Local plastic collapse conditions for a plate weakened by two closely spaced collinear cracks

Engineering Fracture Mechanics 127 (2014) 1–11

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Local plastic collapse conditions for a plate weakened by two closely spaced collinear cracks A. Kotousov, D. Chang ⇑ School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia

a r t i c l e

i n f o

Article history: Received 13 December 2013 Received in revised form 22 April 2014 Accepted 26 May 2014 Available online 24 June 2014 Keywords: Multiple site damage (MSD) Strip yield model Distributed dislocation technique Plastic collapse Through-the-thickness crack Crack interaction

a b s t r a c t Fracture assessment of structures weakened with multiple site damage (MSD), such as two or more interacting cracks, currently represents a challenging problem. Strength calculations of structural components with MSD are still largely based on two-dimensional solutions for single (or isolated) cracks available in various texts and handbooks or derived from a simplified finite element analysis. Such simplifications could often result in nonconservative predictions. Therefore, there is a strong motivation for the development of more advanced modelling approaches, which could incorporate effects of the interaction between multiple cracks, thickness-induced constraints and other nonlinear phenomena. The primary purpose of this work is to develop and validate a simplified three-dimensional analytical model for the evaluation of the residual strength of two through-the-thickness cracks in a plate of finite thickness subjected to uni-axial loading. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Multiple site damage (MSD) is a type of structural damage characterised by the presence of mutually-interacting multiple cracks. Typical MSD is often found in aging aircraft fuselages where fatigue cracks grow from rivet holes, and some of them can eventually coalesce into a major crack [1]. The importance of MSD was first recognised as early as 1978 when an increasing number of aircraft were forced to operate beyond their original design life [2]. However, this special type of structural damage did not receive much attention until the catastrophic in-flight failure of the fuselage of an Aloha Airlines Boeing 737 in 1988 [3]. The subsequent investigation into this accident found that a sudden coalescence of small cracks emanating from the collinear rivet holes in the fuselage structure has led to this catastrophic failure. This accident investigation also revealed the lack of understanding of MSD as well as possible severe consequences of mechanical failures associated with MSD. It is now well-recognised that the presence of MSD can pose a serious threat. Even relatively small multiple cracks have a potential for a substantial reduction of the residual strength of the structure [4]. This is because of (i) the possibility of a sudden coalescence of the cracks resulting into a continuous crack with the length greater than the critical one and (ii) the link-up of multiple cracks can significantly accelerate the crack growth and shorten the fatigue life [5–7]. There have been many studies to address the MSD problem since the Aloha Airlines accident [3]. These vast research efforts mainly focused on the development of analytical and numerical tools capable of providing an accurate assessment of the integrity and life-time of aging structural components containing MSD by taking into account the crack interaction ⇑ Corresponding author. Tel.: +61 8 8313 6385. E-mail address: [email protected] (D. Chang). http://dx.doi.org/10.1016/j.engfracmech.2014.05.009 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.

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A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

Nomenclature 2a crack length 2b length between outer tips of two collinear cracks 2c length between inner tips of two collinear cracks 2d centre-to-centre distance between cracks g(x) crack opening displacement 2h plate thickness s, t transformed coordinates w tensile plastic zone size x, y Cartesian coordinates By(n) density of continuously distributed y-dislocations E Young’s modulus G(x, n) Cauchy kernel KI stress intensity factor in mode I K0(), K1() modified Bessel functions Wi weight function r1 remotely applied stress in y-direction yy rab(x) stress components rf flow stress rY yield strength eY yield strain ru ultimate strength  iÞ /ðs non-singular function m Poisson’s ratio

