A phenomenological form of the q2 parameter in the Gurson model

doi:10.1016/j.ijpvp.2007.10.009

International Journal of Pressure Vessels and Piping

Volume 85, Issue 4, April 2008, Pages 199-210

https://doi.org/10.1016/j.ijpvp.2007.10.009 Get rights and content

Abstract

Continuum damage mechanics is applied to compute fracture J-initiation and the J-resistance (J-R) curve for SA333 Gr.6 carbon steel material. A parametric study is carried out to assess the effects of variation of Gurson material parameters on the J-R curve. A difficulty is experienced to obtain a set of Gurson parameters that can compute J-initiation as well as the entire J-R curve close to the measured values. It has been found that a phenomenological form of the q₂ parameter, used in the Gurson constitutive model, based on experimental observation helps to overcome this difficulty. The new form of q₂ has an exponential spatial variation near the crack tip and has two constants. These additional constants help analysts to compute the complete J-R curve (including J-initiation) close to experimental data. The same set of constants has also been found to be useful for other fracture specimens, materials and two piping components of a nuclear plant.

Introduction

Micro-mechanical modeling of material damage helps to overcome some of the inherent limitations of conventional fracture mechanics. A number of models exist within the framework of micro-mechanical modeling, which address various phenomena of material behavior during material damage ranging from ductile to cleavage fracture. Ductile rupture follows the nucleation, growth and coalescence of micro-voids with significant plastic deformation. This consumes much more energy due to plastic deformation. Modeling is done using two independent methodologies. The model of Rice and Tracey [1] for ductile crack initiation is based on a critical cavity growth. This is calculated by using stresses and strains surrounding the crack tip obtained from a conventional elastic–plastic finite element analysis. Many authors used a similar concept by using different damage potentials to characterize crack initiation [2], [3], [4], [5]. An effort has been made in [5], [6] to overcome mesh dependency of such models by using a modified form of the Rice and Tracey integral.

On the other hand, the “coupled” constitutive models of Gurson [7] and Rousselier [8] affect the yield behavior. Specific material subroutines are required to perform FE analysis. In order to account for the effects of void nucleation and coalescence, and to obtain better agreement between the experimental results and numerical simulations, the original Gurson model was modified and extended into a semi-phenomenological form by Needleman and Tvergaard [9]. The material input parameters of the micro-mechanical models should be obtainable from metallurgical observations. However, the determination of damage parameters is still predominantly a phenomenological fitting procedure and is done by combining test results and numerical simulations. An international numerical round robin exercise on micro-mechanical modeling was conducted by the European Structural Integrity Society (ESIS), Phase-II, Task-A [10]. A number of participants (including the present authors) made efforts to come up with an appropriate combination of damage parameters for 22NiMoCr37 material, which can compute the complete J-R curve. The damage parameters adjusted for this purpose are f₀, f_n, f_c and element size. The conclusions drawn through this international exercise have been summarized in [11], which concludes that “no generally accepted recommendations are available for the choice of q parameters or the parameters specifying the void nucleation. It is not even clear, if an optimized parameter set is unique and transferable to all other geometries”.

As an extension to this work, the present authors continued the investigations to ascertain Gurson material parameters for SA333 Gr.6 material used in the primary heat transport (PHT) piping system of Indian nuclear reactors. The results obtained from such a study and a suggestion to introduce an extra constant in the q₂ parameter of the Gurson model to facilitate the agreement between computed and experimental results are shown below.

