Singular Stress Field Research Articles

Singularity of Electromechanical Coupling Field in Piezoelectric Composite Junctions

The double singularities including singular stress field and singular electric displacement field, in the tips of piezoelectric composite junctions, are analyzed by the interpolation matrix method (IMM). The double singularity analysis problem of piezoelectric composite junctions is converted into eigenvalue solution problem of ordinary differential equations with variable coefficients under corresponding boundary conditions. In numerical examples, the first couple of singularity orders and the corresponding characteristic angular functions of displacement and electric potential for the electromechanical coupling field are obtained and comparisons are presented to validate the accuracy of the proposed method. The singularity of the electromechanical coupling field at the tip of piezoelectric composite material junctions is closely related to the bonding angle and fiber direction. According to the numerical results, the best scheme can be configured for the combination of dissimilar materials.

Adhesive strength evaluation for three-dimensional butt joint in terms of the intensity of singular stress field on the interface outer edge

Our previous studies analyzed the adhesive strength of butt joint systematically in terms of the intensity of singular stress field (ISSF) by using two-dimensional modeling. In consideration of the actual specimen geometry, in this paper, the adhesive strength of butt joint is analyzed by using three-dimensional modeling. A mesh-independent numerical method combined with the finite element method (FEM) is used to calculate the ISSF. The ISSF distribution on the interface outer edge is analyzed, the effect of adhesive thickness is investigated. The ISSF at the interface vertex is analyzed by using a small fillet instead. The difference between 2D and 3D models for adhesive strength evaluation is discussed. The usability of 2D modeling is investigated.

Intensity of stress singularity for the circumferential V‐shape corner front of a three‐dimensional diamond‐like defect

AbstractThree‐dimensional diamond‐like defects with circumferential V‐shape corner fronts are often contained in engineering materials. In this paper, generalized stress intensity factors are calculated for this type of defect using a modified advanced finite element method. A super corner front element model in the global coordinates is established to capture the stress singularities along the circumferential corner front. Three‐dimensional numerical series eigen‐solutions in the element have been transformed from asymptotic expressions in the local curvilinear coordinates. The element is suitable for a sharp V‐shape corner with arbitrary opening and inclination angle. Singular stress fields near various shapes of diamond‐like defects are systematically investigated. The interaction of an embedded defect with free surface or another identical defect is also investigated. The numerical results can be used as stress intensity parameters to predict fatigue strength at circumferential corner front of a diamond‐like defect.

General Stress Fields of Orthotropic Plate with Mode II Crack

The plane crack problem of the orthotropic plate is discussed for the shear loading. The solution of this problem is mainly achieved by the complex variable method. Essential equations are introduced fully on the basis of linear elastic mechanics. The stress functions have been given to meet the needs of shear mode II crack states for orthotropic materials. The formulae for stress fields are derived by the complex functions in two cases. With the simplification of relative functions, the expressions of the singular stress fields in the crack-tip are determined by the real functions.

Reduction of intensity of singular stress field at interface end in lap joint

Reduction of intensity of singular stress field at interface end in lap joint

Characteristic Tensor for Evaluation of Singular Stress Field under Mixed-mode Loadings

A characteristic tensor is defined using stress tensor averaged in a small circular domain at the crack tip and multiplied by the root of domain radius. It possesses the original stress tensor characteristics and has a simple relationship with conventional fracture-mechanics parameters. Therefore, it can be used to estimate stress intensity factors (SIFs) for cracks of arbitrary shape subjected to multiaxial stress loads. A characteristic tensor can also be used to estimate SIFs for kinked cracks. This study examines the relation between a characteristic tensor and SIFs to demonstrate the correlation between the characteristic tensor and fracture-mechanics parameters. Consequently, a single straight crack and a kinked crack of finite length existing in a twodimensional, infinite isotropic elastic body in a plane stress state, were considered to investigate the properties of the characteristic tensor under mixed-mode loadings. To demonstrate the practical utility of the characteristic tensor, the stress distribution obtained through finite element analysis (FEA) was used to estimate mixed-mode SIFs, and the values of estimated SIFs were compared with those obtained using an analytical solution. Results demonstrate that SIFs estimated under mixed-mode loadings exhibit a good agreement with the analytical values. This indicates that the proposed characteristictensor-based approach is effective in extracting features of singular stress fields at crack tips, and can be employed to estimate values of fracture-mechanics parameters, such as SIFs. Owing to its simplicity, the proposed approach can be easily incorporated in commercial FE codes for practical applications to simulate the crack-growth problem under both static and dynamic loading scenarios. The excellent applicability of the characteristic tensor greatly contributes to efficiency of the design process in industries.