effects. This common research tend towards the development of more advanced assessment tools and is well justified because of its large impact on the maintenance cost and efficiency of safe operation [8]. MSD is a highly complex phenomenon [9]. The non-linear crack interaction, amongst others, is the key factor which complicates the development of accurate mathematical models. The presence of interaction between closely located cracks can have a significant impact on the plastic zone formation, fracture controlling parameters and the stress/strain fields around the crack tip [6] . As a result, fracture and fatigue behaviour of this type of damage can be very different from non-interactive or isolated cracks. The intensity of the crack interaction can change substantially during fatigue crack growth depending on various factors, such as the relative location, the relative size and the shape of the cracks [5]. A quantitative analysis of the crack interaction is therefore essential to provide a reliable estimate of the structural integrity of plate and shell components weakened with MSD. It was reported in many studies that the application of conventional failure assessment methods for MSD often has many limitations and often results in non-conservative predictions [2,10]. This is because many of the conventional prediction tools are usually based on solutions obtained from the analysis of non-interactive crack problems. Hence, incorrect conclusions can be made if the crack interaction effects are ignored. As an example, Kuang and Chen [11] conducted a case study related to MSD, which clearly demonstrated that a failure evaluation method that did not consider the crack interaction, over predicts the residual strength as compared with the corresponding experimental results. They also indicated that the residual strength of a panel with MSD may be overestimated up to 40% if the crack interaction is not incorporated into the predictive model. Advanced analytical approaches for the analysis of multiple crack problems are often based on the strip yield model, which was originally introduced by Dugdale [12] and Barenblatt [13]. The popularity of this model is due to a relative simplicity of the mathematical formulation, which enables closed-form analytical solutions. In the strip yield model, it is assumed that plastic deformation ahead of a crack tip is confined to an infinitesimally thin strip along the crack line. Therefore, the calculation of the fracture controlling parameters, such as the crack tip opening displacement, can be greatly simplified [14]. The analytical solutions can often be obtained by using the superposition principle and utilising well-known results for crack problems obtained within the linear theory of elasticity [15]. Various criteria for the assessment of the residual strength of plate and shell components with MSD have been proposed in the past. The most popular amongst them is the plastic zone coalescence criterion [11,16–21], also known as the ligament yield criterion. In many practical situations it produces conservative predictions of the residual strength. According to the plastic zone coalescence criterion, the ligament between cracks is assumed to fail if the plastic zones at the crack tips contact with each other. Other approaches for the assessment of the residual strength of MSD structures include the classical elastic– plastic fracture mechanics criteria developed for a single crack, such as crack tip opening angle or crack tip opening displacement [21–23]. Labeas and co-workers [24,25] developed a crack link-up criterion based on the strain energy difference before and after the ligament failure. In this approach, the increase of the ‘specific’ total strain energy, which is the total

