Crack Boundary Conditions Research Articles

A concerted experimental and numerical approach for frequency based non-destructive analysis of Bi-directional cracked laminated composite beams

ABSTRACT The effects of single and multiple transverse open cracks on the dynamic properties of woven fibre laminated composite beam (LCB) under free vibration by experimental and numerical approaches are explored in this research work. The experimental modal testing is performed n laminated glass/epoxy composite beams using vibration Fast Fourier Transformation (FFT) analyser system. The finite element simulation software ABAQUS is used as the platform for numerical modal analysis to arrive at the natural frequencies. Within the test domain of modal analysis, the simulated results are finely agreed with the empirical test results. The effects of crack location, relative crack depth, boundary conditions and fibre orientations on the modal frequency of the cracked LCB are explored under free vibration. The results of the study conclude that the maximum decrease in frequencies is observed for 0° ply oriented single and multi-cracked LCB. Furthermore, the results elicit that the presence of crack positioned nearer to the restrained end exhibits more reduction in vibration frequencies for any particular crack depth. Within the frequency domain, the present research approach can be employed as an effective tool for the detection and analysis of crack or crack-like defects in LCB structures in any applied field of engineering.

Transient thermoelastic fracture analysis of functionally graded materials using the scaled boundary finite element method

To model fracture in functionally graded materials (FGMs), the scaled boundary finite element method (SBFEM) is extended to examine the effects of fully coupled transient thermoelasticity. Previously developed SBFEM supplementary shape functions are utilized to model thermal stresses. The spatial variation of thermal and mechanical properties of FGMs are approximated by polynomial functions facilitating the semi-analytical evaluation of coefficient matrices. The dynamic stress intensity factors (SIFs) are also evaluated semi-analytically from their definitions without the need for additional post-processing. Scaled boundary polygon elements are employed to facilitate the meshing of complex crack geometries. Both isotropic and orthotropic materials with different material gradation functions are considered. To study the transient effects of thermoelasticity on fracture parameters, several numerical examples with different crack configurations and boundary conditions are considered. The current approach is validated by comparing the results of dynamic SIFs with available reference solutions.

The Study of Localized Crack-Induced Effects of Nonlinear Vibro-Acoustic Modulation

The nonlinear interaction of longitudinal vibration and ultrasound in beams with cracks is investigated. The central focus is on the localization effect of this interaction, i.e., the locally enhanced nonlinear vibro-acoustic modulation. Both numerical and experimental investigations are undertaken. The finite element (FE) method is used to investigate different crack models, including the bi-linear crack, open crack, and breathing crack. A parametric study is performed considering different crack depths, locations, and boundary conditions in a two-dimensional beam model. The study shows that observed nonlinearities (i.e., nonlinear crack-wave modulations) are particularly strong in the vicinity of the crack, allowing not only for crack localization but also for the separation of the crack-induced nonlinearity from other sources of nonlinearity.

The jumping dielectric breakdown behavior induced by crack propagation in ferroelectric materials: A phase field study

Ferroelectric materials often experience fracture and dielectric breakdown under large mechanical and/or electrical loads. These two failure behaviors can affect each other and may lead to new phenomena. This study predicts a new type of jumping dielectric breakdown during crack propagation in ferroelectric materials using a phase field model involving multiple order parameters. The driving forces for the evolution of polarization, fracture, and dielectric breakdown were analysed based on the concept of configurational forces. The jumping dielectric breakdown was attributed to the competition between crack propagation and breakdown initiation at the moving crack tip. The breakdown pattern is related to the polarization domain structure and is significantly influenced by the applied electric field and crack boundary conditions. Furthermore, the jumping breakdown associated with possible mechanical degradation of the material can change the driving and resistance forces for crack propagation. This study introduces a novel dielectric breakdown behavior during crack propagation and provides new insights into the coupling failure of ferroelectric materials.

