Bi-material V-notch Research Articles

An analytical stress field for bi-material V-notches with end hole: New solution and effects of higher order terms

Current manuscript develops an analytical stress solution for the bi-material V-notches with an end hole (VO). To do so, based on the Kolosov-Muskhelishvili’s approach, while appropriate potential functions are employed, the boundary conditions of the problem are imposed to the solution to reduce the number of unknown parameters. Subsequently, the analytical stress field is derived as an asymptotic series solution, where each term possessing a constant coefficient and an order of singularity. The order of singularity for each term is obtained from the characteristic equations of the problem which is dependent on the notch geometry and material combinations. The so-called least square method (LSM) is then used to compute the constant coefficients of the asymptotic series for several case studies. Special attention is given to the bi-material notch stress intensity factors (BNSIFs) and the coefficients of the higher order terms (HOTs) in the stress series expansion. The accuracy of the presented stress solution is verified by benchmarking the results with numerical values obtained from finite element (FE) method. In this process, several notch opening angles and notch radii are simulated using the three-point bend (3PB) specimen. The developed asymptotic stress solution is demonstrated to be capable of accurately evaluating the stress field around bi-material VO-notched structures.

The calculation of singular orders for composite material anti-plane propagating V-notches

Because of geometric or material discontinuities, stress singularities can occur around the vertex of a V-notch. The singular order is an important parameter for characterizing the degree of the stress singularity. The present paper focuses on the calculation of the singular order for the anti-plane propagating V-notch in a composite material structure. Starting from the governing equation of elastodynamics and the displacement asymptotic expansion, an ordinary differential eigen equation with respect to the singular order is proposed. The interpolating matrix method is then employed to solve the established eigen equation to conduct the singular orders. The effects of the material principal axis direction, shear modulus, and propagation velocity and acceleration on the singular orders of the V-notch are respectively investigated, and some conclusions are drawn. The singular orders of the V-notches decrease with an increase in the material principal axis direction angle, except for the crack, whose singular orders do not change with the principal axis direction. The singular orders increase with an increase in the shear modulus $$G_{13} $$ , while they decrease with an increase in the shear modulus $$G_{23} $$ . The singular orders increase as the magnitude of the propagation velocity increases, while they decrease as the direction of the propagation velocity increases. The singular orders increase with an increase in the value of the propagation acceleration, while they decrease with an increase in the direction of the propagation acceleration. The singular orders become larger when the ratio of $$G_{13}^{(2)} /G_{13}^{(1)} $$ increases, while they become smaller when the ratio of $$G_{23}^{(2)} /G_{23}^{(1)} $$ increases, and they increase with an increase in the mass density ratio for the bi-material V-notch.

Influence of geometry and material on the stress intensity of an interfacial crack propagating from a bi-material notch

The stress intensity factor (SIF) is an important parameter to characterize the stress intensity near a crack tip. In order to successfully evaluate the SIFs for the interfacial crack emanating from a bi-material notch, a method coupling the boundary element method with the singularity asymptotic expansion technique is introduced. Then, the crack initiating from a sharp V-notch tip and the one emanating from a semicircle notch root are respectively taken into consideration, to investigate the influence of notch geometry shape and material property on the stress intensity for the interfacial crack from a bi-material notch. It is found that the notch opening angle has great influence on the SIFs of an interfacial short crack, while it has little influence on the SIFs of an interfacial long crack propagating from a bi-material V-notch. The value of KI for the interfacial crack initiating from a V-notch tip increases with the notch depth, while KII decreases with it. The value of KI and absolute value of KII for the interfacial crack initiating from a V-notch tip increase with the elasticity modulus ratio. The elasticity modulus ratio has little influence on KI, while it has important influence on KII for the interfacial crack emanating from a semicircle notch root. The value of KI and absolute value of KII for the interfacial crack initiating from a semicircle notch root increase with the notch radius.

Multi-parameter failure assessment of a bi-material V-notch – Crack initiation from a free-edge singularity

Multi-parameter fracture mechanics criteria are used to describe crack initiation conditions in general singular stress concentrators. In the cases of stress concentrators described by only one singular stress term, multi-parameter description provides a qualitative and quantitative increase of accuracy. In this paper multi-parameter criteria, based on an average tangential stress and average strain energy density factor, are described and used to determine a crack initiation direction and the critical load in the case of a free-edge singularity. The critical load expresses a failure/stability criterion for the studied general singular stress concentrator. Contrary to criteria employing singular stress terms only, a criterion of this kind is able to cover the influence of the direction of the external load.