A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

3

strain energy divided by the ligament area, due to the ligament failure in the absence of plastic deformation in the ligament, is considered a critical value for the ligament fracture. T* integral criterion was also employed by Wang, Brust and Atluri [8] for the assessment of the residual strength of MSD structures. In this work, the crack is assumed to grow when the T* integral attains a critical value, i.e. T* integral resistance, which is calculated from tests. A similar criterion was proposed by Duong et al. [26] based on the concept of total work of fracture and assuming that the specific work required to cause ligament failure is a linear function of the normal extent of the plastic region. A considerable effort have been directed to the implementation of numerical techniques, such as the finite element method [5,7,27] and dual boundary element method [28–30] for the solution of non-linear crack problems. However, these techniques require a lot of care and an extensive experimental validation as they often produce inconsistent results due to unavoidable variations between different studies, specifically, in the mesh density, adopted constitutive relationships, crack advance scheme, identification of crack opening and closure conditions as well as contact stresses in the case of fatigue crack growth modelling [31]. Many experimental studies confirmed that the fracture conditions and fatigue crack growth can be significantly influenced by the three-dimensional effects such as plate thickness [32]. It has been shown in a number of articles that the fatigue crack growth rates increase significantly with an increase in the specimen thickness [33–38]. This highlights the importance of accounting for the thickness effect in assessing structural integrity of mechanical components. Although the thickness effect has been largely investigated for isolated cracks, it is likely that it has the same or even greater influence on the behaviour of mutually interactive cracks, or in the case of MSD. Moukawsher, Grandt and Neussl [6] discussed the importance of including the effect of three-dimensional thickness amongst other factors in the analysis of MSD problems. However, there were no systematic theoretical or experimental studies on the effect of plate thickness or other 3D effects on the residual strength and fatigue life of plate components subjected to MSD. The overall objective of this paper is to develop a simplified three-dimensional model of residual strength of two throughthe-thickness collinear cracks of equal length in a plate of finite thickness. The selected crack geometry represents one of the simplest types of MSD and has a rather limited practical application. However, the investigated mechanisms of crack coalescence (local plastic collapse) and the various nonlinear effects can provide a vital insight into more practical problems involving multiple cracks. Moreover, the developed methods for the assessment of fatigue crack growth can be readily generalised for more complicated geometries of MSD. In accordance with the overall objective of this study, a theoretical model for the assessment of residual strength and local plastic collapse in plates subjected to MSD will be developed first. This model is based on the classical strip-yield model, plasticity induced crack closure concept and fundamental three-dimensional solution for an edge dislocation in an infinite plate [39]. The numerical procedure for obtaining two-dimensional and three-dimensional solutions utilises the distributed dislocation technique (DDT) and Gauss–Chebyshev quadrature method, which normally provides an effective way for obtaining highly accurate solutions to various types of problems with singularities [40–42]. To support and validate the theoretical modelling, an experimental program on the investigation of the residual strength and plastic collapse of aluminium plate specimens weakened with two collinear cracks were conducted. The residual strength and the conditions leading to the local plastic collapse of two collinear cracks of equal length were investigated with an experimental method utilising the measurement of the strain in the ligament. The outcomes of the experimental study were used to (1) evaluate and confirm experimentally the effect of the plate thickness and crack interaction on the residual strength of two interacting cracks and (2) validate the outcomes of the three-dimensional analytical model developed in the current study. 2. Problem formulation and theoretical approach The following section will describe the problem formulation and briefly outlines the theoretical approach. Fig. 1 shows the problem geometry of two collinear through-the-thickness cracks of identical length, 2a, with centre-to-centre crack spacing, 2d, located in an infinite plate of finite thickness, 2h. The plate is subject to a remotely applied tensile stress, r1 yy . When a cracked plate is loaded, plastic zones are formed at the crack tips. In this work the plastic zones are modelled with the help of the classical Dugdale strip yield model. In the proposed theoretical model we also adopt the rigid perfectly plastic material model and assume the uniform length of the plastic strip across the crack front [43]. As it is mentioned in the introduction, these assumptions are quite common in analytical studies and allow the analytical treatment of crack problems. It was also demonstrated that, in many cases, the analytical solutions correlate rather well with the results obtained with experimental or numerical methods [44]. The inner and outer plastic zone sizes are denoted by wi and wo, respectively. The effective crack length is defined as the sum of sizes of an actual crack and plastic strip. The distance between the inner effective crack tips of two cracks is denoted as 2c, whereas the distance between the outer effective crack tips is 2b. The origin of the coordinate system is set at the middle point of the two collinear cracks (at the symmetry point) as shown in Fig. 1. The distributed dislocation technique can be effectively applied to investigate this problem [41,44]. In this technique the crack and plastic zone line are represented by an unknown distribution of dislocation to simulate strain nuclei. As a result, a set of governing integral equations for x, y and z stress fields along the positive x-axis can be obtained through use of the distributed dislocation approach. By taking advantage of the symmetry of the problem under consideration, these governing equations can be written as:

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A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

y b

b d

d

x w

a

a

w

z

w

a

a

2h

w

Fig. 1. Problem geometry and coordinate system.

p

b

By ðnÞGxx ðx; nÞdn;

ð1aÞ

By ðnÞGyy ðx; nÞdn þ r1 yy ;

ð1bÞ

By ðnÞGzz ðx; nÞdn;

