A higher-order approximation for the T-criterion of fracture in biaxial fields

The T-criterion of fracture is based on the principle that crack propagates when the maximum value of the distribution of the dilatational component of strain energy density T_v, evaluated along contour lines of constant distortional energy density T_D around the crack tip, attains a limiting value T_vo The angle of this maximum defines also the direction of initiation of crack propagation. Then, the study of the distribution of T_v around the crack tip presents a special interest for understanding mechanisms of fracture.

In this investigation an exhaustive theoretical analysis of the distribution of t_v-component around the tip of crack under in-plane modes of loading was undertaken. The T_v-distribution was evaluated along the elastic-plastic boundary, developed around the crack tip for impending plasticity, according to the Mises yield condition (T_D = T_D0 = const.). The mode of loading of the cracked plate was assumed biaxial with different biaxiality ratios k and a two-term approximation for the respective complex stress function was considered, according to the studies of Liebowitz et al.[1], instead of only the singular term considered up-to-now.

It was found that the T_v-distribution along the Mises initial elastic-plastic boundary presents always a maximum in front of the crack tip, whose position and magnitude depend on the biaxiality factor k and the angle of loading β. The position and the magnitude of this maximum for the two-term approximation of φ(z) showed differences in some regions with the respective values for the singular solution.     

A fracture criterion for three-dimensional crack problems

A criterion for predicting the growth of three-dimensional cracks is developed on the basis of the strain energy density concept which has been used successfully for treating two-dimensional crack problems. Fracture is assumed to initiate from the nearest neighbor element located by a set of spherical coordinates (r, θ, φ) attached to the crack border. The new fracture surface is described by a locus of these elements whose locations correspond to the strain energy function, dW/dV, being a minimum. The function dW/dV is found to be singular of the type 1/r and is of quadratic form in the three stress intensity factors k₁, k₂ and k₃ expressed through the strain energy density factor S. It is postulated that unstable crack propagation initiates from a region where S reaches a critical value S_cr = r₀(dW/dV)_cr. The locations of failure lying on the fracture surface is determined by holding (dW/dV)_cr = S_min/r₀ constant. The quantity S_min stands for the value of S minimized with respect to θ and φ and r₀ is a radial distance measured from the crack border.

An example of failure prediction for an embedded elliptical crack subjected to both normal and shear loads is presented. According to the S-criterion, fracture initiation takes place at the ends of the minor axis. An unexpected result is that for a narrow elliptical crack and Poisson's ratio of 1/3 the lowest failure load occurs when the uniaxial tensile load makes an angle of approximately 60° with the crack surface and is in the plane of the major axis. This is in contrast to the expectation that the lowest critical load occurs when the uniaxial tension is perpendicular to the crack surface. In the limit as the elliptical crack becomes increasingly narrower, the result reduces to the two dimensional line crack case of Mode I and III loading. The S-criterion is also applied to the failure prediction of three dimensional cracks under compressive loads.     

Some remarks on the Det.-criterion and caustics

A comprehensive study is made of the Det.(σ_ij)-criterion that involves the weighted difference of the dilatational and distortional component of the strain energy density function. Certain inconsistencies and contradictions when applied to the Mode I crack extension problem are pointed out. Clear distinctions are also made on the meaning of the core region in contrast to the loci of yield initiation and the initial curve of the caustics. Examples are provided to illustrate how overspecification in the initial assumption could lead to obscure results.     

Properties of eigenfunction expansion form and relative integral for a crack with completely closed surfaces

The important property of the eigenfunction expansion form found by Chen [Engng Fracture Mech. 22, 673–686 (1985)] is found as the pseudo-orthogonal property. The analysis of this property and relative integral for a semi-infinite crack with completely closed surfaces in homogeneous plane elasticity is studied in detail. It is found that the integral proposed by Bueckner [in Mechanics of Fracture, Vol. 1, pp. 239–314 (1973)] and Chen is no longer path-independent in this circumstance. The differences between the present investigation and that of Chen are discussed.     

