Eigenvalue method for computing size effect of cohesive cracks with residual stress,with application to kink-bands in composites

The previously developed eigenvalue method for computing the size effect of cohesive crack model is extended to the cohesive crack model with a finite residual stress. In this model, the structure size for which a specified relative length of kink-band corresponds to the maximum load is obtained as an eigenvalue of a homogeneous Fredholm integral equation. This new method is direct and much more efficient than the classical finite element approach in which the entire load-deflection history must be computed to obtain the maximum load. A secondary purpose of the paper is to apply the new method to the effect of structure size on the compressive strength of unidirectional fiber–polymer composites failing by propagation of kink-band with fiber microbuckling. The kink-band is simulated by a cohesive crack model with a linear compressive softening law and a finite residual stress. The simulation shows that the specimens tested have a negative–positive geometry, i.e., the energy release rate of the kink-band for a unit load first decreases but at a certain length of propagation begins to increase. Finally the effect of shape of the softening law of cohesive crack on the size effect curve is studied by using the new eigenvalue method. It is shown that, for a negative–positive geometry, the size effect on the peak load depends on the entire softening curve if the specimens is not too small.     

Effect of acid corrosion on crack propagation of concrete beams

The effect of acid corrosion on crack propagation of concrete beams was theoretically studied by the method of crack extension resistance curve. Based on this method, a calculation approach was proposed to determine fracture stress intensity factors in crack propagation of concrete beams. Loop iteration analysis was carried out to calculate maximum bearing capacity load, unstable crack toughness, resistance toughness curve, cohesive toughness curve and load–crack mouth opening displacement. Both bilinear and nonlinear softening traction–separation curves were adopted for each of these calculation parameters. The analysis results of each showed the effect of acid corrosion degrees. The influence of acid corrosion on fracture properties was discussed through the calculated results of cohesive toughness curves. These five kinds of simulated results were basically consistent, before the load attained the maximum value. However, with further crack propagation, cohesive toughness of nonlinear softening model was significantly larger than that of bilinear softening model, and the descending branch of P–CMOD curve by nonlinear law is higher than that by bilinear law. To validate the approach, tests of specimens under six different corrosion periods were experimentally studied, using three-point bending notched concrete beams soaked in sulphuric acid solution. The Double-K fracture parameters were investigated based on the test results, and load–crack mouth opening displacement curves for different acid conditions were obtained using synchronous sampling of a load sensor and clip-gauge. Numerical results by bilinear softening model showed a good correlation with the experimental ones.     

GENERALIZED SIZE EFFECT EQUATION FOR QUASIBRITTLE MATERIALS

Abstract— Structures made of quasibrittle materials show a marked decrease in strength as their size increases; this is the well known size effect on strength. This contribution proposes a general formula to predict with great accuracy the size effect of pre-cracked specimens. Such a formula provides the basis to determine experimentally the fracture parameters of a cohesive crack model from the measurement of the peak load in specimens with an initial crack. The formula is applicable whenever the fracture of the material can be described by a cohesive crack model with initial linear softening, as is generally accepted for concrete. Experimental evidence is presented showing that the formula is reasonably accurate also for rock.     

Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I. Mathematical Model and Scaling

The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the softening law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two influences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). Thus it is unnecessary to deal mathematically with a moving crack tip, which keeps the formulation simple because the compliance functions for the surface points of such an imagined preexisting unopened crack do not change as the actual front of the opened part of the cohesive crack advances. First the compliance formulation of the cohesive crack model is generalized for aging viscoelastic material behavior, using the elastic-viscoelastic analog (correspondence principle). The formulation is then enriched by a rate-dependent softening law based on the activation energy theory for the rate process of bond ruptures on the atomic level, which was recently proposed and validated for concrete but is also applicable to polymers, rocks and ceramics, and can be applied to ice if the nonlinear creep of ice is approximated by linear viscoelasticity. Some implications for the characteristic length, scaling and size effect are also discussed. The problems of numerical algorithm, size effect, roles of the different sources of time dependence and rate effect, and experimental verification are left for a subsequent companion paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of softening function on the cohesive crack fracture parameters of concrete CT specimen

