A micromechanical model for the prediction of the temperature fracture behaviour dependence in metallic alloys

Available online at www.sciencedirect.com

Engineering Fracture Mechanics 75 (2008) 3646–3662 www.elsevier.com/locate/engfracmech

A micromechanical model for the prediction of the temperature fracture behaviour dependence in metallic alloys Andrea Carpinteri, Roberto Brighenti *, Alan Davoli, Sabrina Vantadori Department of Civil and Environmental Engineering and Architecture, University of Parma, Viale G.P. Usberti 181/A, 43100 Parma, Italy Received 10 May 2007; received in revised form 17 September 2007; accepted 25 September 2007 Available online 10 October 2007

Abstract In the present paper, a micro-mechanical model based on energetic considerations is developed to simulate the effect of environmental temperature on the fracture toughness of metallic alloys. By considering a reference elementary volume (REV) with the same composition of the real material, the stress–strain field inside such a volume and the corresponding strain energy due to a temperature variation is determined. The energy balance to determine the material fracture toughness is generalised in order to take into account the temperature effects. The proposed micro-mechanical model is governed by few parameters which can be simply estimated, and allows us to determine the fracture toughness for any temperature below the room temperature. Such a model is applied to three metallic alloys which show a ductile–brittle transition temperature: ASTM A471, Carbon Steel D6ac, Steel S275 J2. From the comparison of theoretical results with experimental data, it can be concluded that the model seems to be able to correctly predict the fracture toughness at low temperatures.  2007 Elsevier Ltd. All rights reserved. Keywords: Low temperature; Metallic alloys; Stress-intensity factor (SIF); Fracture toughness; Energy balance; Micro-mechanical model

1. Introduction The evaluation of the fracture toughness under different environmental conditions is very important for many engineering applications in order to quantify the structural safety according to the so-called damage tolerant design. In particular, the assessment of the structural safety at low temperatures should be done with extreme care since the fracture toughness is generally lower than that at room temperature. The effects of low temperatures should be accurately examined especially in the near-threshold region [1–3]. Recent studies [3–5] have shown that metals could be roughly divided into two groups, that is to say, materials with a ductile– brittle transition (DBT) and those without a DBT at low temperature. When T > TDBT (where TDBT is the temperature at which the ductile–brittle transition occurs), the Crack Growth (CG) mechanism consists in forming ductile striations, and the CG rate decreases by decreasing T. When T is equal to about TDBT, the

*

Corresponding author. E-mail address: [email protected] (R. Brighenti).

0013-7944/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.09.009

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3647

CG mechanism becomes micro-cleavage and the fracture toughness KIC sharply decreases. The CG rate tends to increase as the temperature is further reduced. In the present paper, a micro-mechanical model based on energetic considerations is developed in order to take into account the effect of the environmental temperature on the fracture toughness of metallic alloys. By considering a reference elementary volume (REV) with the same composition of the real material, the stress– strain field inside such a volume and the corresponding strain energy due to a temperature variation can be determined. A generalisation of the classical energy balance (firstly introduced by Griffith) is discussed by including the temperature effects, in order to determine the material fracture toughness at low temperatures. The expression obtained for KIC (T) contains some parameters which can be determined if the values of the fracture toughness at two different temperatures are known. Such an expression allows us to estimate the fracture toughness for any temperature T below the room temperature RT. The proposed model is applied to three metallic alloys which show a ductile–brittle transition temperature: ASTM A471, Carbon Steel D6ac, Steel S275 J2. From the comparison of theoretical results with experimental data, it can be concluded that the model seems to be able to correctly predict the fracture toughness at low temperatures. 2. Micro-mechanical model for metallic materials at low temperatures Metallic structural materials are usually composed by more than one component (phase), and the corresponding material is called metallic alloy. In a metallic alloy, all the phases are crystalline and solid. Metallic alloys are usually classified as mono- or multi-phase depending on the number of constituents. From a chemical point of view, generally metallic alloys can be distinguished in substitutional (Fig. 1a) or interstitial (Fig. 1b, typical of austenitic steel alloys), depending on the fact that the secondary phases are placed in positions corresponding to the main crystalline lattice vertices or are placed between the standard positions of the crystalline lattice vertices. The most common mono-phase materials are copper and zinc, while steel and aluminum alloys can be mono- or multi-phase materials depending on their chemical composition and heat treatment. By observing the microstructure of a metallic alloy, it can be assumed that the matrix with different materials dispersed inside (secondary components) can be modelled by considering a spherical reference elementary volume (REV) with one spherical equivalent inclusion which presents the same volume fraction content of the real dispersed materials (Fig. 2). The whole material can be considered to be composed by many REVs assembled in reciprocal contact. By considering the mass composition of the metallic alloy, the volume fraction of the equivalent inclusion to be assumed in the REV can be determined (Fig. 3). Let M be the total mass of a portion of the considered alloy, whereas MB and MI, j are the matrix’ mass and the jth inclusion’s mass, respectively. Further, MI indicates the mass of the single equivalent inclusion Fig. 4.

Fig. 1. Schematic structure of a substitutional (a) or interstitial (b) alloys.

3648

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

Fig. 2. Micrograph of pure iron: ferrite with pointwise shape inclusions. Cristals can be assumed to have polygonal shape (a) and simplified spherical model of a single crystal (b).