Strain energy density fracture criterion in elastodynamic mixed mode crack propagation

Sih's fracture criterion based on strain energy density, S, for mixed mode crack extension under static loading is extended to dynamic mixed mode, K_I and K_II, crack propagation. Influence of the second order term, σ_ox, which represents the non-singular constant stress acting parallel to the direction of crack propagation, on the S distribution surrounding the crack tip, is demonstrated. Numerical studies show that positive σ_ox enhances the fracture angle and negative σ_oxreduces the fracture angle irrespective of the sign of K_II/K_I, when S is measured at a critical distance r_c from the crack tip. This fracture criterion is verified by the crack curving results of dynamic photoelastic fracture specimens. Omission of σ_ox term leads to predicted fracture angles which are at variance with experimental data.     

半圆盘构件冲击断裂特性的动态焦散线实验研究

采用动态焦散线实验系统,对有机玻璃（PMMA）在冲击载荷下的I型和I-II混合型裂纹在起裂和扩展时的动态断裂特性进行了研究。结果表明：随着PMMA由I型断裂转变为I-II混合型断裂,从落锤作用在试件上到裂纹起裂所需时间不断增加,说明裂纹起裂需要的能量有所增加,同时从裂纹起裂到最终贯通所需时间不断减少,说明裂纹平均扩展速度也不断增大;在I型断裂中,PMMA的断裂韧度KIC为2.04 MN/m3/2,而在I-II混合型断裂中,PMMA的断裂韧度KIC低于I型断裂时的断裂韧度KIC,但是KIIC有所增大;对于I-II混合型断裂,PMMA极限扩展速度约为366m/s,当达到极限扩展速度后,裂纹尖端出现微裂纹增韧现象,使裂纹的表面能迅速增大,随后裂纹的扩展速度迅速减小。     

The nt-criterion for predicting crack growth increments

A new approach is presented to determine the crack propagation increment after the direction of crack propagation has been predicted. The maximum dilatational strain energy density (NT-criterion) is employed in the derivation for predicting both direction and increment of the propagating crack. The crack propagation path predicted by the NT-criterion is compared to the one predicted by the S-criterion and to some available experimental data.     

An attempt to separate elastic strain energy density of linear elastic anisotropic materials based on strains considerations

A promising new effort toward the decomposition of the elastic strain energy density of linear elastic anisotropic materials into a dilatational and a distortional part is presented. By assuming that volume changes must keep the material symmetries unchanged, a new physical perspective is presented and interesting definitions are drawn. This new perspective necessitates the introduction of a strain parameter m characteristic of the material’s anisotropy. This strain parameter besides easing the calculation of the dilatational and distortional energetic terms additionally accounts for the directional sensitivity of anisotropic materials.     

Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads

In this paper, the brittle fracture initiation characteristics under general combination of the opening mode (Mode I), sliding mode (Mode II) and tearing mode (Mode III) were investigated both theoretically and experimentally.

First, the perfectly brittle fracture tests were conducted on specimens of PMMA (Polymethylmethacrylate) for all possible combinations of the fracture modes including respective pure modes. The experimental fracture strengths were compared with those predicted by the fracture criteria which are represented in terms of: (1) maximum tangential stress, [σ_gq]_max, extended to general combined modes, (2) maximum energy release rate at the propagation of a small kinked crack, [G^k(γ)]_max, and (3) newly derived maximum energy release rate at the initiation of a small kinked crack, [G(γ)]_max. It was found that the [G^k(γ)]_max or [G(γ)]_max criterion was very effective to predict both the direction of initial crack propagation and the fracture strength. These energy release rates are expressed in closed forms, and the interaction curves of the brittle fracture strength under arbitrary combinations of Modes I, II and III were derived.

Next, for fracture accompanied by plastic deformation, tests were carried out on specimens of mild steel (SM 41) imposing bi-axial tensile loads at various low temperatures. Then, brittle fracture with plastic deformation occurs under a combination of Modes I and II. In the case of brittle fracture with small scale yielding, the [G(γ)]_max criterion predicts well the direction of initial crack propagation but estimates only lower fracture strength than the experimental one. In the cases of brittle fracture with large scale yielding and under general yielding, it was found from the fracture tests that the direction of initial crack propagation was nearly normal to the resultant vector of the crack opening displacements in the opening and sliding modes at the notch tip. To this type of fracture, the modified COD criterion predicts well the direction of initial crack propagation, but lower fracture strength.

