Modeling dynamic crack propagation in fiber reinforced composites including frictional effects

Dynamic crack propagation in a unidirectional carbon/epoxy composite is studied through finite element analyses of asymmetric impact (shear loading) of a rod against a rectangular plate. A finite deformation anisotropic visco-plastic model is used to describe the constitutive response of the composite. Crack propagation is simulated by embedding zero thickness interface element along the crack path. An irreversible mixed-mode cohesive law is used to describe the evolution of interface tractions as a function of displacement jumps. Contact and friction behind the crack tip are accounted for in the simulations. The failure of the first interface element at the pre-notch tip models onset of crack extension. Crack propagation is modeled through consecutive failure of interface elements. The dynamic crack propagation phenomenon is studied in terms of crack initiation time, crack speed, mode I and mode II displacement jumps and tractions associated with the failure of interface elements, effective plastic strain at the crack tip and path independent integral J^′. Analyses are carried out at impact velocities of 5, 10, 20, 30 and 40 m/s, assuming the crack wake is frictionless. Moreover, analyses at impact velocities of 30 and 40 m/s are also carried out with a friction coefficient of 0.5, 1, 5 and 10 along the crack surfaces. The analyses show that steady-state intersonic crack propagation in fiber reinforced composite materials occurs when the impact velocity exceeds a given threshold. A steady-state crack speed of 3.9 times the shear wave speed and 83% of the longitudinal wave speed is predicted in the cases in which the impact velocity is above 10 m/s. Detailed discussion is given on the features of sub-sonic and intersonic crack propagation. It is shown that friction effects, behind the crack tip, do not have a significant effect on maximum crack speed; however, they do on characteristics of the shock wave trailing the crack tip. The analyses also show that the contour integral J^′, computed at contours near the crack tip, is indeed path independent and can serve as a parameter for characterizing intersonic crack propagation.     

Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method

Dynamic crack growth is simulated by implementing a cohesive zone model in the generalized interpolation material point (GIMP) method. Multiple velocity fields are used in GIMP to enable handling of discrete discontinuity on either side of the interface. Multilevel refinement is adopted in the region around the crack-tip to resolve higher strain gradients. Numerical simulations of crack growth in a homogeneous elastic solid under mode-II plane strain conditions are conducted with the crack propagating along a weak interface. A parametric study is conducted with respect to varying impact speeds ranging from 5 m/s to 60 m/s and cohesive strengths from 4 to 35 MPa. Numerical results are compared qualitatively with the dynamic fracture experiments of Rosakis et al. [(1999) Science 284:1337–1340]. The simulations are capable of handling crack growth with crack-tip velocities in both sub-Rayleigh and intersonic regimes. Crack initiation and propagation are the natural outcome of the simulations incorporating the cohesive zone model. For various impact speeds, the sustained crack-tip velocity falls either in the sub-Rayleigh regime or in the region between (c _S is the shear wave speed) and c _D (c _D is the dilatational wave speed) of the bulk material. The Burridge–Andrews mechanism for transition of the crack-tip velocity from sub-Rayleigh to intersonic speed of the bulk material is observed for impact speeds ranging from 9.5 to 60 m/s (for normal and shear cohesive strengths of 24 MPa). Within the intersonic regime, sustained crack-tip velocities between 1.66 c _S (or 0.82 c _D ) and 1.94 c _S (or 0.95 c _D ) were obtained. For the cases simulated in this work, within the stable intersonic regime, the lowest intersonic crack-tip velocity obtained was 1.66 c _S (or 0.82 c _D ).     

Intersonic delamination in curved thick composite laminates under quasi-static loading

Dynamic delamination in curved composite laminates is investigated experimentally and numerically. The laminate is 12-ply graphite/epoxy woven fabric L-shaped laminate subject to quasi-static loading perpendicular to one arm. Delamination initiation and propagation are observed using high speed camera and load–displacement data is recorded. The quasi-static shear loading initiates delamination at the curved region which propagates faster than the shear wave speed of the material, leading to intersonic delamination in the arms. In the numerical part, the experiments are simulated with finite element analysis and a bilinear cohesive zone model. Cohesive interface elements are used between all plies with the interface properties obtained from tests. The simulations predict a single delamination initiating at the corner under pure mode-I stress field propagating to the arms under pure mode-II stress field. The crack tip speeds transition from sub-Rayleigh to intersonic in conjunction with mode change. In addition to intersonic mode-II delamination, shear Mach waves emanating from the crack tips in the arms are observed. The simulations and experiments are found to be in good agreement at the macro-scale, in terms of load-displacement behavior and failure load, and at the meso-scale, in terms of delamination initiation location and crack propagation speeds. Finally, a mode dependent crack tip definition is proposed and observation of vibrations during delamination is presented. This paper presents the first conclusive evidence of intersonic delamination in composite laminates triggered under quasi-static loading.     