Fig. 3. Scheme of the process to determine the RVE composition.

Equivalent inclusion

RB

RI Base material Fig. 4. Scheme of the volumes of the base material and of the equivalent inclusion in the REV.

If only the most significant inclusions (n inclusions) are considered (i.e. those with a most significant mass), the following equation holds: M ¼ MB þ

n X

M I;j ¼ M B þ M I

ð1Þ

j¼1

and dividing all terms by M, the previous equation can be written in a dimensionless form: gB þ

n X j¼1

gI;j ¼ gB þ gI ¼ 1

ð2Þ

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3649

where gB = MB/M is the mass fraction of the matrix, and gI, j = MI, j/M and gI = MI/M are the dimensionless mass fraction of the jth inclusion and that of the single equivalent inclusion, respectively. Analogously, we can determine the corresponding mass density fraction qI of the equivalent inclusion as follows: Pn Pn n n X V Ij X j¼1 qIj V Ij j¼1 qIj V Ij ¼ ¼ qIj ¼ qIj gIj ð3Þ qI ¼ Pn VI VI j¼1 V Ij j¼1 j¼1 Finally, the corresponding volume fractions (referred to the REV volume) can be obtained: VI ¼

M I gI ¼ M; qI qI

VB ¼

M B gB ¼ M qB qB

and the corresponding radii RB and RI of the REV’s spheres can be expressed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi  4  3 g 3 g g 3 B þ I M V B ¼ p RB  R3I ¼ B M ) RB ¼ 3 4p qB qI qB sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 3 gI 3 3 gI V I ¼ pRI ¼ M ) RI ¼ M 3 4p qI qI

ð4Þ

ð5Þ

It should be noted that only the relative volume contents gI, gB of the different constituents are significant for the present micro-mechanical model, while the corresponding absolute quantities VB, VI are not needed. 2.1. Thermo-mechanical properties of the REV The mechanical and thermal properties of the equivalent homogenised material can be obtained by simply evaluating weighted mean values from the effective mechanical and thermal properties of the components; as an example some parameters are reported in the following expressions: Pn Pn n n X M Ij X j¼1 EIj M Ij j¼1 E Ij M Ij E I ¼ Pn ¼ ¼ EIj ¼ EIj gIj ð61 Þ MI MI j¼1 M Ij j¼1 j¼1 Pn Pn n n X M Ij X j¼1 mIj M Ij j¼1 mIj M Ij mI ¼ Pn ¼ ¼ mIj ¼ mIj gIj ð62 Þ MI MI j¼1 M Ij j¼1 j¼1 Pn Pn n n X M Ij X j¼1 aIj M Ij j¼1 aIj M Ij aI ¼ P n ¼ ¼ aIj ¼ aIj gIj ð63 Þ MI MI j¼1 M Ij j¼1 j¼1 where EI, mI, aI are the Young’s modulus, the Poisson’s ratio and the thermal expansion coefficient of the equivalent inclusion, respectively. 2.2. Thermo-mechanical analysis of a sphere with a spherical inclusion In order to analyse the mechanical effect of a temperature variation on the considered spherical REV, a simple mechanical model can be introduced (Fig. 5). The elastic problem is characterised by the (internal and external) boundary conditions which can be described as follows: the opposite pressures between the matrix and the spherical inclusion must be equal to each other on the internal boundary, whereas the outside pressure must be determined by assuming zero radial displacement on the outside surface of REV, since each REV is assumed to be in contact with other REVs and the domain is assumed to be unbounded. Analytically such conditions can be written as follows: uB ðRI Þ ¼ uI ðRI Þ;

uB ðRB Þ ¼ 0

ð7Þ

Such a problem can be analysed by considering a hollow sphere (representing the matrix) and a smaller sphere (with the same size of the cavity in the matrix) subjected to a temperature variation which causes a system of pressures under the above outside boundary displacement condition.

3650

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

po RI pi

RB

RI pi

Fig. 5. Pressure acting on the outside and inside boundary of the spherical REV with a spherical inclusion.

The reference state is assumed that at the room temperature (RT, typically equal to 293 K), whereas the final state is characterised by a generic temperature T (the temperature variation is DT = T  RT) which, in the following, is assumed lower than the room temperature. A steady state condition is considered. The solution of the above problem is given by the following expressions depending on the pressures at the outside surface (p0) and at the base-inclusion interface (pi) [6]:    2EB R3B  R3I pi pi R3I 1 R3B p0 ¼ ðaI  aB ÞDT þ ð1  2mI Þ þ 1  2mB þ ð1 þ mB Þ 3 2 3ð1  mB Þ R3B EI EB R3B  R3I RI ð8Þ  3 3 3 3  2EB RB  RI p0 RB 1 RI ð1 þ m pi ¼ a DT þ 1  2m þ Þ B B B 2 3ð1  mB Þ R3I EB R3B  R3I R3B where the indexes B and I refer to the base and the inclusion material, respectively. By defining the following four constants     R3B 1 R3I R3I 1 R3B A1 ¼ 3 1  2mB þ ð1 þ mB Þ 3 ; A2 ¼ 3 1  2mB þ ð1 þ mB Þ 3 2 2 RB  R3I RB RB  R3I RI ð9Þ 3 3 2 RB  RI EB A3 ¼ ; A4 ¼ ð1  2mI Þ þ A2 3 1  mB EI the Eq. (8) can be rewritten as follows: 8 < p0 ¼ RA33 ½ðaI  aB ÞEB DT þ pi A4  B

