Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity

In a remarkable series of experiments, Elssner et al. (1994) and Korn et al. (2002) observed cleavage cracking along a bimaterial interface between Nb and sapphire. The stress required for cleavage cracking is around the theoretical strength of the material. Classical plasticity models fall short to reach such a high stress level. We use the conventional theory of mechanism-based strain gradient plasticity (Huang et al., 2004) to investigate the stress field around the tip of an interface crack between Nb and sapphire. The tensile stress at a distance of 0.1 m to the interface crack tip reaches 13.3_Y, where _Y is the yield stress of Nb. This stress is nearly 4 times of that predicted by classical plasticity theory (3.6_Y) at the same distance to the crack tip, and is high enough to trigger cleavage cracking in materials and interfaces. This is consistent with Elssner et al.'s (1994) and Korn et al.'s (2002) experimental observations.     

Calibration of Weibull stress parameters using fracture toughness data

The Weibull stress model for cleavage fracture of ferritic steels requires calibration of two micromechanics parameters . Notched tensile bars, often used for such calibrations at lower-shelf temperatures, do not fracture in the transition region without extensive plasticity and prior ductile tearing. However, deep-notch bend and compact tension specimens tested in the transition region can provide toughness values under essentially small-scale yielding (SSY) conditions to support Weibull stress calibrations. We show analytically, and demonstrate numerically, that a nonuniqueness arises in the calibrated values, i.e., many pairs of provide equally good correlation of critical Weibull stress values with the distribution of measured (SSY) fracture toughness values. This work proposes a new calibration scheme to find which uses toughness values measured under both low and high constraint conditions at the crack front. The new procedure reveals a strong sensitivity to m and provides the necessary micromechanical values to conduct defect assessments of flawed structural components operating at or near the calibration temperature in the transition region. Results of a parameter study illustrate the expected values of m for a typical range of material flow properties and toughness levels. A specific calibration is carried out for a mild structural steel (ASTM A36). This revised version was published online in August 2006 with corrections to the Cover Date.     

The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity involves two material length parameters, namely the intrinsic material length land the mesoscale cell size l , which are on the order of a few microns and 0.1 m, respectively. Prior studies suggest that l has essentially no effect on the macroscopic quantities, but it may affect the local stress distribution. We demonstrate in this paper that there is a boundary layer effect associated with l in MSG plasticity, and the thickness of the boundary layer is on the order of   l ² big/l. By neglecting this boundary layer effect, a stress-dominated asymptotic field around a crack tip in MSG plasticity is obtained. This asymptotic field is valid at a distance to the crack tip between l and l(i.e., from 0.1 m to a few microns). The stress in this asymptotic field has an approximate singularity of r ^–2/3, which is more singular than not only the HRR field in classical plasticity but also the classical elastic Kfield (r ^–1/2). The stress level in this asymptotic field is two to three times higher than the HRR field, which provides an alternative mechanism for cleavage fracture in ductile materials observed in experiments.     

Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.     

Influence of threshold parameters on cleavage fracture predictions using the Weibull stress model

This study explores applications of three-parameter Weibull stress models to predict cleavage fracture behavior in ferritic structural steels tested in the transition region. The work emphasizes the role of the threshold parameters (_th and _{w – min}) in cleavage fracture predictions of a surface crack specimen loaded predominantly in tension for an A515-70 pressure vessel steel. A recently proposed procedure based upon a toughness scaling methodology using a modified Weibull stress (^* _w) extends the calibration scheme for the Weibull modulus, m, to include the threshold parameters. The methodology is applied to calibrate the Weibull stress parameter for the tested material and then to predict the toughness distribution for the surface crack specimen. While the functional relationship between ^* _w and m suggests a strong effect of the threshold stress, _th, on the calibrated m-parameter, the results show a remarkably weak dependence of fracture predictions on _th as does the dependence of fracture predictions on _w–min for this specimen.     

