By

Mahdi Ashrafi

Supervisor

Prof. Mark Tuttle

A Study of Strain Gradients in Notched Composite Laminates

March 2013

Outline

● Introduction to Lehkhnitskii/Savin Solution

● Strain gradients in composite laminates:

● Future Work:

○ Residual Stresses in composite Laminates

○Strain gradients for rectangular hole case

○ [90/±45/90]↓2𝑠  ○ [10]↓16 

Lehknitskii / Savin Solution For Anisotropic Plate Containing a Hole:

1 2

1. Lekhnitskii, S. G., Anisotropic Plates, translated by S.W. Tsai and T. Cheron, Gordan and Breach Pub Co, ISBN 0-677-20670-4 (1968). 2. Savin, G. N., Stress concentration Around Holes, Pergamon Press, New York (1961). See also NASA Technical Translation TT F-607, Nov 1970.

● This solution is based on 2-D anisotropic elasticity theory in absence of body forces. ● By applying 2-D equilibrium equations, strain/displacement relations, compatibility equations, and Hooke’s law the characteristic equation in terms of Airy stress function U(x,y) is produced:

Where:

Lehknitskii / Savin Solution For Anisotropic Plate Containing Circular Holes:

● By solving the characteristic equation and applying the boundary conditions the in-plane stresses can be expressed as follows:

Where: are remote in-plane stresses, and are a function of characteristic equation roots, remote stresses, and geometry of the hole.

● By applying conformal mapping, a closed-form solution for various hole shapes (elliptical, rectangular, triangular, etc) can be formed.

Strain Gradients in a Notched Composite Laminate :

● Classical Lamination Theory (CLT) was used to calculate the [𝑎↓𝑖𝑗 ] matrix for any laminate type.

● displacement-, strain-, and effective stress- fields around circular as well as elliptical holes can be predicted by this code.

A Matlab code was developed to implement Lehknitskii/ Savin Solution, producing plots of predicted displacement, strain, and stress fields subjected to arbitrary in-plane loadings.

Implementation :

Experimental Measurements: ● Digital Image Correlation(DIC) is an optical method that employs tracking & image registration techniques for whole field 2D and 3D measurements of displacements . ● Expected Error: From a similar study on Aluminum samples the scatter of data is expected to be ±150 µε.

IM7/5260 Graphite/Bismalemide: Tested    

𝐸↓1    192  (GPa)  𝐸↓2    11.9  (GPa)  𝐺↓12    6.89  (GPa)  υ↓12    0.33  

Tested Samples:

○ [𝟗𝟎/±𝟒𝟓/𝟗𝟎]↓𝟐𝒔 

○ [𝟏𝟎]↓𝟏𝟔 : [10/10/10/10/10/10/10/10/ 10/10/10/10/10/10/10/10]

[90/45/-45/90/90/45/-45/90/ 90/-45/45/90/90/-45/45/90]

Residual Stress Analysis of Composite Laminates:

The hole-drilling method is a well-established popular technique for measuring residual stresses in wide range of engineering materials.

Future Work:

The main goal of this study is to produce calibration coefficients to calculate residual stresses from experimental strain measurements for any composite material layup.

Strains obtained by Savin/Lehknitskii solution

Strains obtained by Classical Lamination Theory

Residual Strains

Future Work:

○ Strain Gradients Around a Rectangular Hole: With numerous applications in aerospace, automotive, and naval industries, a similar Matlab routine will be created for a rectangular hole case as well as DIC measurements.

Thank you for your attention

Questions?

For further information/questions please do not hesitate to contact me via my email:

ashrafi@uw.edu