: pi ¼ A33 ðaB EB DT þ p0 A1 Þ R

ð10Þ

I

and, solving with respect to p0 and pi, we get: h

i h

i1 0 A3 A4 A3 EB DT aI  aB 1  AR3 A3 4 R3I A A E DT a  a 1  1 3 B I B aB EB DT R3I I A p0 ¼ ; p i ¼ A3 @ þ 2 3 2 3 3 3 3 RI RI R B  A 1 A3 A4 R I RB  A1 A3 A4

ð11Þ

By calling r the generic distance from the centre of the (spherical) REV, the following quantities (in the radial and tangential directions) can be computed (see Fig. 5): (a) Domain inside the inclusion, r 2 [0, RI] rr ðrÞ ¼ rh ðrÞ ¼ pi ; p ð1  2mI Þ þ aI DT ; er ðrÞ ¼ eh ðrÞ ¼ i EI   p uðrÞ ¼ ð1  2mI Þ i þ aI DT r EI

ð12Þ

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3651

(b) Domain inside the matrix, r 2 [RI, RB]  3    R3 RB R3B R3I rr ðrÞ ¼ pi 3 I 3  1 þ p 1  ; 0 3 r3 RB  RI r 3 RB  R3I  3   3  R3I RB R3B RI rh ðrÞ ¼ pi 3 þ 1 þ p0 3 þ1 RB  R3I 2r3 RB  R3I 2r3      1 R3B R3I R3B R3I er ðrÞ  aB DT ¼   pi 3 ð1 þ mB Þ 3  ð1  2mB Þ  p0 ð1 þ mB Þ 3  ð1  2mB Þ EB R3B  R3I r r R  B 3    3 3 1 RB RI RB R3I eh ðrÞ  aB DT ¼   pi 3 ð1 þ mB Þ 3 þ 1  2mB þ p0 ð1 þ mB Þ 3 þ 1  2mB EB R3B  R3I 2r 2r R  B    1 R3B 1 R3I R3I 1 R3B uðrÞ ¼ aB DT  r þ   p0 1  2mB þ ð1 þ mB Þ 3  pi 3 1  2mB þ ð1 þ mB Þ 3 r EB R3B  R3I 2 2 r r RB ð13Þ The mean elastic density energy hREV(T) of the whole REV in the considered temperature steady state can be obtained as follows (the room temperature RT state is assumed to be the reference state characterised by zero energy content): Z Z H I ðT Þ þ H B ðT Þ 1 ¼ hREV ðT Þ ¼ hI ðT Þ dV I þ hB ðT ÞdV B ð14Þ V V VI VB where hI(T), hB(T) are the elastic energy density – corresponding to the temperature T – for the inclusion and the matrix, respectively. It should be noted that, since the stress state is constant inside the inclusion (see Eq. (121)), also the elastic energy density hI(T) must be constant, and can be explicitly expressed as: 8 9T 2 38 9 1> 1 mI mI > > = = 3 p2 < <1> 2 1 1 pi 6 7 T i ð15Þ 1 hI ¼ frI ðT Þg ½AI frI ðT Þg ¼ 1 mI 5 1 ¼ ð1  2mI Þ 4 mI > 2 2> ; EI ; 2 EI : > : > 1 mI mI 1 1 The mean energy density value  hB of the matrix can be deduced by averaging the effective energy density distribution on the corresponding hollow sphere: Z Z 1 1 1  frB ðr; T ÞgT ½AB frB ðr; T ÞgdV B hB ðr; T ÞdV B ¼ ð16Þ hB ¼ V B VB V B VB 2 In the previous expressions, [AI], [AB] are the compliance matrices of the inclusion and the matrix, respectively, whereas {rI(T)} and {rB(r,T)} are the array representations of the stress tensor for the inclusion and of that for the matrix, respectively. In order to obtain a simple (approximate) closed-form expression for the energy density in Eq. (16), a  ¼ 1=2ðpi þ p0 Þ can be assumed to exist in the matrix. Therefore, the mean constant stress state equal to r energy density  hB can be computed: 8 9T 2 38 9 > > r 1 mB mB > > = = 3 ðp þ p Þ2 < > > 2> E 8 E B B : ; : ;   r mB mB 1 r Finally, the approximate total energy in the REV volume can be obtained: hB  V B H ðT Þ ¼ H I ðT Þ þ H B ðT Þ ¼ hI ðT Þ  V I þ 

ð18Þ

and, by introducing the radii RB, RI of the spheres representing the REV and the equivalent inclusion, the approximate mean energy density in the REV is given by:   Z  Z 1 hI ðT Þ  V I þ hB  V B H ðT Þ hI  R3I þ hB  R3B  R3I   ¼ hI dV I þ ¼ ð19Þ hB dV B ¼ h¼ V V VIþVB R3B VI VB