When brittle fracture occurs under the influence of plastic deformation, in such cases as the last three mentioned above, the actual fracture strength is higher than what the most reliable criterion predicts and it increases as deformation in Mode II becomes larger.     

A strain energy criterion for mixed mode crack propagation

The strain energy criterion for crack propagation proposed in the paper is based on the principle that the direction of crack propagation takes place along the direction where the distance from the crack tip to a certain contour line of constant distortional strain energy density is minimum, and the crack will begin to propagate when the total strain energy in the region surrounded by the contour line reaches a critical value. In the paper, predicted was compared with the measured results.     

How cracks propagate in a vitreous dense biopolymer material

Crack propagation in vitreous biopolymer material is studied considering numerical and experimental aspects. Finite Element computation is performed to predict crack propagation in starch material. Maximum dilatational strain energy criterion is used as a criterion for crack propagation. Model validation is undertaken considering notched specimens with a hole under tensile loading. The experimental results show that the crack path is affected by the presence of the defect, which allows adequate testing of the crack propagation criterion under a mixed mode. The predicted results show that the selected criterion anticipates adequately the crack propagation in vitreous starch under linear elastic conditions.     

On the use of the T-criterion in fracture mechanics

The applicability of the maximum dilatational strain energy density criterion, T-criterion, as proposed by Theocaris and Andrianopoulos, is investigated for use in a crack propagation study. It is found that the T-criterion has to be modified to correctly determine the fracture load. A detailed comparison between the minimum strain energy density criterion, S-criterion, and the new modified T-criterion is given for generalized plane stress and plane strain conditions.     

Fracture and fatigue of bonded rubber blocks under compression

Fracture and fatigue failure of bonded rubber cylinders are discussed. Under large compressive forces, two modes of fracture are possible: splitting open of the free surface and tearing at or near the bonded edges; tearing energy T for the latter case is estimated. Under cyclic compression, the probable fracture mode of rubber is by crack propagation, leading to the bulged volume breaking away. The corresponding tearing energy is calculated. To predict the fatigue life, the rate of crack growth dc/dn is assumed to be proportional to T². A life prediction equation is thus obtained, of the form: load cycle N = (K/g)⁵, where K is a constant, about 10 for a typical soft natural rubber compound, and g is the maximum shear strain set up at the edges of the bonded surfaces.     

Effect of mode III fracture on composite laminates

The digital speckle correlation method (DSCM) is preformed to study the inplane displacements along the x and y direction of laminated composite panels containing preset elliptical damage. The different principle axis of the ellipse and the different sequence of laminates are considered in the experiments. It is shown that the method is very useful to get the displacements on the laminate surface and between the adjacent plies. According to the experiment results, the deformations of x and y direction can be obtained. The conclusion that the mode III fracture may exist and may have an effect on the crack growth is formed. The total strain energy release rate is calculated by the finite element method. Using Mindlin plate theory and the virtual crack closure technique, the energy release rate of mode III fracture can be calculated by FEM. The results show that mode III fracture has an influence on the total energy release rate and also on the delamination growth. The energy release rate of mode III fracture cannot be ignored. The delamination growth also is influenced by the stacking sequence.     

Plane-stress crack-tip fields for pressure-sensitive dilatant materials

In this paper we present plane-stress crack-tip stress and strain fields for pressure-sensitive dilatant materials. A hydrostatic stress-dependent yield criterion and the normality flow rule are used to account for pressure-sensitive yielding and plastic dilatancy. The material hardening response is specified by a power-law relation. The plane-stress mode I singular fields are found in a separable form similar to the HRR fields (Hutchinson, J. Mech. Phys. Solids 16, 13–31 and 337–347, 1968; Rice and Rosengren, J. Mech. Phys. Solids 16, 1–12, 1968). The angular variations of the fields depend on the material hardening exponent and the pressure sensitivity parameter. Our low-hardening solutions for different degrees of pressure sensitivity agree well with the corresponding perfectly plastic solutions. An important aspect of the effects of pressure-sensitive yielding and plastic dilatancy on crack-tip fields is the lowering of the opening stress and the hydrostatic stress directly ahead of the crack tip. This effect, similar to that under plane-strain conditions (Li and Pan, to appear in J. Appl. Mech. 1989), has implications in the material toughening observed in some ceramic and polymeric composites.     