Section snippets

A brief theoretical background of the Gurson material constitutive model

A short review of the Gurson-based material constitutive model is presented here, which will be referred to subsequently in the present paper. Based on the work of Berg [12] (which shows that a porous medium is governed by the normality rule), Gurson [6] derived a constitutive model “φ” for materials containing either cylindrical or spherical voids. $φ (σ^{ij}, σ_{M}, f) = \frac{σ_{eq}^{2}}{σ_{O}^{2}} + 2 f^{*} q_{1} \cosh {\frac{q_{2} σ^{kk}}{2 σ_{O}}} - {+ q_{3} (f^{*})^{2}} = 0 .$

Here $σ_{eq} = (\frac{3}{2} s_{ij} s_{ij})^{1 / 2}$ is the macroscopic effective stress, s_ij represents the stress deviator

Characterization of SA333 Gr.6 carbon steel material

The SA333 Gr.6 carbon steel material is used as the piping material for the PHT system of Indian pressurized heavy water reactors. Therefore, material characterization is mandatory for demonstrating integrity of piping systems under all possible scenarios [14] and to qualify the system for leak-before-break. Table 1 shows the chemical composition and Table 2 shows the metallography data of this material.

Tensile tests were conducted on three tensile specimens. The tests were performed using a

Analysis of a TPB specimen using the Gurson constitutive model

A European numerical round robin was carried out to simulate tearing of ferritic steel using constitutive equations for ductile damage. The results of this exercise are reported in [10], [11], [15], [16]. The present authors participated in the exercise using their in-house damage mechanics code MADAM [17]. Following the procedure of the round robin exercise, analyses have been carried out on a TPB specimen to ascertain Gurson parameters for SA333 Gr.6 material.

Analysis of CT specimen of SA333 Gr.6 material using the present modification

A CT specimen has been analyzed to verify the above damage parameters along with the new form of q₂. The experimental load–displacement and J-R curves are available in [14]. The salient dimensions of this specimen are shown in Fig. 6. The same figure also shows the finite element mesh used in this analysis. The size of the elements along the crack line is taken as twice the characteristic length l_c. This ensures the average distance between the two Gauss points as l_c. The analysis has been

Analysis of CT specimen of 22NiMoCr37 material

To check the suitability of the present modification in Gurson parameter q₂, another analysis has been carried out on a CT specimen made of 22NiMoCr37 material. The load–displacement and J-R curves of this 1T-CT specimen were provided to all the participants of the ESIS international bench mark study [10]. The dimensional details of this specimen are shown in Fig. 16. The finite element mesh used for this purpose is the same as shown in Fig. 16. The Gurson parameters used are shown in Table 4.

Test specimens

An experimental database has been generated over the years in the present organization by testing a number of reactor-grade piping components with flaws. The details of these experiments are available in [18], [19]. Test specimens consist of straight pipes and elbows made of SA333 Gr.6 carbon steel material with through-wall circumferential cracks. These pipe specimens are subjected to four-point bending load: in the case of straight pipes and closing bending moment in the case of elbows. The

Conclusions

The following conclusions may be drawn from the present study:

1.
Difficulty is encountered while computing J-initiation and J-R curves of fracture specimens using the Gurson model in agreement with the experimental data.
2.
Such an anomaly may be due to the requirement of much smaller elements (in comparison to the element size based on characteristic length “l_c”) near the crack tip to model the sharp stress gradient.
3.
A new phenomenological form of the q₂ parameter is suggested, which helps the

References (19)

J.R. Rice et al.
On the ductile enlargement of voids in triaxial stress fields
J Mech Phys Solids
(1969)
B. Budiansky et al.
B.K. Dutta et al.
A modified damage potential to predict crack initiation: theory and experimental verification
Eng Fract Mech
(2004)
B.K. Dutta et al.
Application of a modified damage potential to predict ductile crack initiation in welded pipes
Int J Press Vessels Piping
(2005)
G. Rousselier
Ductile fracture models and their potential in local approach of fracture
Nucl Eng Des
(1987)
A. Needleman et al.
An analysis of ductile rupture in notched bars
J Mech Phys Solids
(1984)
V. Tvergaard et al.
Analysis of the cup–cone fracture in a round tensile bar
Acta Metall
(1984)
J. Chattopadhyay et al.
Experimental and analytical study of three point bend specimens and through-wall circumferentially cracked straight pipe
Int J Press Vessels Piping
(2000)
J. Chattopadhyay et al.
Load bearing capacity of flawed piping components—comparison of experiment with calculation
Int J Press Vessels Piping
(2004)

There are more references available in the full text version of this article.

