Singular Stress Concentrations Research Articles

3D assessment of surface influence on crack initiation in sharp notches under a mixed mode of loading

Sharp notches occur in constructions and lead to stress concentrations. Thus they are dangerous points of constructions and must be properly evaluated. In the present study the effect of a free surface on crack initiation conditions is observed and evaluated by means of a 3D model of a specimen with a sharp notch and by means of the finite fracture mechanics of general singular stress concentrators (GSSCs). The effect of a free surface on crack propagation and the crack front shape is described in many publications (Hutař et al., 2010; He et al., 2016; Oplt et al., 2019). Concerning GSSCs (sharp notches, bi-material notches, sharp material inclusions, etc.) crack initiation conditions can also be influenced by the presence of the free surface. In our study we suppose the case of a sharp notch in a homogeneous material with a straight notch front perpendicular to the free surface. In this case a crack can be initiated either from the central region of the notch front or at the vertex point (the point where the notch front intersects the free surface). In order to assess the crack initiation conditions in the 3D model, the critical quantity criterion is employed (Klusák et al., 2019). This criterion can be easily used with the knowledge of basic material parameters and with the use of results of finite element calculations of the assessed notch. It is shown that the place of crack initiation depends on the direction of external loading and thus on a mixity of normal and shear modes of loading. A comparison of results of 2D and 3D models is discussed.

An easy and engineering stability criterion of general singular stress concentrators

General singular stress concentrators often occur in engineering components and they represent places prone to crack initiation. Estimation of crack initiation conditions is usually performed by means of stability criteria of generalized fracture mechanics. These stability criteria require knowledge of asymptotic description of the stress state around the general singular stress concentrator. In cases of some concentrators, this analytical-numerical description can be quite difficult and does not belong among straightforward engineering approaches. This paper introduces an easy and engineering way to assess conditions for crack initiation and propagation in general singular stress concentrators using standard numerical approaches. The paper handles well-known criteria of generalized fracture mechanics of sharp and bi-material notches and simplifies them to a practical form usable for engineers. The resulting criterion fits the frame of criteria of finite fracture mechanics and extends their applicability.

Nonlinear elastic finite fracture mechanics: Modeling mixed-mode crack nucleation in structural glazing silicone sealants

Requiring both stress and energy conditions to be met simultaneously proved key to modeling brittle crack formation at singular and nonsingular stress concentrations in linear elastic materials. The present work extends this so-called coupled stress and energy criterion to brittle crack nucleation in hyperelastic media using the example of silicone adhesives. For this purpose, we provide a comprehensive constitutive as well as fracture mechanical characterization of the structural silicone adhesive DOWSIL™ 993 using a large set of experiments and propose a mixed-mode failure model for crack initiation in nonlinear elastic materials. Characterized in independent experiments, the model is used to determine critical loads of hyperelastic adhesive bonds in both shear and tension dominated configurations. For any of the examined adhesive joint configurations the model predicts and explains size effects and agrees well with experimental findings. We study stable and unstable crack propagation observed in video recordings of our experiments. It is shown that crack initiation, crack growth and crack arrest are caused by nonmonotonic energy release rates and can be predicted. Effects of excess energy available after crack nucleation and initial unstable crack growth are discussed.

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.

Multi-parameter average strain energy density factor criterion applied on the sharp material inclusion problem

The tip of a sharp material inclusion (SMI) embedded in a parent material is considered as the singular stress concentrator. The SMI tip is modelled as a special case of a multi-material junction. The singular stress behaviour is caused by the material mismatch and geometrical discontinuity. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress terms exponent. The exponents are determined as a solution of eigenvalue problem. The factors are calculated by combination of analytical and numerical approaches. The terms can be either singular or non-singular depending on the stress term exponent. When approaching the concentrator, the singular terms become unbounded while the non-singular terms vanish. The non-singular terms increase precision of stress description on larger distances from the point of singularity. In some cases they provide the only means to describe the stress field well. Because of the singular stress behaviour near SMI tip, this location is prone to crack initiation. The crack initiation conditions are calculated by the average strain energy density factor (SEDF) criterion. Contrary to the case of a crack, the direction of minimum of SEDF changes with distance from the singular point. Therefore, an averaged value over specific distance is used. If the specific distance is relatively large, the employment of non-singular terms in multi-parameter criterion can significantly improve the critical parameters prediction. Thanks to that e.g. the particle composite design can be optimized.

Generalised fracture mechanics approach to the interfacial failure analysis of a bonded steel-concrete joint

Steel-concrete joints are often made by welded shear studs. However, this connection reduces the fatigue strength, especially in situations where locally concentrated loads occur with a large number of load cycles e.g. in bridge decks. In this paper the shear bond strength between steel and ultrahigh performance concrete (UHPC) without welded mechanical shear connectors is evaluated through push-out tests and a generalized fracture mechanics approach based on analytical and finite element analyses. The connection is achieved by an epoxy adhesive layer gritted with granules. In the tests, specimens made with various manners of preparation of the epoxy interlayer are tested experimentally. Numerical-analytical 2D and 3D modelling of a steel-concrete connection is performed without and with the epoxy interlayer. The model of a bi-material notch with various geometrical and material properties is used to simulate various singular stress concentrators that can be responsible for failure initiation. Thus conditions of crack initiation can be predicted from knowledge of the standard mechanical and fracture-mechanics properties of particular materials. Results of the fracture-mechanics studies are compared with each other and with experimental results. On the basis of the comparison, the 2D simulation of the steel-concrete connection without the epoxy interlayer is shown to be suitable for the estimation of failure conditions.

Crack onset assessment near the sharp material inclusion tip by means of modified maximum tangential stress criterion

In the case of particle reinforced composites, where the particles are in a form of sharp material inclusions, singular stress concentration exists on each tip of each inclusion. This is due to the geometric and material discontinuities between matrix and particle. These points of stress concentration are susceptible of crack initiation and thus often responsible for failure of the whole structure. The modified maximum tangential stress criterion is employed in order to predict crack onset conditions.

Crack onset assessment near the sharp material inclusion tip by means of modified maximum tangential stress criterion

In the case of particle reinforced composites, where the particles are in a form of sharp material inclusions, singular stress concentration exists on each tip of each inclusion. This is due to the geometric and material discontinuities between matrix and particle. These points of stress concentration are susceptible of crack initiation and thus often responsible for failure of the whole structure. The modified maximum tangential stress criterion is employed in order to predict crack onset conditions.

Application of nonlocal criteria for destruction in problems with a nonuniform stress field

The dependence of the critical load on the degree of nonuniformity of the stress field is considered on the example of polymethylmethacrylate beams and plates with a round hole or a sharp notch. It is proposed to use nonlocal criteria for brittle fracture to describe the behavior of a nonuniformly stressed material. Among the criteria considered, the structural criterion is the most universal, because, in this case, the distance parameter d is least dependent on the test configuration. The distance parameters in problems with a regular concentrator are of the same order of magnitude, as in problems with a singular stress concentrator. A further modification of the criterion that takes into account the properties of the fracture process will probably make it possible to obtain a unique value for the length parameter for any degree of nonuniformity of the stress field.

The influence of non-singular terms on the precision of stress description near a sharp material inclusion tip

A theoretical elastic stress field in the vicinity of a sharp material inclusion tip has a singular character. The power of a stress singularity, characterized by the exponents of the singularity, is different from the power of a singularity in the case of a crack in homogenous media. Stress distribution near the singular point can be described by an asymptotic expansion. The series consists of singular and non-singular terms, depending on the eigenvalue λ in the exponent of each term. The singular terms are characterized by 0<R(λ)<1, while the relation 1<R(λ) applies to non-singular terms. A sharp material inclusion is modelled as a special case of a multi-material junction, a bi-material junction. A method to calculate eigenvalues for a problem of a bi-material junction of given boundary conditions is described. Then based on the knowledge of eigenvectors, the eigenfunctions can be formed and Generalized Stress Intensity Factors (GSIFs) obtained by the Overdeterministic Method (ODM). The ODM returns GSIFs as a least square solution of a system of linear equations, which consists of analytical relations and a large amount of Finite Element Method (FEM) displacement results. In the majority of fracture mechanics analyses of cracks and notches, only the singular terms are considered to assess the stability of these general singular stress concentrators. This article presents results for four bi-material combinations of a sharp inclusion and a matrix in the form of stress plots. Stress plots show an analytical solution with the use of (i) singular terms, and (ii) singular and non-singular terms. Analytical solutions are then compared to pure FEM results. Further, the absolute and relative errors between both cases of analytical description and an FEM solution are calculated and plotted. The effect of an elastic moduli mismatch and the distance from the inclusion tip is investigated and quantified by mean absolute error of tangential stress.

Mechanobiology of the Ligament to Bone Insertion

The interface between materials with mismatched elastic moduli is failure-prone due to singular stress concentrations that could develop at the interface. Yet most ligaments tear at mid-substance and not at the ligament-bone interface. Current physiological understanding of these interfaces centers around the graded material properties of the ligament as it nears the bone, with increasing mineralization of the ligament. Open questions remain on which features of the gradation, or more generally of the ligament-bone interface, significantly improve the robustness and eliminate the stress singularity. To address these questions, we develop a 2D elasticity model of the interface and analyze it using a combination of numerical simulations and asymptotics. Increasing size of the gradation zone yields diminishing returns once the zone gets longer than the width of the ligament. Moreover, we apply a previous result in elasticity and show that the frequently-observed fanned geometry of ligaments near the interface can also eliminate the singularity. Previously, the functional advantage of the fanned geometry was thought simply to increase the cross-sectional area. However, our analysis reveals a different and more significant functional benefit to this geometry. The contribution of this work is to identify scaling laws for the region with graded material properties and a fanned geometry. Based on the observation that neonatal ligament tears often occur at the ligament-bone interface, while injuries in adults primarily occur mid-substance, we propose that the graded interface develops in response to the stress state experienced at dissimilar material interfaces.

Multi-Parameter Based Stress Distribution in Vicinity of Sharp Material Inclusion Tip

General Singular Stress Concentrators (GSSCs) which exhibit singular stress concentration are often responsible for crack initiation and thus failure of the component. The GSSC of the type of bonded bi-material junction occurs in a variety of technical applications including but not limited to sharp material inclusions, silicate based composites and electronic components. The GSSC cannot be assessed by means of standard fracture mechanics. Approaches of generalized fracture mechanics require precise description of stress distribution near the stress concentration points. In order to determine the stress field accurately, the paper incorporates the multi-parameter based description.

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0&lt;Re (λ)&lt;1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1&lt;Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.

Reconstruction of a 2D stress field around the tip of a sharp material inclusion

The stress distribution in the vicinity of a sharp material inclusion (SMI) tip exhibits a singular stress behavior. The strength of the stress singularity depends on material properties and geometry. The SMI is a special case of a general singular stress concentrator (GSSC). The stress field near a GSSC can be analytically described by means of Muskhelishvili plane elasticity based on complex variable function methods. Parameters necessary for the description are the exponents of singularity and generalized stress intensity factors (GSIFs). The stress field in the closest vicinity of an SMI tip is thus characterized by 1 or 2 singular exponents (λ-1), for which 0<Re(λ)<1, and corresponding GSIFs. In order to describe a stress field further away from an SMI tip, the non-singular exponents, 1<Re(λ), and factors corresponding to these non-singular exponents have to be taken into account. For given boundary conditions of the SMI, the exponents are calculated as an eigenvalue problem. Then, by formation of corresponding eigenvectors, the stress or displacement angular functions for each stress or displacement series term are constructed. The contribution of each stress or displacement series term function to the total stress and displacement field is given by the corresponding GSIF. The GSIFs are calculated by the over deterministic method (ODM), which finds a solution of an over-determined system of linear equations by the least squares method. On the left-hand side of the system are the displacement series term functions multiplied by unknown GSIFs, while the right-hand is formed by results of finite element analysis (FEA). Thus the results of FEA, namely nodal displacements in the radial and tangential direction, are employed in order to obtain the GSIFs. In the numerical example, the stress field for particular bi-material configurations and geometries is reconstructed using i) singular terms only ii) singular and non-singular terms. The reconstructed stress field polar plots are compared with FEA results.

Comparison of different failure approaches in knotty wood

This article presents and assesses 64 different ways for predicting the failure onset in knotty wooden beams. The aim is to provide engineers and modellers a general view of how to evaluate the failure in wooden structural members with knots. The studied criteria included both the conventional point-based and average stress theories. Special attention was paid to the effect of the elements of the wood mesostructure, i.e. knots and fiber deviation, which can generate singular stress concentrations as notches or cracks would do in fracture mechanics. The case study consisted of predicting the failure onset of bending in structural wooden beams. A previously validated finite element model was used in order to compute the hete rogeneous stresses. It was found that the knots caused considerable stress singularities so that the size of the average stress theory influenced the failure predictions by up to 23%. However, the variations generated by distinct phenomenological criteria were in general much smaller. The application of the average stress theory in large stress integration volumes is strongly recommended when predicting the failure in wood members.