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (c _R ) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between c _R and the shear wave speed, c _s , do not exclude intersonic (c _s <υ<c _l ) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above c _s and subsequently accelerate rapidly to the longitudinal wave speed (c _l ) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2–c _s . The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2–c _s is discussed. Received: 13 September 1999 / Reviewed and occerted: 19 November 1999     

Effect of loading and geometry on the subsonic/intersonic transition of a bimaterial interface crack

An experimental investigation was conducted to study the nature of intersonic crack propagation along a bimaterial interface. A single edge notch/crack oriented along a polymer/metal interface was loaded predominantly in shear by impacting the specimen with a high velocity projectile fired from a gas gun. The stress field information around the propagating crack tip was recorded in real time by two different optical techniques--photoelasticity and coherent gradient sensing, in conjunction with high speed photography. Intersonic cracks on polymer/metal interfaces were found to propagate at speeds between the shear wave speed (c_s) and of the polymer. The nature of the crack tip fields during subsonic/intersonic transition and the conditions governing this transition were examined. Experimental observations showed the formation of a crack face contact zone as the interfacial crack speed exceeds the Rayleigh wave speed of the polymer. Subsequently, the contact zone was observed to expand in size, shrink and eventually collapse onto the intersonic crack tip. The recorded isochromatic fringe patterns showed multiple Mach wave formation associated with such a scenario. It is found that the nature of contact zone formation as well as its size and evolution differ substantially depending on the sign of the opening component of loading.     

Steady motion of a crack parallel to a bond plane—further results

The present analysis concerns the steady propagation of a crack of length 2a, parallel to a bond plane between two half-planes having different material properties. The crack speed is less than the transverse wave speed of the half-plane in which it is located, but the material properties of the uncracked material are such that the shear or dilatational wave speeds may be exceeded. A transition speed is seen to exist above which the uncracked medium tends to act as a stiffened material.     

Mode II intersonic crack propagation in poroelastic media

A crack is steadily running in an elastic isotropic fluid-saturated porous solid at an intersonic constant speed c. The crack tip speeds of interest are bounded below by the slower between the slow longitudinal wave-speed and the shear wave-speed, and above by the fast longitudinal wave-speed. Biot’s theory of poroelasticity with inertia forces governs the motion of the mixture. The poroelastic moduli depend on the porosity, and the complete range of porosities n ∈ [0, 1] is investigated. Solids are obtained as the limit case n = 0, and the continuity of the energy release rate as the porosity vanishes is addressed. Three characteristic regions in the plane (n, c) are delineated, depending on the relative order of the body wave-speeds. Mode II loading conditions are considered, with a permeable crack surface. Cracks with and without process zones are envisaged. In each region, the analytical solution to a Riemann–Hilbert problem provides the stress, pore pressure and velocity fields near the tip of the crack. For subsonic propagation, the asymptotic crack tip fields are known to be continuous in the body [Loret and Radi (2001) J Mech Phys Solids 49(5):995–1020]. In contrast, for intersonic crack propagation without a process zone, the asymptotic stress and pore pressure might display a discontinuity across two or four symmetric rays emanating from the moving crack tip. Under Mode II loading condition, the singularity exponent for energetically admissible tip speeds turns out to be weaker than 1/2, except at a special point and along special curves of the (n, c)-plane. The introduction of a finite length process zone is required so that 1. the energy release rate at the crack tip is strictly positive and finite; 2. the relative sliding of the crack surfaces has the same direction as the applied loading. The presence of the process zone is shown to wipe out possible first order discontinuities.     

Molecular dynamics investigation of the effect of the interatomic potential on steady‐state crack propagation

Abstract

We present molecular dynamics simulations examining the effect of the interatomic potential on steady‐state mode I crack propagation in a two‐dimensional triangular lattice as a function of applied strain. The interatomic potential is the Morse potential whose failure strain exhibits linear variation with its exponential parameter. The limiting crack speed is defined to be the steady‐state crack velocity observed at the onset of instability in steady‐state crack propagation leading to dislocation nucleation or crack branching. For all systems studied, the limiting crack speed is observed to be less than one third the Rayleigh wave speed. The fastest crack propagation in these ideal systems is associated with a material's long‐wavelength elastic properties being dominated by the strength of the nearest‐neighbor bond.     