ð1cÞ

c

Z

1

ryy ðxÞ ¼ rzz ðxÞ ¼

Z

1

rxx ðxÞ ¼

p

Z

1

p

b c b

c

where By(n) is an edge dislocation density function which corresponds at the location n (between c and b) on the x-axis to Burgers vectors in the y-direction, and Gxx(x, n), Gyy(x, n) and Gzz(x, n) are kernels in the x, y and z directions, respectively. The kernels can be considered as induced stresses in a given direction at an arbitrary point x, due to an unit Burgers vector in the y-direction located at a point n; they become singular at the point where x = n. The dislocation density function, By(n), can be determined by enforcing boundary conditions such as traction-free on crack faces and material yielding in plastic zones. In this study the Tresca yield criterion is employed, assuming that ryy P rxx P rzz, which has been checked to occur for all considered cases. In accordance with this criterion the stress components in the plastic zone must satisfy the following equation (or yielding criterion):

jryy  rzz j ¼ rf

ð2Þ

where rf is the material’s flow stress. The crack opening displacement, g(x), is associated with the distributed dislocation density, By(n), through the notion that the sum of negative infinitesimal Burgers vectors, By(n)dn, from the point c to any arbitrary point x positioned between c and b; it leads to the crack opening displacement at point x, such that:

gðxÞ ¼ 

Z

x

By ðnÞdn;

ð3Þ

c

or, alternatively,

By ðnÞ ¼ 

dgðnÞ : dn

ð4Þ

Therefore, the physical meaning of the dislocation density function can be regarded as the negative gradient of the crack opening displacement at a point between two crack tips. Through the use of the traction free condition on crack faces and the Tresca yield criterion in plastic zones, the governing integral equations for problems of two collinear cracks in a plate with plane stress, plane strain or finite thickness can now be reduced to the following equation [34,41,42]:

rðxÞ ¼

1

p

Z c

b

By ðnÞGðx; nÞdn þ r1 yy ;

ð5Þ

where r(x) and G(x, n) are a representative stress and a representative kernel, respectively, and are provided for each stress condition of the cracked plate in the following subsections.

A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

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The kernel in Eq. (5) can be expressed using the three-dimensional solution derived by Kotousov and Wang [39] for an edge dislocation. This solution can be represented as 3D Gðx; nÞ ¼ G3D yy ðx; nÞ  Gyy ðx; nÞ

rðxÞ ¼ ryy ðxÞ ¼ 0

) for c þ wi  jxj < b  wo ;

3D 3D 3D Gðx; nÞ ¼ G3D yy ðx; nÞ  Gyy ðx; nÞ  Gzz ðx; nÞ þ Gzz ðx; nÞ

ð6aÞ

)

rðxÞ ¼ ryy ðxÞ  rzz ðxÞ ¼ rf

ð6bÞ

for c  jxj < c þ wi or b  wo  jxj < b; 3D where the three-dimensional kernels in the y-direction, G3D yy ðx; nÞ, and in the z-direction, Gzz ðx; nÞ, are:

G3D yy ðx; nÞ ¼  G3D zz ðx; nÞ ¼

" # E 1 4m2 2ð2 þ k2 q2 Þm2 K 1 ðkjqjÞ 2 2  ð1  m Þ  2 m K ðkj q jÞ  ; 0 4ð1  m2 Þ q ðkqÞ2 kjqj

E q kmK 1 ðkjqjÞ ; 2ð1  m2 Þ jqj

ð7aÞ ð7bÞ

respectively. In the above equations q = x  n, E is the Young’s modulus, k is a parameter, given by the following formulae [43]:

1 k¼ h

rffiffiffiffiffiffiffiffiffiffiffiffi 6 ; 1m

ð8Þ

and K0() and K1() are the modified Bessel functions of the second kind, which are the solutions to the modified Bessel differential equation, and represent the zero-th and the first order solutions, respectively. Further, the Gauss–Chebyshev quadrature method is applied to obtain the solution to the formulated problem. A scale transformation of coordinates is first carried out by introducing new parameters t and s such that:

bþc bc þ t; 2 2 bþc bc n¼ þ s: 2 2

x¼

ð9aÞ ð9bÞ

The integral Eq. (5) is then transformed and evaluated in the range from 1 to 1:

r ðtÞ ¼

1

p

Z

1

BðsÞGðt; sÞ

1

bc ds þ r1 yy 2

ð10Þ

 ðtÞ and where the terms with a bar indicate that they have been transformed through Eqs. (9a) and (9b). The values for r Gðt; sÞ in Eq. (10) can be determined by applying the coordinate transformation Eq. (9a) to Eqs. (6) and (7). In addition, the transformed dislocation density function, BðsÞ, satisfies the following condition:

Z

1

BðsÞds ¼ 0;

ð11Þ

1

which is determined from the requirement that crack faces physically come together at both ends, or, in other words, moving from one end of the crack to the other there should be no net dislocation.  such that: The solution of Eq. (10) can now be obtained by introducing an unknown regular function /ðsÞ

 /ðsÞ BðsÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi : 1  s2

ð12Þ

 i Þ, where si (i ¼ 1; 2 . . . N) This integral Eq. (10) can thus be converted into N  1 algebraic equations with N unknowns /ðs are the integration points, and at collocation points, tk, the following equations can be written:

r ðtk Þ  r1 yy ¼

N bcX  i ÞGðtk ; si Þ; W i /ðs 2 i¼1

k ¼ 1; 2    N  1;

ð13Þ

where

1 ; N   2i  1 ; si ¼ cos p 2N Wi ¼

ð14Þ

i ¼ 1; 2; . . . N;

ð15Þ

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A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

 tk ¼ cos

p

 k ; N

k ¼ 1; 2; . . . N  1:

ð16Þ

The Nth equation, coming from Eq. (11), results in: N pX 

N

/ðsi Þ ¼ 0

ð17Þ

i¼1

Through standard Gauss elimination computer-based procedures, a system of N linear algebraic equations with N unknowns, which is expressed by Eqs. (13)–(17), can be readily solved. Employing an asymptotic analysis [43], the stress intensity factors at the inner (x = c) and the outer (x = b) tips of cracks can be respectively found as follows:

Kc ¼ 

Kb ¼

E 4ð1  m2 Þ

E 4ð1  m2 Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 2

 ðb  cÞ/ð1Þ;

ð18aÞ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 2

 ðb  cÞ/ð1Þ;

ð18bÞ

From the above equations, the dislocation density must be zero at the tips of cracks to ensure the bounded stress field at the end of the plastic region:

 /ð1Þ ¼ 0:

ð19Þ

To determine the inner and the outer plastic zone sizes in this problem, initial (guess) values for these zones are first assumed based, for example, on the corresponding solution for a single crack problem [41] leading to determine the corre sponding /ð1Þ values. The next step is to employ an iterative procedure to alter the initially guessed values until Eq. (19) is satisfied with a desired accuracy. The crack tip opening displacement can be calculated from Eq. (3) once the function BðsÞ is determined. Gauss–Chebyshev quadrature can be applied to convert the integral into a sum of Bðsi Þ at integration points, similar to Eq. (13). More details can be found in Kotousov [41] or Chang and Kotousov [45]. 3. Modelling results and validation The remotely applied tensile stress levels required for the particular case of local plastic collapse where the inner plastic zones fully extend along the ligament (this corresponds to c = 0 in Fig. 1) have been calculated based on the developed threedimensional model. The values of the ratio of the remote stress causing the local plastic collapse to the flow stress, r1 pc =rf , as a function of the ratio of the crack length, 2a, to the spacing between cracks, 2d, is shown in Fig. 2. Analytical results corresponding to the plane stress and plane strain analyses are shown along with the theoretical predictions for two cases of h/a values. Additionally, for validation purposes, previously published analytical results by Collins and Cartwright [16] for plane stress are plotted in the figure. According to Fig. 2, the applied stress level leading to plastic collapse of the ligament between cracks is highly dependent on a/d, which characterises the level of the crack interaction. A lower applied stress is required for a higher value of a/d, i.e. closer cracks, to cause the ligament failure. The figure also shows that the plastic collapse stress is considerably influenced by the plate thickness: thicker plates are characterised by higher plastic collapse stress for a fixed a/d value. This is due to an

1

0.5

h/a=increasing

Symbols: analytical results for plane stress (Collins and Cartwright [16]) Lines: present results

0

0

0.5

1

a/d Fig. 2. Variation of calculated plastic collapse stress levels with the ratio of crack length to centre-to-centre spacing of cracks for different plate thicknesses (c = 0, h/a = plane stress, 0.3, 1.0, plane strain).