Temperature dependence of material length scale for strain gradient plasticity and its effect on near‐tip opening displacement

This paper investigates the temperature dependence of the material length scale in the conventional mechanism‐based strain gradient (CMSG) plasticity theory. The work reported here also examines the plastic strain gradient effect on the opening displacement near a sharp crack tip. The study examines the mechanical properties of two typical structural steels (S355 and S690) in onshore and offshore structures at two different temperatures (20 and 300 °C) through both the uniaxial tension test and the indentation test. The CMSG‐based finite element analysis then confirms a constant material length scale for these two steels at the two tested temperatures, despite the apparent temperature dependence of the macroscopic material parameters measured from the tension test. Using the calibrated material length scale, the subsequent numerical study demonstrates that the magnitude of the near‐tip crack opening displacement computed by the CMSG theory remains significantly lower than that computed from the classical plasticity.     

Temperature dependence of Weibull stress parameters: Studies using the Euro-material

This work demonstrates the temperature invariance of the Weibull stress modulus, m, for a 22Ni-MoCr37 pressure vessel steel through calibrations at two extreme temperatures of the ductile-to-brittle transition. This temperature invariance reflects the characterization of microcrack size distribution in the material described by the modulus. The calibrations performed here also demonstrate the clear dependence of the Weibull scale parameter, σ_u, on temperature. The increase of σ_u with temperature reflects the increase in microscale toughness of ferritic steels. The calibration procedure employs a three-parameter Weibull stress model which includes the effects of a minimum (threshold) toughness, K_min. The calibrations suggest that K_min increases gradually with temperature. Finally, an engineering procedure is presented to enable practical applications of the Weibull stress model for defect assessments. This procedure combines the demonstrated temperature invariance of m, a recently developed method for predicting the variation of σ_u with temperature using the Master Curve, and calibration of the Weibull stress parameters at one temperature. The (calibrated) temperature invariant m and the estimated σ_u as a function of temperature are used to predict the cumulative probability of fracture for several large datasets without direct calibration.     

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

An investigation of the loading rate dependence of the Weibull stress parameters

This paper examines the dependence of the Weibull stress parameters on loading rate for a 22NiMoCr37 pressure vessel steel. Extensive fracture tests, including both quasi-static and dynamic tests, are conducted using deep- and shallow-cracked SE(B) specimens. The fracture specimens are carefully prepared to ensure the crack fronts are placed at the location where the material is homogeneous. Three dynamic loading rates (in terms of the stress intensity factor rate, in the low-to-moderate range are considered. The load-line velocities for the dynamic tests are chosen so that the resulted values for the deep- and shallow-cracked specimens are the same. Independent calibrations performed at each loading rate (quasi-static and the three dynamic loading rates) using deep- and shallow-cracked fracture toughness data show that the Weibull modulus, m, is invariant of loading rate. The calibrated m-value is 7.1 for this material. Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The demonstrated loading rate invariant of m, when combined with the master curve for dynamic loading, can provide a practical approach which simplifies the process to estimate σ_u as a function of loading rate.     

C0 solid elements for materials with strain gradient effects

Experiments conducted by various researchers in the past few decades have shown that materials display strong size effects when the material and characteristic length scales associated with non‐uniform plastic deformation are of the same order at micron and submicron levels. The state of stress under such a condition was observed to be a function of both strain and strain gradient. The meso‐scale constitutive relation taking into account Taylor dislocation theory is briefly described. The conventional theory of mechanism‐based strain‐gradient (CMSG) plasticity incorporating the intrinsic material length scale is adopted in the formulation of a series of C⁰ solid elements of 20–27 nodes. The model is implemented in ABAQUS, a finite element package via a user subroutine. Convergent studies have been carried for the series of elements with classical as well as CMSG plasticity theories. Numerical results on a bar under constant body force and indentation at submicron level reinforce the observation that materials are significantly strengthened for deformation at micron and submicron levels and the effects of strain gradient cannot be ignored without significant loss of the accuracy of the results. Copyright © 2005 John Wiley & Sons, Ltd.     

Loading rate effects on parameters of the Weibull stress model for ferritic steels

This study investigates the effects of loading rate on parameters of the Weibull stress model for prediction of cleavage fracture in a low strength, strongly rate-sensitive A515-70 pressure vessel steel. Based on measured, dynamic fracture toughness data from deep- and shallow-cracked SE(B) specimens, the calibrated Weibull modulus (m) at shows little difference from the value calibrated previously using static toughness data. This newly obtained result supports the hypothesis in an earlier study [Gao X, Dodds RH, Tregoning RL, Joyce JA. Weibull stress model for cleavage fracture under high-rate loading. Fatigue Fract Engng Mater Struct 2001;24:551-64] that the Weibull modulus likely remains rate independent for this material over the range of low-to-moderate loading rates. Additional experimental and computational results for higher rates show that a constant m-value remains applicable up to the maximum loading rate imposed in the testing program . Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m, and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The predicted cumulative probability for cleavage fracture exhibits a strong sensitivity to small changes in σ_u. Consequently, σ_u must be calibrated using dynamic fracture toughness data at each loading rate of interest in an application or selected to make the Weibull stress model predict a dynamic master curve of macroscopic toughness for the material.     

