Fractal dimensions for fatigue fracture surfaces performed on micro- and meso-scale levels

A methodology was developed to use digital photograph fracture for calculating the integral fractal dimension and spectrum fractal dimensions for two perpendicular directions. The fractal dimension value, which was discovered by the variation of the pixel size up until the fractal dimension became unchangeable, i.e., where d _\min,d _\max are the borders of the scale level used. The effectiveness of the methodology performed showed the basis of the fracture surface analyses for Al-based alloys BS L65 and 2024 T351 (analogous to Al alloy D16T), specimens of which were tested by pull-push with imitation fretting corrosion damage of the sur- face and with four-point bending after the laser peening, respectively.     

Fracture toughness and crack morphology in indentation fracture of brittle materials

The relationship between the indentation fracture toughness, K _c, and the fractal dimension of the crack, D, has been examined on the indentation-fractured specimens of SiC and AIN ceramics, a soda-lime glass and a WC-8%Co hard metal. A theoretical analysis of the crack morphology based on a fractal geometry model was then made to correlate the fractal dimension of the crack, D, with the fracture toughness, K _IC, in brittle materials. The fractal dimension of the indentation crack, D, was found to be in the range 1.024–1.145 in brittle materials in this study. The indentation fracture toughness, K _c, increased with increasing fractal dimension, D, of the crack in these materials. According to the present analysis, the fracture toughness, K _IC, can be expressed as the following function of the fractal dimension of the crack, D, such that $$In K_{IC} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\{ In[2\Gamma E/(1 - \nu ^2 )] - (D - 1)In r_L \}$$ Where Γ is the work done in creating a unit crack surface, E is Young's modulus, v is Poisson's ratio, and r _L is r _min/r _max, the ratio of the lower limit, r _min, to the upper limit, r _max, of the scale length, r, between which the crack exhibits a fractal nature (r _min ?r?r _max). The experimental data (except for WC-8%Co hard metal) obtained in this study and by other investigators have been fitted to the above equation. The factors which affect the prediction of the value of K _IC from the above equation have been discussed.     

Correlation between crack tortuosity and fracture toughness in cementitious material

This paper proposes a new technique for the evaluation of fractal dimension (D) of fracture surface and a quantitative correlation between D and fracture toughness of cementitious materials. The experimental program has been performed on compact tension (CT) specimens (600 × 525 × 125 mm) with three different aggregate sizes (d _max=4.7 mm, 18.8 mm and 37.5 mm). The fractal geometry concept is utilized in the evaluation of fracture surface roughness. To avoid indirect or destructive experimental procedures that are prohibitively laborious and time consuming, a new non-destructive technique is presented. Results of the analysis indicate that the concept of fractal geometry provides a useful tool in the fracture surface characterization. The results also suggest that the fracture toughness can be correlated with the fractal dimension of fracture surface.     

