IA'96, Proceedings of International Seminar on Quasi-Impulsive Analysis, Osaka, Japan, pp. A6.1-15, K. Wakiyama, E. Tachibana, K. Imai and T. Kitano Eds. , Nov. 20-22, 1996

DYNA3D ANALYSIS OF DYNAMIC FRACTURE OF WELDMENTS

Mr. Steven W. Kirkpatrick Dr. Jacques H. Giovanola

Dr. Jeffrey W. Simons SRI International, 333 Ravenswood Avenue

Menlo Park, CA, USA 94025

SUMMARY

A local ductile fracture model was developed and implemented into the finite element code DYNA3D. The local fracture model calculates damage in each element as a function of the stress state and plastic strain increments. Crack propagation is modeled by discrete separation of the elements through the fully damaged regions of the mesh. We applied the fracture model to simulate the behavior of dynamic fracture experiments on welded steel specimens. The local damage model successfully reproduced the experimentally observed fracture behavior including following the crack path through the steel plate. We then used the method to illustrate how to design stronger and safer weldments for metal structures subjected to earthquake loadings.

1. INTRODUCTION

This paper presents the results of an on-going research effort in which we use a simple local fracture model in an attempt to understand and predict the fracture behavior of steel weldments of different sizes and geometries, fabricated using different welding processes. The load bearing capability and the fracture resistance of weldments are difficult to predict because weldments have inhomogeneous microstructures, residual stresses, and often complex geometries. As a result of the limited metallurgical control achievable in weldments, their mechanical properties may differ from those of the base metal being joined. In addition, the geometry of the weldment often results in significant stress concentrations in the vicinity of the weld joint. Thus, weldments may often be weak points in a structure, and it is important to evaluate and optimize their resistance to fracture.

Welding is an essential process in assembling large modern steel structures. Ideally, the moment-resistant connections are designed to promote the formation of a plastic hinge in the beam, away from the column connection, during seismic loading. This ductile energy absorbent behavior results in a damage tolerant design for the welded steel frame construction. However, recent evaluation of earthquake damage in Northridge and Kobe has shown that connections do not always perform as desired and damage occurred in many welded beam-column connections.

For most practical load distribution patterns, the bending moment is maximum at the connection. Further, for a given applied moment, the reduction in effective cross section (from the web cutout or the bolted beam web) associated with current designs causes the maximum stresses in the connection to be higher than those that would act in the full beam cross section. As a result, yielding occurs in and around the connection region first. In addition, the geometrical details of the connection (such as web cut out, backing bar, beam flange chamfer) produce stress concentrations which result in plastic strain localization within the connection. The mismatch in material properties for the various regions of the welded joint (column flange, heat-affected zones, weld metal, beam flange) can also contribute to plastic strain localization, elevation of stresses, and consequently damage localization.

The strength and ductility of various welded beam-column joints need to be evaluated both for analysis of structural integrity of existing buildings as well as for design of improved construction processes. Because of the number of variables involved, and because these variables may often have a coupled effect on the fracture behavior of the weldment, a strictly experimental investigation would not be cost effective and may not lead to a sufficient understanding. Therefore, a simulation technique capable of predicting damage and failure of welded connections represents a powerful and cost-effective tool for such an engineering study.

Classical linear elastic and elastic-plastic fracture mechanics (LEFM and EPFM, respectively) have been used to assess the margin of safety against fracture of welded structural joints (Schwalbe, 1992; Harrison and Anderson, 1988; Blauel and Schwalbe, 1987; Toyoda, 1985). In particular, attention has focused on the presence of cracks in the so-called "local brittle zone," to evaluate the toughness of this vulnerable region of the weldment, and the possibility for a crack initiated in this region to arrest in tougher adjacent material. Although classical fracture mechanics approaches have been useful for many weldment applications, they have a number of limitations that hinder their more widespread use and affect the reliability of their predictions. Among the limitations inherent to the fracture theories (constitutive assumption, state of stress assumptions, geometry and dimensional requirements) are that the presence and location of a crack be postulated a priori and the difficulty in accounting for variations in mechanical and fracture properties within the weldment.

