Static crack extension prediction in aluminium alloy at low temperature

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Engineering Fracture Mechanics 75 (2008) 510–525 www.elsevier.com/locate/engfracmech

Static crack extension prediction in aluminium alloy at low temperature Andrea Carpinteri *, Roberto Brighenti, Sabrina Vantadori, Danilo Viappiani Department of Civil and Environmental Engineering & Architecture, University of Parma, Viale G.P. Usberti, 181/A, 43100 Parma, Italy Received 12 December 2006; received in revised form 6 April 2007 Available online 22 May 2007

Abstract The evaluation of the crack initiation and crack extension under different environmental conditions is very important for many engineering applications. Several crack extension criteria have been proposed in the last decades, but each criterion can be employed only for particular materials, loading configurations, environmental conditions. In the present paper, the R-criterion (minimum extension of the core plastic zone) is modified in order to take into account the temperature dependence. The modified criterion is herein employed to predict the crack path and the crack extension force for an edge-cracked finite plate under tension, by using a simplified procedure to determine the stress-intensity factor (SIF) for different initial crack configurations. Then, results concerning some experimental tests performed on edge-cracked aluminium alloy sheet specimens at different temperatures are reported. Finally, the theoretical results are compared (in terms of crack extension force and crack path) with the experimental data.  2007 Elsevier Ltd. All rights reserved. Keywords: Crack extension; Crack path; Low temperature; Aluminium alloy

1. Introduction The evaluation of the crack initiation, the crack extension force and the crack path under different temperature conditions is very important for many engineering applications in order to quantify the structural safety according to the so-called damage tolerant design. Furthermore, other environmental effects should be taken into account to assess the structural component safety: for example, the humidity and salt air content play an important role especially under fatigue loading. In the present paper, only the temperature effect is considered and the static crack propagation is examined. The assessment of the structural safety at low temperature should be done with extreme care since the fracture toughness of the material is generally lower than that at room temperature. The effects of low temperatures should be accurately examined especially in the near-threshold regime [1–3]. Recent studies [3–5]

*

Corresponding author. E-mail addresses: [email protected] (A. Carpinteri), [email protected] (R. Brighenti).

0013-7944/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.05.001

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Nomenclature a 0, a initial and current crack length CI(n, h), CII(n, h) finite geometry correction factors F(I1, J2) = 0 boundary of the plastic domain f crack driving force I1, I2 first and the second stress tensor invariants, respectively J2 second deviatoric stress tensor invariant Kc plane-stress fracture toughness KIC fracture toughness KI,II Modes I and II stress-intensity factors Keq equivalent stress-intensity factor KI(si),II(si) Modes I and II SIFs for a semi-infinite plate Kfc fatigue fracture toughness M = rH/req stress triaxiality ratio Rp distance function from the crack tip of the plastic core region S strain energy density T temperature TFDBT fatigue ductile–brittle transition temperature TD distortional strain energy density TV dilatational strain energy density YI(n, h), YII(n, h) dimensionless Modes I and II SIFs YI(si), YII(si) dimensionless Modes I and II SIFs for a semi-infinite plate wf = cs + cp fracture energy per unit surface area W plate’s width a(T), b(T), k(T) temperature-dependent parameters of the yield function F(I1, J2) = 0 bi orientation angle of the equivalent straight crack at a generic crack propagation stage i c(T) = a(T)/b(T) parameter to quantify hydrostatic dependence of the yield function cp plastic work done per unit crack surface area created cs surface energy per unit crack surface area created j(T) = k(T)/b(T) equivalent temperature-dependent yield stress q = rt/rc tensile-compressive yield stresses ratio rH, req hydrostatic and equivalent stress, respectively r0 applied remote constant stress rt, rc tensile and compressive yield stress, respectively n = a/W relative crack depth h crack orientation angle with respect to the loading direction

have shown that metals could be roughly divided into two groups, that is to say, materials with fatigue ductile– brittle transition (FDBT) and those without FDBT at low temperature. When the temperature T is greater than TFDBT (where TFDBT is the temperature at which the ductile–brittle transition occurs), the fatigue crack growth (FCG) mechanism is forming of ductile striations, and the FCG rate decreases by decreasing T. When T is equal to about TFDBT, the FCG mechanism becomes microcleavage and the fatigue fracture toughness Kfc sharply decreases. The FCG rate tends to increase as the temperature is further reduced. The knowledge of the crack configuration during service life is crucial in order to assess the structural reliability. Several practical cases often involve mixed-mode fracture, and an appropriate method to correctly describe the crack growth process in such situations is needed. Several well-known crack propagation criteria [6–15] have been proposed, and some of them are briefly reviewed in the following. When experimental data are compared with the predictions by different criteria, it is quite difficult to find a criterion which fits the test data for different materials, loading configurations, environmental conditions.