Reduction of intensity of singular stress field in butt joint

Reduction of intensity of singular stress field in butt joint

A semi-analytical stress correction method for bi-material Mode-III V-notches

This article proposes a method to correct the numerical near singularity stress field in vicinity of reentrant corners by a semi-analytical solution. The approach pertains to perfectly bonded bi-material interface V-notches with arbitrary opening angles under Mode-III loading. The method addresses dissimilar joints with sharp corners in large-scale numerical engineering models where high-resolution discretization becomes computationally expensive. The method is therefore particularly useful for fatigue lifetime analysis of shear-bending dominated large utility multi material structures such as compound castings, aircraft wings and wind turbine rotor blades to name a few. The solution for the governing Laplace equation is obtained by separation of variables in conjunction with the eigenvalue expansion method yielding the bi-material notch singularity exponent. The notch stress intensity factor is then used to fit the analytical singular near field to the numerical far field utilizing the fact that the latter is more accurately predicted. The analytical anti-plane shear stress correction method is valid for any geometry subject to arbitrary remote loading conditions. The accuracy of the approach is investigated by numerical stress analysis of a slender beam subject to axial torsion for different notch opening angles.

A wavelet multiresolution interpolation Galerkin method with effective treatments for discontinuity for crack growth analyses

A wavelet multiresolution interpolation Galerkin method (WMIGM) with effective algorithm for treatments on the discontinuity of cracks is developed to handle the growth of cracks. In our WMIGM, two robust nodal refining models of multiresolution enhancement are proposed to effectively control the generation of local nodes around the crack tips for capturing the singular stress fields. With our local multiresolution refinement algorithm, these local nodes are automatically generated and/or moved with ease, which will not affect the other nodes. The formulation of the proposed WMIGM is presented in detailed. Several problems of both stationary and growing cracks are studied to validate the present method. Results demonstrate that the present WMIGM possesses an excellent accuracy in both estimating stress intensity factors and tracing crack growth paths, as well as a strong stability against irregular nodal distribution with extremely large ratio of the maximum grid size to the minimum in the problem domain.

An isogeometric-meshfree collocation approach for two-dimensional elastic fracture problems with contact loading

A strong form-based isogeometric-meshfree moving least-squares collocation (IMMLS-C) approach is developed for two-dimensional linear elastic fracture problems with contact loading. The IMMLS-C approach uses reproducing conditions to establish an equivalence between MLS shape functions and isogeometric basis functions. The advantages of this approach include the exact geometry representation, convenient crack modeling and flexible adaptive refinement. The IMMLS-C approach focuses on solving fracture problems based on strong formulations without requiring the numerical integration of Galerkin weak forms. Traction-free boundary conditions are enforced over a set of collocation points located on both sides of a crack surface. The displacement discontinuity along the crack surface is modeled by the visibility criterion and the singularity of near-tip stress fields is captured by adaptive mesh refinement without adding tip-enrichment functions, thereby reducing the degrees of freedom compared with the extended finite element method. Moreover, contact constraints are enforced by introducing a penalty algorithm to the strong formulations. The numerical results demonstrate that the adaptive refinement is able to achieve a high convergence rate at a low computational cost.

Simulation of fracture of elastoplastic materials in mode III: from brittle to ductile

The initiation of a longitudinal shear crack (mode III fracture) in elastoplastic materials with the limiting strain is considered. The crack propagation criterion is formulated using a modified Leonov–Panasyuk–Dugdale model using an additional parameter—the width of the plasticity zone (the width of the pre-fracture zone). Conditions of a small-scale yielding in the presence of a stress field singularity in the vicinity of the crack tip are considered. A two-parameter criterion for quasi-brittle fracture of an elastoplastic material is formulated for mode III cracks. The proposed fracture criterion includes the deformation-based criterion, which is formulated at the crack tip, as well as the force-based criterion, formulated at the tip of a model crack. The lengths of the initial and model cracks differ by the length of the pre-fracture zone. The application of the proposed strength criterion to the determination of fracture loads for bodies containing longitudinal shear cracks is demonstrated. The diagrams of quasi-brittle fracture are constructed for a strip of finite width with an edge crack in case of the out-of-plane deformation. It is proposed to determine the model parameters using an idealized simple shear diagram and the critical stress intensity factor. For quasi-ductile and ductile types of fracture, the limiting loads are determined numerically. The propagation of plastic zones in the vicinity of the crack tip under quasistatic loading is successively described by the nonlinear finite element method. It is shown that the shapes of the numerically simulated plastic zones differ significantly from the known classical concepts.

