Crack Faces Research Articles

On the solution technique for the interior electric displacement in a piezoelectric crack with the semi-permeable electric condition

In the study of crack problems in piezoelectric materials, the choice of electric conditions on the crack surface is crucial. The semi-permeable condition is believed to be more consistent with the real physical situation than the impermeable and permeable. In the semi-permeable crack condition, one task is to solve the crack interior electric displacement. However, there is no discussion in literature on the possible general method for its solution. A feasible method is proposed in this paper, addressed by taking as an example a penny-shaped crack in a piezoelectric layer with strip-like electrical saturation and mechanical yielding zones. The mixed boundary value problem is transformed into a set of coupling Fredholm integral equations of the second kind by Hankel transform and Copson method. With appropriate numerical discretization, the bisection method is adopted to solve the crack interior electric displacement. The algorithmic flow of the problem is given and some degenerated examples are presented to illustrate the effectiveness of the proposed technique. Finally, numerical results are presented for the relation of external load as well as the crack face electric displacement versus zone-lengths. Comparison and discussion are presented for the results of semi-permeable and impermeable crack condition. Results show that the crack interior electric displacement increases with electrical load but decreases with mechanical load. The electrical saturation zone-length is affected by the mechanical load as well as electrical load, regardless of the thickness of piezoelectric layer as well as the size relation of the mechanical and electrical zone-length. The present solution technique has a direct reference value for solving the problem of semi-permeable crack with distinct strip-like yielding zones in the finite piezoelectric material.

Rapid Healing of Thermal Cracks in Ice

AbstractThe structural integrity of the arctic sea ice cover is under threat owing largely to the combination of thinning and larger waves. Another contributor may be thermal cracking. In concentrating stress, thermal cracks may weaken the cover. Of interest, therefore, is the strength of thermally damaged ice. To that end, new experiments were performed on sea ice and on lab‐grown saline and salt‐free ice that had been cracked by thermal shocking. As expected, the cracks weakened the materials in accord with fracture mechanics. However, within tens to hundreds of seconds of shocking, the strength recovered completely, for the ice had healed. Healing is attributed to thermally activated sintering related to surface diffusion, assisted possibly by the formation of a quasi‐liquid layer on crack faces. Whether behavior on the small scale is indicative of behavior on the large scale remains to be determined.

Characterization of fatigue crack of hydrogen-charged austenitic stainless steel by electromagnetic and ultrasonic techniques

In this study, the feasibility of the fusion sensing of eddy current testing (ECT) and ultrasonic testing (UT) as effective tools to clarify the hydrogen-embrittlement mechanism of austenitic stainless steels was investigated. Fatigue testing was conducted on hydrogen-charged and uncharged AISI 304 specimens. The effect of hydrogen exposure on the martensitic transformation, and crack closure and crack face morphology were investigated by ECT and UT. The results suggest that a comparison of ECT and UT results can evaluate martensitic transformation and crack closure and crack face morphology, which are important in understanding the hydrogen embrittlement of austenitic stainless steels.

Theoretical estimation of the elastic moduli of self-healing concrete relevant to the evolution of cracks closing due to crystallization- and precipitation-based mechanism

A micromechanical model is presented to simulate the physical configuration of self-healing process of concrete due to the mechanism of crystallization and precipitation based on the double inclusion method, by which the homogenized mechanical properties are estimated. The effect of the number of cracks healed and the mechanical nature of healing components are investigated under varying conditions of healing. The results show that 0.2 mm and 0.4 mm are critical crack widths for abruptly change of the recovered modulus for the same age of healing. The moisture supply has a greater effect than temperature on the healing, particularly at higher temperatures. For narrow cracks (e < 0.1), the recovered mechanical properties solely depend on whether the crack faces are attached by healing, irrespective of the moduli of the healing components. However, the mechanical nature of the healing components has a more significant effect on healing of wider cracks. Finally, the experimental results validate the theoretical calculations.

Surface wave across crack-tip in a lattice model.