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In the present paper, a criterion recently proposed by Shafique and Marwan [16,17] (minimum plastic zone extension criterion, R-criterion) is modified in order to predict the static crack propagation of a slant crack at different values of temperature. Some displacement-controlled fracture tests at low temperatures for aluminium alloy plates under mixed mode I + II have been performed. The modified R-criterion is assessed by comparing (in terms of crack path and crack extension force) theoretical results with experimental data. 2. Crack-tip stress field As is well known, the stress field near the crack front can be generally described as follows, according to the linear elastic fracture mechanics [18]: frg ¼ ð2p  rÞ1=2  ½Y   fKg

with fKg ¼ f K I

K III gT

K II

ð1Þ

that is, the stress field components at point P (grouped in the column vector {r}) depend on both a singular function of the distance r from the crack front (Fig. 1a) and the stress-intensity factors (SIFs), which are given by the product of geometric correction factors (grouped in the matrix [Y]) and the Modes I–III SIFs (grouped in the column vector {K}). For a 2-D problem under mixed mode I–II loading, if the crack is assumed to be straight and to lie on the x-axis, the above relationship can be simplified as follows (Fig. 1b): 8  9 8  9 a a 3a > > a a 3a > > > > > > > 9 9 8 I 8 II > > > cos 1  sin sin >  sin 2 2 þ cos 2 cos 2 > > > > > > > > > r r ðr;aÞ ðr;aÞ 2 2 2 > > > > > > > > x = = < < = = < x <   KI K a a 3a II I II a a 3a ry ðr;aÞ ¼ pffiffiffiffiffiffiffi ry ðr;aÞ ¼ pffiffiffiffiffiffiffi ; cos cos sin cos 1 þ sin sin > > > > > 2 2  > 2pr > 2pr > > > > ; ; : I : II 2 2 2 > 2 > > > > sxy ðr;aÞ sxy ðr;aÞ > > > > > > > a a 3a > > > > > > ; : > ; : cos 1  sin sin sin a2 cos a2 cos 3a 2 2 2 2   rIðIIÞ ðr;aÞ ¼ m rIðIIÞ ðr;aÞ þ ryIðIIÞ ðr;aÞ plane strain z x ðr;aÞ ¼ 0 plane stress rIðIIÞ z pffiffiffiffiffiffi pffiffiffiffiffiffi with K I ¼ Y I r0 pa; K II ¼ Y II r0 pa

ð2Þ

where the superscripts I and II stand for Modes I and II, respectively, and m is the Poisson ratio. By writing the displacement fields for a crack under Mode I or Mode II loading: 8 8

9

9 a 3a > a 3a > > > > > > rffiffiffiffiffiffi > r ffiffiffiffiffi ffi > > > >      cos þ sin ð1 þ mÞ ð2j  1Þ cos ð1 þ mÞ ð2j þ 3Þ sin uz uz 2 2 = 2 2 = KI r < K II r <

;

¼ ¼   2E 2p > 2E 2p > a 3a > a 3a > un un > > > > > > > > ; ; : ð1 þ mÞ ð2j þ 1Þ sin  sin : ð1 þ mÞ ð2j  3Þ cos  cos 2 2 2 2

ð3Þ the corresponding SIFs can be evaluated. In Eq. (3), uz and un are the displacements normal and parallel to the crack, respectively, a is the position angle of the considered point P with respect to the crack direction, r is the

(u z ) z

A

rα

Z Y O

P(x,y,z) X

σz

P'(x',y',z') n (u n )

σn t (u t)

τnz

τnz A τtz τtn σt

y (v) σ y r o crack tip

α

τxy σx x (u)

crack front

Fig. 1. Stress field near the crack front in a 3-D (a) and in a 2-D (b) fracture mechanics problem.

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distance from the crack tip (Fig. 1), whereas j = (3  m)/(1 + m) for plane stress and j = 3  4m for plane strain. Fracture failure under Mode I takes place when the critical stress-intensity factor, KIC (fracture toughness of the material for a given temperature), is attained at the crack tip, i.e. when the condition KI = KIC is fulfilled [18], whereas fracture failure under mixed mode occurs when an equivalent stress-intensity factor Keq, evaluated by taking into account simultaneously Modes I and II SIFs, attains the critical SIF KIC at the crack tip, i.e. Keq = KIC. For a tensioned semi-infinite (si) cracked plate (Fig. 2a), the dimensionless Mode I (YI(si)) and Mode II (YII(si)) SIFs have been evaluated by Hasebe and Inohara [19] and Nisitani [20] by using the conformal mapping function technique. By fitting their results (obtained for different values of the angle h) and using a fifthorder polynomial expression, the approximate dimensionless SIFs in Modes I and II can be described by the following relationships: Y IðsiÞ ¼ 7:526  104 þ 1:716  102  h  2:020  104  h2 þ 7:419  106  h3  9:995  108  h4 þ 4:000  1010  h5 Y IIðsiÞ ¼ 1:093  103 þ 2:096  102  h  4:854  104  h2 þ 7:656  106  h3  9:070  108  h4 þ 4:088  1010  h5

ð4Þ pffiffiffiffiffiffi with Y IðsiÞ;IIðsiÞ ¼ K IðsiÞ;IIðsiÞ =r0 pa and h expressed in degrees. The corresponding graphical representations are displayed in Fig. 2c.

a

b

α

α θ

θ a

h a

Dimensionless SIFs, Y I(si), (II(si)) = K I(si), (II(si)) / σ0 ( π a )1/2

W

c

1.20 Y I(si )

1.00

Y II(si)

0.80 0.60 0.40 0.20 0.00 0

30

60

90

Angle, θ (degrees) Fig. 2. Semi-infinite (a) and finite (b) edge-cracked plate with an inclined edge crack under remote tension r0, and the dimensionless Modes I and II SIFs (c) for case (a).

