Dynamic Stress Intensity Factor Research Articles

Dynamic fracture behavior for inclusion-initiated cracks in graded non-homogeneous magnetic-electric-elastic material

In this paper, dynamic anti-plane problem in graded non-homogeneous magnetic-electric-elastic material with a conductive cylindrical rigid inclusion is studied. Cracks are generated from the inclusion edge under the action of SH guided waves. By applying a pair of reversed shear stresses to the crack surface, the mechanical model of the crack is achieved, and Green’s function method is used in this process. The scattering fields of displacement, electric potential and magnetic potential are constructed by the wave function expansion method. Numerical results clarified the factors that affect the dynamic stress intensity factor of the crack tips, including material parameters, inclusion parameters, wave number and incident angle. It is expected that the investigation of magnetic-electric-elastic material with inclusion-initiated crack defects can provide some references for the project designing, manufacture and application of non-homogeneous multi-fields coupling materials.

Intensity of Dynamic Stresses of Longitudinal Shear in a Periodically Layered Composite with Penny-Shaped Cracks

We consider the problem of harmonic torsional loading of an infinite elastic composite formed by alternating plane layers made of two different materials in the presence of a penny-shaped crack in one of the components of periodic structure. By using the integral representations for displacements and stresses in the frequency domain and satisfying the conditions of periodicity and perfect contact on the interfaces, we deduce a system of uncoupled boundary integral equations for the functions of tangential dynamic crack-opening displacement in a two-layered representative element of the composite. We perform the numerical analysis of the mode-III dynamic stress intensity factors in the vicinity of the analyzed crack depending on the wave number, the thicknesses of constituent layers, and the properties of their materials.

Uncertainty qualification in evaluating dynamic and static stress intensity factors using SBFEM based on model order reduction

This paper presents a novel framework for Monte Carlo simulation (MCs) of dynamic and static fracture problems. The stress intensity factors can be directly extracted by using polygon scaled boundary finite element method (SBFEM) to analyze dynamic and static fracture problems. Uncertainty qualification analysis focuses on the output uncertainty caused by the uncertainty of system input. MCs is considered as a general tool to solve complex multi-dimensional uncertainty problems because of its simplicity and directness. However, its applicability is hindered due to the huge calculating cost. The use of singular value decomposition (SVD) and radial basis function (RBF) can reduce the computational cost, improve the computational efficiency, and realize the rapid uncertainty qualification analysis based on MCs. Considering the influence of uncertainty of structural material property parameters and structural shape parameters on the calculation results, MCs is used to analyze the statistical characteristics of structural response under random variables. Finally, several examples are given to verify the correctness and effectiveness of the algorithm.

Study on mode I dynamic fracture characteristics with a mini three-point bending specimen for the split Hopkinson bar technique

In recent years, dynamic fracture experimental techniques have been developed rapidly in the field of engineering and scientific research. However, there still exist some difficulties in dynamic fracture tests, such as the limitation of loading rate, the accuracy of transmitted wave, the difficulty of crack initiation, and etc. To solve such problems, a novel mode I dynamic fracture experimental technique is proposed in this work, with a mini three-point bending(3PB) specimen and two kinds of fixtures specially designed for the split Hopkinson pressure bar (SHPB) system. In this method, the dynamic stress intensity factor curve (DSIF) is obtained by the experimental-numerical method, and the crack initiation time is determined by the strain gauge method. To verify the feasibility and reliability of the proposed method, the dynamic fracture toughness (DFT) of two kinds of ultra-high-strength steel (UHSS) is determined by this new method and then compared with the results of other methods. Meanwhile, the loading rate effect of fracture toughness, the size effect of crack initiation time and the failure mechanisms of the two materials under mode I fracture are also studied. The results show that fracture toughness depends on the energy consumed to form two types of fracture regions within the fracture initiation time. This work provides a new method for studies of the dynamic fracture behaviors of brittle solids.

Extended wavelet Galerkin method for mixed-mode cracked FGM plate under static and dynamic loads

The extended wavelet Galerkin method (XWGM) is presented for fracture mechanics analysis of functionally graded material (FGM) cracked plates under static and dynamic loads. The nonhomogeneous stress distribution and stress wave transition can be approximated by the piecewise truncated polynomial (B-spline) functions. The steep stress gradients around the crack tip can be captured by superposing the higher resolution wavelet functions with the nature of multiresolution analysis (MRA) in the wavelet theory. Additionally, a trigonometric function and Heaviside function are employed to enrich the wavelet bases for the fracture modeling. The interaction integral method is utilized to evaluate the static and dynamic mixed-mode stress intensity factors taking the inertia and material nonhomogeneity into account. Applicability of XWGM to the FGM crack problems is discussed through the numerical examples.

