anti-plane displacement interface and perfect bonding are usually assumed in the phenomenological approaches for the composite satisfies the harmonic equilibrium equation in rubber materials. In recent years, considerable work has matrix (D 1 ) and reinforcing phase(D 2 ), i.e., been focused on the discussion of the imperfect ∇ = = 2 w 0, in D ( i 1,2) (1) interface[3,4]. Based upon the imperfect interface i i and the neutral inclusions that do not disturb the The imperfect interface conditions along the prescribed uniform stress field in the surrounding interface ( Γ ) are given by elastic body, Ru [3] presented the interface design of ∂ ∂ a single neutral elastic inclusion in many typical w w − = μ = μ 1 2 h x y w ( , )( w ) (2) cases. In practice, the neutral inclusion does not ∂ ∂ 1 2 1 2 N N exist if a perfectly bonded interface between where N is the direction of the outward normal to Γ inclusion and elastic body is assumed. Bertoldi et al.[4] concluded that a circular inclusion coated by a and h x y ( , ) the interfacial parameter proportional to continuous structural interface was neutral for a far the density of the adhesive layer. If there existed a broader material parameter range than for the linear neutrality condition for the cord-rubber composites, interface. In this work the interfacial parameters of h x ( , ) y must be non-negative everywhere. the cord-rubber composites are studied by means of Integration of Equ.(2) on the interface yields the concept of neutral inclusion. μ = + 1 w x y ( , ) w x y ( , ) w 0 (3) μ 2 1 2 2 Interface Design for Cord-Rubber in Anti- plane Shear

indicated by the height of green shadow with a w in which is an arbitrary value corresponding to a 0 symbol of plus. w rigid body displacement. For convenience, is In the other case of the displacement field 0 = neglected in the following analysis for it does not w x y 1 ( , ) B xy which also satisfies the harmonic 3 contribute to the deformation. And then, the equilibrium equation, the interfacial parameter is neutrality condition is rewritten as, derived, i.e., μ − = ⎧ − 1 h x y ( , )(1 ) w 2 x a μ μ ≤ ≤ + 1 ,0 x R a ⎪ ⎪ 1 2 Rx (4) = ⎨ ∂ ∂ h x y ( , ) (8) w w + μ + ⎪ 2 x a 1 1 [ cos N x y ( , ) sin N x y ( , )] μ − + ≤ ≤ ∂ ∂ , ( R a ) x 0 1 ⎪ x y ⎩ 1 Rx This equation denotes the relation between the a a − < < In the range of x , the interface parameter h x y ( , ) interface parameter and the shape of cord. 2 2 ( ) , so ( ) < h x y , 0 h x y is out of the non-negative , 2.2 Determination of Interfacial Parameters − < < restriction. If the interfacial boundary at a x a The Nylon66 cord with twisting factor of is replaced by two lines parallel to the principle axis 1400detex/2 is commonly used in a tire. A scanning as shown in Fig.3, the interfacial parameter becomes electron microscopy (SEM) photograph in Fig.1 shows a non-circular cross-sectional construction for μ ⎧ < 1 a single nylon66 cord embedded in the rubber matrix. , x a ⎪ ⎪ R Thus the cross-section of the cord is depicted by = ⎨ h x y ( , ) (9) − 2 x a ⎪ μ ≤ ≤ + ⎧ − + = ≤ ≤ + , a x R a ⎪ 2 2 2 ( x a ) y R ,0 x R a ⎪ 1 ⎩ R x ⎨ (5) + + = − + ≤ ≤ ⎪ 2 2 2 ⎩ ( x a ) y R , ( R a ) x 0 H x y ( , ) Furthermore, a dimensionless parameter is and plotted in Fig.2 where R is the radius of one introduced to illustrate the interfacial parameter Γ strand. distribution along the interface as follows, = ( , ) In the case of the displacement filed w x y B y 1 1 h x y R ( , ) = which satisfies the harmonic equilibrium equation, H x y ( , ) μ the interfacial parameter h x y ( , ) is given by 1 ⎧ < 1, x a (10) μ ⎪ 1 = = ⎨ − 1 h x y ( , ) a a μ ≤ ≤ + R (6) 2 , x R a ⎪ − 1 1 ⎩ x μ 2 H x y ( , ) As illustrated in Fig.5, is positive on the Since the stiffness of cord is much higher than that μ entire boundary. In addition, it is found that the μ ≈ 1 0 of rubber, or namely , the interfacial interfacial parameter is also independent of the 2 prescribed uniform stress field and the cord stiffness. parameter is reduced to μ 3 Interface Design for Cord-Rubber in Plane = 1 h x y ( , ) (7) Deformations R 3.1 Governing Equation Clearly, the interfacial parameter is identical along the interface boundary and independent of the In the analysis of Ru[3], the imperfect interface prescribed uniform stress field. Fig.4 shows the Γ condition along the interface was described in the = h x y R μ dimensionless parameter H ( , ) x y ( , ) / as normal and tangential directions by 1