Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as J_Vρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of J_Vρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, J_Vρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of J_Vρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.     

Optimized butterfly specimen for the fracture testing of sheet materials under combined normal and shear loading

The present paper is concerned with multi-axial ductile fracture experiments on sheet metals. Different stress-states are achieved within a flat specimen by applying different combinations of normal and transverse loads to the specimen boundaries. The specimen geometry is optimized such that fracture initiates remote from the free specimen boundaries. Fracture experiments are carried out on TRIP780 steel for four different loading conditions, varying from pure shear to transverse plane strain tension. Hybrid experimental–numerical analyses are performed to determine the stress and strain fields within the specimen gage section. The results show that strain localization cannot be avoided prior to the onset of fracture. Through-thickness necking prevails under tension-dominated loading while the deformation localizes along a band crossing the entire gage section under shear-dominated loading. Both experimental and simulation results demonstrate that the proposed fracture testing method is very sensitive to imperfections in the specimen machining. The loading paths to fracture are determined in terms of stress triaxiality, Lode angle parameter and equivalent plastic strain. The experimental data indicates that the relationship between the stress triaxiality and the equivalent plastic strain at the onset of ductile fracture is not unique.     

Discontinuous crack model for piezoelectric materials and its application to CED evaluation

In order to study the influence of electric displacement saturation on fracture behavior of a piezoelectric material, the electric displacement strip-saturation model by [Gao H, Zhang TY, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 1997;45:491-510] has been applied. However, this model is only applicable to problems such as that of a crack in an infinite plate for which it provides a singular solution. In order to overcome this situation, we developed a crack model for a piezoelectric material named discontinuous crack model that is presented in this paper, to evaluate crack energy density (CED) considering electric yielding and we studied its applicability through finite element analyses. The model is defined and methods to establish its constitutive equation are discussed. Moreover, it is shown that the model by Gao et al. and the ordinary crack model in continuum can be regarded as special cases of the discontinuous crack model. Subsequently, the CED and its derivatives for the discontinuous crack model are defined and their path independent expressions are also derived based on conservation laws. Finally, a finite element formulation is devised and the applicability of the model to the evaluations of CED and its derivatives is studied through finite element analyses of an example.     

An advanced model for initiation and propagation of damage under fatigue loading – part I: Model formulation

This paper presents a method for modelling initiation and propagation of damage in composites under fatigue loading in a single, coherent analysis using cohesive interface elements. Damage initiation laws based on SN-curves for initiation are applied to all interface elements within a zone of characteristic length, the so-called initiation zone. The formulation allows for a fully automated identification of the location of the initiation zone as well as automatic extraction of the local severity from the identified interface elements. Once a macroscopic crack has been initiated, it is propagated using damage accumulation laws based on the Paris-law. The formulation has been applied to the Short Beam Shear test and the Double Notched Shear test under fatigue loading. For both tests, good agreement was achieved with experimental data from the literature using an initiation zone length of 3.0 mm.     

Strain energy release rate calculation for a moving delamination front of arbitrary shape based on the virtual crack closure technique. Part I: Formulation and validation

This paper proposes a simple, efficient algorithm to trace a moving delamination front with an arbitrary and changing shape so that delamination growth can be analyzed by using stationary meshes. Based on the algorithm, a delamination front can be defined by two vectors that pass through any point on the front. The normal vector and the tangent vector for the local coordinate system can then be obtained based on the two delamination front vectors. An important feature of this approach is that it does not require the use of meshes that are orthogonal to the delaminations front. Therefore, the approach avoids adaptive re-meshing techniques that may create a large computational burden in delamination growth analysis. An interface element that can trace the instantaneous delamination front, determine the local coordinate system, approximate strain energy release rate components and apply fracture mechanics criteria has been developed and implemented into ABAQUS^® with its user-defined element (UEL) feature. In this Part I of a two-part paper, the approach and its implementation are described and validated by comparison to results from existing cases having analytical solutions or other established FEA predictions.     

Relationship between J-integral and averaged strain-energy density for U-notches in the case of large control volume under Mode I loading

The main purpose of this technical note is to present a relationship between J-integral and averaged strain-energy density () in U-notches under Mode I loading for brittle or quasi-brittle materials. In this work, control volume includes the rectilinear edge of the notch in addition to semi-circular arc of the notch root. A dimensionless function (f) between J and has been presented in this paper. Finite element analysis has been used for verification. It is found that this relationship is identical for tension or bending loading.     

The M-integral for calculating intensity factors of an impermeable crack in a piezoelectric material

In this study, a conservative integral is derived for calculating the intensity factors associated with piezoelectric material for an impermeable crack. This is an extension of the M-integral or interaction energy integral for mode separation in mechanical problems. In addition, the method of displacement extrapolation is extended for this application as a check on results obtained with the conservative integral. Poling is assumed parallel, perpendicular and at an arbitrary angle with respect to the crack plane, as well as parallel to the crack front. In the latter case, a three-dimensional treatment is required for the conservative integral which is beyond the scope of this investigation. The asymptotic fields are obtained; these include stress, electric, displacement and electric flux density fields which are used as auxiliary solutions for the M-integral.Several benchmark problems are examined to demonstrate the accuracy of the methods. Numerical difficulties encountered resulting from multiplication of large and small numbers were solved by normalizing the variables. Since an analytical solution exists, a finite length crack in an infinite body is also considered. Finally, a four point bend specimen subjected to both an applied load and an electric field is presented for a crack parallel, perpendicular and at an angle to the poling direction. It is seen that neglecting the piezoelectric effect in calculating stress intensity factors may lead to errors.