The paper presents numerical study on the fracture parameters of concrete compact tension test specimens of different sizes using cohesive crack model. As softening function is the main ingredient of the cohesive crack model, seven numbers of widely used softening functions are incorporated in the model based on enhanced algorithm. It is found that the difference between the highest and the lowest peak loads obtained using various softening functions (except linear) is less than about 9%. The peak load predicted by the linear softening curve is about 16% larger than that of mean peak load predicted by other softening functions. The cohesive crack model with linear softening yields the fracture process zones lower by approximately 30–50% than those obtained by using the other softening relations for specimen size range 200–600 mm. The numerical results are further compared with a reference test result available in the literature. It is observed that some of softening relations (except linear) predict the experimentally obtained peak load up to 6% of accuracy whereas the linear softening curve overestimates it by about 30%. The numerically gained softening branch of load-displacement curves compare well with the experimental observation.     

Nonlinear fracture of cohesive materials

The cohesive crack is a useful model for describing a wide range of physical situations from polymers and ceramics to fiber and particle composite materials. When the cohesive zone length is of the order of the specimen size, the influence method—based on finite elements—may be used to solve the fracture problem. Here a brief outline of an enhanced algorithm for this method is given. For very large specimen sizes, an asymptotic analysis developed by the authors allows an accurate treatment of the cohesive zone and provides a powerful framework for theoretical developments. Some recent results for the zeroth order and first order asymptotic approaches are discussed, particularly the effective crack concept and the maximum load size effect. These methods are used to analyze the effect of the size and of the shape of the softening curve on the value at the peak load of several variables for three point bent notched beams. The results show, among other things, that for intermediate and very large sizes the size effect curves depend strongly on the shape of the softening curve, and that only the simultaneous use of asymptotic and influence methods may give an adequate estimate of the size effect in the intermediate range.     

Size effect on strength of laminate-foam sandwich plates: Finite element analysis with interface fracture

Recent three-point bend tests of size effect on the strength of geometrically scaled sandwich beams of three types – with no notches, and with notches at the upper or lower skin–foam interface, which were previously evaluated using simplified sandwich beam theory and equivalent linear elastic fracture mechanics, are now reanalyzed more accurately by finite elements. Zero-thickness interface elements with a softening cohesive law are used to model fractures at the skin–foam interface, in the fiber composite skins, and in the foam. The fracture energy and fracture process zone length of a shear crack in foam near the interface are deduced by fitting an analytical expression for size effect to the test data. Numerical simulations reveal that small-size specimens with notches just under the top skin develop plastic zones in the foam core near the edges of the loading platen, and that small-size specimens with notches just above the bottom skin develop distributed quasibrittle fracture in the foam core under tension. Both phenomena, though, are found to reduce the maximum load by less than 6%. Further it is shown that, in notch-less beams, the interface shear fracture is coupled with compression crushing of the fiber–polymer composite skin. For small specimens this mechanism is important because, when it is blocked in simulations, the maximum load increases. The size effect law for notch-less beams is calibrated such that beams of all sizes fail solely by interface shear fracture.     

Asymptotic analysis of a cohesive crack: 2. Influence of the softening curve

This paper presents a numerical method well suited to solve the integral equation governing the asymptotic behavior of a cohesive crack, and uses it to analyze the influence of the softening curve on the cracking response of large specimens. The analysis is performed with two main objectives in mind: (1) providing criteria to determine when a simplified linear elastic fracture mechanics (LEFM) approach can be applied, and (2) providing possible procedures of extracting information on the softening behavior from experimental data. The main conclusion is that the effective crack extension prior to peak is nearly determined by the length of the softening curve (the critical crack opening) and so is the deviation from LEFM. Furthermore, a simplified curve approach is proposed as an approximate alternative to solving the governing integral equation.     

