L.R.F. Rose

Wave reflection and transmission in beams containing delamination and inhomogeneity

This paper presents an analytical approach using higher order plate theories to determine wave reflections from and transmissions through a damaged region in a beam. The damaged region is either treated as two split beams or as an inhomogeneity. The reflection ratios and transmission ratios are found to depend strongly on the frequency of the incident flexural waves, as well as the size of the damage, which gives rise to strong stop/pass band behaviour. Using the spectral analysis method, the transient wave propagation in a beam with a part-through delamination is predicted and compared with experimental results, indicating a good agreement in the phases and amplitudes of both the reflected and transmitted waves.

Mindlin plate theory for damage detection: Source solutions

A consideration of the relevant length scales and time scales suggests that Mindlin plate theory provides a judicious model for damage detection. A systematic investigation of this theory is presented that emphasizes its mixed vector-scalar character and analogies with 3D elasticity. These analogies lead to the use of Helmholtz potentials, and to compact statements of the reciprocal theorem and the representation theorem. The plate response for a point moment is derived using a direct source specification, rather than an indirect specification through boundary conditions. Solutions are presented for combinations of such point moments (doublets) that represent, respectively, a center of bending, a center of twist and a center of inplane twist. The flexural response due to finite sources, such as piezoelectric actuators, can be modeled by distributions of centers of bending. Detailed results are presented for a circular, and for a narrow rectangular actuator. The far-field radiation pattern for an array of equally spaced actuators parallel to a straight boundary is derived. The solutions presented for the point moment and the point force constitute the components of a dyadic Green’s function which is required, along with its spatial derivatives, for a representation of plate-wave scattering by flexural inhomogeneities.

Compact solutions for the corner singularity in bonded lap joints

This paper presents compact solutions of the corner singularity at the adhesive/adherend interface in a bonded lap joint. Two configurations of special importance to evaluating joint strength are considered: corners pertaining, respectively, to a square edge and a spew fillet. The stress intensity factors are determined for the limiting case of rigid substrates by means of a numerical matched asymptotic expansion method. The non-dimensional stress intensity factor depends only on the Poissons ratio of the adhesive, and this variation is characterised in a convenient analytical form by polynomial expressions. Comparison with finite-element results of a bonded joint obtained using fine mesh near the corner points confirms that the present analytical solutions provide very good representations of the singular stress fields at adhesive/substrate corners.

A crack bridging model for bonded plates subjected to tension and bending

Abstract A crack bridging model is presented for analysing the tensile stretching and bending of a cracked plate with a patch bonded on one side, accounting for the effect of out-of-plane bending induced by load-path eccentricity inherent to one-sided repairs. The model is formulated using both Kirchhoff–Poisson plate bending theory and Reissners shear deformation theory, within the frameworks of geometrically linear and nonlinear elasticity. The bonded patch is represented as distributed springs bridging the crack faces. The springs have both tension and bending resistances ; their stiffness constants are determined from a one-dimensional analysis for a single strap joint, representative of the load transfer from the cracked plate to the bonded patch. The resulting coupled integral equations are solved using a Galerkin method, and the results are compared with three-dimensional finite element solutions. It is found that the formulation based on Reissners plate theory provides better agreement with finite element results than the classical plate theory.

Determination of triaxial stresses in bonded joints

Abstract Adhesion in a bonded joint is subjected not only to a transverse peel stress and a shear stress, but also to two normal stresses, parallel and perpendicular to the joint, due to the constraint imparted by the adherends. In this paper, analytical solutions have been obtained for these two normal stresses. These solutions have been verified by comparing with finite element results. As an alternative, an approximate method is also developed, which is shown to be in good agreement with these triaxial stresses in a region close to the end of the joint. The triaxial stress state induces a stiffening effect, and hence existing peel stress solutions should be modified to reflect such stress triaxiality. The influences of these additional stress components on the yielding behaviour of adhesives in bonded joints are briefly discussed. Crown copyright

Analysis of out-of-plane bending in one-sided bonded repair

Abstract An un-supported cracked plate repaired with a reinforcement bonded on one side may experience a considerable out-of-plane bending, resulting mainly from the load-path eccentricity. A geometrically linear analysis is presented in this paper for the crack extension force after the application of a one-sided repair. It is found that although the stress intensity factor K for a one-sided repair is higher than in two-sided repairs where there is no bending present, the key feature is retained that K does not increase indefinitely with increasing crack length, but instead approaches asymptotically a finite upper-bound corresponding to the solution for a semi-infinite crack. An analytical expression is derived for this upper bound, which provides a conservative estimate suitable for design purposes and parametric studies. This analytical solution is shown to agree well with a fully three-dimensional finite element analysis. Strategies to minimize the detrimental effect of out-of-plane bending are briefly discussed.