Damage mechanics or so-called local fracture mechanics approaches (LFM) offer an alternative for solving weldment design and reliability problems as well as the possibility of obtaining more detailed and eventually more reliable analysis results than can currently be obtained with classical fracture theories. LFM models the microstructural deformation and failure processes leading to fracture in terms of continuum parameters averaged over a small volume of material (Beremin, 1981; Chaboche et al., 1986; Lemaitre, 1986; Curran et al., 1987; Giovanola and

Rosakis, 1992). In contrast to LEFM or EPFM, which characterize fracture in terms of the boundary conditions of the fracture process zone while ignoring the details of the processes occurring in that zone, LFM focuses on the evolution of the process zone itself. Although LFM may initially seem more complex to formulate and more difficult to apply than LEFM/EPFM, it is potentially more versatile and more general than the latter approaches. Some of the earlier attempts to apply LFM to weldments or inhomogeneous materials include the work of Fontaine and Maas (1987), Matic and Jolles (1988, 1989), and Devaux et al. (1989).

In previous studies, the authors have applied the local damage model for analysis of T-weldments dynamically loaded with explosives (Giovanola and Kirkpatrick, 1992, 1993). In addition, the local damage model has been used to investigate the scaling behavior of relatively small laboratory fracture experiments such as compact tension specimens or blunt notch bend bar fracture specimens (Giovanola, et al., 1996). The purpose of this study is to illustrate both the technical challenges and potential benefits of applying LFM to applications of earthquake type loadings on welded steel beam-column type structures used in the construction of many modern buildings.

In the first part of the paper, we discuss implementation of the local fracture model in the DYNA3D finite element code and summarize previous applications to fracture of weldments. We also illustrate how we can use the model to assess the influence of changes in weldment geometry or material properties on the fracture behavior. In the second part, we present an example simulation of a welded beam column joint under a cyclic plastic loading. This simulation is representative of full scale laboratory experiments used to evaluate the performance of welded beam-column joint geometries used in steel frame construction to earthquake type loads. Finally, we discuss the outstanding issues related to the application of LFM to the analysis of the welded beam-column structures and outline possible solutions.

2. WELDMENT FRACTURE MODEL

Fracture models and testing methods to evaluate the fracture resistance of welded steel joints are needed, for practical and cost saving reasons, such that the fracture response of large welded structures can be evaluated from analyses and experiments on small scale replicas. To meet this objective, we developed (1) a weld fracture model to estimate the fracture resistance of weldments of different sizes and produced with different materials or welding processes, and (2) simple static and dynamic test procedures to evaluate the fracture resistance of various metals and welded T-joints.

As illustrated in Figure 1, the weldment fracture model consists of three components: (1) a material damage model formulated in terms of plastic deformations weighted by a stress state

function; (2) a geometric and strength model of the weldment, based on metallographic observations and hardness measurements; and (3) a finite element formulation that implements these models with an algorithm permitting independent degrees of freedom for nodes on either side of the calculated fracture path. In this approach, each region of the welded joint is modeled separately and can have different fracture and strength properties.

Figure 1. Elements of the weldment fracture model.

As implemented, the fracture model is the simplest form of a ductile fracture criterion (Mudry, 1985). It assumes that failure of a material location occurs when the damage within a surrounding microstructural characteristic volume (VMIC) exceeds a critical value. Mathematically, the damage function in the failure criterion can be written in the form

D = ⌡⎮⌠

dε peq

εc(σmean/σeq) = 1 over VMIC ≈ (RMIC)3 (1)

where D is the normalized damage parameter, dε peq is an increment in plastic strain, and

εc(σmean/σeq) is the critical failure strain as a function of the stress triaxiality, defined as the

ratio of the mean stress to the equivalent stress. This critical strain function can be determined by a series of notched tensile tests with specimens of varying notch radii (see for example, MacKenzie etal., 1977). VMIC is a characteristic volume of the material, which can be interpreted as the critical microstructural process zone. In turn, RMIC, the representative linear dimension of the volume VMIC, can be associated with microstructural dimensions such as grain size or spacing of the microvoid nucleating inclusions. RMIC is therefore a constant length dimension that will introduce a scaling effect in the fracture simulations. In its present form, the model does not account for a possible strain rate sensitivity of the fracture process and has been calibrated using static data for the failure strain as a function of stress triaxiality.

