PSI - Issue 28

Nikolay Dolgov et al. / Procedia Structural Integrity 28 (2020) 1010–1017 Nikolay Dolgov & Tsanka Dikova / Structural Integrity Procedia 00 (2019) 000 – 000

1012

3

coatings by Lyashenko et al. (1996), Jeong and Kwon (1998), Choudhury and Agrawal (2011) and Dolgov et al. (2015). A specific feature of the coated materials is that under uniaxial loading of the substrate a biaxial stress-strain state occurs in the coating due to the difference in the Poisson's ratio of the substrate and the coating by Frank et al. (2009). Therefore, it is necessary to take into account the mismatch of the Poisson’s ratios of the substrate and the coating in order to determine more accurately the mechanical characteristics of the coating. The aim of the present paper is to apply an analytical approach using the shear-lag approximation in determination of the stress fields in the porcelain coatings on Co-Cr dental alloy substrate during tensile tests. 1. Problem formulation An analytical model for the coating-substrate system, based on the shear-lag approximation, is developed earlier by Dolgov et al. (1995), Dolgov et al. (1996), and Dolgov et al. (1997). The shear-lag theory is applied for transferring the stresses from the substrate, subjected to uniaxial tensile loading, to the coating. Besides, the mismatch between the Poisson’s ratios of the substrate and the coating leads to generating of stresses in direction perpendicular to the tensile load. The geometry of the coating-substrate system, considered in this study, is illustrated schematically in Fig. 1. A coating of thickness h , length l and width b is placed on the two sides of substrate with thickness 2 H . At the ends of the substrate, a load P is applied along the z- axis.

Fig. 1. Scheme of the substrate-coating system.

When the coated substrate is subjected to a uniform tensile load, in the coating-substrate interface shear stresses are generated, while in the coating – normal stresses σ z and σ x are initiated. To simplify the derivation of the formulas for the evaluation, it is assumed that the normal stresses σ z and σ x do not change in the thickness of the coating, and the coating material is linearly elastic with elastic modulus E c . The distribution of the normal stresses σ z ( x , z ) and σ x ( x , z ) in two directions in the coating is determined by Dolgov et al. (1995), Dolgov et al. (1996) and Dolgov et al. (1997):

(1)

  

  

  

  

1

ch r b ch r b x n ( / 2) ( ( / 2 −

))

n ( ) 8  

ch r b ch r b x n ( / 2) ( ( / 2 −

))

lk s d n 5

  

  

−

 

x z ( , ) 4 =

sin

k z n

z 

+

3 max

1,3,5...

n

=

2

n

n

   

   

(2)

b p d n  * 5

1   

  

z ( / 2) ( ( / 2 )) −

n ( ) 8  

ch q l ( / 2) ( ( / 2 )) − z n

ch q l ch q l n

ch q l

  

  

−

 

x z ( , ) 4 =

sin

x

x 

n 

+

3 max

1,3,5...

n

=

2

n

n

where d 5 , k n , s 2 ,  max , r n , d 5

* ,  n , p 2 , q n ,  max are constants and parameters given in the Appendix A.