3652

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

2.3. Energy balance for crack propagation instability As is well-known, the fracture toughness can be determined by writing an energy balance in the condition of incipient unstable crack propagation [7]. By assuming that the elastic energy released during an infinitesimal crack extension is completely spent for the formation of the corresponding new fracture surface, the critical energy release rate can be obtained. By considering a circular fracture surface starting from the outside surface of an inclusion (Fig. 6), the energetic balance in the situation of unstable crack propagation can be written in the following way:   1 1 4 T dV  fdeg ½Cfdeg þ dW s ¼ d pb3 deij C ijhk dehk þ dðpb2 Þ  2c P 0 ð20Þ 2 2 3 where d(4pb3/3) is the spherical volume variation of the space influenced by the crack extension with a radius equal to b, whereas c is the energy required to create a new unit crack surface. Therefore:   1 T 2 4pb  db fdeg ½Cfdeg þ 4cpb  db P 0 ð21Þ 2 In the situation of an incipient crack extension, the energy balance becomes:   1 T fdeg ½Cfdeg þ c ¼ 0 b 2

ð22Þ

By assuming that the stress state in the spherical volume is constant, the elastic energy density can be written 2 T as 12 fdeg ½Cfdeg ¼ 32 rE ð1  2mÞ and, by substituting it in Eq. (22), we get: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  3r 2c  E GIC  E ¼ ð1  2mÞ þ c ¼ 0 ) rC ¼ b ð23Þ 2 E 3  ð1  2mÞb 3  ð1  2mÞb where GIC = 2c is the fracture energy per unit crack extension (i.e. upper and lower fracture surfaces are considered simultaneously), and rc is the critical stress value. On the other hand, for an embedded penny-shaped (with a radius b) in an infinite body, the critical pffiffiffiffiffiffifficrack ffi stress-intensity factor can be written as K IC ¼ 2rc b=p [8]. By comparing this expression with Eq. (23), the fracture toughness KIC can be obtained as a function of the energy release rate GIC: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GIC  E K IC ¼ 2 ð24Þ 3pð1  2mÞ The well-known energy balance by Griffith [7] can be generalised by including the above strain energy hRVE(T) due to a temperature variation. The instability criterion states that fracture takes place when the total released elastic energy due to a new crack surface formation is greater than or equal to the energy required to Penny-shaped radius variation, db

2(b+db) Volume involved in the elastic energy evaluation (V) 2b Fracture surface

Fracture surface extension with anular shape db

db Volume V + dV

Fig. 6. Surface crack originated from a spherical inclusion.

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3653

develop such a crack surface. By including the elastic energy density hREV (T) due to the temperature variation with respect to the room temperature in the energy balance we get: 1 T dV  ½fdeg ½Cfdeg  hREV ðT Þ þ dW s ¼ ½dW e  dW e ðT Þ þ dðpb2 Þ  2ðc  DcðT ÞÞ P 0 2

ð25Þ

where it is assumed that the unstable crack propagation takes place by keeping the initial geometry, i.e. the initial assumed penny-shaped crack propagates remaining circular in shape. The elastic energy involved in the energy balance can be calculated in a region assumed having a spherical shape with radius R, which can be considered to be the region where such an elastic energy contributes to the crack propagation (Fig. 7). The term Dc(T) in Eq. (25) is the variation of the surface energy associated with the formation of a new unit fracture surface caused by a temperature variation. The sign ± of the elastic term in Eq. (25) depends upon the positive (dWe(T)) or negative (+dWe(T)) temperature variation. Further, note that the simplified notation hREV(T) = h(T) is used in the following. The above spherical region with radius R is considered since the material can be assumed to contain many micro-cracks and, consequently, their influence on the elastic energy released rate for a single equivalent macro crack must be evaluated in a region with an appropriate extension. In the situation of an incipient crack extension, Eq. (25) becomes:       1 4 3 T 2 4pb  db fdeg ½Cfdeg þ 4cpb  db  d pR  hðT Þ  dð2  DcðT Þ  pb2 Þ ¼ 0 ð26Þ 2 3 In the previous expression of the energy balance in the situation of incipient unstable crack propagation, no inelastic strain contribution have been considered; since the material at low temperature can be considered to show a brittle behaviour, the plastic energy dissipation can be assumed to be negligible and has been omitted in Eq. (26). On the other hand at high temperature, where plastic deformation easily occurs, the energy balance should take into account also the plastic energy. In the present study such a situation will be neglected. By assuming a dependence between the radius R and the penny-shaped crack radius b, the derivative oR3 =ob can be written as: oðR3 Þ oðR3 Þ oR oR ¼ ¼ 3R2 ob ob ob oR

ð27Þ

Therefore, by recalling Eqs. (23), (27) and (26) becomes:  2    r R2 oR 3 ð1  2mÞb  2hðT Þ ¼ GIC ðRT Þ  jDGIC ðT Þj E b ob

ð28Þ

In other words, the fracture energy DGIC(T) can be considered as a correction term (due to temperature variation DT) of the energy release rate GIC(RT) at room temperature RT.

Influence volume Plane of fracture propagation (penny-shaped crack) 2b 2R

Fig. 7. Volumes of the material where the energy balance is written.

