a study of strain gradients in notched composite laminates · mahdi ashrafi supervisor prof. mark...
TRANSCRIPT
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: