Singular Stress Field Research Articles

An Analytical Approach for the Near‐Tip Field Around V‐Notch in Orthotropic Materials

ABSTRACTThis study introduces a novel theoretical framework for determining the notch stress intensity factors (NSIFs) in composite materials. The approach involves utilizing Irwin's complex stress functions for Mode I and Mode II loading conditions. By investigating displacement and singular stress components around V‐notch tips, the research aims to provide a comprehensive understanding of NSIFs in composite materials. Three different composite materials were considered in the study to explore the relationship between stress and material properties. The stress singularity order, denoted as , is determined through eigenequations. Moreover, explicit solutions for displacement and singular stress fields are derived for a more thorough understanding of the behavior around notches in composite materials. To validate the proposed theoretical framework, a rigorous numerical study is conducted using FEM on finite‐size notches. The results obtained from the numerical simulations are compared with theoretical predictions to assess the accuracy and reliability of the developed approach.

Strength analysis due to thermal loading and tensile loading when metals are bonded by heat-curing adhesives

Heat-curing adhesives are widely used after being cured by heating to a temperature higher than room temperature. To evaluate the adhesive strength, therefore, it is necessary to consider both the thermal stress generated during heat curing and external loads such as tensile stress. Butt joint specimens are essential for evaluating tensile adhesive strength but also thermal strength. The interfacial strength can be discussed from the stress intensity factor (SIF) of a fictitious edge interfacial crack assumed at the interface end. This is because the SIF is controlled by the intensity of singular stress field (ISSF) at the crack-free interface end and a constant term associated with the thermal load. In this paper, a useful thermal SIF solution is proposed by superposing the SIF under tensile stress and the SIF under uniform interface stress associated with thermal loading. This general SIF expression provided under arbitrary material combination can be applied for predicting the tensile strength σc and critical temperature change ΔT without performing new FEM calculations. The usefulness of the expression is confirmed through the adhesive strength of Aluminum/Epoxy butt joint experimentally obtained. Once the critical SIF K1C can be obtained from the tensile strength σc and the temperature change ΔT, the adhesive strength can be expressed asK1C = constant of an assumed fictitious interface, and this can be used to predict critical σc for various temperature change ΔT and for various adhesive bondline thickness h.

Adhesive strength improvement by providing steps in joints and differentiating initial and final debonding stresses

Step adhesive joints have a special characteristic quite different from other joints. When initial delamination occurs in other joints such as butt, lap and scarf joints, final failure always occurs. In these cases, the external stress causing initial delamination σcInitial is equal to the final failure stress as σcFinal = σcInitial. However, in step joints, the final failure stress σcFinal can be greater than the initial delamination stress σcInitial < σcFinal. To clarify the adhesive improvement mechanism, first, this paper discusses the ISSF (Intensity of Singular Stress Fields) for fully bonded step joint by varying the number of steps NS. Second, the singular stress field causing 2nd debonding is discussed by analyzing partially delaminated step joint. The results show that 2nd debonding requires larger external load than the initial debonding as σcInitial < σcFinal. This is because under the same external load the singular stress causing the 2nd deboning is smaller than the one causing the initial debonding. When NS≥6 and suitable overlap length, the final bond strength σcFinal can be more than 3.6 ∼ 4.4 times larger than the initial delamination stress σcInitial≪σcFinal resulting in much larger bond strength.

Cancelling the effect of sharp notches or cracks with graded elastic modulus materials

Recent technologies permit to build materials which have elastic spatially varying modulus which can also imitate solutions adopted in Nature to optimize some structures. It has been shown that for example the stress concentration due to a hole in an infinite plate can be cancelled with a radially varying modulus making it similar to load-bearing bones which seem to resist structural failures even in the presence of blood vessel holes (foramina). Here, we attempt to study the classical problem of a sharp wedge (which includes the important case of a crack) looking for stresses varying as power law of the distance from the notch tip, σ∼rα, with a modulus varying as E∼rβ. In the inhomogeneous case the order of singularity of the LEFM case decreases if β>0, as confirmed by FEM investigations. Hence, we can remove stress singularities, which suggests an interesting alternative to the “rounding” of the notch. More in general, since for many materials it has been found that both strength and modulus are power laws of the density, using the so called strength-modulus exponent ratio we can obtain optimal design by keeping the asymptotic stress constantly equal to the strength. The present investigation paves the way for a new optimization strategy in the problems which eliminates size-scale effects due to singular stress fields, with potentially very wide applications.

Nitsche's method enhanced isogeometric homogenization of unidirectional composites with cylindrically orthotropic carbon/graphite fibers

An isogeometric homogenization (IGH) technique is constructed for the homogenization and localization of unidirectional composites with radially or circumferentially orthotropic carbon/graphite fibers. The proposed theory employs multiple non-conforming Non-Uniform Rational B-Splines (NURBS) patches to depict repeating unit cells (RUCs) representative of composite microstructures. Displacements are formulated using a two-scale expansion that integrates macroscopic and microscopic contributions, with the latter addressed through the isogeometric analysis technique. Nitsche's method is utilized to apply the interfacial traction and displacement continuity and periodicity conditions. The capability and accuracy of the IGH theory were validated upon comparison with the elasticity solutions that take into account explicitly fiber morphologies, along with classical micromechanics solutions based on equivalent fiber moduli. A comparative analysis with conventional finite-element techniques showcases the developed theory's ability to accurately replicate the singular stress field at the fiber center and to capture smooth stress distributions where significant stress gradients are encountered.

Analysis of Mode I Dynamic Crack Tip Singular Stress Fields for MEMS Structural Design

Analysis of Mode I Dynamic Crack Tip Singular Stress Fields for MEMS Structural Design

Three-Dimensional Singular Stress Fields and Interfacial Crack Path Instability in Bicrystalline Superlattices of Orthorhombic/Tetragonal Symmetries

First, a recently developed eigenfunction expansion technique, based in part on the separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion technique in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of the front of an interfacial crack weakening an infinite bicrystalline superlattice plate, made of orthorhombic (cubic, hexagonal, and tetragonal serving as special cases) phases of finite thickness and subjected to the far-field extension/bending, in-plane shear/twisting, and anti-plane shear loadings, distributed through the thickness. Crack-face boundary and interface contact conditions as well as those that are prescribed on the top and bottom surfaces of the bicrystalline superlattice plate are exactly satisfied. It also extends a recently developed concept of the lattice crack deflection (LCD) barrier to a superlattice, christened superlattice crack deflection (SCD) energy barrier for studying interfacial crack path instability, which can explain crack deflection from a difficult interface to an easier neighboring cleavage system. Additionally, the relationships of the nature (easy/easy, easy/difficult, or difficult/difficult) interfacial cleavage systems based on the present solutions with the structural chemistry aspects of the component phases (such as orthorhombic, tetragonal, hexagonal, as well as FCC (face-centered cubic) transition metals and perovskites) of the superlattice are also investigated. Finally, results pertaining to the through-thickness variations in mode I/II/III stress intensity factors and energy release rates for symmetric hyperbolic sine-distributed loads and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the bicrystalline superlattice plate under investigation also form an important part of the present investigation.

Experimental studies on the static sharp notch effects during dynamic crack kinking/nucleation at the material interfaces

Dynamic fracture experiments on four bonded polymers using high-speed photography showed that different static sharp notches formed after the incident dynamic cracks met the material interfaces. In one scenario, an incident dynamic crack induced interfacial crack nucleation, and two convex notches formed after the interfacial crack propagated away. In another scenario, interface-induced crack kinking and a “head-to-tail” crack kinking pattern led to a concave notch with a singular stress field, which merits consideration in fracture mechanics modeling. In contrast, load-induced crack kinking did not cause the above scenarios. In another unique experimental phenomenon, a slightly curved/kinked dynamic crack only had a small local crack kinking angle, but its final crack direction was almost 90° to the initial crack direction over a long crack path.

Singular stress field induced two-dimensional topological polar structures in SrTiO3: Topological number manipulation via loading modes

Topological structures in condensed matter attract much attention of the scientific community due to their unique and significant properties. In contrast to the well-explored topological spin structures such as meron, and skyrmion in ferromagnetic materials, ferroelectrics face challenges in forming topological polar structures due to the absence of a spontaneous driving force. Nevertheless, the coupling effect between polarization and strain (gradient), e.g. piezoelectricity and flexoelectricity, provides a mechanical approach to induce topological polar structures that reflect the intricate features of the strain (gradient) field. This study demonstrates the creation of a topological polar structure through a mode II crack-induced singular stress field in SrTiO3 by using phase-field simulations. Furthermore, under mixed-mode loading of mode I and mode II, alterations in the mode ratio induce a variety of topological structures, including a novel structure (winding number nv=4) that is absent under the single-mode loading. Our study demonstrates that topological strain field can be a general and effective tool to generate and control topological polar structures.

Finite element analyses of failure of fiber reinforced polymer-concrete interface considering the concrete tension–compression softening effect

The concrete near the shear failure surface of the FRP (fiber reinforced polymer)-concrete interface is subjected to both tensile and compressive biaxial stress. However, existing finite element methods of the interfacial failure behavior of FRP-concrete interface to describe the tension–compression softening effect of concrete failure are not sufficiently accurate, and some detailed results may not be reflected. To address these issues, a finite element analysis method is proposed that accurately considers the tension–compression softening effect on concrete failure. The effectiveness of the method is demonstrated through analysis of literature data from experiments on plain concrete failure, FRP-concrete interface shear, and three-point bending. The method accurately reflects the generation of interfacial discontinuous micro-cracks, the oscillation singularity of stress field near the interfacial crack tip, the phenomenon of interface mismatch, the dispersion observed in the bond-slip curves, and the phenomenon of interfacial shear stress reversal. Moreover, this finite element analysis method is applied to the double shear experiments of the fully-bonded FRP-concrete interface under different boundary conditions, and the reasons for the different failure modes observed in the specimens are explained. The results indicate that considering the biaxial effect on concrete failure strength in numerical analysis can more accurately and comprehensively describe the shear failure behavior of the FRP-concrete interface.

