Dynamic Stress Field Research Articles

Dynamic Crack Propagation Analysis by Node-Based FEM

Due to a very short time span of dynamic fracture event, it is quite difficult to experimentally measure dynamic stress field near the crack tip. Node-based finite element method (NBFEM) developed by Yagawa et al was extended to solve dynamic crack propagation problems by modifying the original NBFEM and three schemes of crack tip node separation during crack propagation are examined. A dynamic stress intensity factor in mode I, KdynI, is calculated for fast propagating crack at a constant speed. Three different schemes of the crack tip node separation were considered. Comparing the numerical results with those analytically obtained by Freund showed that the condition of zero initial velocity and nonzero acceleration gives the highest accuracy of three conditions with only 0.91% error.

Geometrical effects on residual stresses in 7050-T7451 aluminum alloy rods subject to laser shock peening

Laser shock peening (LSP) is an emerging surface treatment technology for metallic materials, which appears to produce more significant compressive residual stresses than those from the conventional shot peening (SP) for fatigue, corrosion and wear resistance, etc. The finite element method has been applied to simulate the laser shock peening treatment to provide the overall numerical assessment of the characteristic physical processes and transformations. However, the previous researchers mostly focused on metallic specimens with simple geometry, e.g. flat surface. The current work investigates geometrical effects of metallic specimens with curved surface on the residual stress fields produced by LSP process using three-dimensional finite element (3-D FEM) analysis and aluminium alloy rods with a middle scalloped section subject to two-sided laser shock peening. Specimens were numerically studied to determine dynamic and residual stress fields with varying laser parameters and geometrical parameters, e.g. laser power intensity and radius of the middle scalloped section. The results showed that the geometrical effects of the curved target surface greatly influenced residual stress fields.

Dynamic stress field of a kinematic earthquake source model with k-squared slip distribution

SUMMARY Source models such as the k-squared stochastic source model with k-dependent rise time are able to reproduce source complexity commonly observed in earthquake slip inversions. An analysis of the dynamic stress field associated with the slip history prescribed in these kinematic models can indicate possible inconsistencies with physics of faulting. The static stress drop, the strength excess, the breakdown stress drop and critical slip weakening distance Dc distributions are determined in this study for the kinematic k-squared source model with k-dependent rise time. Several studied k-squared models are found to be consistent with the slip weakening friction law along a substantial part of the fault. A new quantity, the stress delay, is introduced to map areas where the yielding criterion of the slip weakening friction is violated. Hisada’s slip velocity function is found to be more consistent with the source dynamics than Boxcar, Brune’s and Dirac’s slip velocity functions. Constant rupture velocities close to the Rayleigh velocity are inconsistent with the k-squared model, because they break the yielding criterion of the slip weakening friction law. The bimodal character of Dc/Dtot frequency‐magnitude distribution was found. Dc approaches the final slip Dtot near the edge of both the fault and asperity. We emphasize that both filtering and smoothing routinely applied in slip inversions may have a strong effect on the space‐time pattern of the inferred stress field, leading potentially to an oversimplified view of earthquake source dynamics.

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A3, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A3 decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A3, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials subjected to harmonic antiplane shear waves is investigated using the Schmidt method. The present problem can be solved using the Fourier transform and the technique of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric, magnetic flux, and dynamic stress fields near crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between the two parallel cracks, and the circular frequency of the incident waves upon the stress, electric displacement, and magnetic flux intensity factors at crack tips.

Numerical Simulation of Rock Stability under Dynamic Blasting Load

In order to study the effect of blast shake on the stability of tunnel during mining process, finite element analysis software for rock and soil engineering is used to simulate the effect of blasting shake. In the simulation, a short-time dynamic load is applied to the rock in the blasting zone. Dynamic stress field in the rock mass and distortion in the surface of the tunnel are calculated with the finite-element method. Equivalent displacement method is utilized to determine the amount of sudden applied load, the actuation duration and the range of action. Consequently, the maximum critical explosive content at the critical shake speed of rock mass can be defined.

The non-local theory solution of a Griffith crack in functionally graded materials subjected to the harmonic anti-plane shear waves

In this paper, the dynamic stress field near crack tips in the functionally graded materials subjected to the harmonic anti-plane shear stress waves was investigated by means of the non-local theory. The traditional concepts of the non-local theory were extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it was assumed that the material properties vary exponentially with coordinate parallel to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable was the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at crack tips, thus allowing us to use the maximum stress as a fracture criterion. The magnitude of the finite dynamic stress field depends on the crack length, the parameter describing the functionally graded materials, the circular frequency of the incident waves and the lattice parameter of materials.