Fractal dimension of metallic fracture surface

In this study, a complete method of determination of the fractal dimension for fracture surfaces of ferrous alloys has been proposed. This dimension is determined for the vertical profile obtained by the profile technique cross-section. The image of the profile, seen through the microscope coupled with a camera, is recorded in a computer, where numerical processing is performed. For calculation of the same fractal dimension, the fd3 program has been used, which is available through the Internet. The essential element of the method is optimisation concerning microscopic magnification (scale of a length), resolution of the recorded image and selection of the grey level threshold at binarization. The tests for the stability of discretization, which enable minimization of the error of the measurement, have also been carried out. These tests consist in checking the difference in fractal dimensions for the same profile obtained in two different methods of contouring as well as the difference between capacitive, informative and correlative dimensions. In both cases, too big difference suggests that the determined dimension is not reliable. This method allows determination of the fractal dimension with an absolute accuracy of 0.05 in non-dimensional units. The method has been employed in many studies. In this paper the following tests have been presented: a “fractal map” of the fracture surface was made, an influence of the mechanical notch radius in a compact specimen on the fractal dimension of the fracture surface, an influence of the distortion rate on the fractal dimension, an effect of fatigue crack propagation rate on the fractal dimension and influence of the stress-intensity factor on the fractal dimension of the fracture surface. The following materials were examined: Armco iron, P355N steel and 41Cr4 steel in different states after the heat treatment. The measurements have been made for the specimens of the compact type. There was considered an influence of location of the place of measurement on the fractal dimension being determined. The dimension was determined on the profiles lying longwise and crosswise the crack propagation direction. It has been found that the fractal dimension of the fracture surface does not depend on a place of measurement. This suggests, among other things, that a distinction between the places, which were created under conditions of the plane stress, and the places, which were created under conditions of the plane strain state, cannot be made with the help of the fractal dimension. When testing an influence of the radius of the mechanical tip notch on the fractal dimension of a fracture surface, this dimension was determined in the places located at different distances from the tip of the mechanical notch. With respect to the radii up to 1.0 mm, no significant differences in fractal dimensions have been found. The fractal dimensions of the fracture surface for all examined materials were practically the same and they ranged from 2.02 to 2.10. However in some ranges of da/dN rate the dimension was changing inversely proportional to da/dN. Obtained results confirm that fractal dimension do not depend on the investigated material.     

Stochastic and fractal analysis of fracture trajectories

Analyses of fracture trajectories are used to investigate structures that fall between “micro” and “macro” scales. It was shown that fracture trajectories belong to the class of nonstationary processes. It was also found that correlation distance, which may be related to a characteristic size of a fracture process, increases with crack length. An assemblage of crack trajectory processes may be considered as a diffusive process. Chudnovsky [Lectures on Fracture, CWRU, Cleveland, 1981–1985] introduced a “crack diffusion coefficient” d which reflects the ability of the material to deviate the crack trajectory from the most energetically efficient path and thus links the material toughness to its structure. For the set of fracture trajectories in AISI 304 steel d was found to be equal to 1.04 microns. The fractal dimension D for the same set of trajectories was found to be 1.133.     

Study on wood fracture parallel to the grains based on fractal geometry

The fracture toughness (K _IC ) parallel to the grains of five kinds of wood was tested by compact tension specimen and the profile contour analysis method was employed to measure fractal dimensions D _s of their fracture surfaces. The results show that fracture toughness parallel to the grains of various woods is different because of their textural diversity and such differences are also shown on the morphology of fracture surfaces. Furthermore, the fractal dimension D _s and fracture toughness ${K_{IC}^{TL} }$ parallel to the grains have evident direct proportional relation, and this helps to reveal the inherent relationship between fracture toughness of wood and its microstructure.     

Fractal effects of crack propagation on dynamic stress intensity factors and crack velocities

Crack extension paths are often irregular, producing rough fracture surfaces which have a fractal geometry. In this paper, crack tip motion along a fractal crack trace is analysed. A fractal kinking model of the crack extension path is established to describe irregular crack growth. A formula is derived to describe the effects of fractal crack propagation on the dynamic stress intensity factor and on crack velocity. The ratio of the dynamic stress intensity factor to the applied stress intensity factor K(L(D, t), V)/K(L(t), 0), is a function of apparent crack velocity V_o, microstructure parameter d/a (grain size/crack increment step length), fractal dimension D, and fractal kinking angle of crack extension path . For fractal crack propagation, the apparent (or measured) crack velocity V_o, cannot approach the Rayleigh wave speed Cr. Why V_o is significantly lower than Cr in dynamic fracture experiments can be explained by the effects of fractal crack propagation. The dynamic stress intensity factor and apparent crack velocity are strongly affected by the microstructure parameter (d/a), fractal dimension D, and fractal kinking angle of crack extension path . This is in good agreement with experimental findings.     