Singularity evaluation for single material and bi-material propagating V-notches

By introducing displacement asymptotic expansion around the vertex of a propagating V-notch into elasticity governing equation, radial boundary conditions and interfacial continuity conditions, the singularity eigen analysis for a propagating V-notch is transformed into solving a set of eigen ordinary differential equations. The singularity orders and corresponding eigen angular functions for a propagating V-notch can be obtained after applying interpolating matrix method in order to solve the established eigen equation. The singularities for single material and bi-material propagating V-notch are respectively analyzed under different material properties, propagating directions and propagating velocities. It is found that the relationship between stress singularity and propagating velocity depends on the propagating direction as well as the singularity variation with respect to propagating velocity is not monotonous. For a single material V-notch, the singularity will be the weakest when the propagating direction is along the bisector of V-notch, and it will be the strongest when the propagating direction is perpendicular to the notch bisector. For a bi-material V-notch, the strongest singularity for a bi-material V-notch occurs when the propagating direction is along the harder material.

Bi-material V-notched SIFs analysis by XFEM and conservative integral approach

In this paper, a new effective approach based on conservative integral approach associated with extended finite element method (XFEM) is developed for evaluating stress intensity factors (SIFs) of bi-material V-notched structures. The XFEM model for bi-material V-notches is established, which owns various features: (a) jump enrichment functions are taken for describing the intersection of notch-faces; (b) eight (real eigenvalue) or sixteen (complex eigenvalue) branch functions are employed for capturing nodes surrounding the notch-tip; and (c) interface enrichment function is used to model the material interface. These enrichments allow the representation of notch-faces and material interface independent of the finite element mesh. The conservative integral approach derived from the Betti reciprocal principle is used for the evaluation of SIFs. The conservative integral approach avoids the complicated stress fields around the notch-tip, so good accuracy of SIFs can be obtained. Also, the proposed XFEM model can easily be used to solve homogenous V-notched structures by setting the same material parameters of two materials. Numerical results of the SIFs calculated by the present method indicate the independence of integral paths. Several bi-material V-notched numerical examples for single and mixed modes fractures are analyzed to demonstrate the accuracy and effectiveness of the developed method.

Stress singularity analysis for orthotropic V-notches in the generalised plane strain state

The singularity for the V-notch under the generalised plane deformation is investigated by the combination of the asymptotic analysis with the interpolating matrix method developed by part of the authors before. The displacement asymptotic expansions at the vicinity of the V-notch vertex are introduced into the equilibrium equations, which are transformed into a set of characteristic ordinary differential equations with respect to the notch singularity orders. The boundary conditions and interfacial compatibility conditions are also represented by the combination of the singularity orders and characteristic angular functions. The determination of the singularity orders and characteristic angular functions are transformed into solving the ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method. The present method is suitable for the singularity analysis for isotropic and orthotropic V-notches. It is versatile for analysing the stress singularity of single material V-notches, bi-material V-notches, interface edges and cracks. The correctness of the results by the proposed method is ensured by the comparison with the published ones.

Analysis of the stress singularity for a bi-material V-notch by the boundary element method

A new algorithm coupling the boundary element technique with the characteristic expansion method is proposed for the computation of the singular stress field in the V-notched bi-material structure. After the stress asymptotic expansions are introduced into the linear elasticity equilibrium equations, the governing equations at the small sector dug out from the bi-material V-notch tip region are transformed into the ordinary differential eigen-equations. All the parameters in the asymptotic expansions except the combination coefficients can be achieved by solving the established eigen-equations with the interpolating matrix method. Furthermore, the conventional boundary element method is applied to modeling the remaining structure without the notch tip region. The combination coefficients in the asymptotic expansion forms can be computed by the discretized boundary integral equations. Thus, the singular stress field at the V-notch tip and the generalized stress intensity factors of the bi-material notch are successfully calculated. The accurate singular stress field obtained here is very useful in the evaluation of the fracture property and the fatigue life of the V-notched bi-material structure.

Strain energy approach to compute stress intensity factors for isotropic homogeneous and bi-material V-notches

A strain energy approach (SEA) is developed to compute the general stress intensity factors (SIFs) for isotropic homogeneous and bi-material plates containing cracks and notches subject to mode I, II and III loading conditions. The approach is based on the strain energy of a control volume around the notch tip, which may be computed by using commercial finite element packages. The formulae are simple and easy to implement. Various numerical examples are presented and compared to corresponding published results or results that are computed using different numerical methods to demonstrate the accuracy of the SEA. Many of those results are new, especially for the cases of bi-material notches where the problem is quite complicated.