- Email: [email protected]

Contents lists available at ScienceDirect

Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal

Fracture analysis and constitutive modelling of ship structure steel behaviour regarding explosion A.-G. Geffroy a,b,⇑, P. Longère a, B. Leblé b a b

UEB/UBS/LIMATB, Centre de Recherche Christiaan Huyghens, Rue de Saint Maudé, 56321 Lorient Cedex, France DCNS, Rue Choiseul, 56311 Lorient Cedex, France

a r t i c l e

i n f o

Article history: Available online 16 October 2010 Keywords: Fracture Non-linear modelling Ship structure Explosion

a b s t r a c t Consistent constitutive modelling of material behaviour and further reliable numerical prediction of the response of structures under severe loading necessitate the knowledge of the microstructural mechanisms at the origin of failure. The present work deals notably with the identiﬁcation of the microstructural damage mechanisms of a high purity (ferritic–pearlitic) mild steel employed as structural material in military ship building. With this aim in view, an extensive campaign of experiments has been carried out, including interrupted and until fracture tests on smooth and notched, axisymmetric and plane specimens. Initial and post-mortem microstructures of the samples have been observed using a scanning electron microscope (SEM) in order to reveal the damage mechanism. The latter is double: quasi spherical cavity nucleation and growth inside the soft ferritic matrix and microcracking at the (soft)ferrite–(hard)pearlite interphase. Conditions for initiation and evolution of these two kinds of damage appear as being different as expected. The various steps of diffuse damage, microcracking and macro cracking yielding ultimate failure are also observed. Fractographies obtained from tensile tested samples and explosion loaded plates are also compared. Moreover, the material behaviour has been modelled, describing the salient effects observed experimentally, namely strain and strain rate hardening, and thermal and damage softening. The parameters identiﬁcation was accomplished using an inverse method based methodology. Finite element numerical simulations involving far-ﬁeld underwater explosion loading implemented as user subroutine in the FE computation code ABAQUS has also been performed, leading to a satisfying agreement between experimental and numerical results. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Designing military ship structures, as surface ships or submarines, imply to consider, in addition to conventional sailing stresses, vulnerability related conditions. The context of the present work is mainly concerned with ship structure vulnerability regarding explosions with the aim to preserve the sailing functions of the ship and the integrity of sensitive areas. A way to minimize the damage induced by underwater explosions is to consume the energy of the explosion by plastic deformation of the structure. The fracture of the external panel is tolerated, but integrity of the internal panel has to be preserved.

⇑ Corresponding author at: UEB/UBS/LIMATB, Centre de Recherche Christiaan Huyghens, Rue de Saint Maudé, 56321 Lorient Cedex, France. Tel.: +33 (0)2 97 87 46 44. E-mail addresses: [email protected] (A.-G. Geffroy), [email protected] (P. Longère), [email protected] (B. Leblé). 1350-6307/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2010.09.038

671

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

This work aims at studying the failure mechanisms of a ship structure material when submitted to severe conditions of strain and strain rate. With this aim in view, laboratory tests and airblast experiments were performed. In a ﬁrst time, microstructural observations were realized on tensile samples at various states of deformation until fracture. They reveal two types of damage, microcavities and microcracks. In a same way, microscopic observations were performed on plates submitted to various loading conditions of airblast. The microscopic failure mechanisms seem to be the same in both cases. A constitutive model has been built in order to describe the sailent effects of strain hardening, thermal softening and viscoplasticity as observed experimentally from laboratory tests. This model, completed by a failure criterion, has been implemented as user material in the computation code ABAQUS. Numerical simulations of airblast conﬁgurations are shown. The present work is divided into three parts. The ﬁrst one details the laboratory experiments and particularly the miscroscopic observations performed on tensile tests. The second part presents brieﬂy the airblast experiments and compares the microscopic observations realized on airblast loaded plates with those performed on the tensile samples. The model is presented in a third section as well as some numerical simulations of airblast loadings. 2. Laboratory experiments Laboratory tests including compressive and tensile experiments have been performed. This part deals with the observation of the evolution of the material microstructure during the process of deformation until fracture. 2.1. Microstructure of the steel at non-deformed state The material of the present study is a low carbon steel constitutive of ship structures. This steel is ﬁrst described through project speciﬁcations. Designated as DH36, its chemical composition is given in Table 1. DH36 is a ferritic–pearlitic mild steel, provided in the form of a cold rolled plate. Considering the metastable cementite phase diagram in Fig. 1, the material is composed of about 87% of ferrite and 13% of pearlite with a lamellar aspect. Both phases are easily observable on the microscopic picture in Fig. 2, which was realized with a scanning electron microscope (SEM). The bands of pearlite suggest an initial anisotropy. As the result of the cold rolling process, the plate exhibits an initial texture, see Fig. 3 (this 3D picture has been built from various 2D micrographs, explaining the apparent discontinuity of the pearlite bands on the sides). The thermomechanical treatment indeed generates a reorganization of the pearlite in the ferritic matrix in the form of plus or minus continuous bands collinear to the rolling plane, visible in the plate thickness. The microscopic analysis of the material has revealed the existence of inclusions as carbon nitride, niobium and zirconium carbide, as well as calcium sulphide, in negligible proportion. The following failure analysis will show that these particlues are not at the origin of the hole nucleation in the present case. Round and plate specimens have been machined in the plate following various orientations with regards to the rolling direction. Quasi static and dynamic tests have been performed at room and elevated temperatures. 2.2. Fracture shapes of the post-mortem tensile samples Considering specimens which were brocken during tensile test, tear shape and fracture facies give information about the fracture mechanisms. Fractographies of round tensile specimens are shown in Fig. 4 for L and TL orientations, respectively. According to the picture (a) (quasi-static test) and (b) (dynamic test), the initially circular L specimen section becomes ellipsoidal during the