Extremum and variational principles for elastic and inelastic media with fractal geometries

This paper further continues the recently begun extension of continuum mechanics and thermodynamics to fractal porous media which are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution lengthscale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the global forms of governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. Here we first generalize the principles of virtual work, virtual displacement and virtual stresses, which in turn allow us to extend the minimum energy theorems of elasticity theory. Next, we generalize the extremum principles of elasto-plastic and rigid-plastic bodies. In all the cases, the derived relations depend explicitly on D, d and R, and, upon setting D = 3 and d = 2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries.     

Analysis of low temperature impact fracture data of thermoplastic polymers making use of an inverse methodology

In this work room and low temperature impact fracture data toughness of rubber modified polypropylene, polyethylene and rubber toughened polymethylmetacrylate have been assessed. In order to minimize the well-known dynamic effects a previously developed inverse methodology was used to treat force-time traces. Fracture parameters, such as K_IQ and J_C were estimated and the benefits of the inverse methodology were evaluated. The suitableness of energetic and force based toughness parameters for estimating a brittle to ductile transition was evaluated. The employment of the inverse methodology allowed us to infer the values of the crack tip loading rate, dK/dt, without the need of cushioning.     

Fractal modeling of the J-R curve and the influence of the rugged crack growth on the stable elastic-plastic fracture mechanics

In this paper, fractal geometry is used to modify the Griffith-Irwin-Orowan classical energy balance. Crack fractal geometry is introduced in the elastic-plastic fracture mechanics by means of the Eshelby-Rice J-integral and the influence of the ruggedness of the crack surface on the quasistatic crack growth is evaluated. It is shown that the rising of the J-R curve correlates to the topological ruggedness dimension of the crack surface. Results from fracture experiments are shown to be very well fitted with the proposed model, which is shown to be a unifying approach for fractal models currently used in fracture mechanics.     

Micropolar continuum mechanics of fractal media

This paper builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a technique based on a dimensional regularization, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established.     

Fractal structure of precursors and phase transformations in the sol-gel synthesis of nanoparticulate <Emphasis Type="Italic">M</Emphasis>-type barium hexaferrite

We have studied the influence of sol-gel process conditions on the fractal structure of precursors, the phase transformations during the synthesis process, and the morphology of the resulting barium hexaferrite particles and determined the optimal fractal structure of sol-gel derived precursors for the solid-state synthesis of barium hexaferrite. The activity of the precursors is shown to correlate with the fractal dimension of their first structural aggregation level. Nanoparticulate barium hexaferrite can be prepared in the form of anisometric particles: platelets (d _av ∼ 65 nm) and nanorods (d _av ∼ 30 nm, l _av ∼ 70 nm).     

Fracture surface topography and fracture mechanism in austenitic SUS316 steel plates fatigued by repeated bending

Fatigue experiments were carried out using the austenitic SUS316 steel plates (the average grain diameter is about 1.3 × 10⁻⁵ m) by repeated bending. The three-dimensional fatigue fracture surfaces were then reconstructed using stereo pairs of scanning electron micrographs by the stereo matching method. Striations were observed on the stage II fatigue fracture surface, while fine slip steps were found on the stage I fatigue fracture surface. The averaged value of the fractal dimension of stage I fracture surface was about 2.2 and was almost the same as that of stage II fracture surface when the fractal dimension was measured in the length scale range smaller than about one grain-boundary length (about 8 × 10⁻⁶ m). This may be attributed to the fact that both fracture surfaces were formed by the same mechanism, namely, slipping-off. According to the two-dimensional fractal analysis, both stage I and stage II fatigue fracture surfaces did not exhibit anisotropy in the length scale range of the fractal analysis smaller than about one grain-boundary length in the SUS316 steel. The fractal dimension of the fatigue fracture surface increased with decreasing the magnification of images when the maximum length scale of the fractal analysis was extended to the size of analyzed area. Magnification dependence of the fractal dimension was associated with large steps and ledges, which were not “typical” fractals.     

