Out-of-plane stretching and tearing fracture in ductile sheet materials

$Page 1: Out-of-plane stretching and tearing fracture in ductile sheet materials$
International Journal of Fracture 84: 297–306, 1997.c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Out-of-plane stretching and tearing fracture in ductilesheet materials

C.M. MUSCAT-FENECH and A.G. ATKINSDepartment of Engineering, University of Reading, PO Box 225, Whiteknights, Reading, RG6 2AY, U.K.+1734 313327 (Fax) e-mail: [email protected]

Received 1 April 1996; accepted in revised form 18 February 1997

Abstract. Pre-cracked ductile steel sheets are fractured by combined in-plane stretching and bending. The defor-mation mimics the mode of fracture when plating is dented and torn as in ship grounding. Fracture toughness isdetermined for this mode of tearing. Values are greater than those obtained with DENT testpieces on the samematerial because of the different mode of crack opening.

Notation� Angle subtended by the triangular tab� Effective plastic work per volume" Strain"y Yield strain�tip Vertical displacement at crack tip (Figure A-1, A-4))�1 Vertical displacement at point G (Figure A-4)�2 Vertical displacement between the horizontal and triangular tab tip below the application of the force

(Figure A-4)� Radius to which the traingular tab bends� Angle between the crack tip and the front edge of the plate at the clamped edges (Figure A-3)�Y Yield stress� Plastic work�tip Angle to which the crack tip is lift up from the horizontal (Figure A-1, A-2)

a Cracked length of DENT SpecimenA Crack AreaB Half width of the test piece (Figure A-1)h Overall length of the test piece (Figure A-1)L0 Starter crack lengthR Fracture toughnesst Thicknessu Displacement of crossheadV Volumew Width of triangular tab (Figure A-1, A-3)W Plastic work done per unit volumeX Forcey Vertical displacement measured relative to the tangent plane of the undeformed test piece (Figure A-4)

1. Introduction

We have an interest in large deformation out-of-plane fracture of ductile materials, in con-nection with ship grounding and collision, crashworthiness in vehicles etc., where a rock,

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298 C.M. Muscat-Fenech and A.G. Atkins

for example, indents and fractures ships’ plating and produces a long tear (Muscat-Fenech,1996). We require to know the appropriate fracture toughnessR for this mode of deformation,which involves out-of-plane stretching and tearing. None of the ‘standard’ Cotterell–Mai tests(Cotterell and Reddel, 1977; Cotterell and Mai, 1981; Mai and Cotterell, 1980; Mai and Cot-terell, 1985; Mai and Cotterell, 1984) for the determination of toughness in the presence ofextensive remote plastic flow reproduces the particular out-of-plane deformation we observe.It occurred to us that a geometry, superficially similar to the loading method employed in theOutwater ‘double-torsion’ test well known in lefm (Figure 1a), would mimic deformation pat-terns found in ship grounding. Figure 1b shows our adaptation of the test to ductile materials.Displacements at fracture in ductile materials are large, consequently it is necessary to clampthe edges of the test pieces. The width of the test piece, 2B, perpendicular to the starter crack,and the overall length, h of the test piece in the direction of crack propagation are determinedby the size of the rig. Owing to the comparatively large out-of-plane displacements at whichcracking occurs, we employ a flexible chain/shackle system to load the test pieces in a tensiletesting machine. The swing of the chain shackle is less than 5� from the vertical as propagationtakes place, and therefore the loading on the specimen remains approximately vertical.

In what follows we describe our experiments in 0.8 mm thick low carbon steel sheets andgive a model for the process. Fracture toughness in this out-of-plane mode is determined fromthe load-displacement diagrams.

Figure 1a. Outer ‘double-torsion’ lefm testpiece. Figure 1b. Adaptation of the Outwater geometry, withclamped edges, for ductile sheets.

2. Experiments

The mechanical properties of the sheet steel investigated, and the geometrical proportions ofthe test pieces, are given in Table 1.