Determination of stress intensity factors of V-notch structures by characteristic analysis coupled with isogeometric boundary element method

To describe the physical fields of the V-notch structure accurately, the structure is first divided into two parts, the notch tip sector containing singular stress field and the non-singular remained part. A twofold procedure is then carried out to obtain the stress intensity factors: (I) the asymptotic expansions are introduced to describe physical fields of the singular region and transform the elastic governing equation to the characteristic ordinary differential equations. The interpolating matrix method is then implemented to provide stress singular orders and related characteristic angular functions of this singular area. (II) The boundary integral equation for the left part is then discretized by the non-uniform rational B-splines (NURBS) elements and is solved under the isogeometric framework. The amplitudes of the asymptotic expansions in process (I) are calculated by coupling procedures (I) and (II) through utilizing the interfacial continuity conditions. The stress intensity factors are finally obtained and compared with the referenced results for both symmetric and inclined V-notch structures. Benefiting from the isogeometric analysis, satisfied results can be provided by discretizing the artificial arc boundary between the singular and non-singular parts with fewer NURBS elements. This method can also be integrated with the preprocessing of computer aided design (CAD) conveniently.

Single-fiber pull-out analysis comparing the intensities of singular stress fields (ISSFs) at fiber end/entry points

This paper deals with a partially-embedded single-fiber under pull-out force in comparison with a single fiber embedded in matrix focusing on two distinct singular stress fields. Then, the intensities of the singular stress fields (ISSFs) are compared at the fiber end named Point A and the fiber/surface intersection named Point E. The results show that if the embedded length lin is shorter, interface debonding may occur at Point A. Instead, if lin is longer, the interface debonding may occur at Point E. To analyze the ISSFs accurately, a mesh-independent technique coupled with the finite element method (FEM) is indicated by applying the same FEM mesh pattern to the pull-out model and the reference model. As the reference solution, a single fiber embedded in matrix is also calculated under arbitrary material combinations by using the body force method (BFM). Stress distributions along the fiber/matrix interfaces are also calculated for carbon and glass fibers.

Adaptive analysis of crack propagation in thin-shell structures via an isogeometric-meshfree moving least-squares approach

This paper reports an isogeometric-meshfree moving least-squares approach for the adaptive analysis of crack propagation in thin-shell structures within the context of linear elastic fracture mechanics. The present approach is developed based on the equivalence of the moving least-squares meshfree shape functions and the isogeometric basis functions, which provides an effective strategy of adaptive mesh refinement for isogeometric analysis (IGA) in a straightforward meshfree manner. The adaptivity of the mesh refinement is achieved by utilizing a gradient-based error estimator to identify the meshes that need to be refined by adding linear reproducing points. The Kirchhoff–Love theory is further applied in the isogeometric-meshfree moving least-squares formulation to simplify the modeling of cracked thin-shell structures by neglecting the rotational degrees of freedom. In this way, the singularity of stress fields near the crack tip and the discontinuity of displacement fields around the crack surface can be efficiently captured by the adaptive mesh refinement to generate accurate results. A series of two-dimensional static and quasi-static crack propagation problems of thin-shell structures are investigated. It is found that the adaptive refinement strategy makes the present approach achieve higher convergence rate and computational efficiency than IGA and the meshfree method. The predicted propagation paths obtained by the present approach are in good agreement with the previously reported results.

New adhesive strength evaluation method based on the singular stress field considering three-dimensional geometry

Adhesive joints are widely used although different materials properties cause the singular stress field whose intensity is controlled by the adhesive joint geometry. Our previous studies showed that debonding strength can be expressed as a constant value of the critical intensity of singular stress field (ISSF) by applying two-dimensional modelling. By considering the real adhesive geometry, in this study, the ISSFs along the interface edge of three-dimensional prismatic butt joints are considered by varying the adhesive thicknesses. It is found that the critical ISSF in 3D modelling is almost constant independent of the adhesive thickness. The magnitude and position of the maximum ISSF are discussed by varying the corner fillet radius in comparison with two-dimensional modelling.

