PSI - Issue 25

Volume 2 5 • 2020

ISSN 2452-3216

ELSEVIER

1st Virtual Conference on Structural Integrity - VCSI1

Guest Editors: Fili pp o Berto

Francesco I acoviello Stavros Kourkoulis P edro Moreira P aulo N o b re B R eis Aleksandar Sedmak L uca Susmel P aulo T avares

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Procedia Structural Integrity 25 (2020) 1–2

1st Virtual Conference on Structural Integrity - VCSI1 Preface Filippo Berto a , Francesco Iacoviello *,b , Stavros Kourkoulis c , Pedro Moreira d , Paulo NB Reis e , Aleksandar Sedmak f , Luca Susmel g , Paulo Tavares d

Norwegian University of Science and Technology, NTNU, Norway a Università di Cassino e del Lazio Meridionale, DiCeM, Italy b* National Technical University of Athens, Greece c d INEGI, Portugal e University of Beira Interior, UBI, Portugal f University of Belgrade, Serbia, g University of Sheffield, UK f

© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the VCSI1 organizers © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the VCSI1 organizers h i en a c n n ( / cr

Keywords: Preface; Fracture; Structural Integrity.

1. Preface The First Virtual Conference on Structural integrity was organized by four active National Groups of the European Structural Integrity Society (ESIS): APFIE (Portuguese Structural integrity Society, Portugal), DIVK (Društvo za integritet i vek konstrukcija „Prof. dr Stojan Sedmak“, Serbia), GSEMM (Greek Society of Experimental Mechanics of Materials, Greece) and IGF (Italian Group of Fracture, Italy), under the aegis of ESIS. These groups have decades of experience in the organization of “traditional” conferences, workshops and summer schools, included some really large events like ECFs (European Conferences on Fracture) with hundreds of participants.

* Corresponding author. Tel.: +39.07762993681 E-mail address: iacoviello@unicas.it

2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the VCSI1 organizers

2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the VCSI1 organizers 10.1016/j.prostr.2020.04.001

Filippo Berto et al. / Procedia Structural Integrity 25 (2020) 1–2

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Author name / Structural Integrity Procedia 00 (2019) 000–000

VCSI1 was organized with the aim to take advantages of the new technologies (web, YouTube channel, Telegram channel etc.), offering a low-cost event to the participants. An innovative and stimulant environment was created with the possibility to have the time to watch all the presentations (more than fifteen days) and prepare the discussion. VCSI was held online on January 16, 2020 using a commercial meeting platform. More than sixty presentations were scheduled in five different sessions:  Numerical modelling and computation (Chairman: Aleksandar Sedmak)  Fatigue (Chairman: Luca Susmel)  Experimental mechanics 1 (Chairman: Pedro Moreira)  Experimental mechanics 2 (Chairman: Paulo Reis)  Structural integrity of structures (Chairman: Stavros Kourkoulis) Considering that all the participants had the possibility to watch the presentations before the conference, the online event was only focused on the discussion. The result has been really exciting, with a really interesting and vibrant discussion that lasted more than six hours. All the presentations and the discussions are now available in a dedicated YouTube channel playlist in the IGF YouTube channel (https://www.youtube.com/playlist?list=PLT1-2PyZ6QrLOo85GLteKsWPJFUrCOM7A), offering the possibility to the fracture and structural integrity community to participate to the conference also in the next years. Considering that this conference was a first experiment, the organizers and the participants declared to be enthusiastic of the result. Many organizational details still need to be optimized (e.g., a dedicated platform for the upload of the videos), but we can be sure that this VCSI1 will be only the first of a long series! This volume collects the papers connected to many presentations, confirming the fracture and structural integrity community as a really active and vibrant “band of scientists”!

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Procedia Structural Integrity 25 (2020) 268–281 Structural Integrity Procedia 00 (2019) 000–000 Structural Integrity Procedia 00 (2019) 000–000