Study the internal cracks effect on vibration of laminated composite square plates

The study of cracks behaviour in a composite plate is of significant importance in the dynamics of the Mechanical parts in order to avoid design failures due to resonance or high amplitude vibrations. In this paper, a square glass-epoxy composite plate is adopted. The plate has four layers with symmetric and asymmetric lamination. Assuming the cracks are profound as defects. The results were obtained by using a numerical solution of mechanical APDL from ANSYS. It has been found for different boundary conditions that the rank of natural frequencies is decreased by increasing the crack ratio due to the reduction of the plate’s stiffness, whereas the crack direction has no mentioned effect for a small angle of rotation. The accuracy of results is verified by comparing a single case of the current work with other previous investigations. Evaluate the influence of the crack length ratio, angle of the crack rotation, boundary conditions and lamination angles on the natural frequencies of the square composite plate with glass-epoxy materials.

A hybrid multi-phase field model to describe cohesive failure in orthotropic materials, assessed by modeling failure mechanisms in wood

Fracture mechanics is crucial for many fields of engineering applications, as precisely predicting failure of structures and parts is required for efficient designs. The simulation of failure processes is, both from a mechanical and a numerical point of view, challenging, especially for inhomogeneous materials, where the microstructure influences crack initiation and propagation and might lead to very complex crack patterns. The phase field method for fracture is a promising approach to encounter such materials, since it is able to describe complex fracture phenomena like crack kinking, branching and coalescence. Moreover, it is a largely mesh independent approach, given that the mesh is homogeneous in the area of the crack. However, the original formulation of the phase field method is limited to isotropic materials and does not account for preferable fracture planes defined through the material’s microstructure. In this work, the method is expanded to take orthotropic constitutive behavior and preferable directions of crack propagation into account. We show that by using a stress-based split and multiple phase field variables with preferable fracture planes, in combination with a hybrid phase field approach, a general framework can be found for simulating anisotropic, inhomogeneous materials. The stress-based split is based on fictitious crack faces and is, herein, expanded to support anisotropic materials. Furthermore, a novel hybrid approach is used, where the degradation of the sound material is performed based on a smooth traction free crack boundary condition, which proves to be the main driving factor for recovering commonly observed crack patterns. This is shown by means of a detailed analysis of two examples: a wooden single edge notched plate and a wood board with a single knot and complex fiber directions. In both cases, the proposed novel hybrid phase field approach is able to realistically reproduce complex failure modes.

Influence of Material Uncertainty on Vibration Characteristics of Higher-Order Cracked Functionally Gradient Plates Using XFEM

In this study, the influence of material uncertainty on the vibration characteristics of the cracked functionally graded materials (FGM) plates is investigated. Extended stochastic finite element formulation is implemented to model the cracked FGM plate with material uncertainty using higher-order shear deformation theory (HSDT). The level set function is employed to track the crack in the FGM domain. The concept of partition of unity technique is implemented to enrich the primary variable with additional functions. The gradation of the material properties along the thickness direction is done using the power-law distribution. The first-order perturbation technique (FOPT) is incorporated in the methodology for stochastic vibration analysis. The convergence and validation study has been performed to verify the efficacy and accuracy of the formulation. Numerical results are obtained to show the effects of various influential parameters like crack length, gradient index, thickness ratio, and boundary condition on the covariance of the square of natural frequencies. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of cracked FGM plates with material randomness.

Analytical solution of T-stresses for an inclined crack in compression

It is widely accepted that T-stress near crack tip has a considerably effect on crack behavior. In the classic fracture mechanics, only Tx (T-stress component in x-direction) was addressed, and recently one researcher pointed out that both Tx and Ty exist for a crack in compression. In order to theoretically study the T-stress for an inclined crack under compression, analytical solution of stresses near crack tip was derived from the complex stress functions which were obtained according to the crack surfaces stress conditions and boundary conditions. The analytical solution of stresses consists of two parts: one is singular term which is the same as the classical stress expressed in terms of stress intensity factor, and the other is non-singularity term which can be expressed in terms of three components of T-stress, i.e. Tx, Ty and Txy, which indicates that the T-stress should have three components. In order to validate the result of T-stress, the crack initiation angle was calculated by the MTS criterion for four cases: (1) without T-stress, (2) Tx only, (3) Tx and Ty, and (4) Tx, Ty and Txy. The prediction results of the crack initiation angles by MTS criterion were compared to the previous test results of PMMA and the test results conducted in this study by using sandstone specimens with an inclined crack under uniaxial compression. The results show that the prediction results by considering the three components Tx, Ty and Txy can better fit both test results of PMMA and sandstone, which indicates that the T-stress near crack tip should have three components Tx, Ty and Txy, and all of them have effect on crack initiation behavior.

