Application of the residual strength degradation model to carbon-epoxy composite
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RESEARCH AND TESTING METHODS
APPLICATION OF THE RESIDUAL STRENGTH DEGRADATION MODEL TO CARBON-EPOXY COMPOSITE S. Putić,1 M. Rakin,1 and S. Bošnjak 2 A residual strength degradation model for two orientations of fibers of carbon–epoxy composite is proposed. The model is constructed on the basis of an analysis of a gradual decrease in the residual strength of the composite and an increase in loading cycles. Two groups of specimens were subjected to tensile tests under static and dynamic loading. It was necessary to determine the relationship between static and dynamic properties of the composite to predict the residual strength for different numbers of loading cycles. On the basis of initial experimental results, the values of the parameters obtained by using the model were computed numerically. The statistical distributions of the corresponding ultimate strengths obtained by using the model were compared with the distributions of experimental results, and their good correlation is demonstrated.
The results of static and dynamic tests of composite materials, according to the analysis given by J. N. Yang [1 – 3], show that the residual strength gradually decreases with an increase in loading cycles: dRm (n) f ( Δσ, R, ω ) = − , dn cRmc− 1 (n)
(1)
where Rm (n ) is residual strength after n loading cycles, c is the material constant, Δ σ is the stress range ( σmax – σmin ), R is the stress ratio (σmin / σmax ), and ω is the loading frequency. The integration of Eq. (1) from n0 = 0 to n1 = n leads to the following equation, which can express the residual strength: Rmc (n) = Rmc (0) − f ( Δσ )n .
(2)
It is assumed that the statistical distribution of the fatigue strength follows the two-parameter Weibull law: α
⎡ ⎛ x⎞ ⎤ FRm (0) ( x ) = P[ Rm (0) ≤ x ] = 1 − exp ⎢ − , ⎣ ⎝ β ⎠ ⎥⎦
(3)
where α is the shape parameter and β is the scale parameter. 1 Faculty of Technology and Metallurgy, University of Belgrade, Serbia. 2 Faculty of Mechanical Engineering, University of Belgrade, Serbia.
Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 3, pp. 122 – 125, May – June, 2008. Original article submitted February 21, 2007. 446
1068–820X/08/4403–0446
© 2008
Springer Science+Business Media, Inc.
APPLICATION OF THE RESIDUAL STRENGTH DEGRADATION MODEL
TO
CARBON-EPOXY COMPOSITE
447
Including different stress ranges and the number of cycles to fatigue failure, one can obtain Woehler’s S – N curve, which can be expressed as follows: b
K S N = 1,
(4)
where N is the number of cycles to fatigue failure, and K and b are material constants. Mathematical transformations of Eqs. (3) and (4) and their substitution into Eq. (2) lead to the following equation for residual strength after n loading cycles: Rmc (n) = Rmc (0) − βc KS b .
(5)
For the sake of simplicity of the model application, it is assumed that the loading frequency and the stress ratio are constants, so that f ( S, r, ω ) = f ( S ) . Experimental Results and Determination of Model Parameters For the determination of the model parameters ( α, β, K, b
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