Issue 65

S. M. J. Tabatabee et alii, Frattura ed Integrità Strutturale, 65 (2023) 208-223; DOI: 10.3221/IGF-ESIS.65.14

So based on the RIS concept, we consider these orthotropic materials as isotropic ones which have reinforced and encounter the effect of these reinforcements on the stress that is applied to the RVE. This reduction in applied stress calculates by using “reinforcement factors” that are defined below mathematically:    . 1 / Comp Matrix xx xx n (1)

Comp

.

   Matrix yy

2 / n

(2)

yy

Comp

.

   Matrix xy xy

6 / n

(3)

where  is the portion of stress reduced by the reinforcements. The reinforcement factor can find with a combination of equilibrium and constitutive relations with compatibility equations for isotropic matrix. The constitutive relation for orthotropic materials is:       xy y x x x y E E (4)       y yx x y y x E E (5)    xy xy xy G (6) . Comp ij is the stress applied to the composite material, and  Matrix ij

with the definition of Airy stress function, we have:

2



 



(7)

x

2



y

2

 

 



(8)

y

2

x

   2







(9)

xy

x y

The compatibility equation for isotropic material can be written as follows:

         2 2 2 2 y x x y

2



xy

(10)

 

x y

So, with the placement of Eq. (4-6) into Eq. (7-9) and substituting the result into the compatibility equation, and using constitutive relations, we have:               4 4 4 22 66 12 11 4 2 2 4 ( 2 ) 0 C C C C x x y y (11)

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