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© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the VCSI1 organizers Abstract Simplified mathematical models are extremely useful for understanding and characterizing physical phenomena; this is the case of plane (stress / strain) elasticity, which has been the main analytical framework of Fracture Mechanics since its origins. Bowed crack fronts in thin sheets and plates, unpredicted fatigue crack paths, and crack initiation as well as brittle fracture location are only examples of fracture mechanisms not completely explained by the classical closed form 2D solutions, e.g. Westergaard’s solution. This paper aims to provide a brief review on 3D e ff ects on the brittle fracture behavior of linear, elastic, homogeneous and isotropic solids in presence of cracks or notches, under the hypothesis of small scale plasticity (LEFM). We overview the main theoretical and numerical results on coupled modes of fracture, i.e. coupling of shear (Mode II) and out-of-plane (Mode III) modes, due to three-dimensional e ff ects and on 3D vertex singularities close to a corner point generated by the intersection of a crack / notch front with free surfaces. We also address recent theoretical-numerical studies and inconsistencies on the relation of crack (notch) tip stress singularities, usually characterized by local (notch) stress intensity factors, in the vicinity of a corner point and far away from it. Despite numerous works have attempted to interpret finite element and analytical results, no consensus exists on the behavior of the 3D stress field near a vertex of cracked / notched solids; a unifying theory, able to guarantee the extension of the results of experimental tests prescribed by standards, based on the 2D theory of elasticity, to real cases, is still required. c 2020 The Authors. Published by Elsevier B.V. T is an open access article under the CC BY- C-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) P er-revie lin : Peer-rev ew und r responsibility of the VCSI1 organizers. Keywords: Fracture Mechanics; 3D e ff ects; coupled modes; stress singularities; vertex singularities; stress intensity factors 1st Virtual Conference on Structural Integrity – VCSI1 3D e ff ects on Fracture echanics Marco Maurizi a, ∗ , Filippo Berto a a NTNU - Norwegian University of Science and Technology –Department of Mechanical Engineering, 7491 Trondheim, Norway Abstract Simplified mathematical models are extremely useful for understanding and characterizing physical phenomena; this is the case of plane (stress / strain) elasticity, which has been the main analytical framework of Fracture Mechanics since its origins. Bowed crack fronts in thin sheets and plates, unpredicted fatigue crack paths, and crack initiation as well as brittle fracture location are only examples of fracture mechanisms not completely explained by the classical closed form 2D solutions, e.g. Westergaard’s solution. This paper aims to provide a brief review on 3D e ff ects on the brittle fracture behavior of linear, elastic, homogeneous and isotropic solids in presence of cracks or notches, under the hypothesis of small scale plasticity (LEFM). We overview the main theoretical and numerical results on coupled modes of fracture, i.e. coupling of shear (Mode II) and out-of-plane (Mode III) modes, due to three-dimensional e ff ects and on 3D vertex singularities close to a corner point generated by the intersection of a crack / notch front with free surfaces. We also address recent theoretical-numerical studies and inconsistencies on the relation of crack (notch) tip stress singularities, usually characterized by local (notch) stress intensity factors, in the vicinity of a corner point and far away from it. Despite numerous works have attempted to interpret finite element and analytical results, no consensus exists on the behavior of the 3D stress field near a vertex of cracked / notched solids; a unifying theory, able to guarantee the extension of the results of experimental tests prescribed by standards, based on the 2D theory of elasticity, to real cases, is still required. c 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the VCSI1 organizers. Keywords: Fracture Mechanics; 3D e ff ects; coupled modes; stress singularities; vertex singularities; stress intensity factors 1st Virtual Conference on Structural Integrity – VCSI1 3D e ff ects on Fracture Mechanics Marco Maurizi a, ∗ , Filippo Berto a a NTNU - Norwegian University of Science and Technology –Department of Mechanical Engineering, 7491 Trondheim, Norway

Nomenclature Nomenclature

2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the VCSI1 organizers 10.1016/j.prostr.2020.04.032 ∗ Corresponding author. Tel.: + 39-327-780-3296. E-mail address: marco.maurizi@ntnu.no 2210-7843 c 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: P er-review under responsibility of the VCSI1 organizers. x , y , z Cartesian coordinates r , θ , z cylindrical / polar coordinates σ i j i , j − th stress component 2 h plate thickness γ supplementary V-notch opening angle and crack surface intersection angle 2 α V-notch opening angle ∗ Corresponding author. Tel.: + 39-327-780-3296. E-mail address: marco.maurizi@ntnu.no 2210-7843 c 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review line: Peer-review under responsibility of the VCSI1 organizers. x , y , z Cartesian coordinates r , θ , z cylindrical / polar coordinates σ i j i , j − th stress component 2 h plate thickness γ supplementary V-notch opening angle and crack surface intersection angle 2 α V-notch opening angle