Figure 2 shows a set of typical force – displacement plots for sheets with different startercrack lengths,L0(30 6 L0 (mm)6 60). The load point displacement u is given by the verticaldisplacement of the flaps. For all specimens, there is an initial elastic displacement (u < 8 mm,

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Out-of-plane stretching and tearing fracture in ductile sheet materials 299

Figure 2. Typical force-displacement (X � u) plots for testpieces with different starter crack lengths.

Table 1. Material properties and geometrical dimensions of thetestpiece.

Yield Stress: �y 175 MN/m2

Youngs Modulus: E 210 GN/m2

U.T.S. 306 MN/m2

Strength Coefficient: �0 574 MN/m2

Work Hardening Index: n 0.26Fracture Toughness (DENT) 240 � 255 kJ/m2

Thickness: t 0.8 mmDepth of Test Piece: h 203 mmWidth of Test Piece: 2B 203 mmCrack Lengths: L0 30, 35, 40, 45, 50, 55, 60 mm

say), followed by a slightly rising load region during which the flaps are being plastically bentand the previously undeformed test piece is being lifted up. The duration of this rising loadregion before cracking starts depends upon the starter crack length L0, e.g. crack propagationcommences at u � 31 mm for L0 = 30 mm, but u � 52 mm for L0 = 60 mm.

Figure 3a shows a stack of test pieces with different starter crack lengths loaded to the pointof crack initiation. We see that the flap bend radii at crack initiation vary with starter cracklength, L0, and therefore upon the width w of the out-of-plane triangular flap formed; theoverall width of all specimens is constant in our experiments. However, at equal propagatedcrack lengths, the bend radii are the same for all starter crack lengths, as shown in Figure 3b

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Figure 3a. Stack of testpieces with different L0 all at the point of crack initiation.

Figure 3b. Stack of testpieces after crack propagation from different L0 all to L = 70 mm.

which portrays a stack of test pieces all propagated to crack length 70 mm from a variety ofL0.

3. Measurement of fracture toughness

The relevant fracture toughness R for crack propagation can be obtained by using a basicrelation of nlefm/Hencky theory:

R = �(@Uinitiation=@A)ju = (1=t)(@Uinitiation=@L)ju;

sinceA = ta = t(h�L). Uinitiation is the total work done only up to fracture initiation, and thepartial derivative is taken at the displacement ufracture at the crack intiation. U is obtained fromthe force-dispacement plots by measuring the energy to the particular point of displacement

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Figure 4. Representative plot of energy U versus crack length L at various u.

ufracture. Figure 4 shows a representative plot of energy U versus crack length, L for a set ofufracture. The values of fracture toughness according to the above are some 400 kJ/m2.

Appendix 1 predicts an equation for the force X , Equation (13) in terms of the flap bendradius �. � is a function of many variables and it is cumbersome to get U in order to workout (@Uinitiation=@A)ju algebraically. For the same reason, there is unfortunately no simplealgebra for this geometry which corresponds with the normalised force versus crack lengthwhich Cotterell and Mai found for mode III tearing of sheet materials [6]. The problem is todo with the fact that deformation is not limited to the triangular flaps (which would produce aCotterell–Mai type plot) but also includes remote deformation at the clamped edges and alongthe line of cracking.

4. Conclusions

Sheets pre-slit down their centre have been subjected to a simultaneous tensile and out-of-plane bending mode of loading. For 0.8 mm thick low carbon sheet we obtained a fracturetoughness of about 400 kJ/m2. For reference, conventional Cotterell–Mai DENT tests gave240–255 kJ/m2. The difference in value reflects the different mode of crack opening.

We embarked on this investigation in order to obtain fracture toughness values from whichwe might predict in-plane failure strains in ship grounding, along the lines described in Atkinsand Mai (1987). As reported elsewhere (Muscat-Fenech, 1996), in that task we have beensuccessful. However we do not recommend this type of fracture toughness test for normal use;‘standard’ Cotterell–Mai tests are simpler to perform and analyse.