A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

7

increase in the out-of-plane constraint around the crack tip with increasing plate thickness and the change of yielding conditions, see Eq. (2). The increased constraint leads to smaller tensile plastic zone and hence larger applied stress level for the complete plastic yielding of the ligament. As a/d changes from 0 (infinite spacing) to 0.5 (moderate spacing), the plastic collapse stress drops by 44% for plane stress and 24% for plane strain. These results demonstrate and confirm the presence of strong crack interaction as well as the plate thickness effect in terms of ligament failure. It is also interesting to point out that the solutions for finite thickness plates recover the plane stress solution as a/d ? 0 while they recover the plane strain solution as a/d ? 1. For validation purposes, analytical results by Collins and Cartwright [16] for plane stress are also plotted in this figure. A general conclusion can be made: the present results, for the case of plane stress, are in a very good agreement with the previously published analytical results. 4. Experimental study Fracture tests were performed to investigate the local plastic collapse phenomenon of the ligament between two collinear cracks subjected to remote loading. The assessment of the collapse conditions is based on the plastic zone coalescence criterion, which was briefly described in the Introduction. The tests were carried out in the LT orientation of specimens with respect to the rolling direction. The following sections provide details of the fracture tests and testing methodology. The material used for the current study is aluminium alloy 5005. This is one of the most commonly used aluminium alloys, and it is prominent for its high corrosion resistance, good machinability and reasonably good mechanical properties. In accordance with ASTM E8M-04, material property tests were initially conducted on standard coupons cut out from the plates in the same orientation as to be used in the investigations of the local collapse conditions. The measured values for Young’s modulus E, 0.2% yield strength rY, corresponding yield strain eY and ultimate strength ru were 62 GPa, 132 MPa, 0.0041 and 152 MPa, respectively. These values were obtained using an Instron tensile machine equipped with an extensometer. Further, six specimens in total were machined by using the water jet cutting technology to avoid any possible formation of heat damage. Each specimen contained two collinear slits of equal length of 10 mm and width of 2 mm. In order to investigate the crack interaction and plate thickness effects, six types of specimens were fabricated with three plate thicknesses (2h = 1.2, 2.0 and 3.0 mm) and two centre-to-centre distances of collinear slits (2d = 20 and 25 mm). The overall specimen design is schematically shown in Fig. 3. To produce sharp cracks, the slits were notched with a 0.5 mm thickness saw, and after that, pre-cracked using a constant amplitude cyclic loading with rmax/rY = 0.3, R = 0.05 and 5 Hz as recommended by the ASTM standard. The measured fatigue produced crack lengths were typically 0.2–1.3 mm. The specimen preparation process was accomplished by attaching a strain gauge (FLG-1-23, gauge width 1.1 mm) in the middle of the ligament between the two collinear cracks (see Fig. 3). The role of the strain gauge was to measure the applied tensile strain level and identify the initiation of the plastic collapse conditions of the ligament based on the plastic zone coalescence criterion. In other words, the local plastic collapse conditions were assumed to occur when the strain level in the middle of the ligament reaches the yield strain of the material measured from coupon tests. The specimen dimensions after fabrication and pre-cracking are shown in Table 1. A minor unavoidable difference between crack lengths can be noted but it is not expected to modify significantly the critical collapse load. The plastic collapse tests were carried out under displacement control (with the elongation rate of 10 mm/min) using an Instron 1342 hydraulic mechanical testing machine. Each specimen was stretched under tensile quasi-static loading until

Grip area

200 mm Strain gauge

2d

220 mm 120 mm

2a1

2a2

Crack1

Crack2

2h

Grip area Fig. 3. Test specimen with two collinear cracks equipped with strain guage.