Mode-II stress intensity factors for special-shaped cylindrical shells in torsion

Special-shaped cylindrical shells in torsion are encountered in engineering applications, which will induce the Mode-II singular stress fields next to circumferential crack tips. The Mode-II stress intensity factor (SIF) represents the strength of these singular stress fields and is an important fracture parameter. The crack problems for the cylindrical shells are typical finite boundary value issues. It is very difficult to get the exact solutions by using the classical theory of elasticity. In present article, based on the concepts of the conservation law and the elementary strength theory, a hybrid methodology to determine the Mode-II SIFs for circumferential cracked special-shaped cylindrical shells in torsion is suggested. Due to the introduction of elementary mechanics, the corresponding errors are also inevitably carried into the Mode-II SIFs. However, the finite element analysis indicates that the error of the Mode-II SIFs calculated by present method is acceptable.

Complexity of crack front geometry enhances toughness of brittle solids

Brittle solids typically fail by growth and propagation of a crack from a surface flaw. This process is modelled using linear elastic fracture mechanics, which parameterizes the toughness of a material by the critical stress intensity factor, or the prefactor of the singular stress field. This widely used theory applies for cracks that are planar, but cracks typically are not planar, and instead are geometrically complex, violating core tenets of linear elastic fracture mechanics. Here we characterize the crack tip kinematics of complex crack fronts in three dimensions using optical microscopy of several transparent, brittle materials, including hydrogels of four different chemistries and an elastomer. We find that the critical strain energy required to drive the crack is directly proportional to the geodesic length of the crack, which makes the sample effectively tougher. The connection between crack front geometry and toughness has repercussions for the theoretical modelling of three-dimensional cracks, from engineering testing of materials to ab-initio development of novel materials, and highlights an important gap in the current theory for three-dimensional cracks.

An adaptive finite element method for coupled fretting wear and fatigue crack propagation simulation

Fretting wear and fatigue crack propagation often occur at the same time in the service of mechanically connected structures. In order to analyze the effect of fretting wear on the process of fatigue crack propagation, a finite element simulation method for fatigue crack propagation considering fretting wear is proposed. In this method, the Archard model is used to predict the wear process, and the conventional Paris equation is modified based on the wear rate. In order to improve the accuracy of crack propagation prediction, a crack-tip super singular element (CSSE) is used to calculate the singular stress field and fracture control parameters of the crack tip. An adaptive meshing technique that automatically adjusts the mesh density according to the simulation process is designed to ensure the computational efficiency of the algorithm and the accuracy of the numerical results. The smoothing technique used in wear simulation can alleviate the influence of contact stress fluctuation on wear profile prediction. The usage of the new simulation method is introduced through benchmark examples. It is proved that the numerical results of crack propagation obtained by the adaptive finite element method (FEM) have high precision, good convergence and high computational efficiency. It is found that wear reduces the crack growth rate compared with the simulation results without considering wear. This method has a promising application in the coupling analysis of fretting wear and fatigue crack propagation of mechanically connected structures.

Influence of Interlayer Thickness on Singular Stress Field in Piezoelectric Bonded Joint

Influence of Interlayer Thickness on Singular Stress Field in Piezoelectric Bonded Joint

Estimation of singular stress field of an interface crack in orthotropic dissimilar material by using the solution for isotropic dissimilar material

Estimation of singular stress field of an interface crack in orthotropic dissimilar material by using the solution for isotropic dissimilar material

Prediction of Crack Initiation at Die Corner of Molded Underfill Flip-Chip Packages Under Thermal Load by New Criteria—Part I: Accurate Formulation of Singular Stress Fields

Prediction of Crack Initiation at Die Corner of Molded Underfill Flip-Chip Packages Under Thermal Load by New Criteria—Part I: Accurate Formulation of Singular Stress Fields