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In the present study, a finite thin-plate with an inclined edge crack is examined (Fig. 2b). The dimensionless Modes I and II SIFs can be evaluated by introducing finite geometry correction factors, CI(n = a/W, h) and CII(n = a/W, h), which can be defined as the ratio between the Mode I (YI) and Mode II (YII) edge-cracked finite thickness plate SIFs and the corresponding values for the semi-infinite case [21]: C I ðn; hÞ ¼ Y I ðn; hÞ=Y IðsiÞ ;

C II ðn; hÞ ¼ Y II ðn; hÞ=Y IIðsiÞ

ð5Þ

By fitting the results from Bowie [21] with a cubic interpolation, the correction factors can be expressed by the following relationships: C I ðn; hÞ ¼ 0:9955 þ 9:2053n  24:1418n2 þ 13:9206n3  0:0918h  0:1077nh þ 0:2645n2 h þ 0:0023h2 þ 7:5777  105 nh2  1:3566  105 h3 C II ðn; hÞ ¼ 1:0870  1:1748n  0:6134n2 þ 0:5308n3  0:0507h þ 0:06493nh

ð6Þ

þ 0:0147n2 h þ 0:0011h2  0:0007nh2  7:4898  106 h3 where n = a/W and h are the relative crack depth and the crack orientation (measured in degrees), respectively (Fig. 2b). In other words, the dimensionless SIFs for an edge-cracked finite plate with a crack characterised by a relative depth n and orientation h can be expressed as: Y I ðn; hÞ ¼ C I ðn; hÞ  Y IðsiÞ ;

Y II ðn; hÞ ¼ C II ðn; hÞ  Y IIðsiÞ

ð7Þ

3. Review of some crack propagation criteria Several criteria for both stable or unstable crack propagation have been proposed during the last decades in order to predict the crack path under simple or mixed mode fracture for different materials. In the following, some of the main criteria are briefly reviewed. According to the MTS-criterion proposed by Erdogan and Sih [6], crack grows in the direction perpendicular to the maximum principal stress (ra) direction or, equivalently, parallel to the direction of the maximum tangential stress (MTS). Analytically, it can be stated as follows: ora ¼ 0; oa

o2 r a <0 oa2

ð8Þ

where the polar coordinate a is used to identify the position vector around the crack tip (Fig. 1). By means of the stress field expression (2), condition (81) can be written as follows: tan2

a l a 1  tan  ¼ 0 2 2 2 2

with l ¼ K I =K II

ð9Þ

The so-called zero shear stress criterion by Maiti and Smith [13] predicts the same results. Following the minimum strain energy density criterion (S-criterion) proposed by Sih [7,8], the crack grows in the direction of minimum strain energy density S around the crack tip. Analytically, the criterion can be stated as follows: oS ¼ 0; oa

o2 S >0 oa2

ð10Þ

where the angle a is measured between the initial crack orientation and the crack growth direction. According to the M-criterion proposed by Kong et al. [9], the crack propagation direction is defined by the maximum value of the stress triaxiality ratio M = rH/req around the crack tip (rH is the hydrostatic stress, whereas req is the equivalent stress which can be assumed to be equal to the Von Mises stress). Analytically, the criterion can be stated as follows: P3 oM o2 M 2ð1 þ mÞ h a ai i¼1 rii p ffiffiffiffiffiffiffiffiffiffi ffi ¼ 0;  K ¼ < 0 with r ðr; aÞ ¼ cos sin ðplane strainÞ ð11Þ K H I II oa oa2 2 2 3 3 2p  r

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By using the stress field expression (2), condition (111) becomes: tan4

a a a 1 a 1  3l tan3  ð1  2l2 Þ tan2 þ ð1  l2 Þl tan  ð1 þ l2 Þ ¼ 0 with l ¼ K I =K II 2 2 2 2 2 2