For a triangular lattice of particles, with nearest neighbour interactions and a traction free boundary, there exists a surface wave band for out-of-plane motion. Interactions between such a surface wave and stationary crack tip in mode III are investigated. The discrete Helmholtz equation for scattered waves, that incorporates an anisotropy parameter for unequal spring constants in horizontal versus slant directions, is solved exactly. The coefficient of transmission from one crack face to another, as well as that owing to reflection on the same face, is obtained in a closed form; the same leads to an estimate of energy fraction of incident wave that is leaked at the crack tip via bulk waves. It is found, in terms of surface wave band, that the transmission coefficient attains its maximum magnitude above the mid-band while the energy leak is minimum at the upper-band limit. Besides surface wave propagation across crack tip, surface wave excitation due to incident bulk wave is also discussed. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Competing effects of current density and viscoplastic deformation on the critical conditions for dendrite growth into solid-state lithium battery electrolytes

All-solid-state lithium (Li) batteries provide a promising pathway toward high energy and power density. Dendrite penetration through the solid electrolyte causing battery short-circuit, however, persists to be one of the challenges impeding their widespread application. Here, considering a pre-existing surface crack in the electrolyte initially filled with an infinitely thin layer of Li, and assuming Li deposit to behave in accordance with rigid-viscoplasticity, we seek for the steady state Li-filled crack opening profile that could potentially form at a given constant current density. Treating the chemical potential of Li ions in the electrolyte and the electric potential to be uniform along the crack face, the model accounts for the coupling between stress buildup in the dendrite, deposition rate, viscoplastic flow of Li deposit, and crack opening induced by electrolyte deformation using singular integral equations of fracture mechanics. The model establishes limiting conditions for crack growth before a steady state dendrite is reached, triggering a cycle of crack growth and dendrite elongation. Using material properties adopted from literature, the model predicts that the critical condition can be met for a microcrack at typical current densities. The effect of pressure applied to the cell is further discussed.

Effect of reinforcement on the crack resistance of concrete slabs

A preliminary analysis of the available publications devoted to the study of crack resistance of reinforced concrete structures showed the absence of established general patterns of influence of important geometric parameters inherent in reinforced concrete elements on the distribution of the characteristics of fracture mechanics along the crack front. Based on the analysis, the purpose of the study was formulated: to establish these regularities for a concrete slab reinforced with a system of longitudinal steel rods. When conducting the research, a linear and elastic model of concrete was used, and the stress intensity factor was considered as a characteristic of the fracture mechanics. A surface crack of constant depth located in the cross-section of the slab was postulated. It was assumed that its faces completely cover the cross-section of reinforcing rods. The crack depth, the depth of reinforcing rods, their diameter, and the distance between adjacent rods were chosen as dimensionless geometric parameters relative to the thickness of the slab. The slab was loaded with two types of loads applied to its ends: constant tensile stresses (pure tension) and linearly variable axial stresses (pure bending). The problem of determining the stress intensity coefficient depending on geometric parameters was reduced to the boundary problem of elasticity theory. The CalculiX finite element analysis package was used to solve it and obtain the stress-strain state of the slab. More than four hundred finite element models were constructed for various combinations of parameters. Based on the known displacements of the crack face points, the values of the stress intensity factor along the crack front were calculated using the relation obtained in the study. It is established that its values significantly depend on the diameter of the reinforcement, and therefore, when conducting practical calculations, it is not recommended to replace the action of reinforcement on concrete with concentrated force. Polynomial approximations with a relative error of 10% are obtained for extreme values of the stress intensity factor. The materials of the study can be useful in the design of reinforced concrete structures, and when studying or teaching a course in fracture mechanics

Effect of reinforcement on the crack resistance of concrete slabs

A preliminary analysis of the available publications devoted to the study of crack resistance of reinforced concrete structures showed the absence of established general patterns of influence of important geometric parameters inherent in reinforced concrete elements on the distribution of the characteristics of fracture mechanics along the crack front. Based on the analysis, the purpose of the study was formulated: to establish these regularities for a concrete slab reinforced with a system of longitudinal steel rods. When conducting the research, a linear and elastic model of concrete was used, and the stress intensity factor was considered as a characteristic of the fracture mechanics. A surface crack of constant depth located in the cross-section of the slab was postulated. It was assumed that its faces completely cover the cross-section of reinforcing rods. The crack depth, the depth of reinforcing rods, their diameter, and the distance between adjacent rods were chosen as dimensionless geometric parameters relative to the thickness of the slab. The slab was loaded with two types of loads applied to its ends: constant tensile stresses (pure tension) and linearly variable axial stresses (pure bending). The problem of determining the stress intensity coefficient depending on geometric parameters was reduced to the boundary problem of elasticity theory. The CalculiX finite element analysis package was used to solve it and obtain the stress-strain state of the slab. More than four hundred finite element models were constructed for various combinations of parameters. Based on the known displacements of the crack face points, the values of the stress intensity factor along the crack front were calculated using the relation obtained in the study. It is established that its values significantly depend on the diameter of the reinforcement, and therefore, when conducting practical calculations, it is not recommended to replace the action of reinforcement on concrete with concentrated force. Polynomial approximations with a relative error of 10% are obtained for extreme values of the stress intensity factor. The materials of the study can be useful in the design of reinforced concrete structures, and when studying or teaching a course in fracture mechanics

