Crack-tip damaged zones in rubber-toughened amorphous polymers: a micromechanical model

The various stages of crack propagation in rubber-toughened amorphous polymers (onset and arrest, stable and unstable growth) are governed by the rate of energy dissipation in the cracktip damaged zone; hence the relationship between the applied stress intensity factorK ₁ and the damaged zone size is of utmost importance. The size of the crack-tip damaged zone has been related toK ₁ via a parameter which is characteristic of the material in given conditions: this factor is proportional to the threshold stress for damage initiation in a triaxial stress field, and has been denoted by ^*. Theoretical values of ^* have been calculated by means of a micromechanical model involving the derivation of the stresses near the particles and the application of damage initiation criteria. The morphology, average size and volume fraction of the rubbery particles have been taken into account together with the nature of the matrix. The calculated values of ^* have been successfully compared with the experimental ones, for a wide set of high-impact polystyrenes (HIPS) and rubber-toughened poly(methyl methacrylate) (RTPMMA).Nomenclature PS; HIPS polystyrene; high-impact polystyrene - PMMA; RTPMMA poly(methyl methacrylate); rubber-toughened PMMA - MI; CS/H; CS/R particle morphologies (multiple inclusion; hard core - rubber shell; rubber core - rigid shell) - K _r;K _g bulk moduli of rubber and glassy materials - G _r;G _g shear moduli of the same materials - v _p particle volume fraction - L mean centre-to-centre distance between neighbouring particles - B; H; W standard names for the dimensions of the compact tension specimen - R _y size of the crack-tip plastic zone in a homogeneous material - h half thickness of the crack-tip damaged zone - r; polar coordinates around the crack tip (Fig. 1) - r;r _p distance from particle centre; particle radius - p normalized distance from the particle (Equation 5) - K ₁;K _1c;K _1p stress intensity factor; critical values ofK ₁ at the onset of and during crack growth - G _1c plane strain energy release rate - _y yield stress in uniaxial tension - _th macroscopic threshold stress for the onset of local damage initiation in a composite material - ^* characteristic parameter (Equation 3) - ⁰; ₁ ⁰ ; ₂ ⁰ ; ₃ ⁰ applied stress tensor and its three principal stresses - ⁰ uniaxial applied stress - ; ₁; ₂; ₃ local stress tensor and its three principal stresses - A tensor which elements are the ratios of those of over those of ⁰ (Equation 4) - v Poisson's coefficient of the matrix - g triaxiality factor of the crack-tip stress field - _e; _p Mises equivalent stress; dilatational stress (negative pressure) - I ₁;I ₂ invariants of the stress tensor - U ₁;U ₂ material parameters for argon and Hannoosh's craze initiation criterion (Equation 12)     

The determination of the critical stress intensity factor in mode II loading and the shear fracture strength of pharmaceutical powder specimens

The shear fracture strength and the critical stress intensity factor in mode II loading of lactose monohydrate and acetylsalicylic acid powder compacts has been evaluated. The experimental results of the shear fracture strength and the critical stress intensity factor in mode II loading appeared to be in good agreement with powder behaviour such as lamination and capping during compaction. Values for the critical stress intensity factor in mode II loading depended on the depth of the crack and hence, any reference of such values or their use to calculate a fracture toughness ratio (K _IC ^I/K _IC ^II) must refer to the notch depth applied. The results confirmed that the failure of such powder compacts occurs mainly in tension, but that lactose monohydrate has a tendency also to fail in shear. The latter does not apply to acetylsalicylic acid. Hence, lactose monohydrate should only be used cautiously in layer or press-coated tablets.     

Steady state crack growth in elastic-viscoplastic materials

A recent theory of Hart for the steady state propagation of a mode III crack in a ductile material is extended to modes I and II. When a crack is moving at non-zero velocity v, it is shown that for a broad class of materials the stress state at the crack tip is characterized by a r ^1/2 singularity and by a local stress intensity factor K. The local K is the sum of the apparent stress intensity factor K _A and a plastic contribution K _P. The value of K _A is calculated from the remote loading and the crack geometry under the assumption of linear elastic response alone. The quantity K _P characterizes stress relief of non-elastic flow. Numerical calculations are made to determine K as a function of K _A and v for elastic-viscoplastic materials. A dependence of v on K _A is obtained by imposing a kinetic law for v as a function of K. The plots of v vs. K _A show that below some critical values of K _A, steady state conditions cannot be sustained. Corresponding to the threshold value of K _A there is a definite value for the velocity.     