Scattering of harmonic anti-plane shear stress waves by a crack in functionally graded piezoelectric/piezomagnetic materials

In this paper, the dynamic behavior of a permeable crack in functionally graded piezoelectric/piezomagnetic materials is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using the Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown is the jump of displacements across the crack surfaces. These equations are solved to obtain the relations between the electric filed, the magnetic flux field and the dynamic stress field near the crack tips using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter and the circular frequency of the incident waves upon the stress, the electric displacement and the magnetic flux intensity factors of the crack.

Damage Localization from the Null Space of Changes in the Transfer Matrix

*A new theorem connecting changes in transfer matrices to the spatial location of stiffness related damage is presented. The theorem states that the span of the null space of the change in the transfer matrix (∆G) contains vectors that are Laplace transforms of excitations for which the dynamic stress field is identically zero over the portion of the domain where the damage is located. An important corollary states that if ∆G proves rank deficient at any s (with the exception of the origin in system with rigid body modes) it is guaranteed to be rank deficient throughout the plane. It is shown that a sufficient condition for rank deficiency in ∆G is that the number of independent measurements be larger than the rank of the change in stiffness resulting from damage. Implementation of the theorem in the time domain involves evaluation of the null space of ∆G along a discretized Bromwich contour, numerical inversion of the resulting s-functions and evaluation of the dynamic response to the computed signals to identify the regions where the stress field is zero. The theorem is most efficiently implemented, however, in the s-domain, where numerical Laplace inversion is avoided and the number of evaluations of the null of ∆G can reduced to one. Application of the theorem to locate damage is illustrated in a simple mass-spring system and in a FEM model of a square plate with free-free boundary conditions.

Lithospheric deformation under pre- and post-seismic stress fields along the Nicobar–Sumatra subduction margin during 2004 Sumatra mega-event and its tectonic implications

A high-resolution study was carried out under pre-seismic (i.e., static) and post-seismic stress fields (i.e., dynamic) in a space–time frame along the Nicobar–Sumatra margin. The study reveals that the descending lithosphere records minimum stress obliquity, predominant thrust movement, and down-dip least compressive stress axis under static stress field in northwest Sumatra (sector III). The imbalance between down-dip component of slab pull force and viscous resistive force possibly caused cyclic stress loading in compressive field around the flexing zone (∼ 25 km), and that undergone brittle failure through generation of mega-thrust event on 26th December' 2004. A sharp decrease in stress obliquity towards north (sector II), redressing of least compressive stress axes from horizontal to down-dip direction, and increasing thrust movements under dynamic stress field account for continued upward shortening of the lithosphere. The weak thinner zone (i.e., between ∼ 159 and ∼ 217 km depth), an age-discontinuity portion, possibly was collapsed through rapid enhancement of stress-induced weakening and strain localization following the 2004 Sumatra mega-shock. It is also well appreciated in the literature that such shallow disruption of the lithosphere is inevitable in the upper mantle, if the slab is weakened or broken there, and this phenomenon is not uncommon below the Sumatra.

Numerically evaluated displacement and stress solutions for a 3D viscoelastic half space subjected to a vertical distributed surface stress loading using the Radon and Fourier transforms

AbstractIn this article a numerical solution for a 3D isotropic, viscoelastic half‐space subjected to vertical rectangular surface stress loading of constant amplitude is evaluated with the aid of the Radon and Fourier integral transforms. Dynamic displacement and stress fields are computed for the half‐space surface as well as for points inside the domain. The analysis is performed in the frequency domain. Viscoelastic effects are incorporated by means of the elastic–viscoelastic correspondence principle. The equations of motion are solved in the Radon–Fourier transformed domain. Inverse transformations to the physical domain are accomplished numerically. The scheme used to perform the numerical inverse transformations is addressed. The solution is validated by comparison with results available in the literature. A sample of original dynamic results for displacement and stress fields for the 3D half‐space is furnished. Copyright © 2006 John Wiley & Sons, Ltd.