Three-dimensional fields in an infinite transversely isotropic magneto-electro-elastic space with multiple coplanar penny-shaped cracks

This paper investigates the Mode-I crack problem that a three-dimensional infinite space of transversely isotropic multi-ferroic composite medium is weakened by multiple coplanar penny-shaped cracks under uniform mechanical, electric and magnetic loadings. These cracks are arbitrarily distributed in an isotropic plane of the material. Four kinds of idealized electro-magnetic crack boundary conditions, which assume a crack to be electrically permeable (or impermeable) and magnetically permeable (or impermeable), are adopted, and all the possible combinations of these electro-magnetic crack boundary conditions are considered. An approximate three-dimensional solution of the field variables and expressions of some important fracture mechanics quantities are proposed, by virtue of Kachanov’s method and the generalized potential theory method. Numerical examples for some particular configurations of cracks are presented. Based on a two-crack system, the validity of the present solution is checked by comparing the results with their counterparts from a finite element simulation. Interaction of cracks is investigated as well, by taking the crack size and distance between two cracks as factors. From the phenomena shown in the numerical examples, a reduced solution and an empirical method for evaluating generalized stress intensity factors are proposed.

Nonlinear analysis of a crack in 2D piezoelectric semiconductors with exact electric boundary conditions

In this paper, taking the exact electric boundary conditions into account, we propose a double iteration method to analyze a crack problem in a two-dimensional piezoelectric semiconductor. The method consists of a nested loop process with internal and outside circulations. In the former, the electric field and electron density in governing equations are constantly modified with the fixed boundary conditions on crack face and the crack opening displacement; while in the latter, the boundary conditions on crack face and the crack opening displacement are modified. Such a method is verified by numerically analyzing a crack with an impermeable electric boundary condition. It is shown that the electric boundary condition on crack face largely affects the electric displacement intensity factor near a crack tip in piezoelectric semiconductors. Under exact crack boundary conditions, the variation tendency of the electric displacement intensity factor versus crack size is quite different from that under an impermeable boundary condition. Thus, exact crack boundary conditions should be adopted in analysis of crack problems in a piezoelectric semiconductor.

General Stress Field for Cracked Orthotropic Plate

The solution of singular stresses is discussed for the crack problem of the orthotropic plate. Essential equations are introduced on the basis of linear elastic mechanics and complex variable method. The complex variable functions including the material parameters are fully analyzed for solving the partial derivation equation and to meet the needs of the crack boundary conditions. By constructing new stress functions, the mechanic analysis of the cracked orthotropic plate is carried out, and also the stress boundary problem is resolved. The expressions of the singular stress fields at the crack-tip are determined.

Three-dimensional closed-form solution to elliptical crack problem in magneto-electro-elasticity: Electrically and magnetically induced Maxwell stress boundary condition

An elliptical mode-I crack problem in an infinite multi-ferroic composite material with transverse isotropy under remote uniform mechanical, electric and magnetic loadings is analyzed. The electro-magnetically semi-permeable crack boundary condition and Maxwell stress on the crack faces caused by the electric and magnetic fields are considered, which makes the problem nonlinear. On the basis of the general solution for multi-ferroic composite materials and modern potential theory method, the nonlinear integro-differential boundary equations are established and solved analytically. The field variables in the three-dimensional space and some important fracture mechanics quantities, are obtained in explicit forms. From numerical examples, it is found that the combination of the applied electric and magnetic loadings should be in a elliptical region in the electric displacement-magnetic induction coordinate plane to keep the crack open. Furthermore, in the vicinity of the center of this region, the electro-magnetically semi-permeable crack boundary condition without the Maxwell stress could provide relatively accurate results. Otherwise, the Maxwell stress would have a significant influence, and thus should not be neglected.

Damageable elastic buckling bodies with a crack

Damageable elastic buckling beams with a crack motivate to formulate a differential equation system which can be extended to higher-dimensional problems. Elastic buckling bodies are assumed to evolve their damage and break apart from an edge crack. A nonlinear variational formulation is derived that is equivalent to a partial differential equation, subject to crack conditions and boundary conditions. Then, time discretizations are applied to the variational formulation and a first-order nonlinear ordinary differential equation which describes the evolution of damage. We pass to limits as time step sizes tend to zero, to prove that numerical trajectories are convergent to solutions. The fully discrete numerical schemes for a one-dimensional problem are proposed, and some numerical simulations are presented as well.

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

A variationally consistent phase-field anisotropic damage model for fracture

In the phase-field modeling of fracture in brittle and quasi-brittle solids, it is crucial to represent the asymmetric tensile/compressive material behavior. Existing phase-field models generally adopt either an intuitive split of the free energy density without capturing the crack boundary conditions properly or an ad hoc hybrid formulation at the loss of variational consistency. To address this issue, this work presents a variationally consistent phase-field anisotropic damage model within the framework of the unified phase-field theory for brittle fracture and quasi-brittle failure Wu [1, 2]. Consistent with the variational approach to fracture, the positive/negative projection of the effective stress in energy norm Wu and Cervera [3] is adopted, minimizing the tensile part of the stored energy that drives crack evolution. A rounded-Rankine criterion naturally emerges to realistically characterize localized failure of brittle and quasi-brittle solids, with no need of ad hoc assumptions. A mixed-mode cohesive zone model is recovered upon strain localization, with the involved parameters determined from the analytical solution of a softening bar under constrained stretching. Representative numerical examples show that the proposed model can capture arbitrary cracks propagation in solids independently of the mesh discretization and length scale parameter. Remarkably, the spurious stress locking, which is notoriously accompanied with classical anisotropic damage models, is not exhibited.

Determination of representative volume elements for small cracks in heterogeneous, linear-elastic domains

A method is introduced for determining the size of a representative volume element for microstructurally small cracks (RVEMSC) in heterogeneous, linear-elastic domains. RVEMSC is defined as the minimum volume required around a crack front such that crack-front parameters are converged with respect to volume size. This method uses finite element models of idealized microstructures, where microstructural constituents are represented by varying the elastic modulus (E). RVEMSC is expressed using two parameters: d1,MSC (the required distance from a crack front to the sides of a volume) and d2,MSC (the required distance from a crack to the top and bottom of a volume), both expressed in terms of the number of microstructural constituents (e.g. grains). RVEMSC is found for cases involving different crack sizes, error tolerances, and boundary conditions. For a given type of boundary condition, a conservative estimate of RVEMSC is provided as a closed-form function of crack size, average grain size, and desired error tolerance. The solutions for RVEMSC are validated numerically using models of synthetic, realistic microstructures, whose volumes are determined from the RVEMSC equations. The equations are shown to be applicable for a variety of crack sizes and error tolerances. The convergence rates for macroscale parameters are compared with convergence rates of crack-front parameters, and RVEMSC is found to lead simultaneously to converged macroscale parameters. Furthermore, a correlation analysis is performed between microstructural features and required volumes for converged crack-front parameters, and the results indicate that highly misoriented neighboring grains (implicitly represented by large differences in E) located near a crack front lead to larger required volumes to guarantee crack-front parameter convergence. A combination of constraint and microstructural effects are found to contribute to RVEMSC, where smaller cracks tend to be more sensitive to microstructural effects while larger cracks tend to be more sensitive to constraint effects.