Dynamic crack propagation in rubber-modified composite models

The dynamic crack propagation behaviour of several rubber-modified composite models has been studied. In all cases the method of high speed photography along with the method of dynamic caustics was used. Results of crack propagation mode observation, fracture toughness and crack propagation velocity measurements are presented here. Especially in the case of two complex inclusions it was found that the crack propagation mode is highly rate dependent. At low test rates the crack growth tends to follow an almost straight crack path while an increase in strain rate in general results in the formation of a kink in the interparticle area. In the same area a crack propagation delay, and in some cases arrest, was observed while both the crack propagation velocity v and dynamic stress intensity factor K _i ^d showed an intense variation. For the sake of comparison, specimens with one and/or two press-fitting inclusions as well as with two holes were fractured under dynamic loads. In all cases both qualitative and quantitative results were obtained.     

Crack growth in a mixed-mode loading on marble beams under three point bending

A combined theoretical and experimental study of a crack growth in a mixed-mode I–II loading is presented. A 160×40×20 mm marble beam, with an artificial crack 8 mm and 10 mm long each, was subjected to three point bending. The crack was located vertically to the beam's lower longitudinal fiber, through the whole width of the beam. The position of the crack was displaced from the center of the beam to one of the supporting points. The vertical force P, placed on the middle of the upper fiber of the beam, imposed the combination of the opening (mode I) and the sliding (mode II) modes on the crack mouth, creating the mixed-mode I–II loading case. The stress intensity factors K _I and K _II, which describe the local stress and strain field around the crack tip, were determined by a suitable finite element program. The crack growth was defined by two classical fracture criteria of LEFM; the minimum strain energy density and the maximum circumferential stress criteria. The initial crack growth angle () was calculated from both criteria and the critical load (P _c) from the minimum strain energy density (SED) criterion. These theoretical predictions were compared with some experimental results found from three marbles with different elastic constants; the Krystallina of Kavala, the Snow-white of Thassos and the White of Piges Drama. The theoretical results showed the same trend of and P _c as the experimental ones and they are in good agreement.Presented at Fourth Greek National Congress on Mechanics, 26–29 June 1995, held at Xanthi, Greece.     

Analysis of brittle fracture under the combined stresses of tension and compression

Brittle fractures occurring under biaxial stress states were analysed based on the weakest link model using the mixed mode fracture criterion. Expressions for the mixed mode fracture criterion were chosen for application to the negativeK _l region, corresponding to the compressive stress for the crack. Calculations for biaxial strength with randomly oriented constant-length cracks from the mixed mode fracture criterion were made in the region ofK _I>0 because an unstable fracture seems to occur in this region. The results indicated that the tensile stress component in the combined tension and compression stress state remains constant when the compressive component is smaller than the critical value, which is given by [1 –(K _c/K _c)²]_t derived from the mixed mode fracture criterion, (K /K _c) + (K /K _c)² = 1. Considering the statistical effects, however, calculation of the biaxial strength is modified to result in: (1) lowering the biaxial tensile strength, and in (2) a smooth transition from the constant tensile strength region to the decreasing strength region under the combined tension and compression stress. This suggests that the highK _IIc/K_Ic ratio results in the increase in the compressive strength relative to the tensile strength.     

A FRACTURE CRITERION FOR CRACKS UNDER MIXED-MODE LOADING

Abstract A fracture criterion is proposed, based on maximum energy release rates at the tips of short kinks when the main cracks are subjected to mixed mode loading. The criterion differs from existing energy based criteria in that the fracture toughness, g_c, is not independent of the stress mode prevailing in the region of the tip of the kink but is a function of the ratio of the mode II to mode I stress intensity factors at the tip of the kink, i.e., g_c is determined directionally by an elliptical region with major and minor axes equal to the fracture resistances of the material, K_Ir and K_IIr, for pure mode I and pure mode II, respectively. Points inside the elliptical region are considered safe. When K_IIr is equal to K_Ir the ellipse degenerates into a circle and the fracture criterion reverts to the existing familiar maximum energy release rate criterion based on a single value of the fracture toughness, irrespective of the active mode prevailing in the region at the tip of the kink. In this case, under pure shear (mode II) applied load, K_II, the angle of inclination of the fracture crack extension to the main crack, α, is in the region of ?76°, in general agreement with previous well established results. However, when the ratio r (=K_IIrK_Ir) is less than r′ (=0.82, approximately) a different pattern emerges and, in particular, under pure mode II load, the crack advance is co-planar with the main crack, i.e., in mode II. A lower transition value r″ (=0.582, approximately) was also detected under pure mode I applied load. Thus for values of r≥r″, the crack extension is in pure mode I and is co-planar with the main crack but when r < r″, the crack branches out at an angle (which can be positive or negative) in mixed modes I/II crack extension. Some implications of these results are discussed.     