Asymptotic stress intensity factor density profiles for smeared-tip method for cohesive fracture

The paper presents a computational approach and numerical data which facilitate the use of the smeared-tip method for cohesive fracture in large enough structures. In the recently developed K-version of the smeared tip method, the large-size asymptotic profile of the stress intensity factor density along a cohesive crack is considered as a material characteristic, which is uniquely related to the softening stress-displacement law of the cohesive crack. After reviewing the K-version, an accurate and efficient numerical algorithm for the computation of this asymptotic profile is presented. The algorithm is based on solving a singular Abel's integral equation. The profiles corresponding to various typical softening stress-displacement laws of the cohesive crack model are computed, tabulated and plotted. The profiles for a certain range of other typical softening laws can be approximately obtained by interpolation from the tables. Knowing the profile, one can obtain with the smeared-tip method an analytical expression for the large-size solution to fracture problems, including the first two asymptotic terms of the size effect law. Consequently, numerical solutions of the integral equations of the cohesive crack model as well as finite element simulations of the cohesive crack are made superfluous. However, when the fracture process zone is attached to a notch or to the body surface and the cohesive zone ends with a stress jump, the solution is expected to be accurate only for large-enough structures.     

Choice of standard fracture test for concrete and its statistical evaluation

The main characteristics of the cohesive (or fictitious) crack model, which is now generally accepted as the best simple fracture model for concrete, are (aside from tensile strength) the fracture energies G _F and G _f corresponding to the areas under the complete softening stress-separation curve and under the initial tangent of this curve. Although these are two independent fracture characteristics which both should be measured, the basic (level I) standard test is supposed to measure only one. First, it is argued that the level I test should measure G _f, for statistical reasons and because of relevance to prediction of maximum loads of structures. Second, various methods for measuring G _f (or the corresponding fracture toughness), including the size effect method, the Jenq-Shah method (TPFM), and the Guinea et al. method, are discussed. The last is clearly the most robust and optimal because: (1) it is based on the exact solution of the bilinear cohesive crack model and (2) necessitates nothing more than measurement of the maximum loads of notched specimens of one size, supplemented by tensile strength measurements. Since the identification of material fracture parameters from test data involves two random variables, f_t (tensile strength) and G _f, statistical regression should be applied. But regression is not feasible in the original Guinea et al.'s method. The present study proposes an improved version of Guinea et al.'s method which reduces the statistical problem to linear regression thanks to exploiting the systematic trend of size effect. This is made possible by noting that, according to the cohesive (or fictitious) crack model, the zero-size limit _N0 of nominal strength _N of a notched specimen is independent of F _f and thus can be easily calculated from the measured f_t. Then, the values of _N0 obtained from the measured f_t values, together with the measured _N-values of notched specimens, are used in statistical regression based on the exact size effect curve calculated in advance from the cohesive crack model for the chosen specimen geometry. This has several advantages: (1) the linear regression is the most robust statistical approach; (2) the difficult question of statistical correlation between measured f_t and the nominal strength of notched specimens is bypassed, by virtue of knowing the size effect trend; (3) the resulting coefficient of variation of mean G _f is very different and much more realistic than in the original version; (4) the coefficient of variation of the deviations of individual data from the regression line is very different from the coefficient of variation of individual notched test data and represents a much more realistic measure of scatter; and (5) possible accuracy improvements through the testing of notched specimens with different notch lengths and the same size, or notched specimens of different sizes, are in the regression setting straightforward. For engineering purposes where high accuracy is not needed, the simplest approach is the previously proposed zero-brittleness method, which can be regarded as a simplification of Guinea et al.' method. Finally, the errors of TPFM due to random variability of unloading-reloading properties from one concrete to another are quantitatively estimated.     

Modelling the fracture of concrete: the cohesive crack

This paper shows the ability of the cohesive crack model to predict reasonably well the behaviour of concrete specimens. To demonstrate this two very different examples are considered. The first is an engineering problem related to the breakage of precase concrete piles; it is shown that the cohesive crack model points to the relevant parameters and suggests ways to improve the behaviour of the concrete. The second example analyses the well known effect of size on the modulus of rupture when measured during the three point bending of beams. The predictions using a simple cohesive model with bilinear softening are very good. The present work shows how this model not only predicts accurately the maximum loads for different geometries and sizes but also is able to make reasonably good predictions of load and displacement at any instant throughout the test.     

A single-domain dual-boundary-element formulation incorporating a cohesive zone model for elastostatic cracks

A cracked elastostatic structure is artificially divided into subdomains of simpler topology such that the well-developed classic dual integral equations can be applied appropriately to each domain. Applying the continuity and equilibrium conditions along artificial boundaries and properties of the integral kernels a single-domain dual-boundary-integral equation formulation is derived for a cracked elastic structure. A cohesive zone model is used to model the crack tip processes and is coupled with the single-domain dual-boundary-integral equation formulation; the resulting nonlinear equations are solved using the iterative method of successive-over-relaxation. The constitutive law used for a crack includes three parts: a law relating cohesive force to crack displacement difference when a crack is opening, a characterization of tangential interaction between crack surfaces when the crack surfaces are in contact, and a maximum principal stress criterion of crack advance. Incorporation of local unloading effect of the cohesive zone material has enabled a simulation of fracture with initial damage, partial development of the failure process zone at structural instability and multiple crack interaction. Some of the features of the method are demonstrated by considering three examples. The first problem is a single-edge-cracked specimen that exhibits a snap-back instability. The second example is the development of wing cracks from an angled crack under compression. The last example demonstrates the capability to consider mixed-mode crack growth and interaction of cracks. Thus, the problem of crack growth has been reduced to the determination of the cohesive model for the fracture process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Reinterpretation of Karihaloo's size effect analysis for notched quasibrittle structures

Karihaloo recently published an analytical study of the size effect in concrete based on large-size asymptotic approximations of the cohesive crack model. From this analysis, he concluded that the nominal strength can be determined only for sizes above a certain lower bound, large enough to invalidate, at least for concrete, all the existing experimental methods based on size effect measurements, such as the size effect method of Baant or the general bilinear fit method of Planas, Guinea and Elices. The purpose of this paper is to show that this conclusion is misleading, and to explain why.     

Fracture Analysis of Concrete Structural Components Accounting for Tension Softening Effect

This paper presents methodologies for fracture analysis of concrete structural components with and without considering tension softening effect. Stress intensity factor (SIF) is computed by using analytical approach and finite element analysis. In the analytical approach, SIF accounting for tension softening effect has been obtained as the difference of SIF obtained using linear elastic fracture mechanics (LEFM) principles and SIF due to closing pressure. Superposition principle has been used by accounting for non-linearity in incremental form. SIF due to crack closing force applied on the effective crack face inside the process zone has been computed using Green's function approach. In finite element analysis, the domain integral method has been used for computation of SIF. The domain integral method is used to calculate the strain energy release rate and SIF when a crack grows. Numerical studies have been conducted on notched 3-point bending concrete specimen with and without considering the cohesive stresses. It is observed from the studies that SIF obtained from the finite element analysis with and without considering the cohesive stresses is in good agreement with the corresponding analytical value. The effect of cohesive stress on SIF decreases with increase of crack length. Further, studies have been conducted on geometrically similar structures and observed that (i) the effect of cohesive stress on SIF is significant with increase of load for a particular crack length and (iii) SIF values decreases with increase of tensile strength for a particular crack length and load.     

A tensile crack in creeping solids with large damage near the crack tip

An asymptotic analysis of stationary mode I crack in creeping solids with large damage near crack tip is conducted. To consider the damage effect, Kachanov damage evolution law is utilized and incorporated into the power-law creep constitutive equation. With the compatibility equation, a nonlinear eigenvalue problem which can be solved by numerical approaches is established. From this result, the distribution of stress and strain rate are obtained with the coupling effect of damage and creep under plane stress condition. Also the influence of material parameters on the stress is examined. According to the result, it is shown that the creep exponent n and damage parameter (=/(1+k)) have a significant effect on determining the eigenvalue s and angular distribution of stress and strain rate near the crack tip. The creep exponent n plays the role to soften and damage parameter plays the role to harden the material near the crack tip. The stress and strain rate show quite different behavior from those of HRR problem.     

Analysis of FRP–concrete debonding via boundary integral equations

The problem of debonding of FRP plates glued over a concrete element is studied making use of boundary integral equations. Mode II cohesive crack model is adopted for the interface, whereas linear elasticity is used for the two materials outside the process zone. Symmetric Galerkin boundary element method is used, adopting the arc-length technique to follow the equilibrium path beyond its critical point. It is shown that, due to the presence of a softening branch in shear stress-slip law, the behavior of a specimen undergoing debonding may be strongly non-linear, and is associated with a very brittle failure mechanism. For bond lengths longer than minimum anchorage length, a snap-back branch typically occurs after the attainment of the maximum force. Two different test setups have been numerically simulated and results in good agreement with experimental tests are found.     

Softening and snap-back instability in cohesive solids

The nature of the crack and the structure behaviour can range from ductile to brittle, depending on material properties, structure geometry, loading condition and external constraints. The influence of variation in fracture toughness, tensile strength and geometrical size scale is investigated on the basis of the π-theorem of dimensional analysis. Strength and toughness present in fact different physical dimensions and any consistent fracture criterion must describe energy dissipation per unit of volume and per unit of crack area respectively. A cohesive crack model is proposed aiming at describing the size effects of fracture mechanics, i.e. the transition from ductile to brittle structure behaviour by increasing the size scale and keeping the geometrical shape unchanged. For extremely brittle cases (e.g. initially uncracked specimens, large and/or slender structures, low fracture toughness, high tensile strength, etc.) a snap-back instability in the equilibrium path occurs and the load–deflection softening branch assumes a positive slope. Both load and deflection must decrease to obtain a slow and controlled crack propagation (whereas in normal softening only the load must decrease). If the loading process is deflection-controlled, the loading capacity presents a discontinuity with a negative jump. It is proved that such a catastrophic event tends to reproduce the classical LEFM-instability (K_I = K_IC) for small fracture toughnesses and/or for large structure sizes. In these cases, neither the plastic zone develops nor slow crack growth occurs before unstable crack propagation.     

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range (\(\Delta \hbox {K})\), load ratio (R), number of cycles (N), plastic compressibility (\({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\), R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study.     

Size effect on compression strength of fiber composites failing by kink band propagation

The effect of structure size on the nominal strength of unidirectional fiber-polymer composites, failing by propagation of a kink band with fiber microbuckling, is analyzed experimentally and theoretically. Tests of novel geometrically similar carbon–PEEK specimens, with notches slanted so as to lead to a pure kink band (not accompanied by shear or splitting cracks), are conducted. They confirm the possibility of stable growth of long kind bands before the peak load, and reveal the existence of a strong (deterministic, non-statistical) size effect. The bi-logarithmic plot of the nominal strength (load divided by size and thickness) versus the characteristic size agrees with the approximate size effect law proposed for quasibrittle failures in 1983 by Bažant. The plot exhibits a gradual transition from a horizontal asymptote, representing the case of no size effect (characteristic of plasticity or strength criteria), to an asymptote of slope -1/2 (characteristic of linear elastic fracture mechanics, LEFM). A new derivation of this law by approximate (asymptotically correct) J-integral analysis of the energy release, as well as by the recently proposed nonlocal fracture mechanics, is given. The size effect law is further generalized to notch-free specimens attaining the maximum load after a stable growth of a kink band transmitting a uniform residual stress, and the generalized law is verified by Soutis, Curtis and Fleck's recent compression tests of specimens with holes of different diameters. The nominal strength of specimens failing at the initiation of a kink band from a smooth surface is predicted to also exhibit a (deterministic) size effect if there is a nonzero stress gradient at the surface. A different size effect law is derived for this case by analyzing the stress redistribution. The size effect law for notched specimens permits the fracture energy of the kink band and the length of the fracture process zone at the front of the band to be identified solely from the measurements of maximum loads. The results indicate that the current design practice, which relies on the strength criteria or plasticity and thus inevitably misses the size effect, is acceptable only for small structural parts and, in the interest of safety, should be revised in the case of large structural parts. This revised version was published online in July 2006 with corrections to the Cover Date.