The damage function in Equation (1) provides the failure criterion based on the accumulation of damage over the history of the material. However, this damage law requires an appropriate definition of a critical failure strain function. In this type of model, the performance of the damage law is typically better if the form of the damage law has a physical basis from observations of the microstructural failure processes. Previous studies have shown that the rate of void growth in the ductile fracture process is an exponential function of the stress triaxiality (Rice and Tracey, 1969) dR

Ro = 0.28 dεeq exp[1.5(σmean/σeq)] (2)

This type of void growth law can be used to develop a critical strain function of the form εc(σmean/σeq) = A

⎩⎪⎨⎪⎧

⎭⎪⎬⎪⎫1.0

exp[1.5(σmean/σeq)] - εshift (3)

where A and εshift are constants. The critical strain function in Equation (3), combined with the damage law in Equation (1), form the basis of the local fracture model.

2.1 APPLICATION OF THE LOCAL FRACTURE MODEL TO WELDMENTS

We implemented the local fracture model into a version of the finite element code DYNA3D, an explicit nonlinear three-dimensional finite element code for analyzing the large deformation dynamic response of solids and structures (Hallquist and Whirley, 1989). The equations of motion are integrated in time using the central difference method. Spatial discretization was achieved with eight-node hexahedron (brick) elements. To allow fracture of the weldment, a tied

node with failure feature was implemented into DYNA3D for the brick elements. This feature requires that node groups have tied degrees of freedom up to the point at which the average damage for the elements associated with the tied node group exceeds the critical value of 1. Following failure, the nodal constraints are removed and the elements can separate. The material model used in our calculations is a piecewise linear isotropic plasticity model. Integration of the damage function, Equation (1), occurred within the material constitutive model subroutine.

In our original implementation of the local fracture model in the DYNA3D finite element code, we associated the microstructural length parameter RMIC with the element size. Therefore, we selected element sizes in the region where we anticipated fracture of the order of the grain size or void spacing, approximately 200 to 700 µm, and applied Equation (1) to individual elements. This range of values for RMIC was adapted somewhat arbitrarily on the basis of the experience of

other authors for different, lower strength steels. The sensitivity of the results to the selection of an appropriate RMIC is a feature of the model that should be studied further.

The model was applied to analyze the dynamic fracture of a welded T-joint specimen as illustrated in Figure 2. This specimen was designed to produce a nearly plane strain response that simplifies the analysis of the fracture process. Additional description of the experimental procedure is given by Giovanola and Kirkpatrick (1992). In the finite element calculations, the explosive loading of the specimen was simulated by imposing the experimentally measured initial velocity to the portion of the plate covered by sheet explosive.

Figure 2. Schematic of the dynamic fracture tests for the welded T-joint specimens.

Vo

base plate

welded stiffenerinitial velocity profile

weld joint

Figure 3 compares the results of a calculation simulating a specimen subjected to an initial velocity sufficient to cause complete fracture with the experimental result for a small-scale specimen. The calculated deformations of the specimen, shown in Figure 3(a) agrees well with the experimental observations. Furthermore, the calculated deflection history agreed well with the experimental measurement. For simulations with a lower initial velocity, we were able to simulate the crack arrest in the base plate as observed in the experiments.

In addition to the global deformation response of the specimen, the model also appears to reproduce the characteristics of the fracture process. The crack paths across the plate for both the experiment and the simulation are shown in Figure 3(b). From the simulation we see that the stronger heat affected zone (HAZ) provides local confinement that increases the stress triaxiality and localizes damage in the base metal adjacent to the HAZ. After initiation, the fracture follows along the outside of the HAZ until the orientation of the crack tip relative to the maximum stresses has sufficiently changed to drive the fracture into the base metal.