Investigation of implicit constitutive relations in which both the stress and strain appear linearly, adjacent to non-penetrating cracks

A novel class of implicit constitutive relations is studied, wherein the stress and the linearized strain appear linearly, that describe material response in elastic porous bodies like rocks, ceramics, concrete, cement, bones and metals. The constitutive relation is applied to a body with a crack subjected to non-penetration conditions between the opposite crack faces. To treat well-posedness of a corresponding variational inequality, we rely on a new approximation by thresholding dilatation and apply the Lions existence theorem on pseudo-monotone variational inequalities. An analytical solution to a specific example (without crack) under uniform triaxial loading is constructed, wherein blow-up can take place at a finite load, and this difficulty is overcome within a thresholding approximation so that blow-up does not occur.

A set of electrically conducting collinear cracks between two dissimilar piezoelectric materials

The plane problem of multiple collinear cracks between two piezoelectric semi-infinite spaces is considered. The cracks can have arbitrary lengths and distances between each other and they are assumed to be electrically conductive. The materials are polarized in the orthogonal to the interface direction. In-plane combined tension-shear loading and the electric field parallel to the crack faces are prescribed at infinity. The Riemann-Hilbert problems of linear relationship are formulated and solved analytically by using the presentations of electro-mechanical quantities via sectionally-analytic functions. The arbitrary constants are found from the conditions at infinity and the system of linear algebraic equations which dimension is equal to the number of cracks. Stresses, electric field outside cracks as well as mechanical and electrical displacement jumps along the crack regions are found in the relatively simple analytical forms. Special attention was devoted to the energy release rate (ERR) determination. Due to analysis of asymptotic fields at the tips of single and multipleinterface cracks, the analytical presentations of the ERR for any crack tips are obtained. Numerical illustrations for several interface cracks are presented. The variations of stresses, electric field and the crack opening along the interface are demonstrated. The ERR are given for different crack tips, loading, crack locations and different physical aspects of the interaction effect are discussed. The particular cases of purely elastic and purely electrically conductive states are considered and good agreement with known results is demonstrated.

An automated mesh generation algorithm for simulating complex crack growth problems

In this manuscript, we expand the conforming to interface structured adaptive mesh refinement (CISAMR) algorithm for modeling complex two-dimensional (2D) crack growth problems involving contact/friction along the crack surface and interaction between multiple cracks. The CISAMR algorithm transforms a structured mesh into a high-quality conforming mesh non-iteratively, which is an attractive feature for modeling the evolution of the crack geometry with minimal changes to the underlying mesh structure. To model such problems, the mesh structure is first adaptively refined and updated near the crack tip to form a spider-web pattern of elements for the accurate approximation of the energy release rate and thereby predicting the new crack path. In each step of the crack advance simulation, a small subset of elements in the vicinity of the crack tip is detected using a tree data structure and then deleted/regenerated to simulate the crack growth. The construction of a high-quality mesh with appropriate element aspect ratios in the algorithm allows the use of an explicit dynamic solver, which is essential to simulate the nonlinear response of the problem caused by contact forces along crack faces. Several benchmark fracture problems are presented to study the accuracy of the proposed algorithm, as well as two more complex problems to demonstrate its ability for modeling interaction of multiple growing cracks with one another and with embedded heterogeneities in the domain.

Fatigue crack growth in bearing steel under cyclic mode II + static biaxial compression

Mode II fatigue crack growth under reversed shear and static biaxial compression was investigated in two bearing steels. Many aborted branches, quasi-orthogonal to the main crack, were observed along the crack face. The compressive stress parallel to the main crack hindered the growth of these branches and favored coplanar mode II crack growth. The crack face sliding displacement profiles measured by DIC were used to derive ΔKII,eff, at the main crack tip, using elastic–plastic FE simulations with crack face friction, by an inverse method. Friction-corrected crack growth kinetics were obtained for mode II crack growth in both steels.