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u displacement field, subscripts x , y , z denote direction w displacement along z -direction dependent only on x and y λ Lame’s modulus; eigenvalue with subscripts 1, 2, 3s, 3a and 0; strength of corner point singularity G shear modulus k shear factor ν Poisson’s ratio N i j i , j − th stress resultant, subscripts x , y , z denote directions; without subscript denotes the mean in-plane g i j angular functions for plane elastic crack problems Φ Airy stress function K N i mode i notch stress intensity factor (NSIF); without superscript stress intensity factor (SIF); with superscript C coupled mode; with ∞ far-field (N)SIF Φ Airy stress function F i dimensionless scale e ff ect function for coupled mode i β crack front intersection angle The history of fracture mechanics starts with the work of Inglis (1913), who provided the analytical exact 2D solution of the stress field around an elliptical hole (notch) in a plate, allowing to extend this result to cracks. Plane elasticity, under the hypothesis of plane stress or plane strain, constitutes the analytical framework under which Linear Elastic Fracture Mechanics (LEFM) has been studied and applied to assess the structural integrity of materials since its emergence. Based on this, Gri ffi th (1995) proposed an energetic failure criterion, Westergaard found out the closed form solution of a center through-the-thickness cracked infinite plate under biaxial tension, Irwin modified the Gri ffi th theory introducing a correction for small scale plasticity at the crack tip and defined the concept of stress intensity factor ( K ) related to that of stress singularities (stress tend to infinite as the crack / corner is approached). These latter have been investigated for angular corners (notches) of plates by the well-known work of Williams (1952), where the 2D stress field σ i j ∝ r λ − 1 has been found to show a singularity for r →− 0, with i , j = 1 , 2 , r the radial distance from the notch tip, and λ the eigenvalue obtained by imposing the boundary conditions and representing the strength of the singularity. These results paved the way for the theoretical generalization of the stress intensity factor (often indicated as K N ) to V-shaped notches (crack is a particular case of it), formally defined by Gross and Mendelson (1972), giving physical meaning to the constants of proportionality of the William’s formulae; subsequently, Lazzarin and Zappalorto (2012) extended the analytical formulation also to blunt notches, for which the singularity disappears. The analytical and numerical 2D results in the field of LEFM have been used as bases for standards, such as ASTM E1820-18 for measurement of fracture toughness and ASTM E647-00 for measurement of fatigue crack growth, which are hence based on the assumption of 2D elasticity. Experimental evidence Pook (2013), Doquet et al. (2009, 2010), Bertolino and Doquet (2009) seems to contradict some 2D analytical results, which for instance do not well predict the site of crack initiation in notched thick members, shifting the failure location from the middle plane to the corner (near the free surface) as the thickness increases. In fact, stress intensity factors and energy release rate have a variable profile along the crack front Berto et al. (2013a,b), and coupled modes (see paragraph 4) exist for shear loading, meaning a breaking down of classical 2D solutions. This could explain the bowed fatigue crack fronts Doquet et al. (2010) as well as the inconsistency of the predicted strength size e ff ect by using the 2D formulation with experimental results Sinclair and Chambers (1987). Crack / notch tip displacements are classically described by three modes of relative motion between the crack / notch surfaces, if considered as points in the 2D frame. However, a 3D description of the surfaces gives rise to regarding them as composed of infinitesimal elements, which can also rotate relative to each other; six distinct movements are hence possible for one element on a crack / notch tip surface, correspondent to Volterra distorsioni (distortions) of dislocation and disclination, as reported by Pook (2013). The former represent the three well-known modes of fracture Mode I (along y axis), II (in-plane shear, along x axis) and III (out-of-plane shear, along z axis), whereas the latter are rotations about the same three axes used to identify the dislocation distortions (Fig.1). 1. Introduction

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(a)

(b)

Fig. 1: (a) Notation for the stress field near the crack tip. (b) Notation for sharp V-notches. Small partial cylindrical zone around a V-notch represented in figure; the dashed red line represents the mid-plane.

Disclinations could be related to the behavior of elements of the crack / notch tip surface near free surfaces, the so-called corner point (vertex) singularity, investigated in detail by Bažant and Estenssoro (1979) and subsequently by Benthem (1980). They found out that in the vicinity of a corner, that is the intersection between a crack and a free surface, the singularity changes. A 3D analytical frame, which reduces the linear elastic problem to a system of one bi-harmonic equation, representing the in-plane problem, and one harmonic equation for the displacement field ( w ) along z axis, has been proposed by Lazzarin and Zappalorto (2012); Zappalorto and Lazzarin (2013). Although it well approximates the finite element solutions at the middle plane of a plate Berto et al. (2016), it is believed to fail on the free surfaces Zappalorto and Lazzarin (2013), where vertex singularities act. Despite this, the 3D frame predicts what have been in the last two decades numerically observed Harding et al. (2010); Berto et al. (2011c); Kotousov et al. (2013), i.e. the coupling between the anti-symmetric modes II and III. They are thought to be related to each other through the Poisson’s ratio and the thickness, even if these two parameters do not have the same influence on the coupled mode induced by in plane or anti-plane loading. Therefore, a shear (II or III) isolated mode cannot exist. Despite the e ff orts made by many researchers, a complete understanding and a theoretical description of 3D e ff ects on fracture mechanics still lacks. In light of this, to clarify what has been already done and to highlight contradictions and zones of no-consensus, we briefly review some 3D analytical formulations, the coupled modes and the vertex singularities.

2. 3D analytical formulations

Kane and Mindlin (1955) have laid the groundwork for the available analytical frameworks for the elastostatic 3D problem ahead of crack / notch tip. By studying the high-frequency extensional vibrations of plates, they proposed the following displacement field:

z h

u x = u ( x , y )

u y = v ( x , y )

u z =

w ( x , y ),

(1)

where 2 h is the plate’s thickness. Based on this assumption and on the Mindlin’s solutions in terms of stress resultants (Lazzarin and Zappalorto (2012)), Yang and Freund (1985) studied the 3D stress field in through-the thickness cracked thin plates. By means of the equilibrium equations, the constitutive model and the compatibility

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equations, they obtained two equations dependent on the material’s properties, that is Lame’s constant λ and shear modulus G , on the shear factor k = 0 . 907, on the mean in-plane stress resultant N = ( N xx + N yy ) / 2, and on w .

N − w = 0 ,

∇ 2

h 2 3 k 2

λ + G 3 λ + 2 G ∇

1 2 kG

λ + 2 G 2 k λ G

λ 3 λ + 2 G

2 w − w =

N

(2)

where the stress resultants are simply the stress components integrated along the thickness. Exploiting of the Fourier Transform and the Wiener-Hopf technique near the crack tip, they derived the generalized plane strain stress resultants in the following form:

h K √ 2 π r

N i j =

g i j ( θ ) i , j = 1 , 2 ,

(3)

where K is the stress intensity factor in Mode I loading, and g i j ( θ ) are the classical angular functions of plane elastic crack problems. From Eq. (3), by means of N = ( N xx + N yy ) / 2, and from the second equation of Eq. (2), the harmonic equation ∇ 2 w = 0 for the out-of-plane displacement ahead the crack tip surface is obtained. Kotousov and Lew (2006) extended the previous approach to the analysis of the 3D stress field ahead of a V-notch, as suming a shear factor k = 1 and introducing an Airy’s stress function Φ , which automatically satysfies the equilibrium equations. For the detailed formulation see Kotousov and Lew (2006); Lazzarin and Zappalorto (2012). Applying the free-free boundary conditions along the edges of the V-notch, two eigenvalue problems have been deduced, symmetric and skew-symmetric, respectively. A recent new frame has been developed by Lazzarin and Zappalorto (2012), who converted any three-dimensional notch problem into a bi-harmonic and harmonic problem. The main assumption is again the Kane and Mindlin’s hy pothesis of generalized plane strain ( ε zz uniform along the thickness) on the displacement field of Eq. (1). It can be deduced that the normal strains ε ii (no repetition rule), and γ xy are not a function of z . Consequently, by means of the stress-strain relationships, also the correspondent stress components are independent on z . The only stress and strain components to be dependent on z are τ xz , τ yz , and γ xz , γ yz , respectively. The equilibrium in z-direction gives rise to the harmonic equation on w :

∇ 2 w = 0 .

(4)

As for the 2D problem, the Airy’s stress function Φ ( x , y ) must satisfies the bi-harmonic equation, as reported in detail in Lazzarin and Zappalorto (2012), that is:

∇ 4 Φ = 0 .

(5)

Hence, any 3D elastostatic notch problem, under the hypothesis of Eq. (1), can be entirely represented by Eq. (4) and (5), which must be satisfied simultaneously. Considering V-notch problems, Eq. (5) and (4) are satisfied by the functions Φ ( r , θ ) and w ( r , θ ), for in-plane (William’s solution Williams (1952)) and out-of-plane shear (Seweryn and Molski (1996)) loading, respectively:

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Φ = r λ 1 + 1 [ A

λ 2 + 1 [ A

s cos (( λ 1 + 1) θ ) + B s cos ((1 − λ 1 ) θ )] + r

a cos (( λ 2 + 1) θ ) + B a cos ((1 − λ 2 ) θ )]

(6) (7)

λ 3 s cos ( λ

λ 3 a sin ( λ

w = D s r

3 s θ ) + D a r

3 a θ ) ,

where λ 1 and λ 2 are the eigenvalues corresponding to Mode I and II, respectively; λ 3 s and λ 3 a are the symmetric and anti-symmetric eigenvalues for the anti-plane shear mode (Mode III), respectively. The constants A s , B s , A a , B a , D s and D a have to be determined by imposing the boundary conditions. Specifically, supposing traction-free conditions on the V-notch flanks ( θ = ± γ in Fig. 1b) and recalling the expressions for the stress components in polar coordinates in relation with Φ and w :

∂ ∂ r −

∂ Φ ∂θ

∂ 2 Φ ∂ r 2

1 r

σ θθ =

τ r θ =

∂ u z ∂θ

z h

1 r

∂ w ∂θ

1 r

= G

τ θ z = G

∂ u z ∂ r

z h

∂ w ∂ r

τ rz = G

= G

,

(8)

two homogeneous systems of equations are obtained (Lazzarin and Zappalorto (2012)), correspondent to the sym metric and skew-symmetric terms. To obtain non-trivial solutions, the determinants are equaled to zero, and the fol lowing equations are deduced:

sin (2 λ 1 γ ) + λ 1 sin (2 γ ) sin ( λ 3 s γ ) = 0 sin (2 λ 2 γ ) + λ 2 sin (2 γ ) cos ( λ 3 a γ ) = 0 .

(9)

(10)

These two equations represent the eigenvalue problem for symmetric (Eq. (9)) and anti-symmetric (Eq. (10)) load ing. It is evident that the terms in square brackets for both equations are the classical in-plane eigenvalue problem (Williams (1952)), whereas the other terms are the out-of-plane eigenvalue equations (Mode III). Moreover, Eq. (9) and (10) match the Kotousov and Lew’s eigenvalue equations (Kotousov and Lew (2006)). Singularities on displace ment fields and strain energy density (SED) averaged in a small volume around the notch tip are not physically possible; hence, the condition Re( λ ) > 0 has to be imposed on the eigenvalues. Besides, singular stress fields are ob tained if and only if Re( λ ) < 1, which together with the previous condition gives rise to the constraint 0 < Re( λ ) < 1. Because λ 3 s = 2 λ 3 a = π / γ > 1 (for 0 < γ < π ), and from Eq. (8), the out-of-plane shear stress singularity occurs only for the skew-symmetric component. Therefore, the anti-plane Mode III shear stresses ahead the notch tip can be written as follows (see Appendix A for details):

K N III √ K N III √

r λ 3 a − 1 sin ( λ

3 a θ )

(11)

τ zr =

2 π

r λ 3 a − 1 cos ( λ

3 a θ ) ,

(12)

τ z θ =

2 π

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where

√ 2 π τ z θ ( r , 0) r

1 − λ 3 a

K N

III = lim r → 0

(13)

is the notch (generalized) stress intensity factor (NSIF). Based on the same definition, the Mode I and II NSIFs can be expressed as follows:

√ 2 π σ θθ ( r , 0) r √ 2 π τ r θ ( r , 0) r

K N

1 − λ 1

I = lim r → 0 II = lim r → 0

(14)

K N

1 − λ 2 ,

(15)

from which the in-plane stress fields around the notch tip, exploiting of the superposition principle (valid in LEFM), have the following representation in polar coordinates, as proposed by Gross and Mendelson (1972):

K N

K N II √

I √

r λ 1 − 1 ψ

r λ 2 − 1 η

i j ( θ, λ 1 , γ ) +

i j ( θ, λ 2 , γ ) ,

(16)

σ i j =

2 π

2 π

where ψ i j and η i j are functions not reported for the sake of brevity (for details Lazzarin and Tovo (1998)). The proposed analytical framework allows to determine the 3D singular stress field near the crack / notch tip as a function of the NSIFs, whenever the Kane and Mindlin’s hypothesis on the displacements is satisfied. Besides, it is worth to be noticed that τ rz and τ θ z are the only shear stress components to have a dependence (linear) on z, fundamental to predict the behavior of coupled modes along the thickness. Obviously, the crack case is recovered for γ = π .

3. General Considerations on 3D E ff ects

Several numerical, analytical and experimental studies (as reviewed by He et al. (2016)) have been conducted to understand the behavior of the 3D singular stress field in the proximity of a crack / notch tip surface. One of the basic results is the radius of influence of 3D e ff ects around the crack tip: they tend to disappear at a radial distance approximately equal to half of the plate thickness (particularly evident in Harding et al. (2010)), converging to a 2D plane stress condition after a distance roughly equivalent to the plate thickness (Nakamura and Parks (1989),Berto et al. (2011c),Berto et al. (2012),He et al. (2016)). It means that for bodies with in-plane dimensions comparable with the thickness (no plates), the 3D e ff ects spread almost all over the body and an asymptotic plane stress field is not possible; by definition of plates, the in-plane dimensions are greater than the thickness, i.e. plane stress conditions can be reached far away from the crack / notch front, giving the possibility to study cracks and notches locally. Indeed, in finite element (FE) simulations the 3D stress state zone is often considered encapsulated by the K-dominance zone (He et al. (2016)), implying that coupled fracture modes must be related to the far-field applied modes. SIFs (or NSIFs) and the strength of singularity (1 − λ ) are the main parameters considered to characterize the 3D singular stress field. The previous described 3D analytical frames predict solutions which di ff er from those of the classical 2D plane elasticity for the presence of along-the-thickness stress and displacement components (along the z-axis), dependent on the z coordinate. In terms of fracture parameters, they provide that only K N III ( z ) is a function of z , i.e. it changes along the plate thickness, whereas K N I and K N II are supposed to be independent of z. However, numerical results of cracked / notched specimens under Mode I and II, such as that of She and Guo (2007) for through-

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the-thickness cracks or Berto et al. (2016) for notched plates, have proved that K N I and K N

II depend on the z-coordinate;

this behavior is shown in Fig. 2 for cracks.

(a)

(b)

Fig. 2: Primary Mode I and II and induced Mode 0 normalized stress intensity factors variation along the thickness for di ff erent Poisson’s ratios ν from He et al. (2016). (a) Mode I SIF variation for ν = 0 . 15 (1), 0 . 30 (2), 0 . 50 (3). (b) Primary Mode II and induced Mode 0 SIF dependence on z for ν = 0 . 1 (1), 0 . 30 (2), 0 . 50 (3) Primary modes term indicates the principal fracture modes obtained directly by the imposed loading conditions, whereas the term coupled modes emphasises that the fracture modes, with the corresponding SIF, are induced by the primary modes. In Fig. 2 the SIFs are normalized with respect to the far-field (asymptotic) stress intensity factors K ∞ I and K ∞ II , which have been adopted to apply the 2D far-field displacement (plane stress) solution (William’s solution) as boundary condition; the distinction between far-field and local modes is necessary, in light of the dependence of these latter from the z-coordinate. Besides, in Fig. 2b the induced coupled Mode 0 K 0 is shown varying with the Poisson’s ratio and achieving the maximum near the free plate surface, approaching zero when this latter is reached (more details in the next paragraphs), as well as K I depends also on the Poisson’s ratio. Defining the local in-plane (notch) stress intensity factors as in Eq. (14) and (15), but introducing the dependence of z in σ θθ ( r , θ, z ) and τ r θ ( r , θ, z ), and considering the linearity of the elastic problem (LEFM), the functional relationship between the local and applied in-plane (N)SIFs reads as: K I ( z ) = K ∞ I F I z h , ν (17) K II ( z ) = K ∞ II F II z h , ν (18) where F I and F II are dimensionless functions dependent on the z-position and Poisson’s ratio, as clear from Fig. 2. These local in-plane (N)SIFs do not depend on the thickness.

