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Continuum Based Micro-Models for Ultra Low Cycle Fatigue Crack Initiation in Steel Structures

A.M. Kanvinde1 and G.G. Deierlein2

1Department of Civil and Environmental Engineering, University of California, Davis, CA 95616. PH (530) 752-2605, FAX (530) 752-7872, email: kanvinde@ucdavis.edu2Department of Civil and Environmental Engineering, Stanford University, CA 94305. PH (650) 723-0453, FAX (650) 725 9755

Abstract

Failure due to fracture or low-cycle-fatigue in steel moment connections is an important limit-state in earthquake-resistant design. Current standards and codes deal with this problem based on semi-empirical, experimental approaches. In reality, such failures involve the complex interactions of material microstructure, irregular loading histories, and uncertainties in crack initiation and subsequent propagation processes. Traditional fracture and fatigue mechanics cannot capture many of these interactions and uncertainties, presenting a significant obstacle to accurate simulation of structural performance. Modern computational technologies coupled with novel micromechanical fracture and fatigue models enable a detailed and realistic simulation of the processes leading to crack initiation and propagation. A recent study by the authors postulated some new mechanisms and proposed micromechanics-based models to predict ductile crack initiation under ultra-low cycle fatigue (ULCF) experienced by structures in earthquakes, which is very different from high-cycle fatigue experienced by bridges. The paper discusses these models and a methodology to apply them by combining basic material microstructural data with complex loading histories to accurately predict seismic performance and failure.

Introduction

Failure due to fracture or fatigue is an important limit state in steel structures and components. Brittle fractures observed in many steel moment resisting frames during the 1994 Northridge earthquake brought this issue to the forefront of steel research. Studies such as the SAC Steel Project resulted in many improvements to the existing connection details by suggesting details to minimize fracture toughness demands, while increasing capacity by using tougher materials, and better quality control for

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welding. Aside from offering solutions to mitigate brittle fracture in pre-Northridge type connections, studies such as the SAC project raise many questions about the state of the art in fracture and fatigue prediction in steel structures.

For example, even the post-Northridge connections that are resistant to brittle fracture, may have potential to develop ductile crack initiation (Stojadinovic, 2000). Traditional fracture mechanics approaches suffer from limitations that hamper their effectiveness to predict fracture in conditions with large inelastic strains. Another issue is that fatigue experienced by structures during earthquakes typically causes failure in less than ten cycles, as compared to hundreds or thousands of cycles for Low and High Cycle Fatigue. Thus, fatigue experienced by moment connections during earthquakes can be termed Ultra-Low Cycle Fatigue (ULCF), and apart from semi-empirical experiment-based predictions, there are limited models available to simulate this phenomenon.

Other structural design conditions, such as gusset plates for braced frames, or even braces themselves, are potential candidates for ULCF or ductile fracture. Some of these situations, such as bolted gusset plates, may not have sharp cracks or flaws, but rather less severe stress concentrations. These situations are unsuitable for the application of conventional fracture mechanics, which typically requires the presence of a sharp crack or flaw.

Conventional fracture mechanics approaches such as the J-integral or the Crack-Tip Opening Displacement (CTOD) can reliably examine the fracture process in situations where the yielding is limited in extent, and when a sharp crack is present.However, they are less suited to conditions of large-scale yielding, cyclic loading, or details with the absence of cracks as discussed above. For such situations, the tendency is require physical testing to help ensure satisfactory performance. Often, such an approach can discourage innovation in design due to the associated project delays and expense of physical testing.

The migration of the engineering and research community towards performance based engineering underlines the importance of fundamental model based simulation. This paper aims to describe one such fundamental approach for modeling the ULCF process in steel structures or components.

The paper begins by introducing the existing approaches in fracture and fatigue mechanics, and highlighting their limitations in the context of earthquake engineering applications. Next, it presents mechanisms and models for ductile fracture as well as ULCF that are based on considerations of micromechanical response. Finally, a methodology for applying these fundamental models to performance assessment of steel moment connections is presented, while outlining the possible issues, limitations and directions for future work.