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A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11

Table 1 Specimen dimensions after saw-cutting and pre-cracking (unit: mm). Specimen no.

2h

2a1

2a2

2d

1 2 3 4 5 6

1.2 2 3 1.2 2 3

11.97 11.6 13.13 12.78 12.64 11.03

11.86 11.86 12.75 12.88 12.55 12.15

24.90 25.16 24.31 19.69 19.86 19.53

complete failure. The applied net stress causing the plastic collapse of the ligament, r1 pc , is specified when the plastically deformed regions developed and the inner crack tips expand toward the centre of the ligament and come into contact. Accordingly, strain values were monitored at the centre of the ligament while the tensile loading increases. When the strain reached the yield strain of the material, the corresponding remotely applied stress value (net-section stress) was considered as the plastic collapse stress for the ligament yielding. The development of plastically deformed regions was visible with even the naked eye. In all six specimens, no crack growth was observed. Because the plastic collapse stress was below the critical applied stress needed for fracture initiation or sub-critical crack growth. The results of the plastic collapse tests are shown in Fig. 4. These results presented in this figure were normalised by the flow stress, or (rY + ru)/2 = 142 MPa, of the material, and the normalised plastic collapse stress, r1 pc =rf , was plotted against the ratio of crack length to centre-to-centre spacing of cracks, a/d, which is an indicator to the crack interaction effect. Stress r1 pc represents the net stress in the specimen. The variation of plastic collapse stress is the result of the combined effects of the crack interaction and plate thickness, which are characterised by different a/d and h/a. It is virtually impossible to fabricate test specimens with the same geometry as the pre-cracking always produces some scatter. To investigate the effect of crack interaction (or influence of a/d on the collapse stress) the data points which have a similar h/a value are paired and connected with a dotted line in Fig. 4. Due to a relatively small number of tests the linear dependences presented in Fig. 4 largely provide a qualitative assessment of the crack interaction effect. The shift of the dotted lines highlighted by an arrow in Fig. 4 demonstrates the effect of plate thickness (or influence of h/a) on the local plastic collapse conditions. This figure also shows the effect of crack interaction on a plastic collapse stress. For the tested specimen geometries, on average, a drop of 20% in the plastic collapse stress was measured with an increase in a/d from 0.49 to 0.62 (27% increase). The conducted experimental study strongly indicates that the plastic ligament collapse conditions are highly dependent on both the crack interaction and the plate thickness. A comparison between the test results and theoretical predictions, made using the theoretical model developed in the previous section, is presented in Table 2. Inspection of the results reveals that the theoretical model leads to a very conservative estimate of the plastic collapse stress of the specimens. The predicted values are substantially lower than the corresponding test results with the relative error being about 21%. These discrepancies are attributed to the approximations and simplifications utilised in the strip yield model and idealised constitutive equation employed in the modelling approach. However, it is highly noteworthy that the plastic collapse stress predictions show the same trends as the experimental results. Furthermore, the differences between the predictions and the experimental results are very consistent throughout

Fig. 4. Plastic collapse stresses of specimens weakened with two collinear cracks.

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A. Kotousov, D. Chang / Engineering Fracture Mechanics 127 (2014) 1–11 Table 2 Predicted and measured plastic collapse stress to yield strength ratios. Specimen no. Prediction

P1

P2

P3

P4

P5

P6

r1 pc =rf

0.60

0.61

0.58

0.45

0.47

0.53

r1 pc =rf

0.75

0.77

0.73

0.56

0.60

0.65

Relative error (%)

20

21

21

20

22

18

r1 pc =rf 0

0.60

0.62

0.58

0.45

0.48

0.52

Relative error (%)

0

1

1

0

2

2

Experiment

rf = 152 MPa rf = 177.5 MPa

the measured data, as shown in Table 2. This demonstrates predictive capabilities of the developed theoretical model if the disparity can be offset by employing an empirical value of the flow stress, rf, in the theoretical model. The introduced empirical flow stress in this context aims to compensate errors associated with the yield strip idealisation, idealised yield conditions and elastic perfectly plastic material behaviour. A remarkable reduction of the discrepancies between the experimental and theoretical results can be observed with this new fitting value of the flow stress (177.5 MPa). The difference between the theoretical predictions and experimental results is now substantially reduced to only 1–2% (see Table 2), which can be considered as an excellent agreement between the theory and experiment. The comparative study has also highlighted the limitations of the classical yield strip concept. Despite the qualitative agreement of the theoretical predictions and experiments, the quantitative agreement has only been achieved with some fitting value of the flow stress. This is not surprising as the theoretical concept provides a very simplistic evaluation the actual elasto-plastic behaviour of the material near the crack tip.

