An Analytical Model for the Effect of Elastic Modulus Mismatch on Laminate Threshold Strength
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Q8.37.1
An Analytical Model for the Effect of Elastic Modulus Mismatch on Laminate Threshold Strength Alok Paranjpye, Glenn E. Beltz and Noel C. MacDonald Department of Mechanical and Environmental Engineering, University of California, Santa Barbara Santa Barbara, CA 93106, U.S.A ABSTRACT A scheme for calculating the stress intensity factor at the crack tip in a two-dimensional bimaterial laminate composite has been developed to model the damage tolerance obtained in brittle ceramics by using this geometry. One limitation of the model is its assumption of homogenous elastic properties throughout the composite, limiting the accuracy of predictions it can make about real material systems. Finite element simulations of the same architecture that allow for elastic modulus mismatch give results that are moderately different from those obtained from the homogeneous model. We present an analytical expression for the stress intensity factor around a crack tip in a laminated composite that can take into account the elastic modulus mismatch. To make the problem tractable, the model is based on the assumption that the system behaves as a homogeneous anisotropic material when the stress field at the crack tip arises out of far field tractions applied away from the crack tip. The model improves upon the homogeneous model, giving results that are closer to those from the finite element simulations. We, however, conclude that more work is required to predict the stresses at the tip as the crack approaches a material interface before a complete analytical model can be obtained. INTRODUCTION A bimaterial laminate architecture (Figure 1) comprised of alternating layers of residual biaxial compressive and tensile stress has been shown [1] to be effective in arresting the propagation of cracks through the structure. This scheme, when applied to ceramics or other brittle materials, effectively changes the strength distribution of these materials from probabilistic to deterministic in a specific applied stress range. The material develops a threshold strength below which it does not fail, and above which the strength is determined by the size of the largest flaw in the structure. A stress intensity function that incorporates the effects of the alternating residual stress was developed to describe the shielding of the crack tip in the compressive layers from the external applied stress, ⎡⎛ t ⎞ 2 ⎛t ⎞ ⎤ K = σ a π a + σ c π a ⎢⎜1 + 1 ⎟ sin −1 ⎜ 2 ⎟ − 1⎥ ⎝ 2a ⎠ ⎦ ⎣⎝ t2 ⎠ π
(1)
where σ a is the applied stress, a is the half crack length, t1 and t2 are the thickness of the compressive and tensile layers. σ c is the magnitude of the residual biaxial stress in the compressive layers and is evaluated from [2] as ⎛ t E' ⎞ σ c = ε E ⎜1 + 1 1' ⎟ ⎝ t 2 E2 ⎠ ' 1
−1
(2)
Q8.37.2
Figure 1. Bimaterial laminate geometry: 2-3 is the plane of transverse isotropy where ε is the residual differential thermal strain, Ei' = Ei /(1 − ν i ) is the modified Young’s modulus for each material component, and ν is Poisson’s ratio. Equation (1) shows that the s
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