4. Coupled Modes

In the last three decades the existence of shear coupled modes, i.e. Mode II and III, has been proven (e.g. Nakamura and Parks (1989),Harding et al. (2010),Berto et al. (2011a),Kotousov et al. (2013)). Additionally, even non-singular shear modes can induce singular coupled modes. Specifically, non-singular terms in the William’s expansion (higher order terms) of the far-field in-plane shear stress can induce the singular out-of-plane mode (Mode 0), as numerically proven by Berto et al. (2011a) for a cracked plate. The same situation also occurs for sharp V-notches with opening

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angles greater than ∼ 102 . 6 ◦ , for which the Mode II eigenvalue λ II > 1, meaning that a non-singular Mode II stress field is obtained, while the Mode 0 eigenvalue still λ 0 < 1 (out-of-plane singularity). This latter has been reported by Harding et al. (2010), as shown in Fig. 3, and extensively investigated by Berto et al. (2011c).

Fig. 3: Eigenvalues as function of the V-notch opening angle for Mode II and 0. Comparison between analytical and numerical (symbols) results by Harding et al. (2010)

The validity of the previous 3D analytical model (Eq. (9) and (10)) for the strength of the stress singularities (1 − λ ) has been confirmed through finite element simulations by Harding et al. (2010); the good match between the analytical and numerical solutions λ (2 α ) (2 α is the opening angle) is highlighted in Fig. 3. Besides, from Eq. (9) and (10), confirmed by 3D finite element models (Berto et al. (2013a)), it can be deduced that the singular Mode II induced by anti-plane loading has the same eigenvalues of the shear Mode II (in-plane loading); vice versa, the characteristic equation ( cos ( λ 3 a = 0)) for the singular Mode 0 induced by in-plane shear loading is the same as for Mode III, even if the two modes have a completely di ff erent nature (next paragraph). Also, non-singular anti-plane loading of cracks, i.e. K III = 0, can cause singular coupled Mode II, which shows a strong K C II variation along z and a dependence on the plate thickness. The possibility to have a singularity at the crack / notch tip surface even if the applied mode is not singular could strongly contribute to failure initiation. The Averaged Strain Energy Density (ASED) approach might well capture the failure location along the thickness of a cracked brittle plate due to a singular coupled Mode 0, despite the non-singular nature of the applied in-plane loading, as documented through finite element modeling by Berto et al. (2011a). One of the first three-dimensional finite element analyses dedicated to studying the 3D e ff ects in fracture mechanics has been conducted by Nakamura and Parks (1989), who investigated the out-of-plane induced mode (Mode 0) on a thin elastic cracked plate under Mode II loading. They noticed that the coupled Mode 0 K 0 is zero at the mid-plane and it increases along the thickness up to a maximum on the intersection of the crack front with the free surface (vertex singularity), assuming higher values as the Poisson’s ratio goes up. This was consistent with the results of Bažant and Estenssoro (1979), who predicted a K 0 approaching infinity, but in contradiction with the boundary condition of zero shear stress ( τ z θ ) at free surfaces. More recent numerical results, such as Harding et al. (2010), Berto et al. (2011b), Berto et al. (2011c) confirmed the latter conclusion, i.e. K 0 →− 0 at free surfaces, arguing that too coarse meshes have been utilized in previous studies. The discussion about the e ff ect and the theoretical description of the corner point singularity still survives in the scientific community, as later briefly reported. An opposite trend occurs for the applied Mode II close to free surfaces. These trends are schematically shown in Fig. 2b and the results of finite element simulations of Kotousov et al. (2013) are reported in Fig. 4b. In Fig. 4a the typical local out-of-plane deformation due to in-plane shear loading close to a crack surface front is highlighted; the displacement field shown is deliberately exaggerated. This behaviour and trend of (N)SIFs around cracks and notches have been observed in several works by FE analyses, citing only as example Harding et al. (2010), Berto et al. (2011b), Berto et al. (2012), Kotousov et al. (2013) for sharp and blunt V-shaped notches, Berto et al. 4.1. Mode 0 of Cracks and Notches Under Shear Loading

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(a)

(b)

Fig. 4: Coupled Mode 0 induced by in-plane shear loading. (a) FE example of local out-of-plane mode (Mode 0) generated by Mode II loading. (b) Induced singular Mode 0 (denoted as K C III ( z )) and primary Mode II stress intensity factor (normalized) variation along the thickness and influence of Poisson’s ratio. The weak e ff ect of Poisson’s ratio on the primary Mode II is not shown.

(2011a) for cracked plates.

4.1.1. Scale e ff ect The 3D stress e ff ect zone is located near the crack / notch tip within a radial distance from it equal about to half of the plate thickness, as previously mentioned. After that, the singular stress field can be described by the 2D William’s solution in plane stress conditions; therefore, a K-dominant zone incorporates the 3D a ff ected area. Based on this consideration and extending the definition of NSIF of Mode III of Eq. (13) to Mode 0, the coupled out-of-plane mode (notch) stress intensity factor K N 0 ( z ) must be necessarily related to that remotely applied K ∞ II such that (He et al. (2016)):

0

, ν ,

z h

K N

( λ 2 − λ 0 ) F

0 ( z ) = K ∞ II h

(19)

where h is the plate thickness and F 0 ( z / h , ν ) is a dimensionless function dependent on z and Poisson’s ratio. For the crack case, λ II = λ 0 , the Mode 0 NSIF coincides with the stress intensity factor and it does not depend on the thickness. The strength of the dependence from the plate thickness reaches maximum for 2 α = 104 ◦ (see Fig. 3), where K N 0 ∝ K ∞ II h 0 . 3 . Moreover, taking into account the asymptotic near crack-tip William’s expansion for the stress field applied remotely as boundary condition, the following expression can be obtained:

f 0 n

, ν b n h

∞ n = 0

z h

K N

n ,

0 ( z ) =

(20)

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where b n are the coe ffi cients for the William’s expansion for the in-plane displacements ( u x and u y ) around a crack tip, whereas f 0 n are analogous to F 0 of Eq. (19). Eq. (20) highlights the existence of the coupled Mode 0 even for non-singular ( K ∞ II = 0) in-plane loading, still depending on the thickness.