Appendix 1: Deformation model and theoretical predictions

From the deformed test pieces a model of the deformation was constructed as shown inFigure A-1. The test piece ACC 0A0 is rigidly clamped on the two sides AC and A0C 0 oflength h. The starter crack lengths AB vary within the lengths 30 6 L0 (mm) 6 60,

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Figure A-1. Model for deformed test piece.

Figure A-2. Deformation down the centre line of the testpiece.

As the edges of the triangular tabs are pulled upwards, the test piece starts to deform bybending up along the clamped edges and forming a ridge down the ligamentDE ahead of thestarter crack. Stretch in the area ABCED also occurs as the crack tip at D moves up by avertical displacement �tip. Along the lengthsAB andBC there is a mean rotation of (�tip�0)=2on each length, and therefore, along the whole length AC of �AC mean = �tip=2 = �tip=2B,since �tip = �tip=B, where �tip is the angle through which the plate bends at the crack tip.The plastic bending work along the length AC is:

�AC = AC Mp=unit length�AC mean

= h(�yt2=4)(�tip=2B): (1a)

The same work is required along A0C 0, so that for both clamped edges:

�clamped edges = 2h(�yt2=4)(�tip=2B)

= h(�yt2=4B)�tip: (1b)

Down the centre line of the specimen, the rotation is 2�tip, at D, Figure A-2, and at E thesheet has remained flat and therefore the rotation is 0. The mean ‘roof’ rotation is therefore�DE mean = (2�tip � 0)=2 = �tip. The plastic bending work along DE is therefore:

�DE = DE Mp=unit length�DE mean

= (h� L0)(�yt2=4)(�tip=B): (2)

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Figure A-3. Bending of triangular tab to mean radius �.

Figure A-4. Total load-point displacement comprises the vertical displacement of the whole sheet at the crack tipplus the bending deflexion of the tabs.

The total plastic bending work if the load were just applied at the crack tip (no moments onthe flaps) for the whole plate would be:

�clamped edges + �DE = h(�yt2=4B)�tip + (C � L0)(�yt

2=4)(�tip=B)

= (�yt2=4B)(2h� L0)�tip: (3)

The associated force is:

XPoint at crack tip d�tip = (�yt2=4B)(2h� L0) d�tip (4)

giving:

XPoint at crack tip = (�yt2=4B)(2h� L0): (5)

The volume of material under stretch is twice ABCED� thickness = [(h � L0)2B +2(L0B=2)]t = B(2h� L0)t. The tensile strain at D is [(B2 + �2

tip)1=2 �B]=B �

12(�tip=B)

2

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Figure A-5. Comparison between experimental X � u data (full line) and predictions of rigid plastic model.

so that the mean tensile strain for the whole stretched area is:

"ABCED = (12(�tip=B)

2� 0)=2

= 14(�tip=B)

2: (6)

The total stretch plastic work per unit volume is �y", where �y is the yield stress of thematerial, giving:

�Stretch = �y"V

= �y14(�tip=B)

2B(2h� L0)t

= (�yt=4B)(2h� L0)�2tip: (7)

It follows that the force associated with the stretch over displacement d�tip is:

XStretch = (�yt=2B)(2h� L0])�tip (8)

from

XStretch�tip = d�Stretch: (9)

The total point force at the crack tip is given by the addition of Equations (5) and (9) to give:

XPoint = (�yt2=4B)(2h� L0) + (�yt=2B)(2h� L0)�tip

= (�yt=2B)(2h� L0)(t=2 + �tip): (10)

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Forces are however not applied at the crack tip. Rather they are applied at the corners of theflaps. Consequently there is bending work on the triangular tabs as the tabs are bent upwardsto a radius �, Figures A-3 and A-4. The radius � to which the triangular tabs bend dependsupon the width, w = L0= cos�, of this tab, where � is the angle between the crack tip and thefront edge of the plate at the clamped sides. The bending work for one tab is:

�Bend = (X=2)� sin� �

= (X=2)L0 sin� sin(L0 sin�=�); (11)

where � is the angle subtended by the triangular tab and the radius �. The magnitude of X inEquation (11) is the same as that in (10). We view the force actually applied to the flaps asbeing equivalent to the same force at the crack tip plus a moment of magnitude (force � leverarm) = (X=2)� sin�. Also the plastic work to cause this bending for the two tabs is given by:

�Bend = WV

= �y"2(1=2BL0t)

= �y(t=4�)BL0t; (12)

where " is the strain to bend the tab to the radius �.Equating Equations (11) and (12) for two tabs gives:

2(X=2)L0 sin� sin(L0 sin�=�) = �y(t=4�)BL0t:

Giving:

X = (�yBt2=4�)(1= sin� sin(L0 sin�=�)): (13)

This takes the same value as Equation (10) and permits � to be determined. The stiffness-corrected machine crosshead displacement u is the addition of the vertical displacement dueto the triangular tab y and the vertical displacement �2 of the tab from the horizontal of theundeformed sheet that is, u = y + �2, Figure A-4.

The crack tip displacement �tip can be related to �2, as a function of the displacement�1, where �1 is the vertical displacement at point G, Figure A-4. Then from geometricalconsiderations:

�1 = �tiph=(h� L0) (14a)

�2 = (�1=(KI + IF ))(KI + � sin(L0 sin�=�)); (14b)

where:

IF = (B2 + �21)

1=2((h2 + �21)

1=2� ((h� L0)

2 + �2tip)

1=2)=(B2 + L20 + �2

tip)1=2 (14c)

JG = [((B2 + �21)

2(B2 + L20)=(B

2 + L20 + �2

tip)2) +B2

�(2B2(B2 + �21)=(B

2 + L20 + �2

tip))]1=2 (14d)

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KJ = ((B2 + �21)�tip=(B

2 + L20 + �2

tip))(IF=IF2� JG2)1=2) (14e)

KI = KJ IF=JG (14f)

y = (u� �2): (14g)

Therefore for a given starter crack length, knowing the experimental instantaneous forcevalues acting up on the system, then the instantaneous values of the crack tip displacement�tip and the bending radius � may be found according to Equations (10) and (13) respectively.

Since the instantaneous values of the crack tip displacement �tip and radius � are known,the displacements �1 and �2 may be calculated according to Equations (14). Knowing theexperimental and theoretically predicted displacements at the instantaneous experimentalforces, comparison of the force-displacement traces may be made. A good comparison betweenthe displacements and the force-displacement traces is obtained, Figure A-5.

References

Atkins, A.G. and Mai, Y.W. (1987). Fracture strains in sheet metal forming and specific essential work of fracture.Engineering in Fracture Mechanics 27, 291–297.

Cotterell, B. and Reddel, J.K. (1977). The essesntial work of plane stress ductile fracture. International Journal ofFracture 13, 267–277.

Cotterell, B. and Mai, Y.W. (1981). Plane stress ductile fracture. Proc. 5th International Conference of Fracture –ICF-5. (D. Francois, editor) Pergamon Press, Oxford 4, 1683–95.

Mai, Y.W. and Cotterell, B. (1980). Effects of pre-strain on plane stress ductile fracture in �-Brass. Journal ofMaterial Science 15, 2296–2306.

Mai, Y.W. and Cotterell, B. (1984). The essential work of fracture for tearing of ductile metals. InternationalJournal of Fracture 24, 229–236.

Mai, Y.W. and Cotterell, B. (1985). Effect of specimen geometry on the essential work of plane stress ductilefracture. Engineering in Fracture Mechanics 21, 123–128.

Muscat-Fenech, C.M. (1996). The tearing of ships’ plating upon grounding. University of Reading, Department ofEngineering Report (6 parts) for EPSRC/MTD/MoD.

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Out-of-plane stretching and tearing fracture in ductile sheet materials