Table 1 Chemical composition of the DH36-type steel. DH36 C (%) 0.126 Ni (%) 0.027

Si (%) 0.443 Cr (%) 0.028

Mn (%) 1.52 V (%) 0.001

P (%) 0.014 Nb (%) 0.033

S (%) 0.001 Ti (%) 0.004

Cu (%) 0.017 Al–T (%) 0.048

Mo (%) 0.009

672

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

Fig. 1. Iron/carbon phase diagram of the steel.

Fig. 2. Micrography of the pearlitic phase in the ferritic matrix (DCNS/CESMAN).

Fig. 3. Bands of pearlite in the ferritic matrix (DCNS/CESMAN).

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

673

deformation process. This is not the case for a sample which has been loaded in the transversal direction (Fig. 4c). The lower deformation along the TL direction is linked to the higher yield stress along this direction. Therefore, one can reasonably conclude to an orthotropic behaviour [1,2]. Moreover, naked-eye observations show the various crack propagation modes. A round sample of a ductile metal generally cracks with a ‘‘Cup & Cone’’ form [3,4], as shown on Fig. 5. Crack ﬁrstly appears in open mode in the center of the sample, and propagates toward the edges with a shear mode. The plate sample presents a cracking form with a mode I in the width and a mode II in the thickness, see Fig. 6. Finally, fracture facies highlight a lunar landscape in the center of the samples (Fig. 7a) and a rugged area on the edges (Fig. 7b), which are typically characteristic of a ductile fracture [5].

2.3. Types of damage Some tests have been interrupted during the deformation process in order to observe the various steps of damage inside the material. Then a section of the sample has been cut in the necking area of a non-brocken specimen. In the central zone, one can distinguish holes in the ferritic matrix, as shown on Fig. 8a. Nevertheless, near the side, microcracks appear, located

Fig. 4. Effects of the initial anisotropy on the section deformation: fracture facies of round L and TL samples (DCNS–CESMAN).

Fig. 5. ‘‘Cup & Cone’’ rupture of the round sample.

Fig. 6. Open I mode and shear II mode of a plate sample.

674

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

Fig. 7. Landscape of the fracture facies of the round and plate samples (DCNS–CESMAN).

in the bands of pearlite or at the ferrite/pearlite interphase (see Fig. 8b). The size of both damage can be compared, with a scale of 10lm here. Consequently hole nucleation seems to result from cavitation inside the ferritic matrix, at the grain boundary or from microcracking at the ferrite/pearlite interphase. Cavitation arround particles or due to particle failure was indeed rarely observed. This implied that a certain amount of plastic deformation is needed before void germination. 2.4. Crack nucleation and propagation In order to observe crack nucleation and propagation, a quasi-static tensile test which has been interrupted just before the ultimate fracture is needed. Fig. 9 shows a non-broken sample where two fragments have been cut with a view to observe the failure in the width (specimen M) and in the thickness (specimen N).

Fig. 8. Presence of two types of damages (DCNS–CESMAN).

Fig. 9. Observation of the failure inside a non-brocken sample.