运用应变梯度塑性理论模拟颗粒增强铝合金强度及延伸率的尺寸效应

运用基于细观机制的应变梯度塑性理论模拟了不同晶粒尺度、不同第二相颗粒直径及体积分数的铝合金应力应变曲线.结果表明,在相同条件下,随着第二相颗粒直径的减小,或随着第二相体积分数的增加,合金的强度明显增强.相反,随着第二相颗粒体积分数的增加,或随着第二相颗粒直径的减小,合金的均匀延伸率均有所下低.同时对不同晶粒尺寸的铝合金应力应变相应的分析表明,第二相颗粒分布的不均匀性对其力学性能也有一定的影响.     

Calibration of the Weibull stress scale parameter, σu, using the Master Curve

This work proposes that the Weibull stress scale parameter, σ_u, increases with temperature to reflect the increasing microscale toughness of ferritic steels caused by local events that include plastic shielding of microcracks, microcrack blunting, and microcrack arrest. The Weibull modulus, m, then characterizes the temperature invariant, random distribution of microcrack sizes in the material. Direct calibration of σ_u values at temperatures over the DBT region requires extensive sets of fracture toughness values. A more practical approach developed here utilizes the so-called Master Curve standardized in ASTM Test Method E1921-02 to provide the needed temperature vs. toughness dependence for a material using a minimum number of fracture tests conducted at one temperature. The calibration procedure then selects σ_u values that force the Weibull stress model to predict the Master Curve temperature dependence of K_Jc values for the material. At temperatures in mid-to-upper transition, the process becomes more complex as fracture test specimens undergo gradual constraint loss and the idealized conditions of high-constraint, small-scale yielding assumed in E1921-02 gradually degenerate. The paper develops the σ_u calibration process to incorporate these effects in addition to consideration of threshold toughness effects and the testing of fracture specimens with varying crack-front lengths. Initial illustrations of the calibration process for simpler conditions, i.e. 1T crack-front lengths, use the temperature dependent flow properties and a range of toughness levels for an A533B pressure vessel steel. Then using the extensive fracture toughness data sets for an A508 pressure vessel steel generated recently by Faleskog et al. [Engng. Fract. Mech., in press], the paper concludes with calibrations of both m and σ_u over the DBT region and assessments of the Master Curve calibration approach developed here.     

Simulation of impact energy in functionally graded steels by mechanism-based strain gradient plasticity theory

Charpy impact energy of functionally graded steels composed of graded ferritic or austenitic layers which were produced by electroslag remelting in both crack divider and crack arrester configurations has been modeled by finite element method. The yield stress of each layer was related to the density of the dislocations of that layer and by assuming Holloman relation for the corresponding stress-strain curve, tensile strength and tensile strain of that layer were determined. Cubic elements were joined together to build the standard Charpy impact specimen. The data used for each cubic element in finite element modeling was the predicted stress-strain curve obtained from strain gradient plasticity theory. After applying the impact loading, a relatively good agreement between experimental results and those obtained from simulation was observed.     

Constraint effects on the elastic–plastic fracture behaviour in strain gradient solids

ABSTRACT Crack‐tip constraint effects (or T‐stress effects) on the elastic–plastic fracture behaviour in strain gradient materials are analysed in the present study. The T‐stress effects on the stress distributions along the plane ahead of the stationary and growing crack tip, respectively, are analysed by using the Fleck and Hutchinson strain gradient plasticity formation. For a steadily growing crack, the T‐stress effects on the steady‐state fracture toughness are analysed by adopting the embedded fracture process zone model. In addition, the analysis for the growing crack is applied to an interfacial cracking experiment for a metal/ceramic system, and the material length‐scale parameter appearing in the strain gradient plasticity theory is predicted. In the present analyses, a new finite element method specially designed for strain gradient problems by Wei and Hutchinson is adopted.     