ð12Þ

According to the maximum dilatational strain energy density criterion (T-criterion) proposed by Theocaris et al. [10–12], crack grows in the direction of the maximum value of dilatational strain energy density TV around the crack tip. The criterion can be written as follows: oT V ¼ 0; oa

o2 T V <0 oa2

ð13Þ

By writing the dilatational strain energy density TV through the stress invariants, the criterion proposed by Ukadgaonker and Awasare [14] can be obtained (it is only a different form of the conditions (13)): oI p ¼ 0; oa

o2 I p <0 oa2

ð14Þ

oRp ¼ 0; oa

o 2 Rp >0 oa2

ð15Þ

P3 P3 P3 with I p ¼ I 21  2I 2 , where I 1 ¼ i¼1 rii and I 2 ¼ 12 ð i;j¼1 rii  rjj  i;j¼1 rij  rij Þ are the first and the second stress tensor invariants, respectively (note that the dilatational strain energy density TV is proportional to Ip). On the other hand, the distortional strain energy TD around the Von Mises plastic boundary is constant and equal to the yield stress and, therefore, shows no minimum. From experimental tests, it has been remarked that the crack propagation direction usually tends to follow the local or global minimum extension of the plastic core region. From a physical point of view, it could be explained by considering that the plastic core region is a highly-strained area, and the crack tends to reach the elastic region of the material outside the plastic zone, propagating through the plastic region which develops around the crack tip. Therefore, it is reasonable to assume that the crack follows the ‘‘easiest’’ path to reach the elastic region. Such a path can be assumed to coincide with the shortest path from the crack tip to the elastic material outside the plastic zone, as is stated by the R-criterion proposed by Shafique and Marwan [16,17] (Fig. 3). Mathematically, the R-criterion can be written as follows:

where Rp is the function which defines the radial distance from the crack tip to a generic point of the plastic zone boundary F(I1, J2) = 0, with J2 = second deviatoric stress tensor invariant (Fig. 3). When the conditions stated in Eq. (15) are fulfilled, the direction of minimum radial distance is determined, and the crack propagation direction vector t is assumed to be coincident with such a direction (Fig. 3). The above criterion can be also justified by considering that the fracture stress rf is proportional to the square root of wf (which is the fracture energy per unit surface area). Such an energy for a quasi-brittle elastic–plastic material is equal to the summation of the surface energy cs and the plastic work cp done per unit surface area created, that is, wf = cs + cp. For structural materials, we typically have cp  cs, so that y

plastic region

Rp

crack t

x elastic region F(I1,J2 )=0

Fig. 3. Graphical representation of the R-criterion.

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the fracture stress rf is primarily dependent on cp only. The shortest distance from the crack tip to the elastic– plastic boundary corresponds to the minimum plastic work which is needed to create a new portion of crack area, that is, such a shortest distance corresponds to the minimum fracture energy and fracture stress. 4. Extension of the R-criterion The R-criterion can be extended in order to take into account the temperature effects on the crack propagation by properly modifying the elastic–plastic boundary which depends upon the yield function F adopted. Since the yield stress and the shape of the yield function F (necessary to identify the plastic core region) are strongly dependent on the environmental conditions such as the temperature, the crack growth criterion can be modified as is hereafter proposed. As is well known, materials usually show a sort of embrittlement by decreasing temperature, while large plastic deformations occur at high temperatures. A Drucker–Prager-like yield criterion can be considered in order to quantitatively describe such a behaviour. A generalisation of the above yield function could be written as follows: pffiffiffiffiffi F ðI 1 ; J 2 Þ ¼ aðT Þ  I 1 þ bðT Þ  J 2  kðT Þ ¼ 0 ð16Þ P3 P3 where I 1 ¼ i¼1 rii , J 2 ¼ i;j¼1 r0ij r0ij =2 are the first stress tensor invariant and the second deviatoric stress P3 invariant (r0ij ¼ rij  dij  k¼1 rkk =3 ¼ rij  dij  p, where p indicates the hydrostatic stress state value), respectively (the following relationship between the stress tensor and deviatoric stress tensor invariants holds: J 2 ¼ 13 ðI 21  3I 2 Þ), whereas a(T), b(T) and k(T) are three temperature-dependent parameters of the material. The parameters a(T) and b (T) respectively measure the hydrostatic and the deviatoric stress dependence of the yield function on the temperature, and the parameter k(T) defines the temperature-dependent yield stress. The generalised yield function can be rewritten in the classical Drucker form: pffiffiffiffiffi F ðI 1 ; J 2 Þ ¼ cðT Þ  I 1 þ J 2  jðT Þ ¼ 0 ð17Þ where c(T) = a(T)/b(T) and j(T) = k(T)/b (T) can be obtained through the following expressions, according to the Drucker formulation: 1 rc ðT Þ  rt ðT Þ ; cðT Þ ¼ pffiffiffi 3 ðrc ðT Þ þ rt ðT ÞÞ