The good agreement between simulations and experiments provides a partial validation of our use of the relatively simple ductile fracture model to make qualitative and semiquantitative analyses of the weldment fracture. Note that in the simulation we did not prescribe the crack path a priori. As a result the agreement in the fracture behavior provides additional confidence in the performance of the model.

2.2 APPLICATION OF THE MODEL FOR OPTIMIZATION OF WELDMENTS

A wide variety of fracture issues for weldments can be addressed with local fracture modeling because of the versatility and generality of the approach. The simulations described in this section illustrate the use of the local fracture model to optimize the design of weldments and investigate the effect of a pre-existing flaw in the weldment on strength (Giovanola and Kirkpatrick, 1993).

To optimize a weldment for a particular application, the primary considerations of the welded joint design may include the strength and ductility of the connection as well as the cost and difficulty of the specified welding procedure. The variables in the weld design may include the choice of weld metal and base metals, the size of the weld bead, the geometry of the weldment, and procedures to affect the development of the HAZ. Experimental investigation of all of these parameters to optimize the weldment is not feasible for most applications, and thus the ability to model these effects can be an important tool for developing better welded joints.

Experiment

Calculation

(a) Final deformation

Experiment Calculation

(b) Details of the crack path

Figure 3. Deformation and fracture behavior in the explosively loaded welded steel specimen.

Figure 4 illustrates the effect the weldment geometry on the fracture behavior of the T-joint connection. Figure 4(a) is the symmetric weldment with weld beads on both sides of the attached stiffener as analyzed previously in Figure 3. The base plate is fully fractured for the initial velocity (Vo) of 180 m/s with this symmetric weldment geometry. The second geometry,

shown in Figure 4(b), has only one weld metal bead deposited asymmetrically on the inside of the stiffener. For this asymmetric weld geometry the width and rotational stiffness of the connection is reduced which results in higher stresses and strains in the stiffener above the connection. As a result, the stiffener fails at an initial velocity of 140 m/s.

(a) Symmetric weldment (b) Asymmetric weldment (Vo = 180 m/s - fully fractured) (Vo = 140 m/s - fully fractured)

Figure 4. Effect of geometry on fracture behavior of the high strength weldment.

The above example illustrates how a small difference in the weld geometry produces initiation and propagation of fracture at different locations and significantly different fracture strengths. Not surprisingly, the asymmetric weldment is less resistant to fracture than the symmetric one (Vo for full fracture of 140 m/s versus 180 m/s), and fracture initiates and propagates entirely in

the weld metal. In contrast, in the symmetric weldment, fracture initiates in the base metal adjacent to the HAZ and then extends through the plate base metal, as observed in the experiments.

To assess the effect of a weld defect, we compared the initial velocity required to initiate and propagate a crack across the base plate in a precracked weldment and in an undamaged one. Note that the simulations performed for this comparison had slightly different loading conditions than the cases discussed above. We imposed an initial velocity Vo on the complete base plate

span between stiffeners, rather than on only the central region. Also, we simplified the boundary conditions by introducing a vertical symmetry line at the mid-thickness of the stiffener. Finally, the dimensions of the specimen were slightly different; in particular, the new geometry had a thicker stiffener to ensure that damage would localize in the base plate.

Figure 5 compares the deformation, fracture damage (fringes of damage D), and crack path in the precracked and uncracked weldments loaded by an initial velocity of 100 m/s. The presence of the small crack (weld defect) causes a complete fracture of the base plate, which still has a residual velocity of 60 m/s. In contrast, although a crack initiates in the weldment without initial defect, it arrests after penetrating about 40% of the plate thickness. The reduced resistance to crack initiation is also demonstrated by the smaller damage zone at the toe of the weld bead in the precracked specimen.

(a) No weld defect (b) 2 mm weld defect

Figure 5. Effect of a weld defect on fracture behavior of the high strength weldment.