Study of fatigue crack propagation by X-ray diffraction approach

X-ray microbeam technique was applied for the observation of change in microstructure at the tip of propagating fatigue crack. Crack propagation rate was discussed in terms of the micro-parameters derived from the observation. The propagation rate of fatigue cracks in annealed carbon steels was found to be uniquely related to each one of the microscopic parameters supplied by the X-ray microbeam observation, i.e. the excess dislocation density D and the subgrain size t, of the grain at the crack tip. These relations were independent of the specimen geometry and the loading mode, although those factors affected the mechanical relation between the propagation rate dl/dN and the stress intensity factor K. In the case of low propagation rates, fatigue cracks will grow along subboundaries when the excess dislocation density becomes a critical value, and this critical value is nearly equal to the value at crack initiation in plain specimens. The equivalent true strain _F at the fracture surface evaluated from the measured value of the excess dislocation density was uniquely related to the propagation without respect to the carbon content of the materials. The local residual stress measured by the X-ray method was compression at the crack tip and gradually changed to tension away from the tip.     

Crack initiation condition in the specimens of ductile materials

In this paper, the elasto-plastic Finite Element Method was used to simulate the ductile fracture initiation of four kinds of specimens made of low-carbon steel No. 20 (Chinese steel), and the damage parameter V_D was calculated for these specimens. The results showed that the damage mechanics criterion V_D proposed by the author and B. X. Yang [Engng Fracture Mech. 27, 371–386 (1987)], could be applied to predict the crack initiation. Also, the relationship between the damage parameter V_Dc and the crack tip opening displacement was discussed in this study.     

A model of brittle fracture propagation part I—continuum aspects

The micromechanism of crack propagation in steel is described and analyzed in continuum terms and related to the macroscopic fracture behavior. It is proposed that propagation of cleavage microcracks through favorably oriented grains ahead of the main crack tip is the principal weakening mode in brittle fracture. This easy cleavage process proceeds in the Griffith manner and follows a continuous, multiply connected, nearly planar path with a very irregular front which spreads both forward and laterally and leaves behind disconnected links which span the prospective fracture surface. A discrete crack zone which extends over many grains thus exists at the tip of a running brittle crack. Final separation of the links is preceeded by plastic straining within the crack zone and occurs gradually with the increasing crack opening displacement. It is suggested that in low stress fracture, straining of the links is the only deformation mode. However, it is recognized that under certain conditions plastic enclaves may adjoin the crack zone. This deformation mode is associated with high stress fracture, energy transition and eventually with crack arrest.

Energy dissipation resulting from the two deformation mechanisms is related to crack velocity, applied load and temperature and the crack velocity in a given material is expressed as a function of the external conditions. Fracture initiation and crack arrest are then discussed in terms of the conditions which are necessary to maintain the propagation process. Finally, the dimensions of a small scale crack tip zone for a steady state, plane strain crack are evaluated as functions of material properties and the elastic stress intensity factor.

The microstructural aspects of brittle fracture will be discussed in a separate Part 2 [1].     

A non-local stress and strain energy release rate mixed mode fracture initiation and propagation criteria

A non-local stress condition for crack initiation and propagation in brittle materials is presented. This condition is expressed in terms of normal and tangential traction components acting on a physical plane segment (damage zone) of specified length. Next, a non-local strain energy release rate criterion is proposed. This condition is based on the assumption that initiation or propagation of cracking occurs when the maximal value of the function of opening and sliding energy release rates reaches a critical value. The value of energy release rates is determined for finite elementary crack growth. Mixed mode conditions are considered, for which both the critical load value and the crack orientation are predicted from the non-local stress and energy criteria, which are applicable to both regular and singular stress concentrations. The effect of non-singular second order term (T_σ-stress) on the crack propagation is discussed.     

Modeling a fracture set in a layered brittle material

A new fracture experimental technique using a brittle coating on PMMA substrates for mode I fractures in layered brittle materials is introduced in this paper. Different strain histories can be applied to the coating including constant strain (relaxation), strain cycling, strain magnitude, and strain rate. Fracture parameters, number of cracks, length of crack, spacing, spatial density, and propagation velocity describe the state of a fracture set.