2 rc ðT Þ  rt ðT Þ jðT Þ ¼ pffiffiffi 3 ðrc ðT Þ þ rt ðT ÞÞ

ð18Þ

with rc(T), rc(T) are the compressive and the tensile yield temperature-dependent stress of the material, respectively. If the tensile strength and the compressive strength are equal to each other and characterised by the same temperature dependence (rc(T) = rt(T) = ry(T)), typical of metal-like materials at ordinary or high temperatures, the yield condition becomes: pffiffiffiffiffi F ðI 1 ; J 2 Þ ¼ F ðJ 2 Þ ¼ þ J 2  jðT Þ ¼ 0

ry ðT Þ with cðT Þ ¼ 0; jðT Þ ¼ pffiffiffi 3

ð19Þ

which corresponds to the classical Von Mises criterion. On the other hand, a brittle behaviour can be observed at very low temperatures: in such a case, the tensile strength can be assumed to be significantly lower than the compressive one (i.e. rt(T)  rc(T) = ry(T)), and the yield condition reduces to: pffiffiffiffiffiffiffi pffiffiffi 1 2 F ðI 1 ; J 2 Þ ¼ I 1 þ 3J 2  3  jðT Þ ¼ 0 with cðT Þ ffi pffiffiffi and jðT Þ ffi pffiffiffi rt ðT Þ ð20Þ 3 3 which is a hydrostatic stress-dependent law (i.e. a particular case of the Drucker criterion). For intermediate values of the temperature, an intermediate behaviour can be assumed (Fig. 4a) between the situations described by Eq. (19) (no temperature dependence of the shape of the plastic core region defined by F(I1, J2) = 0) and (20) (maximum temperature dependence of the shape of the plastic core region). In Fig. 4b, the dependence of the hydrostatic coefficient c(T) (see Eq. (18)) on the tensile-compressive yield stress ratio

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 510–525

0.60

Hydrostatic dependence coefficient, γ

Principal stress, σ2

temperature decreasing

Principal stress, σ1

517

0.40

0.20

0.00 0.0

0.2

0.4

0.6

0.8

Yield stresses ratio, ρ = σt/σc

1.0

Fig. 4. Generalised Drucker criterion (a); hydrostatic coefficient dependence of the yield function with yield stress ratio q (b).

q = rt/rc is displayed. As can be observed from Fig. 4b, the hydrostatic dependence of the yield function vanishes as the yield stress ratio q tends to 1, while it monotonically increases by decreasing q from 1 to 0. In other words, when a hydrostatic stress dependence is assumed, c(T) 5 0 (situation which usually occurs at low temperature), the crack paths should show different patterns since the yield surface shape changes with the temperature T. As a matter of fact, if the ratio rc/rt increases by decreasing temperature (i.e. if rt(T) changes more rapidly than rc(T) for low temperatures), the hydrostatic stress dependence automatically arises from Eq. (181) (the parameter c(T) increases with the ratio rc/rt, whereas it is equal to zero for rc/rt = 1, Fig. 4b). 5. Approximate numerical procedure for crack growth simulation The crack propagation path in an edge-cracked finite plate under remote constant stress r0 is examined. The problem is solved approximately by using incremental steps of crack growth by considering finite size crack extension during the whole fracture process (Fig. 5). Starting from the initial crack configuration defined by the crack length a0 and direction h0 (the crack segment is identified by its two extreme points 1 and 2), the crack growth simulation is performed from point 2 and assuming a crack extension da (where da is a fixed ratio of the initial crack length), which takes place in a direction identified by means of the angle h2 measured with respect to the loading direction. In such a way, the new crack tip (point 3) can be obtained, and the crack propagation can continue in the same way, starting from point 3 to get to a new point which represents the new crack tip. It can be verified that, for a generic edge-cracked plate, the SIFs mainly depend on the local crack extension at the crack tip. Therefore, the effective curved crack in a generic propagation stage (Fig. 5) can be approximated, in order to determine the SIFs, by an equivalent straight crack (see dashed lines in Fig. 5) having the direction of the tangent to the crack at the tip point, and length equal to the distance (measured in such a straight direction) from the current crack tip to the free surface. In order to assess the reliability of such an approximate method, a comparison with Nisitani results for a kinked crack [20] is reported in Table 1. According to the above method, the angle of the first branch of the kinked crack does not influence the SIF results and, therefore, different crack configurations give us the same Mode I SIFs since all of them are characterised by h2 = p/2. Moreover, also the ratio between the lengths of the first and the second branch of the kinked crack does not influence the SIF values, since the effective crack in the present model is substituted by an equivalent straight crack in a semi-infinite plate (the SIF must not depend on the crack length, see Eq. (4)). The relative error between the present results and those by Nisitani is always lower than 2.17% for all the examined cases. More general crack configurations have been considered in order to validate the proposed hypothesis. Fig. 6 shows two FE meshes of a cracked plate characterised by h1 = 45 and h2 = 60 (Fig. 6a and c)

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0

r

2

4

3

1

0 Fig. 5. Finite edge-cracked plate. Geometrical scheme for numerical discrete crack propagation.

Table 1 Dimensionless SIFs obtained from Nisitani results [20] and by the present method for a kinked crack with h2 = const = 90 Crack configurations

da/a0 = 3/2 da/a0 = 1/2 da/a0 = 1/18

Nisitani [20]

Present study

h1 = 30

h1 = 45

h1 = 60

h1 = 30

h1 = 45

h1 = 60

1.1230 1.1220 1.0980

1.1210 1.1210 1.1210

1.1210 1.1210 1.1250

1.1219 1.1219 1.1219

1.1219 1.1219 1.1219

1.1219 1.1219 1.1219

and by h1 = 90 and h2 = 45 (Fig. 6b and d), respectively. Plane stress 8- and 6-noded quadratic finite elements have been used to model the cracked plate, whereas the displacement correlation technique has been employed for the numerical SIF evaluation. In order to verify such an assumption, fully 3-D analyses have been performed by using 20-noded solid elements (the same 2-D mesh patterns have been used): the numerical results have confirmed the suitability of the plane-stress hypothesis since the relative differences between the 2- and 3-D numerical analyses are always less than about 2%. The FE SIFs and theoretical SIFs are displayed in Fig. 7a for h1 = 45 and in Fig. 7b for h1 = 90, by assuming different values of h2 (ranging from 30 to 90). As can be observed, both Modes I and II SIFs are satisfactorily predicted by the simplified method proposed, which tends to slightly underestimate the numerical results. The proposed simplified procedure seems to be appropriate to evaluate the stress-intensity factors for cracks with complex shapes in a semi-infinite plane. By using the above approximation procedure for SIF evaluation, the direction of crack propagation can be obtained at each generic stage of crack growth. In other words, the angle bi of the equivalent straight crack at the generic crack growth stage i (current crack tip) is determined by considering the direction of the straight line defined by the two points i, i  1 (Fig. 5). For example, if we consider a stage of crack propagation characterised by a crack with the tip located at point 3 (Fig. 5), the Modes I and II SIFs are determined from Eq. (7) by considering a straight crack inclined of an angle b3 with respect to the free surface and having length equal to a3. When direction h3 (angle measured with respect to the vertical direction) of the subsequent crack extension is determined by using an appropriate criterion, the actual crack extension da

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519

Fig. 6. FE mesh used for SIFs evaluation: case of a kinked crack composed by two straight segments characterised by h1 = 45 and h2 = 60 (a, c) and by h1 = 90 and h2 = 45 (b, d).

1.60

FEM

Dimensionless SIFs, YI(II) = KI(II) /σ0 ( πa3)1/2

FEM

Present Study

Present Study

YI Y II

YI Y II

1.40 1.20 1.00 0.80

θ1 = 45°

θ1 = 90°

a0 = da

a0 = da

0.60 0.40 0.20 0.00 30

45

60

75

Angle, θ2 (degrees)

90

30

45

60

75

Angle, θ2 (degrees)

90

Fig. 7. Edge-cracked finite plate: dimensionless SIFs obtained from a simplified method (present study) and FE analyses for a two straight segment crack shape characterised by h1 = 45 and h1 = 90.

can be considered to finally obtain the next crack configuration having the tip at point 4. By repeating the described procedure until the desired number of crack extensions has been reached, the whole crack propagation path can be deduced in an approximate way, by considering the propagation pattern defined by the crack tip points 1–2–3–4–   determined during the various crack growth steps. It should be noted that the proposed approximate procedure has been implemented in a numerical code in order to overcome the difficulties arising particularly when the minimum of the plastic zone extension is required: in fact such an extension should be evaluated by considering a complex yield function which must be evaluated along many directions departing from the crack tip. Such a yield function depends on the stress state which is determined by the current crack configuration. Due to such considerations the proposed procedure, although is essentially theoretical (and so analytically applicable), has been implemented in a numerical procedure to avoid very complex analytical expressions.

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6. Experimental tests In order to quantify the effect of temperature, thin edge-cracked aluminium alloy plates have been tested at different temperatures (below and above 0 C), by employing a universal testing machine under uniaxial displacement control conditions (Fig. 8). The specimen material is characterised by the following mechanical and physical parameters: E = 76 GPa, m = 0.34, ry = 330 MPa, k = 3.31 Æ 105 C1 (i.e. Young modulus, Poisson coefficient, yield stress, thermal expansion coefficient, respectively). The plate and crack sizes are as follows: b · h · t = 120 · 240 · 1.3 mm, initial crack length a0 = 60 mm with different values of the initial crack orientation angle (h0 = 30, 45, 90) measured with respect to the loading direction (Fig. 8a). The tests were conducted at a fixed displacement rate equal to 1 mm/s, measured between upper and lower edges of the plate (Fig. 8). Both force and crack length were measured during experimental tests. From the experimental results, it can be deduced that the temperature in the considered range does not heavily influence the crack paths, independent of the initial crack orientation value (for example, see Fig. 9a–c for h0 = 30, and Fig. 10a–c for h0 = 45). Nevertheless, the failure mode changes from brittle-

a

b

actuator

loading direction

link device specimen holder (rigid)

existing pre-crack

θ0 a0 b

specimen h

link device load cell

Fig. 8. Experimental set-up.

Fig. 9. Crack paths and final collapsed configurations for aluminium plates with an initial crack having #0 = 30, for T equal to: (a) 20 C, (b) +20 C, (c) +80 C.

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Fig. 10. Crack paths and final collapsed configurations for aluminium plates with an initial crack having #0 = 45, for T equal to: (a) 20 C, (b) +20 C, (c) +80 C.