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Existing approaches in fracture and fatigue mechanics

Sharp cracks such as the one shown in Figure 1 give rise to singularities in stress fields, meaning that the stress at the crack tip rises to infinity. This precludes the use of a critical stress measure as a criterion for fracture. Traditional fracture mechanics deals with this issue by using the energy release rate (Griffith, 1920) as the driving force behind the fracture phenomenon.

For situations with minimal yielding, this energy release rate can be related to the stress intensity factor IK , which can then be compared to the critical stress intensity factor ICK , which is assumed a material property. This is the basis of Linear Elastic Fracture Mechanics (LEFM). Under situations with somewhat larger plastic zones, the stress intensity factor no longer bears unique correlation with near-tip stresses and strains, invalidating the use of the IK as a reliable fracture predictor.

In such elastic-plastic situations, better estimators of the energy release rate, such as the J-integral (Rice, 1968) or the CTOD (Wells, 1963) are more reliable fracture predictors. These parameters, which are more generally applicable than the IK , constitute Elastic Plastic Fracture Mechanics (EPFM).

Both LEFM and EPFM address the problem of crack initiation and growth based on far-field stresses. In situations of large-scale yielding, these far-field stresses often lose correlation with the near-tip stresses and strains, consequently invalidating the use of either LEFM or EPFM.

Another key requirement for the application of LEFM or EPFM is the presence of an initial sharp crack or flaw in the material. Many realistic situations in structural engineering (e.g. bolted connection details) have no such sharp cracks, forcing ad hocand often unrealistic introduction of hypothetical flaws into analysis models. Current computing technology makes possible the characterization of near-tip stress and strain interactions that drive the micromechanics of the fracture process. The calculated stresses and strains can be combined with fundamental material failure models that account for interactions at a microstructural level to predict ductile fracture. Such models, aiming to simulate the micromechanism of the fracture process are not limited by large-scale yielding or geometric effects, such as specimen thickness – which is a major issue in traditional single parameter fracture mechanics. The application of one such model, the Stress Modified Critical Strain (SMCS), is presented in this paper.

Figure 1. Sharp crack in infinite elastic plate - LEFM

Infinitely sharp crack of length 2a

Remote stress - σ

Singular stressdistribution

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Having outlined the fracture issue, it is important to recognize that structures typically experience large inelastic cyclicdeformations during earthquakes. In fact, such deformations are the primary mode of energy dissipation in structural systems like steel moment frames. Figure 2 shows a schematic of a typical deformation history that structures experience during an earthquake (Kanvinde, 2003). This involves fewer than ten cycles of large inelastic deformations. Conventional fatigue mechanics has not been directed towards this problem, and consequently, the understanding of these processes is limited.

Micromechanical Models for Ductile Crack Initiation – SMCS

Ductile steel details have the potential to exhibitductile fracture accompanied by large-scale yielding. The fundamental mechanism responsible for theinitiation of this type of fracture has been established as one involving microvoid nucleation, growth and coalescence, shown schematically in Figure 3. Most steels contain secondary particles or inclusions such as carbides that sit in the steel matrix, around which voids nucleate under applied stresses. After nucleation, plastic strain and hydrostatic stress cause the voids to grow. Initially, the voids grow independent of one another, but upon further growth, neighboring voids interact to concentrate plastic strain along a certain plane of voids. At this point local necking instabilities cause the voids to grow suddenly, forming the macroscopic fracture surface.

Rice and Tracey (1969) derived an expression for the growth of a single spherical void in an elasto-plastic solid. According to their equation, the growth is governed by the imposed equivalent plastic strain and the triaxiality ratio –

( )exp 1.5 . p

dRC T d

Rε= (1)

Where R is the instantaneous void radius, pε is the equivalent plastic strain, and the

triaxiality m eT σ σ= is a ratio of the mean stress and the effective or von Mises

Figure 3. Mechanism of ductile crack initiation

Figure 2. Displacement time history during an earthquake

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stress, and C is a constant. Based on these concepts, Hancock and Mackenzie (1976),proposed a simplified model – the Stress Modified Critical Strain Model (SMCS) for capturing the void growth and coalescence mechanism in metals. According to this model, ductile crack initiation occurs when the equivalent plastic strain exceeds a critical value of the plastic strain corresponding to the void size exceeding a critical void size. Applying this concept to (1), and simplifying it, the critical plastic strain can be expressed as a function of the triaxiality ratio –