5. Conclusions The primary purpose of the study was to develop computationally efficient and validated three-dimensional theoretical models for the evaluation of structural integrity of plates weakened by two collinear cracks. The problem analysed in the current study represent a simple case of MSD, i.e. two through-the-thickness collinear cracks of equal length. However the general procedure and developed approach can be extended to more complicated and more practical types of MSD. The theoretical modelling of the problem was based on three-dimensional solutions rather than plane stress or plane strain assumptions normally utilised in many previous studies. Subsequently a two-dimensional strip yield model for the analysis of two collinear stationary cracks was extended to accommodate the three-dimensional effects. The analytical modelling of the three-dimensional problem was accomplished by using the three-dimensional fundamental solution for an edge dislocation in an infinite plate of finite thickness. Gauss–Chebyshev quadrature method was applied to obtain a numerical solution to the governing integral equations with singular Cauchy kernel. The three-dimensional strip yield model was utilised to investigate the residual strength of plates containing two collinear cracks. The remotely applied tensile stress causing the local plastic collapse of the ligament was calculated as a function of the spacing between the cracks, yield stress as well as the plate thickness. For the same in-plane geometry, thicker plates have a higher plastic collapse stress. This behaviour is attributed to an increase in the out-of-plane constraint and change of the yield conditions with an increase in the plate thickness. The three-dimensional results were validated against a previously developed two-dimensional model for limiting cases of very thin plate (which is related to plane stress conditions) or very thick plate (where the plane strain conditions dominate). To support the theoretical findings and evaluate the developed models, an experimental program was developed and carried out on aluminium plate specimens weakened by two collinear cracks. The effects of the crack interaction and plate thickness on the plastic collapse of the ligament between two cracks were investigated by using an experimental technique developed based on the plastic zone coalescence criterion. In this technique, the axial strain was monitored at the centre of the ligament with the continuous increase of the tensile loading. When the strain value, measured by a strain gauge attached in the middle of the ligament, reached the yield strain of the material, the corresponding stress (net-section stress) was considered as the plastic collapse stress for the ligament yielding. In the tests, six pre-cracked specimens having various geometries were fabricated to provide a wide range of data. The effect of the crack interaction on the plastic collapse of the ligament was first investigated. On average, a drop of 20% in the plastic collapse stress for the ligament failure was measured with an increase in the ratio of crack length to centreto-centre crack distance from 0.49 to 0.62 (27% increase). The plate thickness effect was also found to be an important influential factor. It was observed, as expected, that an increase in the plate thickness resulted in a higher plastic collapse stress level. A comparison between the experimental data and theoretical modelling results was conducted. The comparison revealed that the theoretical model considerably underestimated the residual strength of the specimens containing two collinear cracks if an average value of the plastic and ultimate stresses is used in the model as flow stress. Even though there was

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a substantial offset between experimental results and predictions, the model exactly predicts the same trends as observed in the experiments. However, when an empirical value of the flow stress is used in the model (to compensate a number of modelling assumptions) a remarkable agreement between the theoretical predictions and experimental results was observed. In conclusion, it can be stated that the developed strip yield model is very effective in predicting the residual strength of plates weakened by two cracks for a wide range of geometries and loading conditions. The theoretical models developed in the research can be employed to develop criteria for the interaction of multiple cracks. The developed models can be further utilised to determine the effective dimensions characterising the interactive and non-interactive crack configurations. The conducted theoretical and experimental work may be considered as an initial study for problems with MSD. 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