4.2. Coupled Mode associated with anti-plane loading

The coupling between Mode II and III is not valid only to one direction: anti-plane loading generates in-plane shear singular mode (Mode II) for cracks and notches, as highlighted in Fig. 5a. The boundary conditions negate the transverse shear stresses at the free surfaces, inducing a coupled in-plane shear mode.

(a)

(b)

Fig. 5: Coupled Mode II induced by anti-plane loading. (a) FE example of local in-plane shear mode generated by Mode III loading. Deflections are expressly exaggerated. (b) Induced singular Mode II and primary Mode III stress intensity factor (normalized) variation along z and Poisson’s ratio influence. This latter is shown only for the primary mode because very weak influence is exhibited for the induced mode. Berto et al. (2013a) and Kotousov et al. (2013) investigated this phenomenon on cracks and sharp V-notches, obtaining the trends on the primary and coupled mode (N)SIFs reported in Fig. 5b. As the free surface is approached, the primary SIF K III ( z ) tends to zero, whereas the coupled in-plane SIF K II increases. Moving towards planes near the mid-plane, the induced SIF K c II ( z ) drops down to zero. Indeed, close to the mid-plane the stress field converges to the plane stress condition and the far-field K ∞ III is recovered, while the coupled mode disappears, leading to K c II →− 0. Analogously to the previous coupling, also non-singular anti-plane loading can give rise to singular coupled in-plane mode Kotousov et al. (2013). Besides, in contrast with in-plane loading, whose coupled Mode 0 vanishes for ν = 0, anti-plane loading conditions lead to coupled in-plane shear mode also when ν = 0. The di ff erence arises from the nature of the coupling: the coupled Mode 0 is caused by the transversal deformation due to the Poisson’s ratio, while the coupled in-plane mode is generated by a mechanism of redistribution of the transversal shear stress components close to the free surfaces, where they assume null values, as imposed by the free-stress boundary conditions. 4.2.1. Scale e ff ect It has been noticed by Berto et al. (2013a) that notched plates loaded in Mode III have a deterministic scale e ff ect, which has a similar form of Eq. (19):

, ν ,

II

z h

K c

( λ 3 − λ 2 ) F

II ( z ) = K ∞ III h

(21)

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where F II ( z / h , ν ) is a dimensionless function dependent on z and Poisson’s ratio, although this latter parameter does not have a great influence as for the Mode 0, as well as the local primary mode K II change with ν , in case of in-plane shear loading, is small compared to the correspondent induced mode K 0 . λ 3 corresponds to λ 3 a of Eq. (10), whereas λ 2 is the eigenvalue of the coupled mode, extending the definition of (N)SIF of Eq. (15) to the induced mode. It is worth noticing that the exponent of h is negative for notch opening angles greater than zero, meaning that brittle failure of thinner plates is more influenced by the in-plane coupled mode. For the crack case, the dependence on h disappears. The eigenvalue problem is indeed the same as before; therefore, the strength of singularity of primary and coupled mode is still governed by Eq. (9) and (10), whose behavior is shown in Fig. 3. The 3D analytical frameworks before mentioned are supposed to be valid in the mid-plane of plates, as proven by finite element calculations (Berto et al. (2016)). However, at the intersection of crack / (sharp)notch front with free surfaces a phenomenon occurs, the so called corner point (or vertex) singularity. The analytical formulation still lacks the understanding of this latter phenomenon. Bažant and Estenssoro (1979) were the first authors to introduce the concept of vertex singularities in crack fronts, later also studied by Benthem (1980) and Pook (1994). In the works of Bažant and Estenssoro (1979) and Pook (1994), the stress field near a vertex singularity is described by a stress intensity measure K λ , defined analogously to the (notch) stress intensity factor; nonetheless, spherical coordinates ( r , θ, Φ ) are used instead of polar coordinates, measuring the angle Φ from the crack front. Additionally, no explicit expressions are available for K λ , which is generically related to the characteristic crack dimension a (e.g. crack length) and the remote applied stress σ ∞ through an unknown dimensionless parameter Y , as K λ = Y σ ∞ ( π a ) λ . Stress and displacement field are proportional to K λ / r λ and K λ r 1 − λ , respectively, where r is the radial distance measured from the corner point in spherical coordinates. The strength λ of the corner point singularity has been found to be function of Poisson’s ratio ν and the crack front intersection angle β , shown in Fig. 6a (Pook (1994)); so, these three parameters are related, and knowing two of them, the third is deduced. 5. Corner Point Singularities

(a)

(b)

Fig. 6: (a) Crack front intersection angle. (b) Crack surface intersection angle.