3654

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

Finally, the total fracture toughness at a generic temperature T can be written as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GIC ðT Þ  E ½GIC ðRT Þ  jDGIC ðT Þj  E ¼2 K IC ðT Þ ¼ 2 3pð1  2mÞ 3pð1  2mÞ vhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u GIC ðRT Þ  2hðT Þ R2 oR  E t b ob 8  hðT Þ R2 oR E ¼2 or K IC ðT Þ ¼ K 2IC ðRT Þ  3pð1  2mÞ b ob 3pð1  2mÞ

ð29Þ

By writing Eq. (29) in the following way: K 2IC ðT Þ  K 2IC ðRT Þ ¼ 

8  hðT Þ R2 oR E 3pð1  2mÞ b ob

ð30Þ

it is possible to appreciate the effect of the temperature variation DT on the critical stress-intensity factor (SIF) variation. At room temperature h(T = RT) is equal to zero and, therefore, we get K IC ðT Þ ¼ K IC ðRT Þ. The sign to be used in the energy balance Eq. (30) depends on the value of T: the sign is ‘‘’’ if T < RT, the sign is ‘‘+’’ if T > RT. 2.4. Energy balance for a material with a distribution of micro-cracks In the case of a real material the above energy balance can be written by assuming the presence of a distribution of micro-cracks; obviously the above formulation must take into account the effective SIF expression (the single penny-shaped crack solution used up to now is only a simplification). By considering an infinite body containing a distribution of equally spaced penny-shaped cracks (Fig. 8), it can be determined an equivalent single penny-shaped crack with radius b, which presents the same SIF (Fig. 8a and b), i.e. an equivalent damaged material having the same safety factor against fracture (no matter of compliance or other mechanical properties have been considered in determining the equivalent penny-shaped crack). By writing: rffiffiffi rffiffiffi b l K I1 ¼ K I2 ¼ 2r ð31Þ ¼ Y 2  2r ) b ¼ Y 22  l ¼ C  l; C ¼ constant p p we can compute b so that the equivalent single penny-shaped crack has the same SIF as that of an array of penny-shaped cracks assumed equally spaced in three dimensions. The correction (or geometry) factor Y2 in Eq. (31) must be determined once we know the vertical and horizontal spacing distance between the cracks. Since such a 3D solution does not exist in a closed-form, we can

a

b

F 2b 2l

K I 1 = 2σ

b

π

K I 2 = Y2 ⋅ 2σ

l

π

Fig. 8. Single penny-shaped crack equivalent to an array of penny-shaped cracks in an infinite 3D body.

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3655

2b 2l

K I 3 = Y3 ⋅ σ π ⋅ l

K I 4 = Y4 ⋅ σ π ⋅ b

Fig. 9. Single straight crack equivalent to an array of straight cracks in an infinite 2D body.

consider a similar 2D equivalent problem (Fig. 9). By imposing the same value for the SIFs of the two cases in Fig. 9, we get: pffiffiffiffiffiffi pffiffiffiffiffi ð32Þ K I3 ¼ K I4 ¼ r pb ¼ Y 4  r pl ) b ¼ Y 24  l ¼ D  l; D ¼ constant Therefore, we can assume C ffi D ¼ Y 24 for the case being examined. In the following, the length b is assumed to be equal to b ¼ Y 22  l ffi Y 24  l at the reference room temperature RT. In order to determine the correction factor Y4, the cracks density and spacing must be known. According to Bristow, Walsh and Kachanov [9–12], the surface and volume crack density in a body with many micro-cracks can be defined as: 1X 2 1 X 3 qA ¼ lj ; qV ¼ l ð33Þ A j V j j where A and V are the surface and volume extension of the reference region of the cracked body, and lj is the length of the jth crack. By assuming that all the cracks have the same length l and that NA, NV represent the numbers of micro-cracks in the reference area and volume, respectively, we get: 1 1 qA ¼ N A  l2 ; qV ¼ N V  l3 ð34Þ A V The two above densities are related by the simple expression qA ¼ p2 qV . By assuming that all the microcracks have the same length 2l, horizontal spacing equal to s0, horizontal distance f and vertical distance w (Fig. 10), the crack density can be finally determined as in Refs. [9–12]: p 4  ð35Þ qA ¼ qV ¼ w s0 2 þ2 l l By assuming an isotropic distribution of micro-cracks, we get s0/l = w/l (i.e. s0 = w) and, by imposing qV = gI (i.e. the crack density is reasonably assumed to be equal to the volume fraction of the equivalent inclusion) the previous equation becomes:

s0 s0 8 þ2 ¼ ð36Þ pgI l l

ζ

s0 ψ 2l

Fig. 10. 2D array of cracks in an infinite body.

3656

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

and the dimensionless spacing can be determined: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s0 8 ¼ 1 þ 1 þ pgI l

ð37Þ

while the horizontal distance becomes: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 8 f ¼ s0 þ 2l ¼ l  1 þ 1 þ pgI

ð38Þ

and the dimensionless parameters l/f and w/f can be finally determined to get the equivalent penny-shaped crack, by only knowing the dimensionless quantity qV = gI: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! l pgI 8 w 1 l 1 pgI 8 ¼ ¼  ¼  1 ; 1 ð39Þ 1þ 1þ f pgI f 2 f 2 pgI 8 8 In the following the required equivalent crack length b is determined. For temperature-sensitive materials, the fracture toughness decreases by decreasing the temperature and, therefore, the equivalent critical crack size can be assumed to decrease, too. Therefore, for a given value of rc, we can write two relationships (in the case of incipient fracture at room temperature RT and at a generic temperature T): rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi bðRT Þ bðT Þ K IC ðRT Þ ¼ 2  rC ; K IC ðT Þ ¼ 2  rC ð40Þ p p In other words, the equivalent crack length b(T) at a generic temperature can be related to those b(RT) at room temperature:  2 K IC ðT Þ bðT Þ ¼ bðRT Þ  6 bðRT Þ ð41Þ K IC ðRT Þ where the SIFs ratio is usually lower than one. Eq. (41) can be interpreted by considering that at a temperature lower than the room temperature, the fracture takes place for crack smaller than that at room temperature (Fig. 11). Finally, the size R of the influence zone where the energy density must be computed for the energy balance is needed. It can be assumed that the radius R of the influence spherical volume can be written as proportional to the equivalent penny-shaped crack size b(T): RðT Þ ¼ b  bðT Þ; b ¼ constant