Shear-thickening behaviour of concentrated polymer dispersions under steady and oscillatory shear

The rheological behaviour of a 58 vol.% dispersion of styrene/acrylate particles in ethylene glycol was investigated using a plate-on-plate rheometer. Experimental results showed that the concentrated polymer dispersion exhibited a strong shear-thickening transition under both steady shear and dynamic oscillatory conditions. The low-frequency dynamic oscillatory behaviour could be reasonably interpreted in terms of the steady shear behaviour. Accordingly, the critical dynamic shear rate [(g)\dot]_{\textc_d} , \dot{\gamma }_{{{\text{c\_d}}}} , agreed well with the critical shear rate obtained in steady flow [(g)\dot]_{\textc_s} , \dot{\gamma }_{{{\text{c\_s}}}} , where [(g)\dot]_{\textc_d} \dot{\gamma }_{{{\text{c\_d}}}} was calculated as the maximum shear rate by the critical dynamic shear strain γ _c and the frequency ω, i.e. [(g)\dot]_{\textc_d} = wg_\textc . \dot{\gamma }_{{{\text{c\_d}}}} = \omega \gamma_{\text{c}} . However, during high-frequency dynamic oscillation, it was observed that the shear thickening occurred only when an apparent critical shear strain was reached, which could not be fully explained by the wall-slipping effect. Based on freeze fracture microscopic observations, the effect of the micro-sized flocculation of particles on the rheology of concentrated dispersions was also discussed.     

Evaluation of Tablet Formation of Different Lactoses by 3D Modeling and Fractal Analysis

ABSTRACT

The aim of this study was to use 3D modeling to differentiate not only among the four different types of lactose α-lactose monohydrate, spray-dried lactose, agglomerated lactose and lactose anhydrous but also between products from different manufacturers. Further “box-counting” fractal analysis of SEM images was done to gain additional information on tableting characteristics and tablet properties which can be found in the fractal structure. Twelve different materials from different manufacturers were analyzed for their powder-technological and physicochemical properties. They were tableted on an eccentric tableting machine at graded maximum relative densities and the recorded data, namely force, time, and displacement were analyzed by the 3D modeling technique. Tablet properties such as, elastic recovery, crushing force and morphology were analyzed. The results show that 3D modeling can precisely distinguish deformation behavior for different types of lactose and also for the same type of material produced with a slightly different technique. Furthermore, the results showed that the amorphous content of the lactose determined the compactibility of the material, which is due to a reversible exceeding of the glass transition temperature of the material. The three fractal dimensions D_BW (box surface dimension), D_WBW (pore/void box mass dimension), and D_BBW (box solid mass dimension) are capable of describing morphological differences in lactose materials. Multivariate regression analysis showed that the fractal surface structure of the lactose-based materials is strongly correlated to tableting characteristics and tablet properties. Especially with regards to 3D modeling, it was found that the fractal indices can describe the parameters time plasticity d, pressure plasticity e, and fast elastic decompression, which is the inverse of ω. In addition, the 3D parameters are able to describe the powder and tablet fractal indices. In conclusion, the 3D modeling is not only able to characterize the compression process but it can also provide information on the final tablet morphology.     

Fractal aspects of ductile and cleavage fracture surfaces

The aim of this paper is to evaluate and interpret the three-dimensional variational fractal dimension of a ductile and a cleavage fracture surface. The fracture surface is acquired by fracturing Charpy impact and static loaded specimens of a low alloy steel in ductile-to-brittle transition temperature range, and reconstructed by a stereoscopic technique. The three-dimensional variational method for measuring fractal dimension is improved by shifting algorithm and tested on the Takagi surface using the local fractal dimension. We find very good fractal behaviour in the ductile area, however, fractal characteristics in the cleavage area are less noticeable. The results are discussed in thermodynamical terms and promote the idea that fractal behaviour reflects the quasi-static process and that the fracture mechanisms in the ductile fracture are independent of strain rate (at least up to 10³ s⁻¹).     