The basic idea of Bažant and Estenssoro (1979) is that, from energetic and other considerations, a crack front tends to intersect the surface at a critical angle β c , such that λ = 0 . 5. Indeed, for small values of β the SIFs tend to zero at the corner point, whereas for large values of β they tend to infinity. Therefore, at the critical angle, SIFs are recovered, i.e. the singularity along the crack front is constant. As reported in the work of Pook (1994), for β < β c , λ < 0 . 5, hence, K I in plane stress conditions tends to zero at the corner point (deduced by considerations on the relation between crack profile and SIF, see Pook (1994)), while for β > β c , λ > 0 . 5 and K I tends to infinity as the vertex singularity is approached. As shown in Fig. 2a from numerical results, K I ( z ) drops down as the corner point is approached, meaning that a strength singularity at the vertex λ < 0 . 5 is suggested, from the previous point of view. Pook (1994) argued that even though the crack front tends to the critical angle, as experimentally proven for bowed crack fronts in plates of constant thickness, the transition between zero and infinity in the behavior of K I at the corner point is not sharp as theoretically predicted, and problems in the experimental measurement of β have been found to be crucial in the validation of the theoretical formulation. Besides, when β = γ = 90 ◦ , with γ defined in Fig. 6b as the crack surface intersection angle, two modes of the stress intensity measure are exhibited: the symmetric mode,

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whose crack tip surface displacements are Mode I, and the antisymmetric mode, which is a combination of Mode II and III. Exploiting of the properties of J-integral in LEFM, recent numerical results of He et al. (2015) have confirmed the predictions of Bažant and Estenssoro (1979) and Benthem (1980) under the hypothesis of β = 90 ◦ for the local Mode I and II SIFs in the vicinity of vertex singularities, i.e. the behavior shown in Fig. 2. Bowed crack fronts on fatigue precracks in fracture toughness test specimens is an example of the e ff ect of corner point singularities, since a propagating crack front shape converges to β c . Despite the application of the analysis of Bažant and Estenssoro (1979) can be used to explain the behaviour of a crack front at a corner point, it needs to be revised. For instance, it predicts that, for a crack front intersection angle of 90 ◦ , K III (therefore also K 0 ) would tend to infinity close to the vertex singularity; however, this does not occur, as shown in Fig. 4, due to the free-transversal-shear-stress boundary conditions at free surfaces. On the other hand, also this statement could be confuted, arguing that Mode III (or Mode 0) is basically torsion, therefore, warp of sections leads to values of τ zy di ff erent from zero. However, this e ff ect does not justify high values (to infinity) as those predicted by Bažant and Estenssoro (1979). We think the question to be addressed would be, Can a unique stress intensity parameter completely describe the intensity of the singular stress field around a crack surface, including the corner point singularity ? Indeed, finite element results of Berto et al. (2017) on cracked plates and discs under anti-plane loading highlighted the non linear behavior of τ yz with the distance from the crack tip ( r ) on a log-log plot, leading to apparent values of K III (computed by the definition of SIF of Eq. (13)) dependent on r (Fig. 1). As a consequence, Berto et al. (2017) argued that the current knowledge seems to suggest that the singular stress field in the vicinity of a corner point may be sum of two singularities of di ff erent strength, not theoretically, numerically or experimentally proven yet. The most representative theoretical and numerical results on 3D e ff ects on LEFM have been provided and dis cussed. 3D analytical frameworks have been briefly presented, highlighting their key strengths and limits. These latter have been discussed referring to the finite element results available in literature, which prove the existence of coupled shear modes, as predicted by the theoretical frame. The good match between the analytical and numerical strength of the singularities has been underlined. At the same time, particular attention to the limit of the analytical formulation of (notch) stress intensity factors for Mode I and II, which does not imply a variation of K I and K II along the thickness, as numerically shown, has been payed. Besides, the theoretical problem of vertex singularities has been introduced, highlighting the main contradictions, and providing an interpretation, which might lead to future numerical, theoreti cal and experimental tests. What appears to be only theoretical speculation, it has actually practical implications. Design against brittle failure and fatigue crack growth of cracked and notched components is based on standards, which in turn are written fol lowing the scientific / engineering research. Lack of 3D applicable results has lead to leave the 2D analytical frame as base for tests and measurements. Coupled singular shear modes and their presence also for non-singular primary modes, and the variation of stress intensity factors along the thickness as well as the thickness scale e ff ect on the induced modes could shift the brittle failure location (from the mid-plane to the lateral surfaces or vice versa) or the expected critical point for fatigue crack initiation. Fatigue of welded lap joints is only a practical example for which these phenomena occur. Moreover, local e ff ects of corner point singularities may contribute to global change in the crack front geometry, which in turn provokes a variation in the stress field far away from free surfaces. We argue that one interesting approach to studying the 3D e ff ects on LEFM seems to be the average strain energy density, which has the ability to take into account the multiaxiality and the coupled modes. In scientific literature, it has been shown that this approach is also able to predict the shift in the failure location due to coupled modes. However, a theoretical breakthrough is needed to deeply understand coupled modes and vertex singularities and their e ff ects on the singular stress field ahead of crack tip surfaces. 6. Conclusions

Appendix A. Out-of-plane shear stress singularity

By means of Eq. (7) and (8), the anti-plane singular shear stresses and the correspondent displacement along the z-axis read as: