Some propositions on caustics and an application to the biaxial-fracture problem

Some fundamental problems on the optical method of caustics, which is used to determine the value of the stress-intensity factor, are studied. The paper shows that the radiusr _o of the initial curve considerably affects the results and describes a method to eliminate this effect. This method is also applied to the biaxial-stress fracture problem. It is shown that the biaxial stress affects the fracture toughness of thin specimens of Plexiglas sheets and that the fatigue-crack-growth rate is increased due to the compressive stress along the crack line.Paper was presented at 1982 SESA/JSME Spring Meeting held in Oahu and Maui, HI on May 23–29, 1982.     

Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading

This paper describes the development and application of a general domain integral method to obtain J-values along crack fronts in three-dimensional configurations of isotropic, functionally graded materials (FGMs). The present work considers mode-I, linear-elastic response of cracked specimens subjected to thermomechanical loading, although the domain integral formulation accommodates elastic–plastic behavior in FGMs. Finite element solutions and domain integral J-values for a two-dimensional edge crack show good agreement with available analytical solutions for both tension loading and temperature gradients. A displacement correlation technique provides pointwise stress-intensity values along semi-elliptical surface cracks in FGMs for comparison with values derived from the proposed domain integral. Numerical implementation and mesh refinement issues to maintain path independent J-values are explored. The paper concludes with a parametric study that provides a set of stress-intensity factors for semi-elliptical surface cracks covering a practical range of crack sizes, aspect ratios and material property gradations under tension, bending and spatially-varying temperature loads.     

Phase-stepped ESPI and moiré interferometry for measuring stress-intensity factor andJ integral

Electronic-speckle-pattern interferometry and moiré interferometry have been used to calculateK ₁ andJ for compact tension specimens. Automated-fringe-pattern analysis enables the full-field of data to be used with the minimum of operator intervention. Measurements are shown to be accurate to within 10 percent. TheJ-measurement procedure employed could form the basis of an automatic-fault detection system.     

Dynamic fracture analysis by moiré interferometry

Dynamic moiré interferometry was used to measure separately theu- andv-displacement fields surrouding a rapidly propagating crack tip in Homalite-100 and 7075-T6 aluminum-alloy plates. These transient crack-tip-displacement data were then used to compute the dynamic stress-intensity factor and the remote stress component.J-integral values were also estimated using the static approximate procedure of Kanget al. This static analysis provided the correctJ when the contour integral was taken within 3 mm of the crack tip.Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WI on June 9–13.     

An improved experimental method for the determination of the critical stress intensity factor KIc of brittle materials

The critical stress intensity factor K_Ic is determined by a simple and accurate method, using small test specimens and a simple procedure in this paper.Single edge V-notched tension specimens made of PMMA are subjected to a load which is slowly increased until the crack begins to move from the notch tip. During the crack propagation event shadow patterns at the tip of the crack are recorded in a video recorder. Under these loading conditions, the creating real crack propagate slowly until the crack propagation velocity take an abrupt increase and the entire fracture of the specimen takes place. The stress intensity factor which correspond to the transition from the slow to fast crack speed, is the critical stress intensity factor K_Ic and it can be the fracture toughness of the material.The results are accurate and in good agreement with those values of K_Ic which are calculated by approximate theoretical expressions.The purpose of this paper is to introduce an improved, simple and accurate experimental method for the determination of fracture toughness of brittle materials.     

Characterization of crack-tip strain singularity in brittle-microcracking materials by means of the photoelastic-coating method

The strain fields ahead of crack tips in rock, mortar, and graphite were measured using a photoelastic coating method. A transparent ferroelectric ceramic was used as a coating material to observe the photoelastic fringe pattern. A coating plate of 110–150 μm thick was placed on single-edge-notch specimens, and photoelasticity experiments were conducted in three-point bending under a polarizing microscope. The results show that the principal-strain difference ahead of the crack tip is given by $$\Delta \in = \Delta \in _o [(J/\sigma _{ult} )/r]^m $$ whereσ _ult is the ultimate tensile strength,r is the distance from the crack tip, and9? _o andm are material constants. Based on this observation, the use of theJ _Ic concept in determining the fracture toughness of brittle-microcracking materials is discussed.     

On the determination of the å-K relationship for birefringent polymers

Methods of utilizing dynamic photoelasticity with fracture-type specimens to simultaneously determine the stress-intensity factorK _Iand the crack-tip velocity are reviewed. Problems associated with data analysis to obtainK _Ifrom isochromatic-fringe loops are discussed. Errors resulting from the use of static near-field equations in the method of analysis are considered and a correction method is developed. Finally, the invariance ofK _Im(the minimumK required to maintain a finite crack velocity) is noted and evidence is provided to indicate thatK _Imcan be treated as a material propertly.Paper was presented at 1978 SESA Spring Meeting held in Wichita, KS on May 14–19.     

Use of photoelasticity to determine orthotropicK I stress-intensity factor

A new experimental method of obtaining orthotropic stress-intensity factor,K _I , is presented. The orthotropic photoelasticity and orthotropic linear-elastic fracture-mechanics laws are combined. The combined set of equations is used along with half-fringe photoelasticity to determineK _I in a compact-tension specimen made of a transparent unidirectional fiberglass-epoxy material. The results are compared with finite-element-method solutions.     

Study of pure and mixed-mode fracture of a brittle material

In order to study the behavior of a crack in a linear-elastic material in plane mixed mode (modes I+II), a specimen's shape and loading have been specially adapted. The specimens are first precracked in mode I and then subjected to monotonic loading until instability is reached by an original device which makes it possible to control the nature of the mixed mode applied and which is adjustable from pure mode I to pure mode II. After the specimens are fractured, the lengths of the initial cracks and the kinking angles were measured in the plane-strain area. Then the stress-intensity factorsK _I andK _II at the moment when crack instability appeared were calculated. This made it possible to apply two criteria: maximum principal stress and maximum stressintensity factorK ₁ ^* at the onset of kinking. From comparing the calculated values with the experimental values we may note that there is good agreement with respect to the crackkinking directions. However, for the limit load values considerable divergences have been recorded which are analyzed.     

Elastic-plastic models of surface cracks in tensile panels

The surface crack is a common flaw in structures and vessels, and its elastic characterization has been studied extensively as reviewed in Ref. 1 and its references. Elastic-plastic fracture-mechanics (EPFM) technology can be used to characterize surface cracks in tough materials. Two EPFM aprameters are commonly used: the crack-tip-opening displacement (CTOD) and the three-dimensionalJ integral. This paper draws on a series of studies^2–11 at the National Bureau of Standards related to the development and verification of analyticalmodels for the calculation of EPFM parameters in surface-cracked tensile panels. The models previously verified for pipeline steel plates^4–7 are used to calculate the crack-mouth-opening displacement (CMOD) andJ for surface cracks in welded-steel specimens.     

Transition range drop tower J-R curve testing of A106 steel

Fracture-toughness properties should be measured in the laboratory at loading rates and temperatures similar to those expected in the application of interest. This is not usually the case because of the experimental difficulties involved. This report describes a method being used to obtainJ _Ic, J-R curves, andJ at cleavage for three-point-bend tests conducted at drop tower rates through the ductile to brittle transition regime of the ferritic A106 steel being tested. The major conclusion is that these tests can now be accomplished, though a high degree of expertise and considerable practical experience is necessary to obtain good test results. The steel tested here is quite rate dependent as shown both by tensile tests and fracture-toughness tests. A load elevation of 30 to 50 percent results in the drop tower 100 in./s (2.5 m/s) tests on this material in comparison with static tests when both tests are conducted on the ductile upper shelf. Nonetheless, for this materialJ _Ic andJ-R curves are not elevated by the loading rate. Looking at the elastic and plastic components ofJ one sees that theJ _EL increases with increased loading rate but also thatJ _PL decreases with loading rate. It is also demonstrated that for the high rate tests more crack extension is present at a given bend angle for the rapid tests than with the static tests. Paper was presented at the 1987 SEM Fall Conference on Experimental Mechanics held in Savannah, GA on October 25–28.     

双槽圆形截面管角裂纹应力强度因子

弯曲载荷作用下,双槽圆形截面管的角裂纹具有两个不同的奇异应力场和相应的应力强度因子,针对该异型薄壁管裂纹问题,提出了一种简单实用的应力强度因子求解方法。即利用守恒律,通过选取适当的三维积分路径,并结合初等力学的应力位移计算方法,显化了应力强度因子对J_2积分的贡献,建立了一个求解应力强度因子的方程。由于该方程不足以求解两个应力强度因子,利用材料力学平截面保持平面的变形假设,建立了应力强度因子之间的补充方程。将J_2积分与补充方程联立求解,既可得到弯曲载荷作用下双槽圆形截面管角裂纹的应力强度因子。对于其他异型薄壁管裂纹问题,该方法同样适用,计算过程简单。     

Measurement of energy release rates for delaminations in composite materials

Conventional measurements of energy release rates,G _I andG _II, for delaminations in composite materials, generally utilize loads, crack lengths and simple standard specimen geometries. In this work, a more widely applicable measurement method, using phase shifting moiré and the J integral, is presented. The experimental technique described requires only fringe-pattern information and the elastic constants for the measurements—thus it can be used when the standard methods are inapplicable. Using conventional double-cantilever beam and end-notched flexure specimens, the energy release rate has been measured simultaneously by the moiré method and the standard methods, with good agreement found between the two. This development will for the first time permit the experimental validation of new finite-element routines as they are developed.Paper was presented at the 1993 SEM Spring Conference on Experimental Mechanics held in Dearborn, MI on June 7–9.     

A bar impact tester for dynamic fracture testing of ceramics and ceramic composites

A bar impact test was developed to study the dynamic fracture responses of precracked ceramic bars, Al₂O₃ and 15/29-percent volume SiC_w/Al₂O₃. Crack-opening displacement was measured with a laser-interferometric displacement gage and was used to determine the crack velocity and the dynamic stress-intensity factorK _I ^dyn . The crack velocity andK _I ^dyn increased with increasing impact velocity while the dynamic-initiation fracture toughness,K _Id, did not vary consistently with increasing impact velocities.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas on June 8–11.     

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J₂J₃ class model, where J₂ and J₃ are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.     

Elastic-plastic analysis of biaxially loaded center-cracked plates

Center-cracked panels loaded in biaxial tension are examined in this paper. Calibration relations for the J integral and the Q constraint factor are presented for a Ramberg–Osgood power law hardening material under plane stress and plane strain loadings. Two cases are examined: an isolated crack and a periodic array of cracks both under biaxial loading conditions. The latter has previously been studied for plane stress conditions. A number of different J estimation schemes are proposed based on the remote load and displacement and their dependence on geometry, biaxiality, and material properties is discussed. The variation of constraint, as characterised by Q, is also presented for plane stress and plane strain conditions. Simple slip line field solutions are derived for perfectly plastic conditions and the resulting limit load solutions are compared with numerically determined values. Implications for failure of cracked plates under biaxial loading are discussed.     

The optimal dimensions of convective-radiating circular fins

The optimal dimensions of convective-radiating circular fins with variable profile, heat-transfer coefficient and thermal conductivity, as well as internal heat generation are obtained. A profile of the form y=(w/2) [1+(r _o/r) ⁿ ] is studied, while variation of thermal conductivity is of the form k=k _o[1+ɛ((T−T _∞)/ (T _b−T _∞)) ^m ]. The heat-transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h=K[(r−r _o)/(r _e−r _o)]^λ. The results for λ=0 to λ=1.9, and −0.4≤ɛ≤0.4, have been expressed by suitable dimensionless parameters. A correlation for the optimal dimensions of a constant and variable profile fins is presented in terms of reduced heat-transfer rate. It is found that a (quadratic) hyperbolic circular fin with n=2 gives an optimum performance. The effect of radiation on the fin performance is found to be considerable for fins operating at higher base temperatures, whereas the effect of variable thermal conductivity on the optimal dimensions is negligible for the variable profile fin. It is also observed, in general, that the optimal fin length and the optimal fin base thickness are greater when compared to constant fin thickness. Received on 22 February 1999     

Experimental study of path independence of theJ integral in an aluminum tensile panel

A newly devised video-optical experimental technique allows for the automated determination of the in-plane, plane-stress components of the infinitesimal deformation tensor at discrete locations over an area of interest in a loaded specimen. It was used to evaluate a square area enclosing the central crack in a 5052-H32 aluminum panel loaded in tension. These experimental strain-tensor values were then used to evaluate theJ integral assuming that the mechanical behavior of the plate material is approximated by deformation plasticity theory. TheJ integral was evaluated using two series of rectangular paths. The first was symmetrical about a horizontal axis passing through the notch length, using strain-tensor values at points to the left and right of a vertical axis passing through the center of the notch, with varying breadth and height. The second series were asymmetrical about either axis, with varying breadth and height. The latter paths served to evaluate the experimental error associated with each of the four quadrants of the plate. TheJ integral is found to be path independent within the range of experimental error associated with the video-optical experimental-measurement technique. An independent verification of theJ integral value is obtained using a calculation related to the crack-mouth-opening displacement (CMOD). It is found to agree well with the values obtained by this new procedure. Paper was presented at 1986 SEM Spring Conference on Experimental Mechanics held in New Orleans, LA on June 8–13.     

J resistance curves in aluminum SEN specimens using Moiré interferometry

TheU _y -displacement field obtained by white-light moiré interferometry were used to estimate the approximate far- and near-fieldJ-integral values associated with the subcritical crack growths in fatigue precracked 7075-T6 and blunt notched and fatigue precracked 2024-0 and 5052-H32 aluminum, single-edged notch (SEN) specimens. The initial phases of theJ-resistance curves for the somewhat brittle 7075-T6 and the two ductile 2024-0 and 5052-H32 aluminum SEN specimens are presented. Paper was presented at the 1987 SEM Spring Conference on Experimental Mechanics held in Houston, TX on June 14–19.