Dynamics steady crack propagation of antiplane shear in an elastic perfectly plastic material

Abstract

A theoretical analysis of steady‐state crack growth in an elastic ideally‐plastic material under small‐scale yielding conditions has been carried out for anti‐plane shear. Asymptotic expansion method is used to construct the solutions for the region near the crack line. Exact solutions for the distribution of strain on the crack line within the primary active plastic zone is obtained. It is shown that the solution reduces to the correct asymptotic form as the crack speed approaches zero (quasi‐static) for any point on the crack line. The results are used to discuss the applicability of quasi‐static solutions to moving steady‐state situations. It is found that if the crack propagation speed is less than 0.1 of the shear wave speed, the quasi‐static solutions can be accurately approximated for the steady state solutions.     

S-wave tracing technique to investigate the damage and failure behavior of brittle materials subjected to shock loading

To reveal the intrinsic rules of dynamic damage and failure behavior for brittle materials subjected to shock loading, a so-called transversal shear wave tracing technique (SWT) is proposed and discussed in detail in the present paper based on the wave propagation theory and the combined compression–shear impact technique. The idea of the SWT method is to diagnose the material state at real time by measuring the propagation features of loading and unloading shear waves. By using SWT technique, the preliminary experimental results of fiber-reinforced cement (FCEM) with electromagnetic particle velocity gauges embedded in the sample at different locations are reported and analyzed. By tracing the shear wave propagation, it is found that the amplitude and speed of unloading shear wave (S⁻) are related to the damage degree of the material. Particularly, S⁻ vanishes when impact velocities exceed 197 m/s, which discloses clearly a transition point from damage state to failure state of FCEM. It is significant for there is no obvious cusp to indicate such transition on the Hugoniot of FCEM. Some further studies are needed for understanding and developing of SWT method.     

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. Symposium on Physics and Scaling in Fracture held during the ICF11 (2005) in Turin.     

On the dynamic propagation of an anti-plane shear crack in a functionally graded piezoelectric strip

Summary. In this paper, a finite crack propagating at constant speed in a functionally graded piezoelectric material (FGPM) is studied. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential gradients along the thickness of the strip, and that the strip is under anti-plane shear mechanical and in-plane electrical loads. The analysis is conducted on the electrically unified (natural) crack boundary condition, which is related to the ellipsoidal crack parameters. By using the Fourier transform, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, crack propagation speed, electric field, FGPM gradation, crack length, and electromechanical coupling coefficient. It reveals that there are considerable differences between traditional electric crack models and the present unified crack model.     

Elastic Wave Propagation in a Class of Cracked,Functionally Graded Materials by BIEM

Elastic wave propagation in cracked, functionally graded materials (FGM) with elastic parameters that are exponential functions of a single spatial co-ordinate is studied in this work. Conditions of plane strain are assumed to hold as the material is swept by time-harmonic, incident waves. The FGM has a fixed Poisson’s ratio of 0.25, while both shear modulus and density profiles vary proportionally to each other. More specifically, the shear modulus of the FGM is given as μ (x)=μ ₀ exp (2ax ₂), where μ ₀ is a reference value for what is considered to be the isotropic, homogeneous material background. The method of solution is the boundary integral equation method (BIEM), an essential component of which is the Green’s function for the infinite inhomogeneous plane. This solution is derived here in closed-form, along with its spatial derivatives and the asymptotic form for small argument, using functional transformation methods. Finally, a non-hypersingular, traction-type BIEM is developed employing quadratic boundary elements, supplemented with special edge-type elements for handling crack tips. The proposed methodology is first validated against benchmark problems and then used to study wave scattering around a crack in an infinitely extending FGM under incident, time-harmonic pressure (P) and vertically polarized shear (SV) waves. The parametric study demonstrates that both far field displacements and near field stress intensity factors at the crack-tips are sensitive to this type of inhomogeneity, as gauged against results obtained for the reference homogeneous material case     

Stability of dynamically propagating cracks in brittle materials

Summary Dynamic crack propagation and bifurcation phenomena are investigated analytically by utilizing the strain energy density fracture criterion in the framework of catastrophe theory. The effect of biaxial stress, loading imperfections (mixed-mode loading), Poisson's ratio, state of stress as well as crack tip propagation speed on the crack path directional stability is analyzed. Special crack path stability charts for (un)stably propagating cracks are obtained, and their connection with the experimentally recorded crack tip stress field is addressed. It is shown that a slight change of the normal stress acting parallel to a crack at its tip (crack-parallel stress) may be able to affect the crack surface roughening and/or branching velocity considerably. It is also indicated that under small tensile crack-parallel stress, the crack propagation is stable only when the crack propagation speed is less than about 30% of the relevant shear wave speed. The crack becomes unstable, and its surfaces roughen severely at a higher speed, and the crack bifurcates at the highest propagation speed, some 45% of the shear wave speed. It is suggested that superimposing mode-II (shear) loading will enhance the dynamic crack path stability while increasing crack propagation speed will reduce the stability of crack propagation. It is expected that under compressive crack-parallel stress no crack surface roughening will occur before the crack stably bifurcates.     