АНТИПЛОСКА ЗАДАЧА ДЛЯ ОДНОМІРНОГО П’ЄЗОЕЛЕКТРИЧНОГО КВАЗІКРИСТАЛА З МІЖФАЗНОЮ ТРІЩИНОЮ ПІД ВПЛИВОМ ВНУТРІШНЬОГО ЕЛЕКТРИЧНОГО ЗАРЯДУ

The paper considers two coupled one-dimensional quasicrystalline half-spaces and a tunnel crack along their interface. The stress-strain state in the vicinity of the electrically conductive faces of the crack is investigated. It is believed that the polarization of materials is directed in the direction of the crack front and in the same direction the arrangement of atoms is quasi-periodic, and perpendicular to the crack front the arrangement of atoms is periodic. Uniformly distributed antiplane phonon and phason shear loads parallel to the crack faces are applied. The electric charge on the crack faces also takes place. A matrix-vector representations for the derivatives of displacement jumps and stresses are constructed through a vector function that is holomorphic in the whole complex plane, except of the crack region. Satisfying the boundary conditions on the crack faces, using matrix-vector representations, the Riemann-Hilbert linear conjugation problem with corresponding conditions at infinity is formed. An analytical solution of this problem is constructed. Analyzing the solution, we obtain analytical expressions for the phonon and phason stresses, the jumps of displacements along the interface between the materials in the crack region that has an electric charge. Numerical analysis of the solution demonstrated the essential influence of the electric charge of the crack to the phonon and phason stress-strain state in the vicinity of the crack. The analysis was performed for a combination of different quasicrystalline compounds. The main results of the solutions, i.e. phonon and phason stresses along the materials interface and the phonon and phason displacement jumps are presented in the graphic form. Conclusions are made regarding the influence of the electric charge of the crack on the behavior of both the crack itself and the material in its vicinity.

Fracture, Friction, and Permeability of Ice

Water ice Ih exhibits brittle behavior when rapidly loaded. Under tension, it fails via crack nucleation and propagation. Compressive failure is more complicated. Under low confinement, cracks slide and interact to form a frictional (Coulombic) fault. Under high confinement, frictional sliding is suppressed and adiabatic heating through crystallographic slip leads to the formation of a plastic fault. The coefficient of static friction increases with time under load, owing to creep of asperities in contact. The coefficient of kinetic (dynamic) friction, set by the ratio of asperity shear strength to hardness, increases with velocity at lower speeds and decreases at higher speeds as contacts melt through frictional heating. Microcracks, upon reaching a critical number density (which near the ductile-to-brittle transition is nearly constant above a certain strain rate), form a pathway for percolation. Additional work is needed on the effects of porosity and crack healing. ▪ An understanding of brittle failure is essential to better predict the integrity of the Arctic and Antarctic sea ice covers and the tectonic evolution of the icy crusts of Enceladus, Europa, and other extraterrestrial satellites. ▪ Fundamental to the brittle failure of ice is the initiation and propagation of microcracks, frictional sliding across crack faces, and localization of strain through both crack interaction and adiabatic heating.

Thermal conductivity of crack-containing media: A numerical study

The present study aims to evaluate the effective thermal conduction behavior of crack-containing media through micromechanical modeling. Numerical analyses were performed using the finite element method, first using periodic arrays of elliptical pores and then reducing the elliptical minor axis to reach the crack limit. This parametric approach was seen to be able to capture the effect of discontinuity across the crack face in a straightforward manner. The overall thermal conductivity of the model structures, encompassing a range of crack densities and spatial distributions, was investigated. It was found that, for cracks with mixed orientations, the effective thermal conductivity is insensitive to the actual crack alignment and can be uniquely represented by the crack density. The mixed-orientation periodic crack configurations are also representative of random sizes, orientations, and spatial distributions of cracks, thus exhibiting a high degree of generality. The modeling analyses can also be used to determine simple empirical expressions for two- and three-dimensional media containing random cracks.

Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem

A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities (VI) is studied with respect to its shape differentiability. The specific problem stemming from quasi-brittle fracture describes an elastic body with a Barenblatt cohesive crack under the inequality condition of non-penetration at the crack faces. Based on the Lagrange approach and using smooth penalization with the Lavrentiev regularization, a formula for the shape derivative is derived. The explicit formula contains both primal and adjoint states and is useful for finding descent directions for a gradient algorithm to identify an optimal crack shape from a boundary measurement. Numerical examples of destructive testing are presented in 2D.