( )exp 1.5criticalp p Tε ε α> = ⋅ (2)

The constant α in (2) is a material parameter indicative of the material resistance to ductile crack initiation. The SMCS model captures the ductile crack initiation mechanism by taking into account the adverse effect of stress triaxiality on void growth and coalescence. A higher triaxiality causes a lower value of critical plastic strain. The toughness index α can be calibrated based on notched round bar testssuch as the one shown in Figure 4 and complementary FEM analyses. The notched geometry introduces a triaxial stress state in the material. The SMCS model is also dependent on a length-scale parameter. Kanvinde and Deierlein (2004) describes the calibration procedure in detail. Ductile crack initiation is predicted to occur if the condition in (2) is satisfied over a material volume described by the length scale. The length scale is typically more important in situations with sharp cracks and high stress gradients. One can compare the SMCS model for ductile fracture as being analogous to the Mises model for yielding, in that it considers interactions of various physical quantities to predict fracture.

Mechanisms and Models for Ultra Low Cycle Fatigue (ULCF )

The area of research concerning Ultra-Low Cycle Fatigue (ULCF) is of great importance to structural and earthquake engineers. ULCF involves extremely large plastic strains and very few (of the order of 10) cycles to failure. Being more similar to ductile fracture processes, mechanisms that cause ULCF are very distinct from those that cause traditional low-cycle fatigue, which involves cycles in the range of several hundred to a thousand. ULCF is not very relevant to mechanical or aerospace

Figure 4. Notched Tensile Tests for SMCS Calibration

1

Round Bar

12 ±

38.1 76.2

25.4 mm gage Extensometer

.

All dimensions in mm

25.4

r* ± 0.1

6.3 ± 0.10.7

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applications, where components are controlled by low-cycle or high-cycle fatigue, consequently receiving relatively less attention in research.

As described earlier, for monotonic loading, ductile crack initiation is dependent on the processes of void growth and coalescence. For ULCF, the situation is similar but for two key differences. First, the voids encounter alternating excursions of positive and negative triaxiality, which cause alternate expansion and shrinkage or squeezing of the voids that have already nucleated. Second, due to the cyclicity of loading, damage accumulates in the necked intervoid ligament, which reduces the critical void diameter at which coalescence occurs. In other words, a smaller void ratio (as compared to the monotonic case) might trigger coalescence due to damage accumulation.

Figure 5 compares fractographs from a monotonic failure – Figure 5(a) to a ULCF failure – Figure 5(b)for the same material after five cycles. It is very interesting to note that the monotonic failure has much deeper dimples, indicating a larger critical void ratio. In contrast, the cyclic failure shows shallow dimples with pockets of cleavage type fracture. This supports the postulated mechanism, that under cyclic loading smaller void ratios lead to failure due to cyclically accumulated damage.

Equation (1) was originally developed for monotonic tension only – Rice and Tracey (1969). To account for cycling, the triaxiality term (T) in the exponent needs to be modified to its absolute value. If the triaxiality is negative, the void will shrink under plastic straining, and if it is positive, the void will grow. Thus, to estimate the void fraction due to cyclic loading, we need to integrate the more general equation, where the sign(T)term takes into account the sign of the triaxiality –

Figure 5. (a) Deep dimples from monotonic loading and (b) Shallower dimples from

cyclic loading

All plastic strain during these increments adds up to cε

All plastic strain during these increments adds up to tε

(-)

Tri

axia

lity

(+)

Equivalent Plastic Strain

Positive Triaxiality (Voids Grow)

Negative Triaxiality (Voids Shrink)

Figure 6. Triaxiality versus Plastic Strain for Reversed Cyclic Loading

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( )( ) exp 1.5 . p

dRsign T C T d

Rε= ⋅ ⋅

(3)

Figure 6 shows a schematic plot of the equivalent plastic strain versus the triaxiality at the critical location in a typical specimen, such as the notched round bar during reversed cyclic loading, illustrating that when triaxiality is positive, the voids are assumed to grow, and when it is negative they are assumed to shrink.