Thermal stresses in a plate with a circular reinforcement

Abstract An analytical method is presented based on an inclusion analogy for determining the thermal residual stresses in an isotropic plate reinforced with a circular orthotropic patch. Explicit formulae are obtained for both the elastic properties and the thermal expansion coefficients of the equivalent inclusion. Exact solutions are derived for the thermal stresses in a circular orthotropic composite reinforcement bonded to an isotropic plate. To quantify the finite size effect, approximate solutions have also been obtained for a circular plate reinforced by a concentric circular patch. The present solutions are compared with finite element results, demonstrating a very good agreement with the numerical results. These explicit solutions provide a convenient tool for evaluating the residual thermal stresses when designing bonded repairs.

Self-similar analysis of plasticity-induced closure of small fatigue cracks

An analytical model of plasticity-induced crack closure is presented that is consistent with self-similar fatigue crack growth, where the plastic zone size and the plastic-wake thickness increase linearly with increasing crack length. The Dugdale model is used to represent crack tip plasticity, which allows the problem to be solved analytically. Detailed results are presented for the key model variables as a function of stress range and load ratio, in particular for the cyclic crack-tip opening displacement which can serve as a correlating parameter for crack growth rates, both within and beyond the small-scale yielding limit. For cases where the crack growth rate is proportional to the cyclic crack-tip opening displacement, results are presented for the variation of the exponent n characterising the dependence of growth rate on the stress range. It is noted that while n=2 in the small-scale yielding limit, as required for consistency with linear elastic fracture mechanics in that limit, the value of n deviates significantly from the small-scale yielding limit, even for applied stresses just above one-fifth of material yield stress. The present model can also be applied to fatigue crack growth under fully-plastic remote loading, with an appropriate choice for the limit stress and the modulus. This is shown to be particularly relevant for predicting small-crack growth rates at notch roots, in the presence of substantial notch-root plasticity. It is noted that small-crack growth rates predicted by previous models would be significantly unconservative relative to predictions from the present model.

Bonded repair of cracks under mixed mode loading

The problem of assessing the effectiveness of a bonded repair to a cracked plate can be reduced to a one-dimensional integral equation for the special case when both the plate and the reinforcement are isotropic and have the same Poissons ratio. This special case is used here to highlight some aspects of bonded repair efficiency under mixed mode loading which are not captured by crack bridging models. It is shown that bonded repair is less efficient in reducing the stress intensity factor under mode II than mode I, although the stress intensity factor under the shear mode also asymptotes to a limiting value as has been previously shown for the tensile mode. A closed form solution is derived for the limiting value of the stress intensity factor under shear mode. It is shown that for the long crack limit, the appropriate two-dimensional idealisation of the representative bonded joint corresponds to a plane strain condition, and the existing asymptotic solution for tensile mode needs to be modified to accommodate the effect of Poissons ratio on the stress intensity factor. It is also noted that crack bridging models lead to non-conservative predictions of repair efficiency for short cracks.

Substrate stress concentrations in bonded lap joints

Abstract A method based on the successive boundary stress correction approach is presented for the determination of the stress concentration in substrates of adhesively bonded joints with square edges or spew fillets at the ends of the overlap. The emphasis is given to developing an estimate of the stress elevation at the end of a bonded joint while the issue of corner singularity at the substrate-adhesive interface is not addressed in detail. It is shown that the adhesive shear stress which acts on the substrates is the main cause of the stress concentration; the adhesive peel stress has little effect. To circumvent the deficiencies of existing bonded joint theories which generally predict a maximum adhesive shear stress at the ends of the adhesive layer, an eigenfunction solution has been derived for the shear stress distribution near the ends of the overlap. Based on the improved adhesive shear stress solution presented here, the stress concentration determined from the present theory is found to be in good agreement with finite element results.