Critical Comparison of Phase-Field, Peridynamics, and Crack Band Model M7 in Light of Gap Test and Classical Fracture Tests

Abstract The recently conceived gap test and its simulation revealed that the fracture energy Gf (or Kc, Jcr) of concrete, plastic-hardening metals, composites, and probably most materials can change by ±100%, depending on the crack-parallel stresses σxx, σzz, and their history. Therefore, one must consider not only a finite length but also a finite width of the fracture process zone, along with its tensorial damage behavior. The data from this test, along with ten other classical tests important for fracture problems (nine on concrete, one on sandstone), are optimally fitted to evaluate the performance of the state-of-art phase-field, peridynamic, and crack band models. Thanks to its realistic boundary and crack-face conditions as well as its tensorial nature, the crack band model, combined with the microplane damage constitutive law in its latest version M7, is found to fit all data well. On the contrary, the phase-field models perform poorly. Peridynamic models (both bond based and state based) perform even worse. The recent correction in the bond-associated deformation gradient helps to improve the predictions in some experiments, but not all. This confirms the previous strictly theoretical critique (JAM 2016), which showed that peridynamics of all kinds suffers from several conceptual faults: (1) It implies a lattice microstructure; (2) its particle–skipping interactions are a fiction; (4) it ignores shear-resisted particle rotations (which are what lends the lattice discrete particle model (LDPM) its superior performance); (3) its representation of the boundaries, especially the crack and fracture process zone faces, is physically unrealistic; and (5) it cannot reproduce the transitional size effect—a quintessential characteristic of quasibrittleness. The misleading practice of “verifying” a model with only one or two simple tests matchable by many different models, or showcasing an ad hoc improvement for one type of test while ignoring misfits of others, is pointed out. In closing, the ubiquity of crack-parallel stresses in practical problems of concrete, shale, fiber composites, plastic-hardening metals, and materials on submicrometer scale is emphasized.

The energy release rate for non-penetrating crack in poroelastic body by fluid-driven fracture

A new class of constrained variational problems, which describe fluid-driven cracks (that are pressurized fractures created by pumping fracturing fluids), is considered within the nonlinear theory of coupled poroelastic models stated in the incremental form. The two-phase medium is constituted by solid particles and fluid-saturated pores; it contains a crack subjected to non-penetration condition between the opposite crack faces. The inequality-constrained optimization is expressed as a saddle-point problem with respect to the unknown solid phase displacement, pore pressure, and contact force. Applying the Lagrange multiplier approach and the Delfour–Zolésio theorem, the shape derivative for the corresponding Lagrangian function is derived using rigorous asymptotic methods. The resulting formula describes the energy release rate under irreversible crack perturbations, which is useful for application of the Griffith criterion of quasi-static fracture.

Asymptotic series solution for plane poroelastic model with non-penetrating crack driven by hydraulic fracture

A new class of coupled poroelastic problems describing fluid-driven cracks (called fractures) subjected to non-penetration conditions between opposite crack faces (fracture walls) is considered in the incremental form. The nonlinear crack problem for a plane isotropic setting in a two-phase medium is expressed in polar coordinates as a variational inequality with respect to the solid phase displacement and the pore pressure. Applying nonlinear methods, the asymptotic theory and Fourier analysis, a semi-analytic solution given as the power series in the sector of angle 2π is proven using rigorous expansions with respect to the distance to the crack-tip. Here no logarithmic terms occur in the asymptotic expansion. Consequently, a square-root singularity for the poroelastic medium with a non-penetrating crack is derived, and the integral formulas for calculating the corresponding stress intensity factors are obtained.

A CFD-FEM based partitioned fluid structure interaction model to investigate surface cracks in elastohydrodynamic lubricated line contacts

This paper presents a partitioned, two-dimensional fluid structure interaction (FSI) model developed to investigate the effects of surface cracks in rolling/sliding line contacts operating under elastohydrodynamic lubrication (EHL). The lubricant behavior as it flows over and through the crack is determined by solving the Navier-Stokes equations using the Ansys Fluent computational fluid dynamics (CFD) solver. The structural response of the solid body is governed via an elastic-plastic constitutive relationship and solved using the Ansys Mechanical finite element solver. The exchange of (i) forces due to fluid pressure, and (ii) displacements due to solid deformation across the fluid-solid interface (including the crack faces) is facilitated by an iterative implicit coupling scheme. A smoothing based dynamic meshing technique is utilized to capture the deformation of the fluid region. The FSI model developed for this investigation is capable of modeling surface cracks with inclined geometries, overcoming the limitations of the classical Reynolds based approach. The fluid pressure results are presented for different loads, speeds and slide-to-roll ratios. The effects of crack geometry (i.e. location, dimension, inclination, crack tip radius, etc.) on fluid pressure and structural response are investigated. The FSI model presents the details of pressure distribution, fluid streamlines and stress concentration in the solid simultaneously. The model results can be used to provide accurate boundary conditions on the crack faces for crack propagation modeling in rolling contact fatigue. The results of this investigation identify the crack geometries that affect fatigue life of rolling elements in EHL contact by predicting the location and severity of stress concentrations in the material.