Cited by (29)

Micromechanical modelling of API X80 linepipe steel
2022, International Journal of Pressure Vessels and Piping
Gurson–Tvergaard–Needleman (GTN) model parameters have been estimated for the API X80 linepipe steel; these parameters and mesh size are transferable from one geometry to another. Fracture data are non-transferable from the lab specimen to the component level. These issues are resolved by local approaches to fracture to predict damage. GTN model incorporates the local features of the fracture, i.e., nucleation, growth, and coalescence of voids. The void volume fraction of material at different stages of loading governs the load-carrying capacity. This research has explored the methodology to determine the parameters by metallographic examination, known as micromechanical modelling. Parametric study and calibration of the parameter are done to fit the load-displacement curve of the notched tensile bar. These parameters are validated on the single edge notch bend (SENB) test. Estimated GTN parameters can predict the damage behavior for moderate to low constraints; hence, these parameters and mesh size are transferable from one geometry to another.
Fracture Modeling Of Cracked Pipe Under Monotonic And Cyclic Loading
2020, International Journal of Mechanical Sciences
Citation Excerpt :
For instance, Ruggieri et al. [10] performed crack growth simulation of the crack behavior of the external surface-cracked pipe under internal pressure condition. Dutta et al. [11] simulated the pipe fracture behavior of the straight and elbow with a through-wall crack under bending. They focused on predicting fracture toughness or ductile tearing in the pipe under monotonic loading condition using a damage model.
The present work presents experiment and numerical modelling to quantify the effect of the loading type on ductile crack growth in circumferential through-wall cracked SA508 Gr.1a pipes. In tests, three types of loading conditions are considered; monotonic, displacement-controlled and load-controlled large-amplitude cyclic loading. Experimental data are compared with numerical crack growth simulation results based on the multiaxial fracture strain energy model. Simulated results show overall good agreements with experimental data for all loading types. Based on FE results, the effects of the loading type and cyclic hardening model on ductile crack growth are discussed.
Ductile tearing simulation of STS410 pipe fracture test under load-controlled large-amplitude cyclic loading: Part I—Effect of load ratio
2020, Engineering Fracture Mechanics
Ductile fracture of circumferential through-wall cracked pipe under large-amplitude load-controlled cyclic loading with three different load ratios was simulated using a fracture strain energy damage model. Damage model parameters were determined from a tensile test and a pipe test under monotonic loading. Comparison of simulated and experimental data reveals that the simulation results depend on the cyclic hardening model. Further investigation confirmed that selection of an appropriate cyclic hardening model is more important under load-controlled cyclic loading then under displacement-controlled one.
Hybrid approach for calculation of J-R curve using R6
2019, Engineering Fracture Mechanics
The fracture property, J-R curve is an essential material property for assessing the cracked structures for unstable ductile tearing. For obtaining this material property, fracture test is conducted and resultant test data namely load, load line displacement (LLD) and relevant crack extension data are post processed to obtain J-R curve. However, in this method, geometry functions $η$ and $γ$ are essential which are available for only limited geometries. To avoid this limitation, recently two approaches based on R6 failure assessment diagram were proposed. First was the load based R6 approach where load vs. crack growth data is converted to fracture property J-R curve using R6 Failure Assessment Diagram (FAD). This method was further modified to displacement based R6 approach where LLD vs. crack extension data is converted to J-R curve using R6 FAD. The predicted J-R curves were in good agreement with conventionally calculated values especially for displacement based approach. However, when these load and displacement based approaches are attempted to be used in case of throughwall cracked elbows, the predicted J-R curve deviated significantly with respect to those obtained using conventional method. On further investigation, it is found that the assumption involved in the displacement based approach is violated for cracked elbows because of severe ovalization of cross section. Hence, a modified approach is proposed in this work where both load and displacement parameter are used for calculation of J-R curves and named as hybrid approach. Further, this approach is adopted for the six through wall cracked pipes which are already investigated by load and displacement based R6 approaches for J-R curves. The predicted J-R curves are found to be in good agreement for all the investigated pipes and elbows. Hence, a simpler yet more accurate hybrid approach based on R6 method is proposed in this paper for post processing the test data to obtain, the material J-R curve.