and

oR o ¼ ðb  bÞ ¼ b ob ob

ð42Þ

By inserting the expressions of b(T) (see Eq. (41)), RðT Þ and oR=ob (see Eq. (42)) into Eq. (292), we get: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8  hðT Þ R2 oR K IC ðRT Þ E ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð43Þ K IC ðT Þ ¼ K 2IC ðRT Þ  hðT Þl 8 3pð1  2mÞ b ob E 1  b3 Y 2 2 3p

2 b(RT)

4 ð12mÞK ðRT Þ IC

σ

σ

σ

2 b(T) σ 2 b(RT)

Fig. 11. Interpretation of the temperature effect on the fracture toughness in term of critical crack length 2b.

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3657

In the case the critical SIF values are known for two different temperatures, RT and T*, the characteristic crack length l can be obtained:    K ðRT Þ 2   IC  1  ðT Þ K IC K IC ðRT Þ K IC ðT Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) l ¼ 8 3 2 hðT ÞE ð44Þ 8 b Y 4 ð12mÞK 2 ðRT Þ 1  b3 Y 24 dhðT ÞEl 2 3p 3p

ð12mÞK IC ðRT Þ

IC

and, by substituting such an expression of l in Eq. (43), the final relationship of KIC(T) can be evaluated: K IC ðRT Þ K IC ðT Þ ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

2 ðRT Þ hðT Þ 1 þ KKIC  1  hðT Þ IC ðT Þ

ð45Þ

Note that such an expression does not depend on the characteristic crack length l. It can be remarked that Eq. (442) is a function of the parameter b which defines the extension of the sphere (with radius RðT Þ) around the equivalent single penny-shaped crack (with diameter equal to 2b(T)), where the elastic energy must be considered for the energy balance ðRðT Þ ¼ b  bðT ÞÞ. In first approximation, we can consider the effect of a spherical void in an infinite medium: the stress state is locally perturbed by the presence of the geometrical defect, and RðT Þ can be considered the distance at which the stress state is not practically influenced by such a defect. It can be easily verified that, at a distance equal to about RðT Þ ¼ b  bðT Þ ffi 8  bðT Þ, the stress field is practically identical to that developing in absence of the defect. Therefore, the parameter b is assumed to be equal to 8 in the following (note that the choice of the value of the parameter b does not heavily influence the final results). In order to apply Eq. (45) (to determine the critical SIF at a generic temperature T), we need: (a) To know the critical SIFs at two different temperatures KIC(RT), KIC(T *) (at temperatures RT and T *, respectively). (b) To compute the elastic energies h(T) and h(T *) due to the two temperature variations T  RT and T *  RT, respectively. In other words, the variation of elastic energy density between a generic temperature (T) and the room temperature (RT) can be evaluated as follows:   hI R3I þ  hB R3B  R3I hðT Þ  hðRT Þ ¼ hðT Þ ¼ ð46Þ R3B where the energy associated with the room temperature RT is assumed to be equal to zero. That is, the REV energy density variation (Eq. (46)) is determined by adding together the contributions of the matrix and of the equivalent inclusion. The mean energy densities in the matrix ðhB Þ and in the inclusion ðhI Þ can be defined as follows: hB ðRI Þ þ hB ðRB Þ 3 p2i  and  hI ¼ hI ¼ ð1  2mI Þ hB ¼ 2 2 EI 3 p2i 3 p20 ð1  2mB Þ; hB ðRB Þ ¼ ð1  2mB Þ with hB ðRI Þ ¼ 2 EB 2 EB

ð47Þ

where the quantities hB(RI), hB(RB) are the energy densities in the matrix measured at the distances RI, RB, respectively, from the centre of the spherical REV. By applying Eq. (46) for two different temperatures (T  RT and T *  RT), we can compute h(T) and h(T *) and, finally, the critical SIF KIC(T) at a given generic temperature T (with T < T *) can be obtained from Eq. (45). 3. Numerical applications and discussion In the present section, the proposed model is applied to predict the fracture toughness of three metallic alloys at low temperatures. Since in the authors’ knowledge no similar theoretical models have been proposed,