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

675

Fig. 10. Crack propagation in the width (DCNS–CESMAN).

Looking ﬁrstly at the specimen M, in the width of the sample, one can observe on the Fig. 10a a macrocrack, which starts from the center of the sample and propagates towards the side. This macrocrack may be represented by an elongate ellipse. Upstream from this macrocrack, see Fig. 10b, one can observe a mesocrack, which may be crudely represented by a line. The length of the mesocrack is about equal to ﬁve times the width of the macrocrack tip. If we focus on a part of the mesocrack, Fig. 10c, one can also note a tortuous path of the crack, and on the side the hardened material. On the last picture, Fig. 10d, it is possible to see the low volume fraction of cavities which takes place near the crack.

Fig. 11. Crack propagation in the thickness (DCNS–CESMAN).

676

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

The deformed shape of the same sample could also be observed in the thickness (specimen N). Fig. 11a allows to remark that the fracture propagates with a ﬁxed angle. Once more, the macrocrack gets the form of an ellipse followed by lines for the mesocracks. Fig. 11b is an enlarging of the mesocrack, and particularly at the head of the crack. Cavities sould be observed, constituting a discontinuous way which may be followed by the crack to propagate. 3. Airblast experiments This part deals with the observation of the material microstructure before and after airblast loading. 3.1. Experimental tests Based on tests performed on ship hull structural material [6,7], airblast experiments were carried out. The used material is DH36 steel, in the form of square thin plate. The steel plate is held down by two frames ﬁxed to the structure, as shown on Fig. 12. The explosive is spherical, hung on a post and braces. The mass of the explosive and the distance between the explosive and the plate are controlled parameters. The priming is done in the center of the explosive, and the ignition is activated by remote control. The photographies of Fig. 13 illustrate an explosion. 3.2. Airblast results Some shots have been done, bringing into play two parameters: the mass of the explosive, and the distance between the plate and the explosive. These experiments do not permit any time revolved measurement. But post-mortem results gave a lot of information, as the deﬂection (Fig. 14a), the nucleation of cracks (Fig. 14b), or also the phenomenon of petalling (Fig. 14c), depending on the explosive mass C and on the explosive/plate distance D. The maximal displacement measured on the various plates are noted on the graph, Fig. 14d, as a function of the explosive center/plate distance. Except for one test, the ﬁgure shows that the smaller the distance and/or the higher the mass of the explosive, the higher the maximal displacement.

Fig. 12. Conﬁguration of the airblast experiment (DGA).

Fig. 13. Explosion of a steel plate.

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

677

Fig. 14. Shape of the plate after airblast loading (DCNS–DGA).

3.3. Microscopic observations of an airblast loaded plate One of the airblast loaded plates was cut in order to realize microscopic observations, in view of comparing the mecanisms of failure under explosion to those under tensile loading. In the previous section, it was seen that the tensile sample cracked with a shear mode in the thickness. Naked-eye observation of a plate suject to petalling allows for concluding to a shear mode failure, see Fig. 15. Moreover, if the failure is observed in the thickness of the plate, one can remark that the hull cracks with a certain angle with regards to the normal to the thickness. These observations have been done on the plate samples. This propagation in mode II is conﬁrmed with the fracture surface of the petals’ sides (Fig. 16). It can be observed the same landscape of elongated cupules, which is characteristic of the shear ductile fracture, than on the plate samples (see Section 2.2). Under the explosion effect, the plates begin to damage before cracking with petalling form. Micrographic pictures revealed the presence of two types of damage: cavities and microcracks. And ﬁnally, the state of the ﬁnal tear could be observed. At a crack tip inside the exploded plate, in a plane normal to the thickness, as shown on the picture Fig. 17a, one can once more distinguish a macrocrack followed by a mesocrack (Fig. 17b and c). In the same way than for the tensile sample, the crack could be represented by an elongated ellipse, followed by lines. With an enlarging of the crack, it could be seen again tortuous paths (see Fig. 17d).

Fig. 15. Cracking of the petals in shear mode (mode II).

678

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

Fig. 16. Fracture surface on the petals’ sides (DCNS/CESMAN).

Fig. 17. Cracks propagation: observation of the tortuous crack paths (DCNS/CESMAN).

Fig. 18. Cracks propagation in the thickness of the plate, with a ﬁxed angle (DCNS/CESMAN).