3. SIMULATION OF A CYCLIC LOAD ON A WELDED BEAM-COLUMN JOINT

To demonstrate the applicability of the local damage model and associated numerical modeling approach to structural steel applications, we have performed an example three-dimensional finite element analysis of a beam-column moment-resistant connection. The loading configuration we modeled is similar to tests performed at the University of Texas at Austin (Engelhardt, 1994) and illustrated in Figure 6. The beam-column connection has a 2.44 meter long W21x57 A36 steel beam welded to a 3.66 meter long W12x136 A572-Gr.50 steel column. We selected typical mechanical properties for the A36 steel beam, A572-Gr.50 steel column, and E60 weld metal. The damage model given in Equation (1) was not used in this simulation because the fracture experiments required to determine the critical strain function for these structural materials were not readily available (notched tensile tests with different notch radii).

Figure 6. Schematic of the problem setup for the cyclic loading of a beam-column joint.

The beam-column joint in the example simulation was loaded by a vertical displacement applied to the beam end. The applied displacement history, shown in Figure 7, produced two full cycles of upward and downward bending of the beam. The amplitude of the displacement cycle produced a response well into the plastic range at the connection. The corresponding end displacement is approximately ±7 cm at maximum load. The frequency of the loading was chosen to be sufficiently low to eliminate significant inertial effects in the calculated response, approximating a quasistatic behavior of the connection.

Figure 8 shows the mesh used for the simulation. The mesh has an increase in detail and mesh refinement around the weld connection between the beam lower flange and the column as shown in Figure 8(b). In this more detailed connection region of the model the weld bead and HAZ were included to more accurately model the local deformation field around the connection. To obtain this level of detailed in the connection region the mesh was generated with a minimum element dimension of approximately 1 mm. However, this element dimension is still significantly greater than that used in the detailed local fracture weldment calculations describe above. No attempt was made to model the behavior of a bolted web connection. Instead, the beam web was modeled as perfectly bonded to the column with the exception of the web welding cutouts adjacent to the flanges.

Figure 7. Beam end displacement history used for the cyclic load simulation.

The three-dimensional nature of this problem resulted in approximately 5000 elements with the relatively course mesh shown in Figure 8. For steel, the minimum element dimension resulted in a time step of approximately 0.16 µs in DYNA3D. Thus, on a Silicon Graphics Indigo2 Solid Impact workstation, the simulation of this beam-column response required approximately 100 hours CPU for the full 2 cycle load duration (5000 elements x 1.5 million time steps x 50 µs CPU per element time step).

The calculated effective plastic strain distribution around the lower flange welded connection is shown in Figure 9. The maximum plastic strains in the flange around the connection are approximately 30-40%. This final plastic strain magnitude is the sum of fairly uniform strain increments produced in each of the tension-compression cycles in the flange. The calculated strain distribution shows localization around the connection that is influenced by the details of the weld geometry, HAZ, and weld backing strip. Thus this type of calculation could be used to investigate how modifications to the connection material and geometry could be used to modify the deformations and produce a stronger more damage tolerant connection.

(a) Overall mesh

(b) Detail of the weld joint

Figure 8. Mesh used for the finite element analysis of a cyclic load on a beam-column connection.

(a) Initial state

(b) Final State

Figure 9. Calculated effective plastic strain profile in the beam-column weld joint from the cyclic loading (black = 30% strain).

DISCUSSION AND CONCLUSIONS

In order to avoid fracture of welded moment connections and to design structural connections that will shift the formation of a plastic hinge away from the welded connection, one needs to understand and model the interactions of all the influencing factors. With the current availability of cost-effective high power computing, the advances in fracture mechanics, and novel materials

testing techniques, such modeling is now possible and should be used for the design of improved moment resistant connections.

The weldment fracture model discussed and applied in this paper is very simple and involves many assumptions and simplifications. Among the main limitations are the following;

• The fracture and constitutive models are not coupled, so no softening of the stress strain curve is introduced by the fracture damage.

• We have not carefully validated the model against well-controlled fracture mechanics experiments.

• We have arbitrarily selected the value of the characteristic length parameter RMIC.

