Quasi-static and dynamic fracture of graphite/epoxy composites: An optical study of loading-rate effects

Strain-rate effects on fracture behavior of unidirectional composite materials are studied. Single-edge notched multi-layered unidirectional graphite composites (T800/3900-2) are investigated to examine fracture responses under static and dynamic loading conditions using a digital speckle correlation method. The fracture parameters for growing cracks are extracted as a function of fiber orientation. A 2D digital image correlation (DIC) method is used to obtain time-resolved full-field in-plane surface displacements when specimens are subjected to quasi-static and impact loading. Stress intensity factor and crack extension histories for pure mode-I and mixed mode cases are extracted from the full-field displacements. When compared to the dynamic stress intensity factors at crack initiation, the static values are found to be consistently lower. The stress intensity factor histories exhibit a monotonic reduction under dynamic loading conditions whereas an increasing trend is seen after crack initiation under quasi-static loading cases. This is potentially due to dominant crack face fiber bridging effects in the latter cases.     

用圆孔内单边裂纹平台巴西圆盘和实验-数值-解析法确定砂岩的动态起裂和扩展韧度

在I型(张开型)动态断裂实验中,利用大直径(?100 mm)分离式霍普金森压杆径向冲击圆孔内单边裂纹平台巴西圆盘试样。考虑了材料惯性效应和裂纹扩展速度对动态应力强度因子的影响,用实验-数值-解析法确定了高加载率和高裂纹扩展速度情况下,砂岩的动态起裂韧度和动态扩展韧度。由动态实验获取试样的动荷载历程,采用裂纹扩展计(Crack Propagation Gauge,CPG)测定试样断裂时刻和裂纹扩展速度,获得裂纹扩展速度对应的普适函数值。然后将动荷载历程带入到有限元软件中进行动态数值模拟,求出静止裂纹的动态应力强度因子历程,再用普适函数值对其进行近似修正。最后根据试样的起裂时刻和穿过CPG中点的时刻,由相应的动态应力强度因子历程分别确定砂岩的动态起裂和动态扩展韧度,它们分别随动态加载率和裂纹扩展速度的提高而增加。     

Rate effect of concrete strength under initial static loading

The dynamic strength enhancement of concrete under different initial static loadings is investigated in this paper. The mechanism of the influence of the inertia effect on the dynamic strength of concrete is discussed first by analyzing a single dimension of freedom system, which shows that the inertia effect only influences the dynamic loading part. The dynamic strength of concrete at initial static loading is calculated assuming a single crack model with consideration of free water viscosity using dynamic fracture mechanics. The dynamic stress intensity factor of the crack under linearly increasing loading on the basis of an initial static loading is achieved. The relationship between the dynamic strength increase factor and the static stress level is obtained. The comparison between the results by the model proposed in this paper and those by experiments in some available references indicates a good agreement.     

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Summary The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.     

Dynamic stress intensity factor due to concentrated loads on a propagating semi-infinite crack in orthotropic materials

The elastodynamic response of an infinite orthotropic material with a semi-infinite crack propagating at constant speed under the action of concentrated loads on the crack faces is examined. Solution for the stress intensity factor history around the crack tip is found for the loading modes I and II. Laplace and Fourier transforms along with the Wiener-Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed which lead to a closed-form solution of the dynamic stress intensity factor. It is found that the stress intensity factor for the propagating crack is proportional to the stress intensity factor for a stationary crack by a factor similar to the universal function k(v) from the isotropic case. Results are presented for orthotropic materials as well as for the isotropic case.     

含半椭圆表面裂纹圆柱壳体的三维热弹性动态断裂

研究了含轴向半椭圆表面裂纹的圆柱壳体在热应力与冲击载荷作用下的动态断裂情况,并应用所研制的三维动态断裂有限元程序进行了大规模的数值计算,确定了圆柱壳体的三维温度分布及热-力耦合下的动态应力强度因子,所得结果在一定程度上揭示了热-力作用下圆柱壳体的边界表面、裂纹面、物质惯性和弹性波的相互作用在结构动态断裂中的重要性。     

Theoretical simulations of a propagating crack subjected to in-plane stress wave loading by caustic method

The optical method of caustics for measuring the dynamic stress intensity factor in a transient process is investigated in this study. The transient full-field solutions of a propagating crack contained in an infinite medium subjected to step-stress wave and ramp-stress wave loadings are used to establish the exact equations of the initial and caustic curves. The results of the stress intensity factor obtained from the caustic method are compared with theoretical predictions and some experiments. The results demonstrate that a significant deviation can occur in the determination of the dynamic stress intensity factor from shadow spot measurements. The factors, such as screen distance, magnitude of loading, crack speed and rising time which can influence the accuracy of the experimental measurements are discussed in detail. In addition, the valid region of the dynamic stress singular field for the propagating crack is discussed in detail and it gives a better understanding of the appropriate region of measurements for investigators. This revised version was published online in July 2006 with corrections to the Cover Date.     