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter Q _II is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between Q _II and T/₀ is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q–T diagram for real specimens, the constraint parameter Q _II is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q–T diagram.     

Creep Crack Growth in Polyethylene Under Combined Mode I and Mode II Loading

Creep crack growth characteristics under various combined mode I and mode II loadings were studied using the compact tension shear (CTS) specimens of polyethylene. Creep crack growth rates da/dtunder combined mode I and mode II loading can be correlated with a single effective stress intensity factor K _Ieffderived from the combined — mode fracture toughness envelope. The steady state or constant crack growth rates which occupy the significant part of creep failure life increase with the increasing initial effective stress intensity factor.     

Micro-mechanical analysis and TEM study of crack initiation in dislocation free zone

Dislocation emission, dislocation-free zone (DFZ) formation and crack initiation in the DFZ and/or at the crack tip were analyzed by micromechanics. The results show that a DFZ is formed after dislocation emission under constant load. The DFZ size decreases with increase in the applied stress intensity factor K _Ia or lattice friction stress _f. There are two stress peaks ahead of the crack tip. The first one is located at the blunt crack tip and the second one in the DFZ. With increasing in the applied stress intensity factor K _Ia, the peak stress at the crack tip may decrease while the peak stress in the DFZ increases monotonically. Microcrack will initiate when the peak stress is equal to the cohesive strength. In situ tensile tests in a transmission electron microscope (TEM) show that microcrack initiates in DFZ or/and at a blunt crack tip after dislocation emitting and DFZ formation.     

Effect of crack shape,inclination angle,and friction coefficient in crack surface contact problems

In rolling/sliding contact fatigue, it is known that the crack propagates at a characteristic angle =15–30 deg to the surface. To analyze the mechanism, however, the body force method has been widely used assuming 3D crack models for =45–90. In this study, therefore, the unknown body force densities are newly approximated by using fundamental density functions and polynomials. Then, a semi-elliptical crack model is analyzed for =15–90 under compressive residual stresses and Hertzian contact loads. The stress intensity factors K _II, K _III are calculated with varying the crack shape b/a, inclination crack angle , and crack face friction coefficient . The calculations show that the present method is useful for the analysis for =15–30 deg with high accuracy. It is seen that the K _II-values when b/a0 are larger than the ones when b/a=1 by 0–24% for both under compressive residual stress and Hertzian contact load. Regarding the maximum K _II values under Hertzian contact load, the results of =15 deg are smaller than the ones of =45 deg by 23–34%. Regarding the amplitude of (K _{II max}–K _{II min}), the results of =15 deg are smaller than the ones of =45 deg by 4–24%. With increasing the value of friction coefficient for crack faces the value of K _II decreases significantly. When the crack is short and the inclination angle is small, the value of friction coefficient f for Hertzian contact load largely affect the K _II value.     

Mixed-mode fatigue crack growth behaviour in aluminium alloy

Fatigue crack propagation tests in compact mixed-mode specimens were carried out for several stress intensity ratios of mode I and mode II, K_I/K_II, in AlMgSi1-T6 aluminium alloy with 3 mm thickness. The tests were performed in a standard servo-hydraulic machine. A linkage system was developed in order to permit the variation of the K_I/K_II ratio by changing the loading angle. Crack closure loads were obtained through the compliance technique. A finite element analysis was also done in order to obtain the K_I and K_II values for the different loading angles. Crack closure increases under mixed-mode loading conditions in comparison to mode-I loading due the friction between the crack tip surfaces. Moreover, the crack closure level increases with the K_I/K_II ratio decrease. Correlations of the equivalent values of the effective stress intensity factor with the crack growth rates are also performed. Finally, an elastic–plastic finite element analysis was performed to obtain the plastic zones sizes and shapes and model the effect of mixed-mode loading on crack closure.     

On the elasticT-term of a main crack induced by near tip microcracks

In the expansion form of the stress field near a crack tip, the nonsingular and constant stress acting parallel to the crack plane is called the elasticT-term, which is very important in fracture analysis. The elasticT-term of a main crack induced by near tip microcracks is analyzed. The original problem considered consists of a main crack accompanied by near tip microcracks, which are loaded remotely by the first three terms of the crack tip stress field expansion parameterized by stress intensity factorsK _I ,K _II , and the nonsingular stress, i.e., theT-term,. With the principle of superposition, the problem can be reduced to a system of Fredholm integral equations which can be solved numerically. After obtaining the solution, the induced elasticT-term is evaluated. Several typical numerical examples are given and the results are shown in Figures, from which some useful discussion and conclusions are obtained.     