Critical Value for Crack Propagation from Sharp V-Notch

The contribution is focused on estimation of a critical value of generalized stress intensity factor for crack propagation from sharp V-notches. Stress distribution around the tip of the V-notch is described on the base of generalized linear elastic fracture mechanics, because V-notch is a singular stress concentrator with stress singularity exponent different from 0.5 (depending on V-notch opening angle). Then also stability criteria based on strain energy density factor and average critical stress are generalized for the stress singularity different from 0.5. Using FE analysis the critical stresses for crack initiation was estimated and compared with experimental data from the literature.

The influence of discontinuity and orthotropy of fracture toughness on conditions of fracture initiation in singular stress concentrators

This paper deals with assessment of conditions of fracture initiation in singular stress concentrations that are modeled by a bi-material notch composed of two orthotropic materials. Within the assessment of the stress concentrators, the stability criteria based on standard linear elastic fracture mechanics can lead to incorrect results due to discontinuity and orthotropy of fracture toughness. Procedures in literature suggested for assessing the critical applied loading and fracture initiation direction usually consider only the change of elastic constants of the material and do not consider the change of fracture toughness. The stability criterion considering fracture toughness dependent on the direction of presupposed fracture initiation is shown in the paper. It is concluded that the criterion of this kind can qualitatively and quantitatively influence the estimated fracture initiation conditions, and it contributes to more reliable assessment of components with geometrical and/or material discontinuity.

The Influence of the Epoxy Interlayer on the Assessment of Failure Conditions of Push-Out Test Specimens

Connection between steel and concrete parts is frequently required in constructions where the steel-concrete joints are often realized by welded shear studs. In order to avoid stress concentrations, corrosion proneness, and other negative consequences of the welding process, steel-concrete connection without welded mechanical shear connectors is sought nowadays. Connection can be realized via an epoxy adhesive layer and gritted with granules. In the paper, the assessment of the push-out test configuration was performed from the generalized fracture mechanics point of view. The numerical-analytical modelling of a steel-concrete connection is performed without and with the epoxy interlayer, while 2D and 3D modelling is used. Thus conditions of crack initiation can be predicted from knowledge of the standard mechanical and fracture-mechanics properties of particular materials. The model of a bi-material notch with various geometry, and material properties is used to simulate various singular stress concentrators that can be responsible for failure initiation. Various manners of preparation of the epoxy interlayer are tested experimentally. Results of the fracture-mechanics studies are compared with each other and with experimental results. On the basis of the comparison, the 2D simulation of the steel-concrete connection without the epoxy interlayer is shown to be suitable for the estimation of failure conditions.

Conditions for Crack Initiation in an Orthotropic Bi-Material Notch

In the present contribution the joint of two orthotropic materials is investigated as the singular stress concentrator and it is modelled as an orthotropic bi-material notch. Such places in constructions are usually responsible for crack initiation and consequently for the final failure of the construction. Within the paper the parametric study of the crack initiation conditions is presented for specific geometry and material combinations and for varying loading conditions. The determination of the crack initiation direction and critical loading conditions is a necessary step in the reliable design of structures with singular stress concentrations.

A discrete element model predicting the strength of ballast stones

An extended discrete element method (DEM) is used to describe strength and failure properties of ballast stones. For this purpose, a rock material is modeled using spherical particles bonded by breakable force elements. An inflation procedure to generate dense sphere packings which is based on a particle’s current coordination number is proposed. The particle bonds are enhanced by a progressive failure model that reproduces the effects of singular stress concentrations near crack tips, which are normally not present in DEM, by successive weakening of bonds. The material model is investigated in uni- and triaxial compression, where an efficient approach for the simulation of a flexible confining membrane is applied, and calibrated to granite yielding wide agreement in strength and failure properties. Furthermore, a procedure to model shape and angularity of ballast particles is proposed. Finally, the strength of ballast stones made from bonded particles is subjected to statistical evaluation and compared to published experimental results. Different measures for single particle strength are investigated with respect to the loading state which causes failure.