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

679

In the thickness of the plate, Fig. 18, one can remark the same angle of the fracture propagation than the one of the plate sample. After this microscopic study, one can conclude to a similar behaviour between laboratory tests and airblast experiments, and globally the same damage and failure mechanisms at the microscopic scale. Therefore the use of laboratory tests in the building of a constitutive model remains justiﬁed. 4. Numerical simulations of airblast This part is a ﬁrst attempt to describe the material behaviour and predict numerically the response of a plate submitted to airblast loading. The model is presented brieﬂy in the ﬁrst section. Next section details the numerical procedure, and ﬁnally, the last section gives various numerical results. 4.1. Constitutive equations An important experimental campaign aiming at characterizing the material behaviour in a wide range of strain, strain rate and temperature was conducted. According to the experimental results, a constitutive model has been built to describe the combined effects of strain hardening, thermal softening, and viscoplasticity. The yield function F of the sound (undamaged) material is expressed as:

Fðr; RÞ ¼ J2 ðrÞ Rðj; TÞ ¼ Lj_ ; T P 0

ð1Þ

8 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 > > > J 2 ðrÞ ¼ 2 S : S > > n oh i < m b 1T T Rðj; TÞ ¼ R0 þ R1 ½1 expðkjÞ where melting > > h i1=n > > > : Lj_ ; T ¼ ! j_ exp V a rM kB T

ð2Þ

J2 represent the second invariant of the deviatoric stress tensor S, R the rate independent yield stress, T absolute temperature, j the cumulate plastic strain and L the strain rate induced over stress. The strain hardening is reproduced using the Voce law [8]. A power law has been retained to describe the thermal softening. R0 represents the initial radius of the Huber–Mises cylinder at 0 K, R1 the isotropic hardening force value at saturation. k and b are hardening related constants, m a thermal softening related constant and Tmelting the melting point. Experimental results showed a tension/compression assymetry. The common way of reproducing such a feature in a constitutive model consists in taking into account the third invariant of the stress tensor [9]. This effect of the loading path may be alternatively considered as a thermally activated mechanism involving the mean stress [10]. This approach has been retained in the present work, leading to the expression of the viscoplastic multiplier, and then to the expression of the strain rate induced over stress L, see Eq. (2.3). ! and n represent viscoplasticity model constants. Quantity Va refers to an activation volume, and kB to the Bolztmann’s constant. The model, which is described in this section, is henceforth called ‘‘ETVP’’ for ‘‘Elasto–Thermo–ViscoPlastic’’ and counts 16 constants. The identiﬁcation of material constants has been conducted using the gradient-based optimization method detailed in [11]. The graphs of Figs. 19 and 20 represent experimental and numerical results obtained with the identiﬁed model ETVP. The model is tentatively completed by a failure criterion based on a critical cumulate plastic strain under a tensile stress state. Self heating under adiabatic conditions is assumed to result from plastic dissipation [12]. The model and the failure criterion have been implemented as user material (VUMAT) into the engineering FE computation code ABAQUS.

Fig. 19. Comparison between numerical results obtained from the identiﬁcation and experimental ones, compressive tests.

680

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

Fig. 20. Comparison between numerical results obtained from the identiﬁcation and experimental ones, tensile tests.

4.2. Numerical simulations In order to evaluate the predictive capacities of the constitutive model as user material and to determine the critical shear value at failure, the numerical simulation of the airblast tests was conducted employing the engineering FE computation code ABAQUS, using an explicit scheme. The plate was meshed using 8-node bricks with reduced-integration (C3D8R) and using ﬁve elements in the thickness. The plate is crimped between two rigid body frames, as shown in Fig. 21.

Fig. 21. Mesh of the structure.

Fig. 22. Comparison between experimental (a) and numerical (b) results of airblast experiments.