Ductile fracture behavior of ECAP deformed AZ61 magnesium alloy based on response surface methodology and finite element simulation
2019, Materials Science and Engineering: A
The response surface methodology in the design of experiments (DOE) wizard and Gurson-Tvergaard-Needleman (GTN) model were employed to estimate the optimum GTN damage parameters and to validate their significant effects respectively on the ductile fracture behavior of ECAP deformed AZ61 magnesium alloy. Hollomon's flow stress was applied to identify uniform deformation and non-uniform deformation regions to investigate the void nucleation and coalescence processes separately. The significance of statistical results was evaluated with the analysis of variance (ANOVA) based on P-values and coefficients of determination ( $R^{2}$ ). Correspondingly, the contributions of ECAP plastic deformation and the corresponding local plasticity (damage progresses) on the progress of GTN ductile fracture damage parameters were studied. Desirability function was used to show the significance of individual and interaction effects of optimum initial damage parameters on the respective response variables. By using response surface methodology, the optimum GTN damage parameters were determined at the corresponding positions of each critical response variables. Results also showed that varying both stress triaxiality and damage variables simultaneously can greatly affect the curve fitting process of experimental, simulation and GTN model curves. However, at constant stress triaxiality condition, the GTN model curve was observed perfectly fitting with the tensile test curve.
Determination of J-initiation toughness using pre-cracked small punch test specimens
2019, Procedia Structural Integrity
Determination of fracture properties is important to assess the in-service degradation of nuclear structural materials subjected to thermal fluctuations and irradiation and to calculate the residual life of the component. The pre-cracked small punch test is an alternative method for the determination of fracture properties in case of limited availability of materials and is insufficient for conducting conventional standard tests. In this paper, pre-cracked small punch test (denoted as p-SPT) specimens are used to obtain the fracture J-initiation toughness of nuclear structural steel 20MnMoNi55 and T91.The size of the p-SPT specimen is 10 x 10 x 0.5 mm and wire EDM technique is used to generate through thickness crack profile from mid-point of one side to a point just passing the center of the specimen, producing a crack length to specimen width ratio (a/W) as 0.4. The tests are conducted to get load v/s displacement data. Elastic-plastic finite element analysis of the p-SPT specimen is carried out and numerically obtained load v/s displacement data are compared with the experimental results. This analysis also helped to compute J-integral near the crack tip as a function of load. FE analysis is then repeated using the micromechanical Gurson-Tvergaard-Needleman model to know the load at which crack is initiated. J-initiation is then calculated using J-integral v/s load data. Computed J-initiation using micromechanical GTN model has good matching with the value quoted in the literature, which shows the viability of the method. The methodology described in this paper has the potential to determine fracture initiation toughness of aged nuclear materials using pre-cracked small punch tests.

View all citing articles on Scopus

View full text

A phenomenological form of the q2 parameter in the Gurson model

Abstract

Introduction

Section snippets

A brief theoretical background of the Gurson material constitutive model

Characterization of SA333 Gr.6 carbon steel material

Analysis of a TPB specimen using the Gurson constitutive model

Analysis of CT specimen of SA333 Gr.6 material using the present modification

Analysis of CT specimen of 22NiMoCr37 material

Test specimens

Conclusions

J Mech Phys Solids

Eng Fract Mech

Int J Press Vessels Piping

Nucl Eng Des

J Mech Phys Solids

Acta Metall

Int J Press Vessels Piping

Int J Press Vessels Piping

A phenomenological form of the q₂ parameter in the Gurson model