However, traditional measures of strain that are conventionally used for fracture prediction do not factor in this triaxiality effect. Moreover, faced with a multiaxial stress-strain situation under cyclic loading, it is difficult to distinguish a tensile cycle from a compressive cycle. These limitations of existing strain measures prompt the introduction of a new strain measure, termed the significant plastic strain, which has a closer physical relation to the void growth process.

At any loading stage, we can define tensile equivalent plastic strain (at a given material point) tε , as the integration of equivalent plastic strain when the triaxiality

was positive and the compressive equivalent plastic strain cε as the integration of

equivalent plastic strain when the triaxiality was negative. The difference between these is defined as the significant plastic strain –

ct εεε −=* (4)

The difference in strain quantities is considered “significant” because it is an indicator of how much the material has been “stretched”, somewhat analogous to a residual displacement in a uniaxial tension coupon.

Similar to the stress and strain quantities in the SMCS model, the significant plastic strain is a local quantity, to be evaluated at each material location. In tests of simple notched tensile bars, the significant plastic strain increases during an excursion of global tension, and decreases during an excursion of global compression. After substantial mathematical derivation, which is omitted here for clarity (Kanvinde and Deierlein, 2004) an equation similar to (2) is generated to predict ULCF initiation. This equation predicts that ductile crack initiation will occur if the significant plastic strain exceeded a critical value. This, in concept, is similar to the SMCS equation, which predicts ductile initiation to occur of the equivalent plastic strain was to exceed a critical value. However, there are two key differences between (2) and (5). First, equation (5) employs the significant plastic strain that captures the effects of the alternating effects of void expansion and squeezing. Second, the critical value of the critical plastic strain is based on the deterioration of the significant plastic strain based on a damage function ( )f D , which accounts for the deteriorating effects of cyclic loading discussed earlier. The choice of the damage function and the damage variable D are discussed in detail in Kanvinde and Deierlein (2004).

* * ( ). criticalcritical pf Dε ε ε> = (5)

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Thus, an approach for predicting ductile crack initiation under ULCF is presented. The new approach does not suffer from the assumptions of large scale yielding or monotonic loading; and it does not require the presence of an initial flaw or crack. A methodology for applying this approach to structural components is described in the next section.

Methodology for Implementation

The purpose of the micromechanics-based models presented in this paper is to realistically simulate fracture behavior in steel structures under varied conditions. Due to the computational expense and expertise required to implement these, in their current state, it is difficult to envision the use of these models for design of connections or structures. However, these models can be an effective substitute or complement to pre-qualification tests described in FEMA (2000). For example, models such as these can be used to extend results of experiments to more general scenarios, such as deeper beam sections. Other uses could involve a completely analytical evaluation of the effect of details such as floor slabs on the fracture performance of connections, which might be outside the scope of an experimental study. However, an effective implementation of such models includes many components (Figure 7 illustrates these schematically) –

1. Identification of the appropriate mechanisms of failure – Unlike continuum-based macro-models that deal with fracture and other issues at a macro-level, micromechanics-based models are sensitive to the mechanism of fracture being encountered, and might not be valid under situations that exhibit different mechanisms from those assumed. For example, a high-constraint situation like a deep crack in a thick plate might trigger transgranular cleavage, and an appropriate model might have to be activated in that case. The models described in this paper are restricted only to microvoid coalescence type mechanisms.

2. Calibration of micromechanics-based models – The micromechanics-based models, such as the ones described in this paper, have parameters that require calibration. Often, other tests (such as Charpy tests or tension tests) can be substituted as calibration tests, by running complementary finite element analyses. However, depending on the specific conditions, it might be necessary to design tests specifically geared towards capturing specific behavior that the chosen model is intended to simulate.

3. Calibration of Material Constitutive Relationships – The calibrated parametersof the micromechanics-based models can be dependent on the modeling parameters of the material constitutive relationship, and consequently, the choice of an appropriate constitutive model and its careful calibration is a very important step in the implementation process. Especially in situations of cyclic loading, calibration of these relationships can be an issue, if cyclic calibration data is not available. For the study described in this paper, a combined isotropic-kinematicJ2-plasticity model (Lemaitré, and Chaboche, 1990) was used.