(for T = 20 C) to ductile-type (for T = +80 C) as the deformation in the ligament zone shows (compare Figs. 9a and 10a with Figs. 9c and 10c). Further, the plane-stress fracture toughness Kc(T) of the plate material has been evaluated, and the diagram of this parameter against temperature is displayed in Fig. 11. The obtained results can be interpolated with straight lines whose equations are: K c ðT Þ ¼ þ4:26916E þ 007  75889T ½Pa m1=2 

for h0 ¼ 90

K c ðT Þ ¼ þ3:33242E þ 007  71507T ½Pa m1=2 

for h0 ¼ 45

K c ðT Þ ¼ þ3:82467E þ 007  59778T ½Pa m1=2 

for h0 ¼ 30

ð21Þ

for the three different values of the initial crack angle h0. For the considered range of temperature, the plane-stress fracture toughness Kc(T) tends to slightly increase with temperature decreasing. Nevertheless, since the slopes of the straight lines above are very small and such

Pl.stress fracture toughness, K c [Pa m1/2 ]

8.0E+7 θ 0 = 30º K c (T) = +4.26916E+007 - 75889 T [Pa m1/2 ]

θ 0 = 45º

6.0E+7

θ 0 = 90º

4.0E+7

K c (T) = +3.33242E+007 - 71507 T [Pa m1/2 ]

2.0E+7

K c (T) = +3.82467E+007 - 59778 T [Pa m1/2 ]

0.0E+0 -40

-20

0

20

40

60

80

100

Specimen temperature, [ºC] Fig. 11. Plane-stress fracture toughness Kc for aluminium alloy specimens tested at different temperatures for #0 = 30, 45, 90.

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three lines are very close to one another, the plane-stress fracture toughness Kc(T) can be regarded as practically constant in the temperature range considered. 7. Numerical simulations By employing the numerical simulation procedure discussed in Section 5, some crack propagation paths can be determined. By assuming a plane-stress fracture toughness Kc(T) expressed through a linear relationship (see Eq. (21)) and considering the compressive yield stress equal to the tensile yield stress (rc(T) = rt(T)) at ordinary, moderately low and high temperatures (i.e. for the cases experimentally investigated in the present study), the crack driving force f vs the crack extension a  a0 curves are plotted in Fig. 12. In our experimental tests, the compressive yield stress (rc) and the tensile yield stress (rt) have been observed to slightly increase with decreasing temperature, and the following relationship can describe such a behaviour: rc;t ðT Þ ¼ rc;t ðT 0 Þ  1:974  105  ðT  T 0 Þ

ð22Þ

where rc,t (i.e. rc or rt) is expressed in Pa, and the reference room temperature T0 is expressed in C. In the numerical analysis, the above laws rc,t(T) have been assumed and, therefore, the hydrostatic dependence has been taken into account by means of the corresponding value of c(T) (see Eq. (181)).

Experim. data

Present Study

Experim. data

θ 0 = 30 º

crack driving force, f [N]

crack driving force, f [N]

1E+5

θ 0 = 45 º θ 0 = 90 º

1E+4

1E+3

Temperature T = - 20 º C

1E+2

θ 0 = 30 º

1E+5

θ 0 = 45 º θ 0 = 90 º

1E+4

1E+3

1E+2

Temperature T = + 20 º C

0.01

0.01

crack extension, a-a 0 [m]

crack extension, a-a 0 [m] Experim. data

Present Study θ 0 = 30 º

1E+5

crack driving force, f [N]

Present Study

θ 0 = 45 º θ 0 = 90 º

1E+4

1E+3 Temperature T = + 80 º C

1E+2 0.01

crack extension, a-a 0 [m] Fig. 12. Crack driving force f against crack extension a  a0 obtained from experiments and from the proposed model, for T = 20 (a), +20 (b), +80 C (c).

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In Fig. 12, the numerical (determined through the modified R-criterion) and experimental crack driving forces f are displayed against the crack extension for different values of the initial crack orientation angle (h0 = 30, 45, 90). From Fig. 12 (where bi-logarithmic scale is used to represent both crack extension force and crack extension), it can be observed that the related experimental curve patterns (lines with symbols) are reasonably reproduced by the proposed theoretical model for all the three different temperatures considered. For small values of the crack extension a  a0 (initial stages of crack propagation), the crack extension force slowly decreases. On the other hand, for crack extension greater than about 2.0 cm, the corresponding extension force significantly decreases, approaching about 1/10 of the initial driving force f when the ligament is approximately equal to a value between 1.0 and 1.5 cm. Both experimental and numerical results show such a behaviour. Numerical crack paths are shown in Fig. 13 for three different values of the initial crack orientation angle, at (a) low, (b) room and (c) high temperature. The extensions of the plastic regions for the first two or three steps of crack growth simulation are also reported. It can be remarked that, for the temperature range examined, the numerical crack paths plotted in Fig. 13 are almost independent of the temperature (as has also occurred in the experimental tests) since the yield criterion practically reduces to that in Eq. (19) (Mises criterion), and the yield surface shape is almost independent of temperature. In other words, for the temperature range considered, the ratio rc(T)/rt(T) is equal to about 1 (with rc(T), rt(T) determined by using Eq. (22) reported above) and the hydrostatic dependence on temperature is practically absent.

