Matrix laminates

Let us describe a special class of laminates called matrix laminates. They represent the most investigated class of high-rank laminates.

Matrix laminates are obtained by iterative lamination of an already built composite with the same initial materials at each step of the procedure.

Structure of a second rank matrix laminate

The scheme of assembling a matrix laminate structure: Material 1 is added to the obtained composite in differently oriented layers at each step.

After several steps, we end up with a structure in which the disconnected inclusions of one of the materials is wrapped into another; the first material forms the envelope (matrix), and the second forms the nuclei. (The nuclei can be arbitrary elongated). This structure mimics a composite with inclusions. The effective properties do not depend on the sequence of wrapping; only the relative fractions in the enforced directions are valid.

The symmetric matrix laminates structure has the same properties as the structure of coated spheres, .

An anisotropic matrix laminates structure has the same effective properties as a laminate from the outer material and a matrix laminates made of remaining part of this material with the nucleus from the other one.

The effective properties of the structure are given by the formula:

$$D_{\mbox{\footnotesize {ml}}-k}= D_{1}+ m_{2}\left[ (D_{2}- D_{1})^{-1}+ m_{1} G \right]^{-1},$$ (21)

where the matrix $G$ is responsible for the degree of anisotropy of the composite. It has the structure:

$$G =\sum_{i=1}^{k} \alpha_i q(n_{i}) H_i^{-1}q ^T(n_{i})$$

where

$$\alpha_i \geq 0, \quad \sum_{i=1}^{k}\alpha _i=1, \quad H_i= q^T(n_{i}) D_{1} q (n_{i}).$$ (22)

The matrix laminates are known to have:

• The extremal conductivity in one direction when the conductivity in orthogonal directions is fixed,
• The maximal stiffness under a given load (structure of mutually orthogonal layers)
• The maximal mean stiffness under a sum of any number of loads (structure of 6 layers in 3D, 3 layers in 2D) Structure of a third rank matrix laminate

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