The role of porosity in the dynamic disturbance resistance of water-saturated coal

Porosity significantly affects the dynamic response of water-saturated coal and the safety of coal mine engineering in water-rich areas. For mastering the dynamic response of water-saturated coal to the porosity, this study used the coal with four levels of the porosity for dynamic tests. In this process, the dynamic compressive tests were operated on these coal specimen in both of water saturation and dry states, and the reduction factor of dynamic mechanical parameters of the coal in water-saturated state to these in dry state were focused to eliminate the effect of self dynamic properties of the coal. The results indicate that the dynamic mechanical properties non-linear positively correlate with the porosity, and its sensitivity to porosity gradually weakens as with the increase of the porosity. This effect is intuitively expressed in the dominant fracture propagation features recorded by a high-speed camera, and microscopically attributed by the size and number of microfractures around the dominant fracture. The dynamic stress intensity factor model of the dominant fracture was further established with consideration of total cohesive force for the equivalent particle on the fracture surface. It well reveals the effect mechanism of porosity on the meso-deterioration and macroscopic mechanical properties of the coal around the roadway. On this basis, the applicable conditions and suggested methods were put forward to improve the dynamic stability of water-saturated coal with various porosity.

Scattering of SH waves by cavity with arbitrary shapes and crack at arbitrary positions in anisotropic elastic half-space media

In this paper, the complex function method, Green's function method and coordinate transformation method are used to solve the scattering problem of cavity with arbitrary shape and crack at arbitrary position in the orthotropic elastic half-space under the incidence of SH waves. Firstly, the displacement field and stress field caused by cavity scattering in the orthotropic elastic half-space under the action of incident SH wave are obtained. Secondly, the solution of Green's function is derived when the plane line source load acts on the orthotropic elastic half-space with cavity. Then, the crack is constructed based on the method of crack ‘cutting.’ The total displacement field and stress field can be further obtained. Then, the analytical solutions of the dynamic stress concentration factor (DSCF) around the cavity, the surface displacement amplitude |W| and the dynamic stress intensity factor (DSIF) at the crack tip are obtained. Finally, taking the existence of elliptical cavity and crack in the orthotropic elastic half-space as an example, a large number of numerical calculations were carried out. Numerous numerical examples show that different parameters have a great influence on the DSCF around the cavity, the surface displacement amplitude |W| and the DSIF at the crack tip.

A Domain-Independent Interaction Integral for Dynamic Fracture in Nonhomogeneous Magneto-Electro-Elastic Materials

In practice, magneto-electro-elastic (MEE) equipment are inevitably subjected to impact loading during service, resulting in fracture failure of the devices. This paper proposes a dynamic domain-independence interaction integral (DII-integral) to solve the dynamic intensity factors (IFs) at the crack tip of MEE materials. Here, this is a new expression of the interaction integral (I-integral) for dynamic crack study of nonhomogeneous MEE media, and we further extend it to MEE materials containing complex interfaces. Domain-independent property of the I-integral is theoretically demonstrated and excellent numerical results (relative deviation<1%) obtained from the various integration domains again verify the domain-independence for nonhomogeneous and multi-interface material properties without considering the material continuity requirements. Through the combination of the DII-integral and extended finite element method (XFEM), a good agreement is observed by comparing with the published data subjected to different MEE impact loadings. Typical examples reveal that the magnitude of dynamic mode-I stress intensity factor (SIF), dynamic electric displacement intensity factor (EDIF) and dynamic magnetic induction intensity factor (MIIF) decrease with the increase of the poling angle of the MEE material, while the dynamic mode-II SIF shows the opposite change. The peaks of dynamic EDIF and MIIF are more sensitive to the distance between the parallel cracks than the SIF, and all IFs are smaller than single crack due to the shielding effect. Finally, the significant difference on the dynamic IFs is observed with various material discontinuities.