• We have modeled the properties of the weld metal and HAZ rather crudely, assuming they were homogeneous within each zone and estimating them on the basis of hardness measurements only.

The results obtained with the model demonstrate that, in spite of these limitations, the model can qualitatively predict the correct fracture behavior of the weldment and can yield valuable information about the effect of various weldment parameters on fracture. This information can help define better weldment design practices, develop more precise characterization tests for weldments, and improve weldment models. For instance, it can qualitatively indicate how overmatching or undermatching the strength of the weldment will affect the overall response of the joint. Relative quantitative data can also be obtained by comparison of the results of various weldment designs (such as changes in base metal, chamfer angle, and full penetration versus partial penetration weld). This type of information is difficult, if not impossible, to obtain with classical fracture mechanics approaches and expensive or time consuming to obtain experimentally.

The application of the modeling approach used in this study to earthquake-type loading of welded steel structures can be a powerful tool for the development of improved connections. However, prediction and prevention of the range of failure modes observed in welded beam-column joints presents significant technical challenges. The relatively simple simulation of a beam-column joint presented in this study shows the ability to model the localized deformation and damage development around the welded connection. Similar analyses could be used to investigate how changes to the connection design, such as different materials or reinforced connections, modifies the strength and damage development. In addition, characteristic of the problem such as the static preload effects, residual stresses, material properties, and varying load

amplitude can easily be investigated with the model. These features can be difficult or expensive to determine experimentally.

Application of the local fracture model to calculate the complete failure process of welded beam-column connections would be a difficult task. The level of mesh refinement required in the LFM combined with the large region of material around the connection in which fractures have occurred would result in very computationally intensive models. A more practical approach would be a combination of global simulations to define the displacement history at the welded connection combined with series of more detailed local simulations of the connection to study the fracture initiation and propagation. The combination of these levels of modeling will allow the full range of weldment responses to be analyzed.

The combination of numerical simulations, the local fracture model, and experimental validation results in an efficient complementary approach for evaluating designs, repairs and retrofits of welded steel structures. Once promising designs have been identified with numerical simulations, they can be demonstrated and evaluated further in laboratory experiments. This complementary numerical and experimental approach provides the most cost-effective method for rapidly developing improved welded connections capable of meeting service requirements and providing the desired safety. As an added benefit, it provides a deeper understanding of the parameters influencing the seismic behavior of the connection.

REFERENCES Beremin, F. M., 1981, "Study of Fracture Criteria for Ductile Rupture of A508 Steel," in Advances in

Fracture Research (ICF5), D. François, Ed., Pergamon Press , pp. 809-816.

Blauel, J. G., and Schwalbe, K.-H., Eds., 1987, The Fracture of Welds, Mechanical Engineering Publications, Bury St. Edmunds, United Kingdom.

Chaboche, J. L. et al., Editors, 1986, "Séminaire International sur l'Approche Locale de la Rupture," Conference Proceedings, Centre de Recherches EDF, "Les Renardières," Moret-sur-Loing, France.

Curran, D. R., Seaman, L. and Shockey, D. A, 1987, "Dynamic Failure of Solids," Physics Reports, Vol. 147, pp. 253-388.

Devaux, J.C., Saillard, P., and Pellisier-Tanon, A., 1989, "Elastic-Plastic Assessment of a Cladded Pressurized-Water-Reactor Vessel Strength Since Occurrence of a Postulated Underclad Crack During Manufacturing," in Nonlinear Fracture Mechanics, Vol. II: Elastic-Plastic Fracture, ASTM STP 995, J. D. Landes, A. Saxena, and J. G. Merkle, Eds., American Society for Testing and Materials, Philadelphia, pp. 454-469.

Engelhardt, 1994, "Experimental Evidence on Welded Flange - Bolted Web Connections," Proceedings of AISC Special Task Committee on the Northridge Earthquake Meeting, March 14-15, 1994, AISC Inc., Chicago, IL.