Analysis of a rate-dependent cohesive model for dynamic crack propagation

The effect of including rate-dependence in the cohesive zone modeling of steady-state and transient dynamic crack propagation is analyzed. Spontaneous crack propagation simulations are performed using a spectral form of the elastodynamic boundary integral equations, while the solution to the steady-state problem is obtained by solving the governing Cauchy singular equation on the crack plane. The steady-state analysis shows that the existing techniques for solving the Cauchy singular integral equation are not suitable. A solution technique for the underlying Riemann-Hilbert problem for the chosen rate and damage-dependent cohesive law is presented. Under spontaneous propagation conditions, quasi-steady-state speeds slower than the theoretically predicted shear wave speed are possible. Results also show that, due to the dissipation of energy inside the cohesive zone, the energy required for crack propagation increases with the crack speed.     

Quasi-brittle dynamic fracture with damping due to crack edge failure zones

A model for dynamic crack propagation involving quasi-brittle fracture is studied in which adjacent material points in crack edge failure zones do not break completely apart instantaneously, but first undergo relative motion resisted by relative velocity-dependent stresses. An exact analysis for such a crack extending in an unbounded elastic plane under uniform shear and tension stresses indicates that the stress singularity orders are damped below the brittle fracture square-root value, and vanish at the Rayleigh wave speed. The power generated in the failure zones is also damped below the brittle fracture values, but the effect is not order-of-magnitude. Indeed, for small zone/crack size ratios, the approximate brittle fracture value can be spread out over the zone. In this case, however, the stresses are only weakly singular.     

Fatigue crack propagation along interfaces of selective laser melting steel hybrid parts

Selective laser melting (SLM) is an emerging additive manufacturing technology, capable of producing complex geometry components. The current work studied both the effect of substrate material and mean stress on the fatigue crack growth behaviour along interfaces of bi‐material specimens, substrate, and part by SLM. Fatigue tests were carried out in agreement with ASTM E647 standard, using 6‐mm‐thick compact specimens. The substrate steel has only a negligible effect both on the fatigue crack propagation rate and on the crack path. The failure occurs in the material additively manufactured by SLM, near the interface. The mean stress produced only a reduced influence on the fatigue crack propagation rate in the Paris regime. For larger values of ΔK, where K_max approaches K_Ic, a significant influence of the mean stress was observed. In spite of nondetection of crack closure, the application of overloads promoted significant fatigue crack retardation, quite similar for both substrate materials, probably due to the crack bifurcation during the overload.     

Analysis of a rate-dependent cohesive model for dynamic crack propagation

The effect of including rate-dependence in the cohesive zone modeling of steady-state and transient dynamic crack propagation is analyzed. Spontaneous crack propagation simulations are performed using a spectral form of the elastodynamic boundary integral equations, while the solution to the steady-state problem is obtained by solving the governing Cauchy singular equation on the crack plane. The steady-state analysis shows that the existing techniques for solving the Cauchy singular integral equation are not suitable. A solution technique for the underlying Riemann–Hilbert problem for the chosen rate and damage-dependent cohesive law is presented. Under spontaneous propagation conditions, quasi-steady-state speeds slower than the theoretically predicted shear wave speed are possible. Results also show that, due to the dissipation of energy inside the cohesive zone, the energy required for crack propagation increases with the crack speed.     

宏微观双尺度运动裂纹模型面内拉伸下的解析解*

研究裂纹动态扩展中宏微观因素相互作用机制与微观裂尖区的钝化效应。平面拉伸状态下,宏观主裂纹以恒定速度运动。通过一个介观约束应力过渡区,将宏观主裂纹与微观裂尖区相连接,由此建立了一个宏微观双尺度运动裂纹模型。应用弹性动力学与复变函数理论,分别在宏观与微观尺度下对该模型进行解析求解,获得了解析解。通过裂纹张开位移从宏观到微观的连续性条件与宏微观应力场协调条件,将两个不同尺度下的解相耦合,获得了计算宏微观损伤区特征长度的显式表达式。研究表明,运动裂纹的宏观应力场仍具有通常的r&;#61485;1/2的奇异性。由于微观裂尖的钝化,微观应力场奇异性的阶次有所降低,与宏观应力场相比具有弱奇异性。双尺度运动裂纹模型中,可允许裂纹运动速度达到剪切波速,解除了经典运动裂纹理论中裂纹速度不能超过Rayleigh波速的限制。数值结果表明,介观损伤过渡区与裂尖微观损伤区尺寸,及裂纹张开位移等,与裂纹运动速度、材料性质、约束应力比、裂尖钝化角度等因素有关。