Scattering of Love wave from an interface crack in a viscoelastic waveguide layer bonded to a piezoelectric substrate: an analytical estimate and numerical validation

This study uses an analytical method to examine the scattering of a Love wave from an interfacial crack in a thin lossy media (viscoelastic material) that serves as a waveguide layer attached to an unbounded loss-less media (piezoelectric material). The viscoelastic material is modeled using the Kelvin–Voigt equation. Phase velocity and attenuation have been calculated by equating the real and imaginary parts of the dispersion relation to zero, respectively, due to the complex wave number of viscoelastic material. Using the integral transform approach and the dislocation density function, the scattering field close to the crack tip is determined, resulting in a Cauchy singular integral equation of the first type. It has been transformed into a system of linear algebraic equations using Gauss-Chebyshev numerical integration. It has been investigated how frequency, waveguide layer thickness, and viscoelastic material loss factor affect phase velocity and Love wave attenuation. The mode-III dynamic stress intensity factor at the left and right crack-tip is investigated for various thicknesses of the waveguide layer and loss factor of viscoelastic material. The outcomes were verified using numerical data from the COMSOL Multiphysics 5.6 FEA software. The findings are fundamental and could be used in the design of particular sensors.

Exploring the dynamic fracture performance of epoxy/cement based piezoelectric composites with complex interfaces

Dynamic intensity factors (IFs) are the key parameters in comprehending and forecasting dynamic fracture behavior of piezoelectric composites. A domain-independent interaction integral (DII-integral) is developed to extract the dynamic stress intensity factors (SIFs) and electric displacement intensity factor (EDIF). The DII-integral is theoretically proved that there is no need to consider the electromechanical material derivatives and complicated material interfaces have no effect on the application of the method. Evaluating the results with the published data yields a good agreement and good domain-independence of the proposed I-integral is checked for multi-interface piezoelectric composites. The numerical simulations show that the amplitudes of the dynamic IFs decrease as the size of piezoelectric particle increases and crack damage is prevented as the number of piezoelectric particles increases. The peak values of the dynamic IFs are not monotonically related to the distance between the particles due to the superposition of elastic waves. The dynamic IFs increase when the volume percentage of piezoelectric fiber rises, while the opposite change is observed in laminated composites. Generally, the dynamic IFs of fiber-reinforced composites are smaller (larger) than that of laminated composites when the volume fraction of piezoelectric ceramics is higher (lower) than 40 percent.

Improved XFEM (IXFEM): 3D dynamic crack propagation under impact loading

An improved XFEM (IXFEM) for three-dimensional non-planar crack propagation under impact loadings is developed. The proposed method couples the IXFEM (Tian and Wen, 2015; Wen and Tian, 2016) and the B-spline ruled surface method (BRSM) proposed previously by authors (Xiao, Wen and Tian (2021)). The issues of the standard XFEM for dynamic crack propagation, such as energy inconsistency, “null” critical time step size and optimal mass lumping at crack tip, are fundamentally avoided with an extra-DOF free PU approximation (Tian, 2013). The tip singularity enrichment is introduced to assure high accuracy through the extra-DOF free PU approximation. The BRSM, which is designed to solve the challenge about robust and efficient geometry representation of 3D non-planar surface crack growth, is applied to model the geometry of the 3D dynamic crack propagation in this paper. The interaction integral for 2D dynamic stress intensity factor (SIF) evaluation is extended to 3D. Efficiency of the proposed approach is demonstrated via numerical tests.

An investigation on the effect of crack lengths and wavelengths on the dynamic cracking behaviours of brittle materials using the improved XFEM

In this paper, the effects of crack lengths and wavelengths on dynamic cracking behaviours are investigated using the improved extended finite element method (XFEM), in which the absorbing boundary condition is coupled into the XFEM. Subsequently, to study the evolution of the displacement and the energy affected by the cracks, numerical tests in a plate containing a crack are conducted under dynamic loads with different wavelengths. Finally, the effects of the crack lengths and wavelengths on the dynamic cracking behaviors are investigated by analysing the evolution of the dynamic stress intensity factor and the energy release rate. The numerical results show that a sudden drop in energy can be found when the angle between the incident dynamic load and the crack varies from a vertical angle to an acute angle. Additionally, it is found that the crack length rarely affects the dynamic cracking behaviours, while the wavelength does significantly affect the dynamic cracking behaviours. When the wavelength is smaller than the critical wavelength, tensile-shear cracking behavior occurs. Otherwise, the cracking behaviour consists of a compression-shear cracking pattern and a tension-shear cracking pattern.

Analysis of mode III interface fracture for hole-initiated cracks in magnetic-electro-elastic bimaterials

The analytical solution of two magnetic-electro-elastic materials with hole-initiated cracks are investigated under anti-plane shearing. The modeling of the cracks is achieved by applying a pair of negative shear stresses in the region where the cracks occur. During this investigation, the fundamental solution of the displacement field is expressed by Green’s function method. The expressions of the magneto-electric-elastic wave fields around the interface can be obtained by using the wave function expansion method and Sommerfeld radiation conditions. Then, combined with permeable electromagnetic boundary conditions, the crack problem is reduced to a system of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tip is obtained. Finally, various examples are provided to illustrate the effects of the geometrical parameters, material properties, wave number, and incident angle on the DSIF. Compared with previous traditional numerical and analytical methods, the methods proposed in this article are more applicable in practical engineering and theoretical analysis.