An energy approach to crack closure

Crack closure is analyzed using an energy approach whereby it is shown that crack closure does not completely shield the input mechanical energy to the crack tip at a load below the crack opening load P _op if the compliance below P _op is non-zero. An equivalent shielding stress intensity range is defined by the energy release rate against crack closure. From this energy standpoint, the true effective stress intensity range should be defined as K _eff=K _max–K _op, where is the shielding factor. The conventional definition (K _eff=K _max–K _op) is equivalent to the new definition only when the compliance below P _op is zero such that =1, i.e., for a fully closed crack. The corrected K _eff is found to be effective in correlating fatigue crack growth rates (FCGRs) generated in 8090-T8771 aluminum-lithium alloy with and without crack closure. In contrast, the conventional K _eff fails to reconcile the FCGR data within an acceptable scatter band.The Canadian Government's right to retain a non-exclusive, royalty-free licence in and to any copyright is acknowledged.     

Physical meaning of ΔKRP and fatigue crack propagation in the residual stress distribution field

Previous papers have shown ΔK_RP to be a useful parameter describing fatigue crack propagation behavior, where ΔK_RP is an effective stress intensity factor range corresponding to the excess RPG load (re-tensile plastic zone's generated load) in which the retensile plastic zone appears under the loading process. In this paper, the relationship between ΔK_RP and the zone size () (which is smaller between the tensile plastic zone at maximum load and the compressive plastic zone at minimum load) was investigated using a crack opening/closing simulation model so as to consider a physical meaning of ΔK_RP. As a result, it becomes clear that ΔK_RP dominates the zone size where fatigue damage mostly occurs. This result supports the following crack propagation equation

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Fracture toughness of Si3N4 measured with short bar chevron-notched specimens

The short bar chevron-notched specimen was used to measure the plane strain fracture toughness of hot-pressed Si₃N₄. Specimen proportions and chevron-notch angle were varied, thereby varying the amount of crack extension to maximum load (upon which K_ic was based). The measured toughness (4.68 ± 0.19 MN m^3/2) was independent of these variations, inferring that the material has a flat crack growth resistance curve.Nomenclature a crack length - a _A crack length at arrest of unstable crack advance - a ₁ length of chevron notch at specimen surface (distance from line of load application to point of chevron emergence at specimen surface) - a ₀ initial crack length (distance from line of load application to tip of chevron) - a _R crack length at ending of stable crack extension (conversely, crack length at onset of abrupt, unstable crack advance) - B specimen thickness - H specimen half-height - K _1A stress intensity factor at arrest of unstable crack advance - K _IR stress intensity factor at end of stable crack extension (crack growth resistance) - K _IC plane strain fracture toughness - P _max maximum applied load in fracture toughness test - W specimen width - Y ^* dimensionless stress intensity factor coefficient for chevron-notched specimen - Y ^* _m minimum value ofY ^* as a function of - a/W - ₀ a ₀/W - ₁ a ₁/W     

Strength Assessment for Butt Joints of Steel 50 by the Fracture Toughness Criteria

We propose an experimental-theoretical engineering procedure for assessing the strength of butt-welded joints using the force (K_Ic) and strain (_c) criteria of the brittle and quasibrittle fracture mechanics, respectively. For this purpose, beam specimens with square and bevel welds and an initial edge crack or notch in the weld metal are tested under three-point bending. The parameter K_Imax controlling fracture of a bent beam with an inclined (Mode I + Mode II) crack is assessed taking into account the values of the stress intensity factors K_I and K_II, and the crack inclination angle ^*. We also studied the plastic zone at the crack tip and the crack propagation kinetics depending on the weld geometry and the V-notch tip radius for butt-welded joints. The data obtained allow one to rate such joints by their strength according to the fracture toughness criteria K_Ic and _c.     

Fracture Toughness and Creep Crack Growth in Polypropylene Under Combined Modes I and II Loading

Fracture toughness and creep crack growth characteristics under combined mode I and II loadings were studied using the compact tension shear (CTS) specimens of polypropylene. The K _I - K _II envelope for crack initiation was obtained under various combined mode loadings. The creep crack growth rates da/dt under combined mode I and mode II loadings can be correlated with a single effective stress intensity factor K _Ieff based on the combined mode fracture envelope.