Notched strength of carbon fibre/epoxy composite laminates with a circular hole

This paper improves the two stress fracture criteria proposed by Whitney and Nuismer (known as the point stress criterion and the average stress criterion) to predict the strength of composite laminates with a circular hole. In the point stress criterion, it is assumed that the failure occurs when the stress over some distance (d ₀) away from the notch is equal to or greater than the unnotched laminate strength. In the average stress criterion it is assumed that failure occurs when the average stress over some distance (a ₀) ahead of the notch equals the unnotched laminate strength. Both stress fracture criteria are two parameter models based on the unnotched strength (σ₀) and a characteristic dimension (d ₀ or a ₀). A simple relation is used for the characteristic length to improve the accuracy while evaluating the notched strength of carbon/epoxy composite laminates. The analytical results are compared well with the existing test results of AS4-carbon/948 A1 epoxy [0/90]_{4 s} and [0/±45/90]_{2 S} composite laminates with various hole diameters and specimen widths.     

Notched strength of carbon fibre/epoxy composite laminates with a circular hole

This paper improves the two stress fracture criteria proposed by Whitney and Nuismer (known as the point stress criterion and the average stress criterion) to predict the strength of composite laminates with a circular hole. In the point stress criterion, it is assumed that the failure occurs when the stress over some distance (d ₀) away from the notch is equal to or greater than the un-notched laminate strength. In the average stress criterion it is assumed that failure occurs when the average stress over some distance (a ₀) ahead of the notch equals the unnotched laminate strength. Both stress fracture criteria are two parameter models based on the unnotched strength (σ ₀) and a characteristic dimension (d ₀ ora ₀). A simple relation is used for the characteristic length to improve the accuracy while evaluating the notched strength of carbon/epoxy composite laminates. The analytical results are compared well with the existing test results of AS4-carbon/948 Al epoxy [0/90]_4s and [0/ ± 45/90]_2S composite laminates with various hole diameters and specimen widths.     

A mechanical model to predict elastic-plastic fracture toughness in high strength materials

A mechanical model of crack initiation and propagation, which is based on the actual mechanism of ductile fracture in high strength materials, is proposed. Assuming that a crack initiates when the equivalent stress at a distance ρ from the crack tip reaches a critical value $\text{}\text{?}$gsf, an equation for predicting fracture toughness J_IC is obtained. From comparison between the predicted values and the experimental results, it is found that the distance ρ corresponds to the spacing of micro-inclusions. The temperature dependence of fracture toughness J_IC estimated according to the derived equation is given in an Arrhenius form of equation and is nearly consistent with the experimental results.     

Correlation of fracture behavior in high pressure pipelines with axial flaws using constraint designed test specimens. Part II: 3-D effects on constraint

This study describes an extensive set of 3-D analyses conducted on conventional fracture specimens, including pin-loaded and clamped SE(T) specimens, and axially cracked pipes with varying crack configurations. The primary objective is to examine 3-D effects on the correlation of fracture behavior for the analyzed crack configurations using the J-Q methodology. An average measure of constraint over the crack front, as given by an average hydrostatic parameter, denoted Q_avg, is employed to replace the plane-strain measure of constraint, Q. Alternatively, a local measure of constraint evaluated at the mid-thickness region of the specimen, denoted Q_Z0, is also utilized. The analysis matrix considers 3-D numerical solutions for models of SE(T) fracture specimens with varying geometries (i.e., different crack depth to specimen width ratio, a/W, as well as different loading point distance, H) and test conditions (pin-loaded ends vs. clamped ends). The 3-D numerical models for the cracked pipes cover different crack depth to pipe wall thickness ratio, a/t, and a fixed crack depth to crack length ratio, a/c. The extensive 3-D numerical analyses presented here provide a representative set of solutions which provide further support for using constraint-designed SE(T) specimens in fracture assessments of pressurized pipes and cylindrical vessels.