3658

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

the present model will be applied to experimental data in order to assess its capability in metallic alloys’ SIFc prediction at low temperature. 3.1. Steel ASTM A471 The first material considered is the Steel ASTM A471 [13–15] which has two main secondary elements characterised by the chemical composition and physical–mechanical parameters given in Table 1. For this material: gB = 93.54% and gI = 5.22%. The weighted mean characteristics of the matrix and the equivalent inclusion are summarised in Table 2. The diagram of the experimentally determined fracture toughness against the environmental temperature is plotted in Fig. 12. The experimental room temperature fracture toughness of the material is equal to KIC(RT) = 135 MPa m1/2. In order to apply Eq. (45), the fracture toughness and the strain energy density for a generic temperature T * < RT are needed. For the present material: KIC(T * = 250 K) = 90 MPa m1/2, h(T *) = 0.19 MJ m3, and the characteristic length determined by using the above temperatures is equal to l = 4.48 · 104 m. By using such parameters and those referred to the standard room temperature, the fracture toughness for T < T * can be determined. The obtained values are shown in Fig. 13. As can be observed the experimental and Table 1 Physical and mechanical parameters of the main elements is the steel ASTM A471

Nickel Cromo

Element

gIj (%)

qIj (kg/m3)

EIj (GPa)

mIj

aIj (K1)

Ni Cr

3.59 1.63

8880 7190

207 248

0.31 0.3

1.30E05 6.20E06

Table 2 Mean physical and mechanical parameters of the main elements is the steel ASTM A471 aI (K1)

qI (kg/m3)

EI (GPa)

mI

Equivalent inclusion 8352

220

0.31

1.09E05

qB (kg/m )

EB (GPa)

mB (kg/m )

aB (K1)

Base material 7870

200.0

0.29

1.20E05

3

3

Temperature, T [K] 100

150

200

200

250

300

350

400

450

Considered temp. range

Fracture toughness, KIC [MPa m1/2 ]

240

160 120 80 40 0 -200

-150

-100

-50

0

50

100

150

200

Temperature, T [°C] Fig. 12. Experimental fracture toughness against the environmental temperature for ASTM A471.

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3659

Temperature, T [°C] -100

Fracture toughness, KIC [MPa m1/2 ]

140

-75

-50

-25

0

25

275

30 300

KIC model

120

KIC experim. 100 80 60 40 20 150

175

200

225

250

Temperature, T [K] Fig. 13. Experimental and theoretical fracture toughness against the environmental temperature for ASTM A471 in the temperature range 153 K < T < 293 K.

theoretical curves matches exactly at the two reference temperatures RT and T * used in the model while slightly increasing differences can be observed for lower temperatures. From the comparison with experimental data, it can be deduced that the maximum relative error for the fracture toughness predicted by the proposed model is lower than about 5% at T = 153 K. 3.2. Carbon Steel D6ac medium toughness The second material considered is the Carbon Steel D6ac medium toughness [13–15] which has two main secondary elements characterised by the chemical composition and physical–mechanical parameters given in Table 3. For this material: gB = 95.74% and gI = 2.10%. The weighted mean characteristics of the matrix and the equivalent inclusion are summarised in Table 4. The diagram of the experimentally determined fracture toughness against the environmental temperature is plotted in Fig. 14. The experimental room temperature fracture toughness of the material is equal to KIC(RT) = 50 MPa m1/2. In order to apply Eq. (45), the fracture toughness and the strain energy density for a generic temperature T * < RT are needed. For the present material: KIC(T * = 250 K) = 42 MPa m1/2, h(T *) = 0.19 MJ m3, and the characteristic length determined by using the above temperatures is equal to l = 2.02 · 105 m. Table 3 Physical and mechanical parameters of the main elements is the Carbon Steel D6ac

Molibden Cromium

Element

gIj (%)

qIj (kg/m3)

EIj (GPa)

mIj

aIj (K1)

Mb Cr

1.05 1.05

10220 7190

330 248

0.38 0.30

5.35E06 6.20E06

Table 4 Mean physical and mechanical parameters of the main elements is the Carbon Steel D6ac qI (kg/m3)

EI (GPa)

mI

aI (K1)

Equivalent inclusion 8705

289

0.34

5.78E06

qB (kg/m3)

EB (GPa)

mB (kg/m3)

aB (K1)

Base material 7870

200.0

0.29

1.20E05

3660

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

300

350

400 125

100 80 60 40 20 -100

-50

100

75

50

0

50

100

150

200

250

25 300

Temperature, T [°C]

Fracture toughness, KIC [MPa m1/2]

250

Considered temp. range

Fracture toughness, KIC [ksi in1/2]

Temperature, T [K] 200 120

Fig. 14. Experimental fracture toughness against the environmental temperature for Carbon Steel D6ac.

Temperature, T [°C]

Fracture toughness, KIC [MPa m1/2 ]

50

-100

-75

-50

-25

0

25

45

40

35

KIC experim. 30

25 200

KIC model

225

250

275

300

Temperature, T [K] Fig. 15. Experimental and theoretical fracture toughness against the environmental temperature for Carbon Steel D6ac in the temperature range 200 K < T < 250 K.