A dynamical (time-harmonic) axisymmetric stress field in a finitely prestretched multilayered slab on a rigid foundation

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the dynamical (time-harmonic) axisymmetric stress field in a finitely prestretched multilayered slab resting on a rigid foundation is studied. It is assumed that the slab consists of two-layer packets. The elasticity of layer materials is described by the Treloar potential. It is assumed that the material of the lower layer in the packets is more rigid than that of the upper one. Numerical results are presented for the cases where the number of layers (packets) in the slab is 2 (1), 4 (2), or 6 (3). These results concern the normal stresses acting on the interface between the layers of the first, upper packet and on the interface between the first and second packets. The influence of the number, prestretch level, and thickness of the layers on relation ships between the stresses and the frequency of the external force is analyzed.

Sliding along frictionally held incoherent interfaces in homogeneous systems subjected to dynamic shear loading: a photoelastic study

An experimental investigation was conducted to study dynamic sliding at high strain rates along incoherent (frictional) interfaces between two identical plates. The plates were held together by a uniform compressive stress, while dynamic sliding was initiated by an impact-induced shear loading. The case of freely-standing plates with no external pressure was also investigated. The dynamic stress fields that developed during the events were recorded in a microsecond time scale by high-speed photography in conjunction with classical dynamic photoelasticity. Depending on the choice of experimental parameters (impact speed and superimposed static pressure), pulse-like and crack-like sliding modes were observed. Visual evidence of sub-Rayleigh, intersonic and even supersonically propagating pulses were discovered and recorded. Unlike classical shear cracks in coherent interfaces of finite strength, sliding areas in frictional interfaces seem to grow at various discrete speeds without noticeable acceleration phases. A relatively broad loading wave caused by the interference between the impact wave and the preexisting static stress field was observed emanating from the interface. There was a cusp in the stress contours at the interface, indicating that the propagation speed was slightly faster along the interface than in the bulk. The observed propagation speeds of the sliding tips were dependent on the projectile speed. They spanned almost the whole interval from sub-Rayleigh speeds to nearly the sonic speed of the material, with the exception of a forbidden gap between the Rayleigh wave speed and the shear wave speed. Supersonic trailing pulses generating Mach lines of different inclination angles, emanating from the sliding zone tips, were discovered. In addition, behind the sliding tip, wrinkle-like opening pulses were observed for a wide range of impact speeds and confining stresses. They always traveled at speeds between the Rayleigh wave speed and the shear wave speed of the material.

Mixed Mode Dynamic Fracture in Particulate Reinforced Functionally Graded Materials

A detailed analytical and experimental investigation is presented to understand the dynamic fracture behavior of functionally graded materials (FGMs) under mode I and mixed mode loading conditions. Crack-tip stress, strain and displacement fields for a mixed mode crack propagating at an angle from the direction of property gradation were obtained through an asymptotic analysis coupled with a displacement potential approach. This was followed by a comprehensive series of experiments to gain further insight into the behavior of propagating cracks in FGMs. Dynamic photoelasticity coupled with high-speed photography was used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the FGMs. Dynamic fracture experiments were performed using different specimen geometries to develop a dynamic constitutive fracture relationship between the mode I dynamic stress intensity factor (K ID ) and crack-tip velocity ( $${\mathop a\limits^ \cdot }$$ ) for FGMs with the crack moving in the direction of increasing fracture toughness. A similar $${\mathop a\limits^ \cdot }$$ -K ID relation was also obtained for matrix material (polyester) for comparison purposes. The results obtained show that crack propagation velocities in FGMs were about 80% higher than the polyester matrix. Crack arrest toughness was found to be about 10% lower than the value of local fracture toughness in FGMs.

Particle Velocimetry and Photoelasticity Applied to the Study of Dynamic Sliding Along Frictionally-Held Bimaterial Interfaces: Techniques and Feasibility

A laser interferometry-based technique was developed to locally measure the in-plane components of particle velocity in dynamic experiments. This technique was applied in the experimental investigation of dynamic sliding along the incoherent (frictional) interface of a Homalite–steel bimaterial structure. The bimaterial specimen was subjected to uniform compressive stress and impact-induced shear loading. The evolution of the dynamic stress field was recorded by high-speed photography in conjunction with dynamic photoelasticity. The combination of the full-field technique of photoelasticity with the local technique of velocimetry was proven to be a very powerful tool in the investigation of dynamic sliding. A relatively broad loading wave with an eye-like structure emanated from the interface. The particle velocity measurements established that sliding started behind the eye-like fringe pattern. It propagated with supershear speed with respect to Homalite. A shear Mach line originating from the sliding tip is visible in the photoelastic images. A vertical particle velocity measurement revealed the existence of a wrinkle-like pulse traveling along the bimaterial interface. The wrinkle-like pulse followed the initial shear rupture tip and propagated at a specific subshear speed.