Моделирование упругопластического разрушения компактного образца

The strength of a compact specimen under normal fracture (fracture mode I) is studied within the framework of the Neuber-Novozhilov approach. A model of an ideal elasto-plastic material with limiting relative elongation is chosen as the model of the deformable solid. This class of materials includes low-alloy steels that are used in structures operating at temperatures below the cold brittleness threshold. The crack propagation criterion is formulated using the modified Leonov-Panasyuk-Dugdale model, which uses an additional parameter, i.e., the diameter of the plasticity zone (the width of the prefracture zone). In the case of stress field singularity occurring in the vicinity of the crack tip, a two-parameter (coupled) criterion for quasi-brittle fracture is developed for type I cracks in an elastoplastic material. The coupled fracture criterion includes the deformation criterion attributed to the crack tip and the force criterion attributed to the model crack tip. The lengths of the original and model cracks differ by the length of the prefracture zone. Diagrams of the quasi-brittle fracture of the specimen under plane strain and plane stress conditions are constructed. The constitutive equations of the analytical model are analyzed in terms of the characteristic linear dimension of the material structure. Simple formulas applicable to verification calculations of the critical fracture load and pre-fracture zone length for quasi-brittle and quasi-ductile fractures are obtained. The parameters in the presented model of quasi-brittle fracture are analyzed. It is proposed to select the parameters of the model according to the approximation of the uniaxial tension diagram and the critical stress intensity factor.

A limit analysis-based CASS approach for the in-plane seismic capacity of masonry façades

The present paper proposes a new methodology for the load-bearing capacity analysis of 2D masonry constructions by extending and reformulating the Continuous Airy-based for Stress Singularities (CASS) method, with a particular focus on the incorporation of volume forces, crucial to solve mechanical problems involving masonry constructions accurately. The masonry material is modelled using a Normal, Rigid, No-Tension (NRNT) model, largely adopted by the scientific community as material parameters, often unknowable, are not needed.The load-bearing capacity problem is derived from an energy-based formulation and framed as a classic limit analysis approach, allowing for the direct determination of the maximum incremental load that a masonry structure can withstand, along with the corresponding internal stress pattern.The structural domain is discretised with a simple finite element mesh, and the Airy stress potential is adopted to enforce the internal equilibrium directly. The boundary value problem is then solved as second-order cone programming (SOCP).The CASS method is shown to offer modelling and computational advantages. Indeed, it accurately describes the mechanical response, including the ability to capture singular stress fields typically exhibited by masonry structures, particularly where cracks appear. The adoption of the Airy potential allows for a reduction in the number of explicit constraints from the problem. Moreover, the formulation of the boundary value problem as a SOCP ensures the existence of a unique load multiplier while offering computationally fast solutions even for large problems.Several numerical examples are presented to demonstrate the CASS potential. Specifically, the ability to capture singular stress patterns diagonally crossing the finite elements showcases the CASS mesh independence, providing a straightforward approach to modelling complex geometries and loading conditions.

A novel elastic strain energy density approach for fatigue evaluation of welded components

The averaged elastic strain energy density (SED) within a characteristic volume radius R has been investigated and applied for characterizing the fatigue strength of welded joints for decades. However, engineering applications are still relatively rare. The main reason is that the calculation process is cumbersome due to fine mesh size and circular pattern requirements. This study proposes a novel approach of SED without requiring special finite element size and special layout requirements. First, a novel approach of the notch stress solution is presented. As a result, the present solution differs from the conventional approach which has been focused on notch tip singular stress field. Then, an averaged strain energy density can be obtained analytically by taking advantage of the new notch stress solutions presented in this study. Both the closed-form notch stress solutions and resulting expression for average strain energy density have been validated by FEA results. Subsequently, a large number of fatigue tests published in the literature are selected to examine the effectiveness of the evaluation approach of SED on interpreting fatigue behaviors of welded joints. Finally, a procedure for practical engineering application is established and verified by a group of fatigue tests of 3D gusset welded components.

A new semi-analytic method for calculating the thermal–mechanical coupling stress intensity factor of interfacial crack

The interfacial crack thermal–mechanical coupling stress intensity factor (SIF) calculation plays an important role in anti-cracking design and safety assessment of compressed air energy storage (CAES) multi-layer cavern. In current SIF calculation method, there is no consideration of thermal–mechanical coupling effect caused by the difference between the singularity of stress field (λ) and temperature field (λT). In this study, a new semi-analytic method for calculating the thermal–mechanical coupling SIF of interfacial crack is established based on a new thermal–mechanical coupling Williams expansion for the cases of both insulated and constant temperature. The singularity index of temperature field at the crack-tip is dependent on the material parameters (heat conduction coefficient ki) for the crack perpendicular to and terminated at the interface, but independent on the material parameters (ki) for the crack located at the interface. The larger the difference of bi-material's heat conduction coefficient (ki), the larger the value of stress field when the crack is located on the side of larger Young's modulus (Ei) or on the side of lower ki. The semi-analytic method is degenerately verified by available literature results (without thermal–mechanical coupling term) and completely verified by finite element results (with thermal–mechanical coupling term), which can provide a theoretical basis for anti-cracking optimal design of CAES multi-layer caverns under thermal–mechanical loading condition.