Fontaine, A., and Maas, E., 1987, "Analyse de la Rupture par Clivage d'un Métal Fondu à l'Aide d'une Approche Locale," in The Fracture of Welds, J. G. Blauel and K.-H. Schwalde, Eds., Mechanical Engineering Publications, Bury St. Edmunds, United Kingdom.

Giovanola, J. H., and Kirkpatrick, S. W., 1992, "Applying a Simple Ductile Fracture Model to Fracture of Welded T-Joints," in Advances in Local Fracture/Damage Models for the Analysis of Engineering Problems, AMD Vol. 137, J. H. Giovanola and A. J. Rosakis, Eds., American Society of Mechanical Engineers, New York, pp. 285-303.

Giovanola, J.H., and Kirkpatrick, S.W., 1993, "Methodology for Evaluating Strength and Fracture Resistance of Weldments Using a Local Approach to Fracture," in Pressure Vessel Integrity-1993, PVP-Vol. 250, W. E. Pennel, S. Bhandari, G. Yagawa, Editors, American Society of Mechanical Engineers, New York, pp.157-171

Giovanola, J.H., and Kirkpatrick, S.W., and Crocker, J.E., 1996, "Investigation of Scaling Effects in Elastic-Plastic Ductile Fracture Using the Local Approach," Proceedings of the First European Mechanics of Materials Conference on Local Approach to Fracture, Euromech-Mechamat '96, Fontainebleau-France, Sept. 9-11.

Giovanola, J. H., and Rosakis, A. J., Editors, 1992, Advances in Local Fracture/Damage Models for the Analysis of Engineering Problems, AMD-vol. 137, American Society of Mechanical Engineers, New York.

Hallquist, J. O. and Whirley, R. G., 1989, DYNA3D User's Manual (Nonlinear Dynamic Analysis of Structures in Three Dimensions), Lawrence Livermore National Laboratory, Report UCID-19592, Rev. 2.

Hancock, J. W. and Mackenzie, A. C., 1976, "On the Mechanisms of Ductile Failure in High-Strength Steels Subjected to Multiaxial Stress States," Journal of the Mechanics and Physics of Solids, 24, pp. 147-169.

Harrison, J. D., and Anderson, T. D., 1988, "Developments in the Application of the CTOD Approach to Fracture Assessment of Welded Constructions," in Fracture Mechanics, Eighteenth Symposium, ASTM STP 945, D. T Read and R. P. Reed, Eds., American Society for Testing and Materials, Philadelphia, pp. 468-484.

Lemaitre, J., 1986, "Local Approach of Fracture," Engineering Fracture Mechanics, Vol. 25, pp. 523-537.

Mackenzie, A. C., Hancock, J. W., and Brown, D. K., 1977, "On the Influence of State of Stress on Ductile Failure Initiation in High-Strength Steels," Eng. Fracture Mechanics, 9, 167-188.

Matic, P., and Jolles, M. I., 1989, "Defect, Constitutive Behavior, and Continuum Toughness Considerations for Weld Integrity Analysis," in Nonlinear Fracture Mechanics, Vol. II: Elastic-Plastic Fracture, ASTM STP 995, J. D. Landes, A. Saxena, and J. G. Merkle, Eds., American Society for Testing and Materials, Philadelphia, pp. 82-92.

Mudry, F., 1985, "Methodology and Applications of Local Criteria for Prediction of Ductile Tearing," Elastic-Plastic Fracture Mechanics, L. H. Larson, ed., ECSC, EEC, EAEC, Brussels and Luxembourg, Belgium, pp. 263-283.

Rice, J. R., and Tracey, D. M., 1969, "On the Ductile Enlargement of Voids in Triaxial Stress Fields," J. Mech. Phys. Solids, 17, 201-217.

Schwalbe, K.-H., 1992, "Effect of Weld Metal Mis-Match on Toughness Requirements: Some Simple Analytical Considerations Using Engineering Treatment Model (ETM)," Int. J. Fracture, 56, 257-277.

Toyoda, M., 1985, "Fracture Toughness Requirement and Fracture Toughness of Weldments" (Parts 1 and 2), J. Jpn. Welding Soc., 54, 10-131.