The scattering of the harmonic anti-plane shear stress waves by two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material half-infinite planes dynamic loading

In this paper, the dynamic behaviour of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material half-infinite planes subjected to the harmonic anti-plane shear stress waves is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By using the Fourier transform technique, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the jump of the displacements across the crack surfaces. These equations are solved by using the Schmidt method. The relations among the electric field, the magnetic flux field, and the dynamic stress field near the crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between two interface cracks, and the circular frequency of the incident waves upon the stress, the electric displacement, and the magnetic flux intensity factors of cracks.

On the dynamical axisymmetric stress field in a finite pre-stretched bilayered slab resting on a rigid foundation

The dynamic axisymmetric stress field in an initially finite pre-stretched bilayered slab resting on a rigid foundation is studied within the framework of the piecewise homogeneous body model. The three-dimensional linearized theory of elastic waves in initially stressed bodies is used. It is assumed that a time-harmonic point-located normal force acts on the free face plane of the slab. The considered problem is solved by employing the Hankel integral transformation. The materials of the layers are assumed to be incompressible, and the elastic relations are given through the Treloar potential. The formulation and solution to the problem coincide with the corresponding ones of classical linear theory of elasticity for an incompressible body in the case where there is no initial stretching in the layers. Numerical results are presented and these results involve stresses acting on the interface planes. In particular, it is established that stresses on the interface planes decrease as the pre-stretching is increased.

Dynamical (time-harmonic) axisymmetric interface stress field in the finite pre-strained half-space covered with the finite pre-stretched layer

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), the dynamical (time-harmonic) stress field in the initially finite strained half-space covered with an initially and finitely stretched layer is investigated. It is assumed that on the upper free face of the covering layer the point located force which acts is harmonic with respect to time. The corresponding boundary contact problem is solved by employing the Hankel integral transformation. Moreover, it is assumed that the material of the layer and half-space are incompressible and elastic relations for those are given through the Treloar’s potential. In the case where the initial strains are absent in the layer and half-space the considered problem formulation and solution to that coincide with the corresponding ones of the classical linear theory of elasticity for an incompressible body. The algorithm for obtaining numerical results is proposed. The numerical results regarding the stresses acting on the interface plane are presented. These results are obtained for the case where the stiffness (distortion wave velocity) of the covering layer material is less (greater) than that for the half-space material. In this case, the main attention is focused on the dependencies between the values of the stresses and frequency of the external force and also the influence of the initial strains on these dependencies. In particular, it is established that the “resonance” values of the frequency of the external force increase, but the absolute maximum values of the stresses decrease significantly with the amount of the initial tension of the covering layer.

Two Collinear Interface Cracks between Two Dissimilar Functionally Graded Piezoelectric/Piezomagnetic Material Layers under Anti-Plane Shear Loading

In this paper, the behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material layers subjected to an anti-plane shear loading is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved using the Schmidt method. The normalized stress, the electrical displacement and the magnetic flux intensity factors are determined for different geometric for the permeable electric boundary conditions. The relations among the electric filed, the magnetic flux field and the dynamic stress field near the crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter and the thickness of the strip upon the stress, the electric displacement and the magnetic flux intensity factors of the crack.

The influence of the third order elastic constants on the dynamical interface stress field in a half-space covered with a pre-stretched layer

The influence of the third order elastic constants on the dynamical (time-harmonic) axisymmetric interface stress field in the system which comprises the half-space and the pre-stretched covering layer is investigated within the framework of the piecewise homogeneous body model by employing the three-dimensional linearized theory of elastic waves in initially stressed bodies. The elasticity relations for the layer and half-space materials are given through the Murnaghan potential. It is assumed that the force acting on the free face plane of the covering layer is a time-harmonic point-located normal force. The influences due to both of the quantities of the pre-stretching of the layer and the third order elastic constants of the layer material on the interface normal stress are analysed. The numerical results are presented for concrete selected materials such as steel, aluminium and acrylic plastic.