Computations of fatigue crack growth with strain gradient plasticity and an irreversible cohesive zone model

Computations of fatigue crack growth with a first-order strain gradient plasticity (SGP) model and an irreversible cohesive zone model are reported. SGP plays a significant role in the model predictions and leads to increased fatigue crack growth rates relative to predictions with classical plasticity. Increased magnitudes of tractions and material separation at the crack tip together with reduced crack closure appear as the cause for accelerated crack growth in SGP. Under plane strain conditions SGP appears as an essential feature of the development of the crack closure zone. Size effects are explored relative to changes in internal material length scale as well as to structural length scales.     

Size effects in phenomenological strain gradient plasticity constitutively involving the plastic spin

In the small deformation range, we consider and discuss the phenomenological (or isotropic) “higher-order” theory of strain gradient plasticity put forward in Section 12 of Gurtin [1], which includes the dissipation due to the plastic spin through a material parameter called χ. In fact, χ weighs the square of the plastic spin rate into the definition of an effective measure of plastic flow peculiar of the isotropic hardening function. Such a model has been identified by Bardella [2] as a good isotropic approximation of a crystal model to describe the multislip behaviour of a single grain, provided that χ be set as a specific function of other material parameters involved in the modelling, including the length scales. The main feature of the underlying gradient approach is the accounting for both dissipative and energetic strain gradient dependences, with related size effects. The dissipative strain gradients enter the model through the definition of the above mentioned effective measure of plastic flow, whereas the energetic strain gradients are involved in the modelling by defining the defect energy, a function of Nye’s dislocation density tensor added to the free energy to account for geometrically necessary dislocations (see, e.g., Gurtin [1]). By exploiting the deformation theory approximation, we apply the model to a simple boundary value problem so that we can discuss the effects of (a) the criterium derived by Bardella [2] for choosing χ and (b) non-quadratic forms of the defect energy. We show that both χ and the nonlinearity chosen for the defect energy strongly affect quality and magnitude of the energetic size effect which is possible to predict.     

Simulation Charpy impact energy of functionally graded steels by modified stress-strain curve through mechanism-based strain gradient plasticity theory

In the present work, Charpy impact energy of functionally graded steels produced by electroslag remelting composed of graded ferritic or austenitic layers in both crack divider and crack arrester configurations has been modeled by finite element method. The yield stress of each layer was related to the density of the statistically stored dislocations of that layer and assuming by Holloman relation for the corresponding stress-strain curves, tensile strengths of the constituent layers were determined via numerical method. By using load-displacement curves acquired from instrumented Charpy impact tests on primary specimens, the obtained stress-strain curves from uniaxial tensile tests were modified. The data used for each layer in finite element modeling were predicted modified stress-strain curves obtained from strain gradient plasticity theory. A relatively good agreement between experimental results and those obtained from simulation was observed.     

A micro‐mechanical damage model based on gradient plasticity: algorithms and applications

As soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computation is significantly affected by the element size. Without remedying this effect in the constitutive model one cannot hope for a reliable prediction of the ductile material failure process. In the present paper, a micro‐mechanical damage model coupled to gradient‐dependent plasticity theory is presented and its finite element algorithm is discussed. By incorporating the Laplacian of plastic strain into the damage constitutive relationship, the known mesh‐dependence is overcome and computational results are uniquely correlated with the given material parameters. The implicit C¹ shape function is used and can be transformed to arbitrary quadrilateral elements. The introduced intrinsic material length parameter is able to predict size effects in material failure. Copyright © 2002 John Wiley & Sons, Ltd.     

The effect of the threshold stress on the determination of the Weibull parameters in probabilistic failure analysis

In the analysis of brittle materials and components the probability of failure is commonly modelled using a two-parameter Weibull distribution. Occasionally, a three-parameter model is used when the material shows significant threshold behaviour. In this paper two methods for determining the three-parameter constants are discussed. Two theoretical two- and three-parameter distributions are then analysed to examine the number of samples needed to determine the parameters accurately. The two-parameter models are the best fits of the three-parameter models and their failure distributions are very similar to the three-parameter distributions. It is concluded that far more specimens need to be tested than is usually the case to be confident that the correct distribution has been found.