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4. Finite Element Modeling – For effective and economic implementation of these models, special consideration needs to be given to certain issues. Though not described in detail in this paper, the characteristic length of the steel material can be an important variable in situations with sharp cracks, where many element lengths might often need to be included within this length scale to effectively capture the stress and strain gradients. If the probable location of fracture is not known beforehand, it might be necessary to use a fine mesh over large areas of the finite element model, significantly adding to the computational cost.

5. Post Processing – Due to the large size of the problem, and the somewhat complex criteria for ductile crack initiation, a comprehensive post-processing program or framework must be implemented to interpret the cyclic stress strain histories at every material point to predict ULCF fracture.

Limitations and Future Research Needs

The micromechanics-based modeling approach provides an attractive, physics-based approach for predicting ULCF. However, because the implementation techniques for such a methodology are in their infancy, the approach has some limitations. Some of the important ones are –

Material Microsrtucture

ε

σ

Constitutive Relationships

Tests

FEM

Identification of Micromechanism and Calibration of Constitutive and

Micromechanical Models

Application of micromechanical

initiation models to structural

components through detailed finite

element meshes.

1. Accurate, assumption-free evaluation of performance

2. Better connection guidelines

3. Safer structures

Figure 7. Methodology of application of Micromechanics-based models to predict performance

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1. Sensitivity to length scale issues – Determination of the characteristic length for the micromechanics-based models is a more subjective issue than calibration of the other parameters, and there are many approaches for this purpose. A unified, unambiguous approach for the length scale determination is required.

2. Computational Expense – For large-scale problems, such as beam column connections, 3-dimensional cyclic finite element models are likely to require extensive computational resources. Application of newer technologies involving parallel and cluster computing have the potential of addressing this issue.

3. Crack Propagation Modeling – Though ULCF crack initiation is an important event, often it is crack propagation and transition to brittle failure that causes catastrophic failure of moment connections. Therefore, the tools for predicting initiation need to be combined with models for predicting this ductile-brittle transition to evaluate potential of catastrophic failure in the event of ductile initiation.

Conclusions

This study describes a relatively new approach to predicting ductile crack initiation due to Ultra Low Cycle Fatigue (ULCF). ULCF involves very large strains and few cycles; and, it exhibits mechanisms distinct from Low Cycle Fatigue which has been extensively investigated for mechanical and aerospace applications. Traditional fracture and fatigue mechanics are based on assumptions that may not necessarily hold during ULCF-type situations. Based on tests, finite element analysis, micrographs and intuitive reasoning, new mechanisms and simulation models are proposed for ULCF.

Continuum based micro-models, that have the potential to overcome some of these inadequacies of the traditional models are presented. These models aim to capture the fundamental processes of void growth and coalescence that cause ductile fracture. Based on experimental and fractographic studies, it is postulated that mechanisms causing ULCF are different from those causing Low-Cycle Fatigue, and involve microvoid growth, coupled with a damage accumulation process. A new strain measure – the significant plastic strain is suggested as a predictor of ULCF. Experimental data shows good agreement with predictions of ULCF based on such an approach.

A methodology is presented to simulate performance of steel moment connections subjected to earthquake type loadings. It is recognized that the new models and the methodology for implementing them to large-scale problems are still developing, and some of the key current limitations are identified. These include the high computational demand, and research needs in the areas of crack propagation modeling and length scale determination.

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Overall, the continuum based micro-models offer an exciting alternative to traditional models, with the potential to take advantage of rapidly improving computing hardware to provide engineering solutions based on the simulation of fundamental physical processes.

Acknowledgements

The authors are grateful to the National Science Foundation (Grant No. CMS 9988902), for funding this research. Additional support was provided by the Steel Structures Development Center of the Nippon Steel Corporation (Futtsu, Japan), which provided steel materials, machining services, and fracture data, and by donations of steel material from the Garry Steel Company (Oakland, CA) and the ATLSS Engineering Research Center (Bethlehem, PA). All of these are greatly appreciated.

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