0.06

0.04

θ 0 = 30 º

Coordinate Y [m]

Coordinate Y [m]

0.06

θ 0 = 45 º

0.02

θ 0 = 90 º

0.00

0.04

0.02

0.04

θ 0 = 45 º

0.02

θ 0 = 90 º

0.00

Temperature T = + 20 º C

Temperature T = - 20 º C

-0.02 0.00

θ 0 = 30 º

0.06

0.08

0.10

-0.02 0.00

0.02

0.04

0.06

0.08

0.10

Coordinate X [m]

Coordinate X [m] 0.06

θ 0 = 30 º

Coordinate Y [m]

0.04

θ 0 = 45 º

0.02

θ 0 = 90 º

0.00

Temperature T = + 80 º C

-0.02 0.00

0.02

0.04

0.06

0.08

0.10

Coordinate X [m] Fig. 13. Numerical crack paths for initial cracks having #0 = 30, 45, 90 and for temperature T equal to: 20 C (a), +20 C (b), +80 C (c).

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Finally, it can be remarked (see Figs. 9, 10, and 13) that, under mixed mode I + II (h0 < 90), the propagation mode reduces practically to a pure Mode I (opening mode) even from the early stages of the crack growth, that is, Mode II quickly disappears from the mechanical process. 8. Conclusions As is well known the environmental effects negatively affects the safety of structural components under static or cyclic loading: among them it must be reminded the humidity, salt air content (which play an important role especially under fatigue loading) and the environmental temperature. In the present paper, the influence of the temperature on the crack growth in aluminium alloy specimens is analysed. An extension of the so-called R-criterion is proposed by taking into account hydrostatic stress dependence of the yield surface, in order to consider the effects of the temperature (on crack propagation) which usually produces an embrittlement of the material under tensile stress. Some experimental tests on aluminium alloy edge-cracked plates (with different values of the initial orientation angle with respect to the loading direction) have been conducted at different temperatures in order to examine the related crack paths and the corresponding crack extension forces. It can be remarked that, for the temperature range considered, the crack paths are practically the same for all the cases examined. Some numerical simulation of the crack growth in variously edge cracked thin plates under mixed mode of fracture, performed by applying the modified criterion, are shown and compared with the experimental data in order to assess the reliability of such an approach. In the numerical simulation a new simple procedure for the Modes I and II SIFs evaluation is proposed in order to consider complex crack pattern. The crack extension forces and the crack paths predicted by the model are in satisfactory agreement with those experimentally determined. The proposed theoretical model seems to be able to tackle crack propagation problems involving different environmental temperatures, as often occurs in practical applications of structural components under static or fatigue loading. Acknowledgement The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR). References [1] Liaw PK, Logsdon WA. Fatigue crack growth threshold at cryogenic temperatures (review). Engng Fract Mech 1984;22: 585–94. [2] Zheng XL, Lu¨ BT. Fatigue crack propagation in metals at low temperatures. In: Andrea Carpinteri, editor. Handbook of fatigue crack propagation in metallic structures. Elsevier Science B.V.; 1994. p. 1385–412. [3] Lu¨ B, Zheng X. Predicting fatigue crack growth rates and threshold at low temperatures. Mater Sci Engng 1991;148A:179–88. [4] Lu¨ B, Zheng X. A model for predicting fatigue crack growth behaviour of a low alloy steel at low temperatures. Engng Fract Mech 1992;42:1001–9. [5] Gasqueres C, Sarrazin-Baudoux C, Petit J, Dumont D. Fatigue crack propagation in an aluminium alloy at 223 K. Scripta Mater 2005;53:1333–7. [6] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Engng 1963;85:519–27. [7] Sih GC. Some basic problems in fracture mechanics and new concepts. Engng Fract Mech 1973;5:365–77. [8] Sih GC. Strain-energy-density factor applied to mixed mode crack problems. Int J Fract 1974;10:305–21. [9] Kong XM, Schulter M, Dahl W. Effect of triaxial stress on mixed mode fracture. Engng Fract Mech 1995;52:379–88. [10] Theocaris PS, Adrianopoulos NP. The Mises elastic–plastic boundary as the core region in fracture criteria. Engng Fract Mech 1982;16:425–32. [11] Theocaris PS, Kardomateas GA, Adrianopoulos NP. Experimental study of the T-criterion in ductile fratcure. Engng Fract Mech 1982;17:439–47. [12] Theocaris PS, Adrianopoulos NP. The T-criterion applied to ductile fracture. Int J Fract 1982;20:R125–30. [13] Maiti SK, Smith RA. Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress–strain field, Part I: slit and elliptical cracks under uniaxial tensile loading. Int J Fract 1983;23:281–95. [14] Ukadgaonker VG, Awasare PJ. A new criterion for fracture initiation. Engng Fract Mech 1995;51:265–74. [15] Richard HA, Buchholz FG, Kullmer G. Key Engng Mater 2003;251–252:251–60.

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