D. Munz

On the calculation of crack opening displacement from the stress intensity factor

Abstract For the application of the weight function method the crack opening displacements for a reference case have to be known. An approximate method to derive the crack opening field from the stress intensity factor was proposed by Petroski and Achenbach [ Engng Fracture Mech. 10 , 257 (1978)]. The limited accuracy of their method becomes evident in cases where the stresses differ strongly from the homogeneous loading case (σ = const.). By expanding the crack opening displacement field in a power series it is demonstrated here how the approximative solutions can be improved by simple additional conditions.

Estimation procedure for the Weibull parameters used in the local approach

The local approach was recently proposed by Beremin and Mudry for evaluating the statistical behaviour of toughness results of materials. This approach introduces a stress parameter σw, termed the Weibull stress, as a measure of the fracture resistance of materials instead of the conventional toughness parameters such as Kc, δc and Jcl (critical stress intensity factor, CTOD and J-integral, respectively). The Weibull stress σw obeys the Weibull distribution with the two parameters m and σu (the shape and the scale parameter, respectively). The first parameter m is normally estimated to be 22 irrespective of the kind of material. In this paper a procedure for the determination of the Weibull parameters m and σu is developed. This procedure consists of the determination of the plastic zone ahead of the crack tip, from which cleavage fracture originates, and of the maximum likelihood estimation of the parameters m and σu based on the stress distribution in the plastic zone. Calculations using this procedure confirm that the distribution of the Weibull stress σw is a material property independent of specimen thickness, and in particular that the shape parameter m depends on the material, e.g. m≃12 for a German reactor pressure vessel steel (20 Mn Mo Ni 5 5). Using these parameters for the distribution of the Weibull stress the size effect in fracture toughness values is predicted and an improved agreement between theory and experiments is obtained compared to the Weakest Link model.

The regular stress term in bonded dissimilar materials after a change in temperature

Abstract The stresses near the free edge of the interface of a joint of dissimilar materials after a change of the temperature can be described as the sum of one or two singular terms and a regular term which is independent of the distance r and only dependent on the angle θ. A relation is given between the regular term, the Dundurs parameters and the geometry angles θ1, and θ2. For a combination of the elastic constants leading to a stress exponent ω = 0, the regular term approaches infinity.

Numerical investigation of residual stress fields and crack behavior in TBC systems

Abstract Due to thermal expansion mismatch and bond coat (BC) oxidation, high residual stresses are induced in the thermal barrier coating (TBC), leading to failure by spalling and delamination. Using the finite element method (FEM), an analysis of the stress distributions in TBC systems, which is a prerequisite for the understanding of failure mechanisms, was performed. As cracking usually occurs at or near the interfaces between BC/thermally grown oxide (TGO) and TBC/TGO depending on the processing mode of the TBC, cracks in the interface region were considered in the FE models in order to determine the loading conditions for their propagation and, thus, the failure criteria of the TBCs. Due to the mode mixity of these cracks, suitable methods are required for the determination of the fracture mechanics parameters needed for their assessment, such as the strain energy release rate G. The modified crack closure integral method (MCCI) was found to be a very efficient tool which can be combined easily with an FE analysis and leads to highly accurate energy release rate values. Moreover, this method enables the determination of mode-dependent energy release rates. Using this tool and appropriate crack propagation criteria, TBC failure models could be developed and verified.

Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium

In this paper, the influence of the electric boundary conditions on cracks in piezoelectric components shall be studied. Several electric boundary conditions have been proposed in the literature. Here, influence of the permeability of the crack on electric and mechanical fields near the crack tip is considered. Cracks of lower permeability lead to stronger electric singularities. Furthermore, the influence on the stress intensity factors and energy release rate will be discussed. Finally, an experiment with piezoceramic CT specimens, which was performed by Park and Sun, will be evaluated taking into account the permeability of the crack.

Stress intensity factor for semi-elliptical surface cracks loaded by stress gradients

Abstract The stress intensity factor at the deepest point of a semi-elliptical surface crack is calculated for stress gradients in direction of depth. The method is based on weight functions. The crack opening displacement for the reference problem is calculated with a method proposed by Petroski and Achenbach. The results are compared to finite element solutions given in the literature. As an example, the stress intensity factor is calculated for a crack in a thermally shocked pipe.

Determination of da/dN-ΔK1 curves for small cracks in alumina in alternating bending tests

Crack-growth relations under cyclic fatigue conditions are mostly determined for long cracks. In order to determine da/dN-ΔK curves for small cracks from lifetimes under cyclic load a procedure has been derived which is based on a method usually applied to subcritical crack growth. To prove the cyclic effect and to demonstrate the procedure in detail, measurements were carried out on an Al2O3-ceramic in bending with anR-ratio ofR=−1 and two types of relatively small cracks, namely natural cracks and Knoop-cracks. It was found that both crack types exhibit the same da/dN-ΔK relation. The exponent of the Paris law for fatigue crack growth is significantly different from the exponent of the power law for subcritical crack growth.

Calculation of the second fracture parameter for finite cracked bodies using a three-term elastic-plastic asymptotic expansion

A three-term asymptotic expansion which is controlled by two amplitude parameters is used to describe the stress field in the vicinity of the crack tip in a power-hardening material. The first parameter is the well-known J-integral. The second parameter (amplitude A) characterizes the following terms. A least squares procedure is developed for the determination of the amplitude parameter A by fitting of finite element data. The convergence of computed A values is investigated for a small scale yielding modified boundary layer problem. It is shown that the three-term expansion has certain advantages over the Q-stress approach. Values of the amplitude parameter A are determined for an edge cracked plate, center cracked plate, three-point bend specimen and compact tension specimen.

Evaluation of R-curve effects in ceramics

In coarse-grained alumina the crack growth resistance increases with increasing crack extension due to crack-border interactions. The crack shielding stress intensity factor can be calculated from the relation between the bridging stresses and the crack opening displacement. The parameters of this relation can be obtained from experimental results on stable or subcritical crack extension. Finally the effect of the R-curve on the behaviour of components with small cracks is discussed.

Determination of Strength

The ‘strength’ of ceramics is commonly defined as the material resistance against tensile stresses. The compressive strength is of minor importance.