A.-G. Geffroy et al. / Engineering Failure Analysis 18 (2011) 670–681

681

The airblast conditions were the same than the experimental ones. The load, which is a spherical pressure applied on the top of the plate, is applied with the CONWEP card. The boundary conditions consist in two contact interfaces between the steel plate and the clamp. Friction is accounted for. Both frames are clamped. The ﬁnite element is eroded as soon as the shear failure criterion is satisﬁed. For loading conditions without failure, the shape of the deformed plate is fairly well reproduced. For loading conditions with failure, the numerical critical shear value at failure was then adjusted in order to reproduce the deterioration states obtained experimentally, see Fig. 22. Some discrepancies between experimental and numerical results can be observed: the number of petals is different and the numerical deterioration is more important. This implies enriching the constitutive model by accounting for e.g. the void growth induced progressive damage and further failure. 5. Conclusions Microscopic observations were performed on tensile samples and revealed the anisotropic structure of the material. Two types of damage, induced by cavities and microcracks, take place during the deformation process leading to the ultimate fracture. The void nucleation seems to require acertain amount of plastic deformation. A comparison with airblast results conclude to similar microscopic failure mechanisms between laboratory tests and explosion experiments. Finally, a ﬁrst constitutive model has been built describing the combined effect of strain hardening, thermal softening, and viscoplasticity. The model was completed by a shear failure criterion and implemented as user material in the engineering FE computation code ABAQUS. The critical cumulate plastic strain at failure was determined by attempting to reproduce numerically the experimental failure states of airblast loaded plates. Now, the application of a failure criterion based on a critical cumulate plastic strain, as used in the present work, is very limited in the sense that it does not take into account the delaying effects of temperature and strain rate and the accelerating effect of stress triaxiality [13]. Furthermore, it is not able to describe the progressive feature of void growth induced damage and cracking before ultimate failure. These effects may be, at least partially, described by the damage-plastic yield functions proposed by e.g. Gurson [14] or Rousselier [15]. The improvement of the model presented in this paper is in progress. Acknowledgments All the work on the airblast experiments would not happened without the precious help of any persons. The authors would like to acknowledge Patrick Sanson (DCNS/ING/DCSE/CSB) for the manufacturing quality of the structure. Thanks to Jean-Luc Rouget and his crew (DGA TN/SDT/M3E/NP) for their pyrotechnics work. And thanks to Bernard Auroire (DGA/CTSN) for his help about the airblast experiments. Finally, the authors would also like to acknowledge Thierry Millot and his crew (DCNS/CESMAN) for their help about the microscopic analysis. References [1] Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. J Engng Fract Mech 1985;21:p. 31–4. [2] Besson J, Steglich D, Brocks W. Modeling of crack growth in round bars and plane strain specimens. Int J Solids Struct 2001;38:8259–84. [3] Lorentz E, Besson J, Cano V. Numerical simulation of ductile fracture with the rousselier constitutive law. Comput Meth Appl Mech Engng 2008;197:1965–82. [4] Le Saux M, Besson J, Carassou S, Poussard C, Averty X. Behavior and failure of uniformly hydrided zircaloy – 4 fuel claddings between 25 °C and 480 °C under various stress, states including ria loading conditions. Engng Fail Anal 2010;17:683–700. [5] Nahshon K, Xue Z. A modiﬁef gurson model and its application to punch-out experiments. Engng Fract Mech 2009;76:997–1009. [6] Luyten J, Tyler-Street M, Predicting deformation and fracture initiation of steel panels subjected to close-in explosions. In: DYMAT 2009, no. 214, EDP sciences; 2009. p. 1515–21. [7] Tyler-Street M, Luyten J. Developing failure criteria for application to ship structures subjected to explosive blast loadings. In: 7th European LS-DYNA conference; 2009. [8] Voce E. The relationship between stress ans strain for homogeneous deformation. J Inst Metals 1948;74:537–62. [9] Cazacu O, Barlat F. A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals. Int J Plast 2004;20:2027–45. [10] Graff S, Forest S, Strudel J-L, Prioul C, Pilvin P, BTchade J-L. Strain localization phenomena associated with static and dynamic strain ageing in notched specimens: experiments and ﬁnite element simulations. Mater Sci Engng A 2004;387–389:181–5. [11] Cailletaud G, Pilvin P. Identiﬁcation and inverse problems: a modular approach. Mater Parameter Estimat Modern Constitut Equat, ASME ASME 1993;43:33–45. [12] Longère P, Dragon A, Deprince X. Numerical study of impact penetration shearing employing ﬁnite strain viscoplasticity model incorporating adiabatic shear banding. J Engng Mater Technol, ASME 2009;131:1–14. [13] Bao Y, Wierzbicki T. On fracture locus in the equivalent strain and stress triaxiality space. Int J Mech Sci 2004;46(81):81–98. [14] Gurson AL. Continuum theory of ductile rupture by nucleation and growth: part I – yield criteria and ﬂow rules for porous ductile media. J Engng Mater Technol 1977;99(44):2–15. [15] Rousselier G. Ductile fracture models and their potential in local approach of fracture. Nucl Engng Des 1987;105(105):97–111.