Effectiveness of the stress solutions in notch/crack tip regions by using extended boundary element method

In this paper, the extended boundary element method (XBEM) is employed to evaluate the singular stress fields of plane V-notched/cracked structures based on the linear elastic theory. Firstly, the V-notched/cracked structure is divided into two parts, which are a small region around the notch tip and the outer region without the tip region. Then, the expressions of the asymptotic series expansion of the singular stress fields in the notch tip region are assembled with the boundary integral equation of the outer region without stress singularity. Hence, the displacement and stress fields of both the tip region and outer structure are determined by the XBEM. Three benchmark examples are presented to show that XBEM can obtain the accurate displacement and stress fields from the tip region to the whole region for the V-notched/cracked structures.

On the Crack‐Tip Region Stress Field in Molecular Systems: The Case of Ideal Brittle Fracture

AbstractContinuum‐based fracture mechanics breaks down at the nanoscale where the discrete nature of atoms cannot be neglected. Intriguingly, this work shows that the concept of stress intensity factor is still valid if the atoms are modeled. Molecular statistics simulations are conducted on single‐edge cracked samples of ideal brittle silicon, varying the size until few nanometers. The local virial stress, derived as the functional derivative of the free energy of a molecular system with respect to the deformation tensor, is used as a measure of the mechanical stress at the atomic level. Then, stress intensity factor at failure is evaluated. The results show that regardless of the size, the atomistic stress field varies according to the classical 1/r0.5 relation, and discrete stress intensity factors can be derived for all the geometries. Continuum values, in contrast, fail to describe the fracture when the length of the singular stress field is smaller than 4–5 times the fracture process zone. Thus, this work shows that the stress intensity factor from atomic stress may be useful to describe the fracture criterion at extremely small dimensions, provided that virial stress is accepted as a representation of mechanical stress at the atomic level.

Stress intensity factor solution for edge interface crack based on the crack tip stress without the crack

In this paper, a new solution is discussed for the stress intensity factor of an edge interface crack. The intensity of the singular stress field (ISSF) at the interface end before appearing the edge interface crack is distinguished from the stress intensity factor (SIF) of the interface crack itself. By considering those two distinct double singular stress fields, the ISSF and the SIF are discussed under arbitrary material combinations. By focusing on the crack tip stress without the interface crack σy(a), the SIF of the edge interface crack can be expressed conveniently. This is because the crack tip stress without the interface crack is reflecting the ISSF at the interface end. The SIFs based on the crack tip stress without the interface crack are indicated by varying the crack length and material combination. It is found that the normalized SIFs (F1,int and F2,int) based on σy(a) are independent of the geometrical condition for the wide range of the crack length.

Intensity of singular stress field (ISSF) variation as a function of the Young's modulus in single lap adhesive joints

Employing mixed adhesive joints has been proven to be very useful. This type of joint leads to improved performance by increasing strength and decreasing stresses in critical areas of the joint. In the same way, the use of the Intensity of Singular Stress Field (ISSF) has been shown to be suitable for adhesive joint calculation, since the adhesive strength can be controlled by the ISSF at the interface end. Four finite element models have been created by combining two epoxy adhesives with different mechanical properties, and therefore with different Young's moduli. New mixed adhesive joints have been compared with respect to only-one adhesive joints in terms of the ISSF. The results show a clear improvement with one of the configurations of mixed adhesive joints. A significant decrease of 35.64% in the ISSF is obtained compared to the only-one adhesive configuration.

On the role of anisotropic expansion and yield stress on crack tip precipitates

This pilot study uses a Landau-Lifshitz phase field model to study the growth of a precipitate originating from a crack tip. A boundary layer giving a remote square root singular elastic stress field is applied. The materials are assumed to be elastic-plastic with linear strain hardening. The precipitate is assumed to expand in a simplified anisotropic manner, meaning that the hydride is assumed to grow with an orientation giving maximum expansion perpendicular to the crack plane. The results show that the anisotropic expansion has a strong effect on the shape of the hydride. For anisotropic materials the hydride becomes wedge like. The resulting shapes for the anisotropic cases are in line with observations of internal crack tip hydrides in zirconium. The shapes resemble earlier results for Einstein-Smoluchowski stress driven diffusion using discrete models and simplified modelling of the growth process. The crack tip shielding caused by both precipitation and plastic yielding is examined using the J integral. It shows that the near tip value is a around a quarter of its remote value for elastic isotropic materials and around a half for anisotropic materials.