Experimental investigation on internal offset crack of PMMA beam using caustics

In this paper, the dynamic stress intensity factor (DSIF) at the crack tip of a PMMA beam containing an internal offset crack under impact loading was investigated by using a transmission digital caustic test system combined with high-speed camera. Firstly, the stress intensity factor, as well as velocity, of the mixed-mode crack tip under static and dynamic loading were derived based on the transmission caustic method; secondly, a drop hammer loading test was conducted on the different specimens with biased cracks in the beam using the transmission caustic test system to obtain the dynamic fracture mechanics parameters of internal crack initiation and expansion; then, the crack initiation angle and expansion path analysis of the impact-generated expansion cracks were performed based on the theoretical analysis and the differences between the fracture process and that of the specimens containing internal symmetric cracks were discussed. The results show that the different locations of the internal off-set cracks make the stress redistribution process inside the beam and the stress state at the crack tip during the crack extension process different, and the fracture process and fracture mechanics parameters are different from those of the specimens with internal symmetric cracks, the results of the caustic test are in good agreement with the numerical analysis. The research results can provide guidance for the study of the fracture process of brittle materials such as PMMA containing internal defects.

Scattering of SH wave by a crack in a functionally graded one dimensional hexagonal piezoelectric quasicrystals

AbstractIn this paper, the dynamic interaction between SH wave and a crack in functionally graded one‐dimensional hexagonal piezoelectric quasicrystals (PQCs) is studied by using integral transform technique. To make the analysis executable, the material properties are assumed to vary partially in the direction perpendicular to the crack face following an exponential function. The problem was transformed through Fourier transform into three pairs of dual integral equations, which are solved numerically by the Copson method. The explicit expressions of phonon field, phason field and electric field on crack surface are determined. The dynamic field intensity factors of crack tip are given and some special cases are discussed. Numerical examples are then conducted to reveal the influences of amplitude, incident angle and wave number of SH wave and crack length, gradient parameter on the normalized dynamic stress intensity factors (DSIFs). The results of this paper lay a theoretical foundation for the design and nondestructive testing of QCs structures.

A two-level nesting smoothed extended meshfree method for static and dynamic fracture mechanics analysis of orthotropic materials

This paper presents a two-level nesting smoothed extended mesh-free method (S-XMM-N) for the static and dynamic fracture mechanics analysis in orthotropic materials. To capture the jump of displacement fields across the crack surface, as well as singularities of stress fields in the vicinity of the crack tip, both Heaviside function and asymptotic solutions from the orthotropic fracture modeling are incorporated into the radial point interpolation shape functions associated with the partition of unity approach. To improve the accuracy and accelerate the computation of the extended mesh-free method, the two-level nesting smoothing integration scheme based on the first and second-level triangular smoothing sub-domains is developed to perform the stiffness integration. Compared to the extended mesh-free method using the conventional Gaussian quadrature, the presented S-XMM-N uses even less integration sampling points. At the same time, the complex and time-consuming derivative calculation of extended shape functions is completely avoided. This leads to the dramatically improvement of efficiency. Moreover, the presented method gives very exactly numerical solutions by optimally combining the contributions from the two-level nesting smoothing sub-domains based on the Richardson extrapolation method. We use the interaction integral method in conjunction with the asymptotic near crack tip fields of orthotropic materials to extract the static and dynamic stress intensity factors. The numerical solutions obtained from the S-XMM-N are further compared with reference solutions from the literature to verify the accuracy of the proposed method.

Dynamic responses and damage mechanism of rock with discontinuity subjected to confining stresses and blasting loads

The propagation of seismic waves induced by blasting changes significantly at rock discontinuities such as joints and faults, many studies have focused on the mechanism of wave propagation at joints. However, in deep rock masses, high in-situ stress is non-negligible, and the interaction between confining stress and discontinuity under blasting disturbance remains poorly understood. To explore the role of confining stress and discontinuity on the dynamic responses and damage mechanism of rock masses, this paper employed the dynamic finite element method and a series of numerical models were subsequently developed. The Riedel-Hiermaier-Thoma (RHT) model was used to simulate the blast-induced damage of rock. Under blasting disturbance, the propagation of stress waves and damage patterns within the rock at different discontinuity orientations and different confining stress magnitudes were presented. Around the discontinuity, the principal stress distribution and displacements were presented and the dynamic stress intensity factor (DSIF) of the discontinuity tip was obtained to analyze the extension mechanism of the discontinuity. Numerical results show that the discontinuity, the magnitude and direction of confining stress significantly contribute to the damage patterns of rock masses, and the guiding effect of high confining stress on cracking will be weakened due to the existence of discontinuity.

