Upper and Lower Bounds for the Parameters of One-Dimensional Theories for Sandwich Fracture Specimens

Abstract

Upper and lower bounds for the parameters of one-dimensional theories used in the analysis of sandwich fracture specimens are derived by matching the energy release rate with 2D elasticity solutions. The theory of a beam on an elastic foundation and modified beam theory are considered. Bounds are derived analytically for foundation modulus and crack length correction in single cantilever beam sandwich specimens and verified using accurate finite element results and experimental data from the literature. Foundation modulus and crack length correction depend on the elastic mismatch between face-sheets and core and are independent of the core thickness if this is above a limit value, which also depends on the elastic mismatch. The results in this paper clarify conflicting results in the literature, explain the approximate solutions and highlight their limitations. The bounds of the model parameters can be applied directly to specimens satisfying specific geometrical/material ratios, which are given in the paper, or used to support and validate numerical calculations and define asymptotic limits.

On elastic clamping boundary conditions in plate models describing detaching bilayers

“…The elastic clamping conditions are typically described through compliance coefficients relating crack tip forces and displacements, and solutions have been obtained in the literature only for some of the coefficients, for specific problems and mostly through approximate approaches based on structural theories to describe one or both layers and their interactions, e.g. beam and plate theory, beams on half-planes, beams on elastic foundations [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47]. Elasticity solutions have been derived for some coefficients through, numerical [5,10,38,39,40,41,42,48] and analytical solutions [36,25,32,33].…”

The effective crack extension and its implications on the single cantilever beam test for sandwich composites

“…Elasticity solutions which are relevant for the SCB specimen, in the two limiting cases of thick-and thin-core sandwiches, have been derived in ref. 22 using the theory in refs. 28,29 for isotropic layers with zero second Dundurs' parameter,…”

“…The relative error in using equation (8) with Δ ub to describe the thick-core sandwich limit in equation ( 11) is below 1% for all admissible E f =E c when a=h f ≥ 5. 22,33 A first validation of the accuracy of the bounds was presented in ref. 22 Comparing the load point displacement in equation ( 9) with the newly derived elasticity solution in the limit of a thick-core specimen, equation (12), shows that the estimate of Δ ub in equation ( 3) derived by matching the energy release rates, also ensure matching of the terms of approximate and exact displacements depending on a 2 and a 3 .…”

“…22,33 A first validation of the accuracy of the bounds was presented in ref. 22 Comparing the load point displacement in equation ( 9) with the newly derived elasticity solution in the limit of a thick-core specimen, equation (12), shows that the estimate of Δ ub in equation ( 3) derived by matching the energy release rates, also ensure matching of the terms of approximate and exact displacements depending on a 2 and a 3 . The relative error in using equation (9) with Δ ub to describe the thick-core sandwich limit in equation ( 12) is below 1% for all admissible E f =E c when a=h f ≥ 5.…”

“…The verification of the effective crack extension defined in ref. 22 and Eq. (3) will be then presented using a rich data set of numerical and experimental results from the literature and novel finite element results.…”