Computation of T-stresses for multiple-branched and intersecting cracks with the numerical manifold method

Due to its dual cover systems (i.e., the mathematical cover and the physical cover), the numerical manifold method (NMM) is outstanding in crack modeling, especially in the simulation of complex cracks. In this paper, the NMM is extended to calculate the T-stresses for two-dimensional multiple-branched and intersecting cracks. The displacement jump across crack surface is essentially portrayed by the NMM, and the stress singularity at the crack tip is represented through the incorporation of related asymptotic basis in the NMM local functions. The T-stresses are extracted with the domain-form interaction integral in the NMM postprocessing. To validate the proposed scheme, three numerical examples are tested on uniform mathematical covers nonconforming with all domain boundaries, and the computed results are in nice consistence with the existing solutions. Besides, the effects of crack configurations and boundary conditions on the T-stresses are also revealed.

Modeling geometrically nonlinear large deformation behaviors of matrix cracked hybrid composite deep shells containing CNTRC layers

Stiffness degradation due to matrix cracks is the main initial form of damage in composite laminates. This paper presents a framework to model geometrically nonlinear large deformation behaviors of matrix cracked hybrid composite double-curved deep shell containing carbon nanotube reinforced composite (CNTRC) layers. Two types of structures, namely Structure-I and Structure-II, are investigated. The CNTRC layers in Structure-I are considered with CNTs arranged in uniformly distributions, while Structure-II is arranged in functionally graded distributions. The degraded stiffness of cracked layers is modeled via the self-consistent model (SCM) micromechanical framework. To describe the geometrically nonlinear large deflection behaviors and account for deep and moderate thick shells, the von Kármán geometric nonlinearity assumptions and the term 1/(1+ς/R) are considered in the relationship between displacement and strain. The IMLS-Ritz method is employed to discretize the non-linear partial differential equations. The modified Newton–Raphson method in combination with the arc-length iteration technique is adopted to solve the discretized equations. Comparison studies indicate that the proposed predictive model can furnish very accurate results for linear and nonlinear behaviors of thin to moderately thick as well as shallow and deep laminated doubly-curved shells. Parametric studies on the effect of CNT distribution, matrix crack density, load type, length-to-thickness ratio, radius-to-length ratio, aspect ratio, boundary condition, and fiber ply-angle on the geometrically nonlinear large deformation behaviors of spherical hybrid composite shells are investigated.

Fracture analysis of an infinite 1D hexagonal piezoelectric quasicrystal plate with a penny-shaped dielectric crack

This article studies the electroelastic behavior induced by a penny-shaped dielectric crack lying at the midplane of a one-dimensional (1D) hexagonal piezoelectric quasicrystal plate of finite thickness. Semi-permeable crack boundary conditions are given. By using the Hankel transform technique, an associated mixed boundary value problem is reduced to two coupled Fredholm integral equations of the second kind. Singular electric and elastic fields around the crack front along with their intensity factors in the phonon and phason fields near the crack front are obtained. The intensity factors of crack opening displacement (COD) across the crack faces are calculated and presented graphically. The influence of electric field and phason field on the COD intensity factors is analyzed. Permeable and impermeable cracks are two limit cases of the present.

Vibratory characteristics of cracked non-uniform beams with different boundary conditions

Non-uniform beams with bending moment of inertia and mass per unit length varying as I(x) = α1(1+βx)λ+4 and m(x) = α2(1+βx)λ are widely used in various engineering fields, such as the civil and mechanical engineering etc. This paper presents an exact method to investigate the free vibration of cracked non-uniform beams with different conditions. Firstly, the closed form solution for the mode shape functions of the non-uniform beam is obtained based on the Euler-Bernoulli beam theory. Secondly, the beam is divided into several segments according to the different variable form, and each segment is further divided into many sub-segments by cracks. Four undetermined coefficients could represent the mode shape function of each sub-segment by simulating crack with the massless rotational spring. The undetermined transfer relationship in the same segment is obtained based on the principle of the transfer matrix method. The fourorder undetermined coefficient matrix is obtained by using continuity and equilibrium conditions between adjacent segments, and then the characteristic equation of the entire cracked beam is obtained after that. Finally, the results obtained from the finite element method and published papers are used to validate the correctness and reliability of the proposed method. The influences of crack depth, location and boundary conditions on natural frequencies of cracked non-uniform beams are discussed.