Dynamic fracture analysis in nonhomogeneous piezoelectric materials with a new domain-independent interaction integral

In order to comprehend and forecast the dynamic fracture behaviors of piezoelectric materials, dynamic intensity factors (IFs) are crucial fracture parameters. It is a challenge to effectively calculate the dynamic IFs of the piezoelectric materials containing complicated elastic and/or electric interfaces. For piezoelectric materials with nonhomogeneous properties or even random interfaces, a dynamic domain-independent interaction integral (DII-integral) is established to assess the dynamic stress intensity factors (SIFs) and the dynamic electric displacement intensity factor (EDIF). Furthermore, it is theoretically demonstrated that random interfaces in the integration domain have no effect on the efficiency of the DII-integral and it does not include any derivatives of electromechanical characteristics. The extended finite element method (XFEM) is integrated with the dynamic DII-integral method to study typical cracked piezoelectric specimens exposed to electromechanical impact loads. A wonderful consistency is obtained by evaluating the current results with the relevant literature. Good domain-independence of the proposed dynamic I-integral is verified for nonhomogeneous and discontinuous piezoelectric properties (relative deviation < 1 %). The numerical simulations show the amplitudes of the dynamic SIFs and EDIF are highly affected by the polarization direction. In general, the density has an obvious influence on the peak occurrence time of the dynamic SIF and EDIF. The dielectric permittivity impacts the EDIF evidently, however, the SIF marginally. On the contrary, the piezoelectric coefficient obviously impacts the SIF and EDIF. The elastic stiffness has a considerable impact on the SIF but a minor one on the EDIF.

A new enrichment scheme for the interfacial crack modeling using the XFEM

A new enrichment scheme is proposed for the interfacial crack modeling in the framework of extended finite element method (XFEM), in which the approximation of displacement in the element including the crack tip is reconstructed by only the Heaviside function and the material interface enrichment function. The proposed enrichment scheme can model not only the discontinuity of displacement across the crack surface but also the discontinuity of displacement gradient across the material interface in the crack tip element with the weight functions. In addition, no blending elements are introduced and it is unnecessary to define the ramp function in the new XFEM formulation. The static and dynamic stress intensity factors (SIFs) of interfacial cracks are analyzed by the new enrichment scheme and the results show that the effect of material interface in the crack tip element plays a significant role in the accuracy of fracture mechanics parameters and the proposed method can predict the deformation more accurately and obtain a more stable dynamic SIFs. The proposed new enrichment scheme can be extended to three dimensional crack modeling in future works.

Experimental study on the energy evolution law during crack propagation of cracked rock mass under impact loads

Rock dynamic fracture is usually accompanied by numerous energy loss, to reveal energy evolution laws during rock dynamic fracture, the single cleavage triangle (SCT) specimens made by two types of sandstone are implemented to conduct dynamic fracture tests by using the split Hopkinson pressure bar (SHPB) device. The dynamic stress distribution law and the energy evolution law can be obtained according to the strain history of cracked rock mass subjected to impact loads. Crack propagation gauges (CPG) were applied to measure crack propagation speed, dynamic stress intensity factor (DSIF) and dynamic energy release rate can be determined by using an experimental–numerical method with the help of the software COSMOL. The experimental results show a physical pseudo-healing of the crack due to the short closure of the crack after its arrest under impact loads. The crack propagation speed and the DSIF show a certain oscillatory with the increase of crack length, the DSIFs at crack initiation are 0.2 ∼ 0.8 of the maximum value. The history of dissipated energy has a slow decrease after they reach a peak during 300 μs ∼ 350 μs, whose curves after the peak contain two stages: stage I is that the absorbed energy of the SCT specimen is mainly transformed into the transmitted energy; stage II is that it will be reflected energy. The input energy overtakes the consumed energy by new crack formation to drive the crack propagation and the test results of sandstone specimens show that 2.11 ‰ to 4.44 ‰ dissipated energy is implemented to drive crack propagation. The energy and stress are released more significantly around the crack tip before the crack arrest.