Elastic properties of composite materials

Multiscale hierarchical modeling of fiber reinforced composites by asymptotic homogenization method

“…To describe a geometric composite structure with an arbitrary number of levels n, where each the lower nth level is included in the higher (n − 1)th level, we apply the method suggested in [5] (Fig.1).…”

“…N β = 0, e j · T N β ) T is transposition of the fourth-order tensor relative to the first two indices; ∆ is the unit fourth-order tensor [2,3]. The solution N (1) N β (ξ N ) is continued into the remaining part of PC V N β in a symmetric or antisymmetric way (for different components of tensor N (1) N β we use either one or another way) described in [5]. The last two conditions in (14) are the consequences of continuation and periodicity conditions at the PC boundary.…”

“…To solve the problems L nβ and A n,α numerically, we apply the finite-element method [4,5]. As an example of the method developed above, let us consider a two-level textile composite, fibers of which are curved in accordance with the sinusoidal law in planes Ox 1 x 3 (the first fibers' system is a warp) and Ox 2 x 3 (the second fibers' system is a weft) and are represented by bundles of a great number of fibers.…”

“…Computations gave the following effective technical elastic constants: E 1 = 20.1 GPa, E 3 = 4.3 GPa, ν 12 = 0.26, ν 13 = 0.28, G 13 = 1.94 GPa. To solve the local elasticity problems of the nth structural level L 21 and A 11 we applied the modified finite-element method with tetrahedral finite elements (FE) and linear approximation of displacements in the FE [4,5]. For finite-element solving the problems and visualization of computed results we developed the specific software [5].…”

“…Moreover, calculations of microstresses (stresses in components of a composite) by the methods are rather approximate because an actual geometrical shape of components is not taken into account. The considerable success [1,[7][8][9] in calculations of composite characteristics was achieved after creation of the homogenization (asymptotic averaging) method and development of effective computational algorithms of solving the local problems appearing in the method [4,5]. However, the method was created only for a classical one-level structure: matrix+fiber is a composite.…”