By using such parameters and those referred to the standard room temperature, the fracture toughness for T < T * can be determined. The obtained values are shown in Fig. 15. In the present case, the equivalent inclusion’s relative volume gI = 2.10% is much lower than in the previous material considered but this seems to not affect the model capability in fracture toughness prediction; once again experimental and theoretical curves matches exactly at the two reference temperatures RT and T * while slightly differences can be observed for lower temperatures. From the comparison with experimental data it can be deduced that the maximum relative error for the fracture toughness predicted by the proposed model is lower than about 3.5% at T = 200 K. 3.3. Steel S275 J2 The third material considered is the Steel S275 J2 [13–15] which has one main secondary element characterised by the chemical composition and physical–mechanical parameters given in Table 5. For this material: gB = 98.10% and gI = 1.60%. The weighted mean characteristics of the matrix and the equivalent inclusion are

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

3661

Table 5 Physical and mechanical parameters of the main elements is the Steel S275 J2

Manganese

Element

gIj (%)

qIj (kg/m3)

EIj (GPa)

mIj

aIj (K1)

Mn

1.60

7440

159

0.35

2.28E05

Table 6 Mean physical and mechanical parameters of the main elements is the Steel S275 J2 aI (K1)

qI (kg/m3)

EI (GPa)

mI

Equivalent inclusion 7440

159

0.35

2.28E05

qB (kg/m )

EB (GPa)

mB (kg/m )

aB (K1)

Base material 7870

200.0

0.29

1.20E05

3

3

summarised in Table 6. The experimental room temperature fracture toughness of the considered material is equal to KIC(RT) = 140 MPa m1/2. In order to apply Eq. (45), the fracture toughness and the strain energy density for a generic temperature T * < RT are needed. For the present material: KIC(T * = 243 K) = 60 MPa m1/2, h(T *) = 0.27 MJ m3, and the characteristic length determined by using the above temperatures, is equal to l = 1.22 · 103 m. By using such parameters and those referred to the standard room temperature, the fracture toughness for T < T * can be determined. The obtained values are shown in Fig. 16. From the comparison with experimental data, it can be deduced that the relative error for the fracture toughness predicted by the proposed model is lower than about 6% at T = 193 K. For Steel S275 J2 the equivalent inclusion’s relative volume gI = 1.60% is lower than in the previous cases. Furthermore its Young modulus is lower than that of the base material, (EI/EB)S275 = 0.795 while for Steel ASTM A471 and Carbon Steel D6ac such a ratio is equal to (EI/EB)A471 = 1.100 and (EI/EB)D6ac = 1.445, respectively; this weaker inclusion can give rise to some inaccuracies in the strain energy evaluation which can be overestimated and consequently in the SIFc prediction gives lower value than that of the experimental data.

Temperature, T [°C]

Fracture toughness, KIC [MPa m1/2 ]

160

-100

-75

-50

-25

0

25

275

300

KIC experim. KIC model

120

80

40

0 175

200

225

250

Temperature, T [K] Fig. 16. Experimental and theoretical fracture toughness against the environmental temperature for the Steel S275 J2 in the temperature range 193 K < T < 293 K.

3662

A. Carpinteri et al. / Engineering Fracture Mechanics 75 (2008) 3646–3662

As in the previous cases experimental and theoretical curves matches exactly at the two reference temperatures RT and T * while some differences can be observed for the only one lower temperature available. 4. Conclusions In the present paper, a micro-mechanical model based on energetic considerations has been proposed in order to take into account the effect of the environmental temperature on the fracture toughness of metallic alloys. By considering a reference elementary volume (REV) with the same composition of the real material, and evaluating the strain energy variation due to a temperature variation with respect to a reference temperature (e.g. the room temperature, RT), such a contribution can be inserted into an energy balance, which can be considered as a generalisation of the classical Griffith approach. In such a way, the fracture toughness of a material can be evaluated by including the temperature effects. Once the parameters of the model have been determined (this can be done if the critical SIFs at two different temperatures are known), the model allows us to estimate the fracture toughness for any temperature below the room temperature. The proposed model has been here applied to three different metallic alloys which show a ductile–brittle transition temperature: ASTM A471, Carbon Steel D6ac and Steel S275 J2. From the comparison of theoretical results with experimental data, it can be concluded that the model seems to be able to correctly predict the fracture toughness at low temperatures when a brittle behaviour can be expected. Acknowledgements The authors gratefully acknowledge the research support for this work provided by the Italian Ministry for University and Technological and Scientific Research (MIUR) and the Italian National Research Council (CNR). Further, the authors gratefully acknowledge Dr. D. Magagnini of RTM Breda srl, V.le Sarca 336, 20126 Milan, for providing experimental data related to steel S275 J2. 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: Carpinteri Andrea, 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 aluminum alloy at 223 K. Script Mater 2005;53:1333–7. [6] Timoshenko SP, Goodier JN. Theory of Elasticity. 3 rev ed. USA: Mc Graw-Hill Book Co; 1970. [7] Griffith AA. The theory of rupture, Proceedings of the first international conference of Applied Mechanics, Delft; 1924. p. 55–63. [8] Sneddon IN, Lowengrub M. Crack problems in the classical theory of elasticity. Wiley; 1969. [9] Bristow JR. Microcracks and the static and dynamic elastic costants of annealed and heavily cold-worked metals. Br J Appl Phys 1960;11:81–95. [10] Walsh JB. The effect of cracks on the compressibility of rocks. J Geophys Res 1965;70:381–9. [11] Walsh JB. The effect of cracks on uniaxial compression of rocks. J Geophys Res 1965;70:399–411. [12] Kachanov M. Elastic solids with many cracks and related problems. Adv Appl Mech 1993;30:259–445. [13] Ashby, Jones. Engineering materials. Pergamon; 1980. [14] ASM Handbook. Fatigue and Fracture, vol. 19. USA: American Society of Metals; 1996. [15] Databook on Fatigue Strength of Metallic Materials. The Society of Materials Science, vol. 2. Japan: Elsevier; 1996.