Environmental dynamic fatigue crack propagation in high density polyethylene: an empirical modelling approach

Part of a programme to study environmental dynamic fatigue crack propagation in engineering thermoplastics is presented. High density polyethylene has been studied in terms of its stress cracking properties under dynamic loading conditions in detergent. This work complements previous investigations on stress cracking in detergent under static loading. The stress cracking was found to vary according to loading conditions and in conclusion, a dependence of crack growth rate on test frequency, amplitude and level of stress intensity factor is reported. An emperical model describing environmental fatigue crack propagation is proposed which adequately represents the experimental results of high density polyethylene, Nylon 66 and which, it is suggested, may be suitable for use with other polymers.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

Experimental Determination of Dynamic Crack Initiation and Propagation Fracture Toughness in Thin Aluminum Sheets

An experimental investigation was undertaken to characterize the dynamic fracture characteristics of 2024-T3 aluminum thin sheets ranging in thickness from 1.63–2.54 mm. Specifically, the critical dynamic stress intensity factor Kdc was determined over a wide range of loading rates ( expressed as the time rate of change of the stress intensity factor KdI ) using both a servo- hydraulic loading frame and a split Hopkinson bar in tension. In addition, the dynamic crack propagation toughness, KD, was measured as a function of crack tip speed using high sensitivity strain gages. A dramatic increase in both Kdc and KD was observed with increasing loading rate and crack tip speed, respectively. These relations were found to be independent of specimen thickness over the range of 1.5 to 2.5 mm.     

Application of fracture mechanics to plastics deformed at high strain-rates

The effects of wall thickness and tube length on the fracture strength of thin walled cylindrical specimens of poly(methylmethacrylate) containing artificial flaws is examined. Data obtained are compared with that predicted by fracture mechanics theory developed for quasi-static conditions. Special attention is given to corrections for finite width and bending effects under dynamic loading conditions. It is concluded that the fracture mechanics relationships derived for quasi-static conditions, with the exception of the bending correction factor, are applicable to the dynamic situation with reasonable accuracy. It is further concluded that, for accurate analysis, the effect of dynamic loading on the stress field, and possibly the stress intensity factor, must be taken into consideration.     

Dynamic stress intensity factor for an unbounded plate having collinear cracks

The steady-state vibration of an infinite plate with collinear cracks is considered for low frequency cyclic loading. The formulation of the mixed boundary value problem leads to a dual trigonometric series. The Schwinger's method gives an automatic perturbation scheme. The dynamic stress intensity factor is found to be higher than the corresponding static one. The inertial effect on the stress intensity factor becomes significant only when the frequency of the external load is close to that of the shear wave.     

Dynamic crack propagation behaviour of rubber-toughened poly(methyl methacrylate)

Dynamic crack propagation has been studied in detail for a series of transparent rubber-toughened samples of poly(methyl methacrylate) using a combination of high-speed photography and the optical method of transmitted caustics. The dynamic stress intensity factor has been measured as a function of rubber content, crack length and loading rate. The dynamic stress intensity factor is found to increase significantly as the rubber content increases, which is consistent with the improvement in impact behaviour found on the addition of rubber particles. It is proposed that the toughening takes place through crack tip blunting caused by localized shear yielding induced by the presence of the rubber particles.     

Anti-plane transient analysis of planes with multiple cracks

In this paper, the transient dynamic stress intensity factor is determined for multiple curved cracks under impact loading. The dislocation method has rarely been applied to the problems involving dynamic loading. The transient response of Volterra-type dislocation in a plane is obtained by means of the Cagniard-de Hoop method. The distributed dislocation technique is used to construct integral equations for an infinite isotropic plane weakened by cracks. These equations are of Cauchy singular type at the location of dislocation which are solved numerically to obtain the dislocation density on the faces of the cracks. The dislocation densities are employed to determine stress intensity factors for multiple smooth cracks. Numerical results are obtained to validate the formulation and illustrate its capabilities.     

Investigations on the stress intensity factor field for unstable dynamic crack growth

In this study, the unstable dynamic crack propagation due to static loading in an elastic material is analyzed for both antiplane and inplane conditions. Of particular concern is the investigation of limitations on the assumption that the stress intensity factor field is fully established over a region of given size near the tip of a growing crack. The transient analysis of the stress for a material particle at a small fixed distance from the moving crack tip is examined in detail. Some estimations are made of the time required for the stress at a point near the moving crack tip to reach the value it would have if the stress field were actually given by the near tip stress intensity factor field. In addition, a simple formulation obtained from the equivalent static problem is proposed which can be used as a good approximation to the associated complicated dynamic transient problem.     

On the dynamic crack propagation in an interface with spatially varying elastic properties

This article provides a comprehensive theoretical treatment of a finite crack propagating in an interfacial layer with spatially varying elastic properties under antiplane loading condition. The theoretical formulations governing the steady state solution are based upon the use of an integral transform technique. The resulting dynamic stress intensity factor of the propagating cracks is obtained by solving the appropriate singular integral equations, using Chebyshev polynomials, for different inhomogeneous materials. Numerical examples are provided to verify the technique and to show the effect of the thickness of the interfacial layer and the material properties upon the dynamic stress intensity factor of the crack and the associated singularity transition.     

A solution method for finding dynamic stress intensity factors for arbitrarily oriented cracks in transversely isotropic materials

The transient elastodynamic response of a transversely isotropic material containing a semi-infinite crack under uniform impact loading on the faces is examined. The crack lies in a principle plane of the material, but the crack front does not coincide with a principle direction. Rather, the crack front is at an angle to a principle direction and thus the problem becomes more three-dimensional in nature. Three loading modes are considered, i.e., opening, in-plane shear and anti-plane shear. The solutions for the stress intensity factor history around the crack tip are found. Laplace and Fourier transforms together with the Wiener-Hopf technique are employed to solve the equations of motion directly. The asymptotic expression of stress near the crack tip leads to a closed-form solution for the dynamic stress intensity factor for each loading mode. It is found that the stress intensity factors are proportional to the square root of time as expected. Results given here converge to known solutions in transversely isotropic materials with a crack oriented along a principle direction and isotropic materials as special cases. The results of this analysis are used to find approximate strain energy release rates for dynamically loaded penny shaped cracks.     

Time domain boundary element method for dynamic stress intensity factor computations

This paper presents a procedure for transient dynamic stress intensity factor computations using traction singular quarter-point boundary elements in combination with the direct time domain formulation of the Boundary Element Method. The stress intensity factors are computed directly from the traction nodal values at the crack tip. Several examples of finite cracks in finite domains under mode-I and mixed mode dynamic loading conditions are presented. The computed stress intensity factors are represented versus time and compared with those obtained by other authors using different methods. The agreement is very good. The results are reliable and little mesh dependent. These facts allow for the analysis of dynamic crack problems with simple boundary discretizations. The versatile procedure presented can be easily applied to problems with complex geometry which include one or several cracks.     

裂纹止裂技术及裂纹动态扩展规律研究

为了对动态荷载作用下水泥粉煤灰砂浆的裂缝动态扩展行为进行研究,提出了一种大尺寸带V型底边的半圆边裂纹(SECVB)试件,其V形底部具有止裂功能。SECVB试件的V形底部设计为180°,150°和120°三个角度。采用落锤冲击装置进行了冲击试验,并使用裂纹扩展计(CPG)用于测量裂纹扩展的相关参数。利用有限差分程序AUTODYN对裂纹扩展行为进行了数值模拟,并用有限元程序ABAQUS计算了裂纹的动态应力强度因子(DSIF);根据CPG测量的裂纹萌生时间和扩展时间来确定临界应力强度因子。试验和数值模拟结果表明,SECVB试件适合于研究动态荷载作用下水泥粉煤灰砂浆的裂纹扩展行为和止裂行为。在裂纹扩展过程中,裂纹可能在一段时间内止裂,并且裂纹在起始时刻的断裂韧度高于裂纹扩展时的断裂韧度。     

Dynamic antiplane behaviour of interacting cracks in a piezoelectric medium

In this article, we examine the dynamic interaction between two cracks in a piezoelectric medium under incident antiplane shear wave loading. The theoretical formulations governing the steady-state problem are based upon the use of integral transform techniques and a self-consistent iterative method. The resulting dynamic stress intensity factors at the interacting cracks are obtained by solving the appropriate singular integral equations using Chebyshev Polynomials at different loading frequencies. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constants of the material and the frequency of the incident wave upon the dynamic stress intensity factor of the cracks. This revised version was published online in August 2006 with corrections to the Cover Date.