PSI - Issue 26

Volume 2 6 • 2020

ISSN 2452-3216

ELSEVIER

1st Mediterranean Conference on Fracture and Structural Integrity, MedFract1

Guest Editors: Stavros Kourkoulis Francesco I acoviello D imos T riantis Fili pp o Berto D imitrios Karelakas Giuse pp e Ferro

Available online al www.sciencedirect.com ScienceDirect

Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2019) 000–000 il l li t . i ir t. i i Structural Integrity Procedia 00 (2019) 000–000

www.elsevier.com/locate/procedia .elsevier.co /locate/procedia

ScienceDirect

Procedia Structural Integrity 26 (2020) 1–2

The 1 st Mediterranean Conference on Fracture and Structural Integrity, MedFract1 Preface Filippo Berto a , Dimitrios Karalekas b , Dimos Triantis c , Giuseppe Ferro d , T st it rr f r r t r tr t r l I t rit , r t ilippo Berto a , i itri r l b , i ri ti c , i rr d ,

Francesco Iacoviello *,e , Stavros Kourkoulis f Norwegian University of Science and Technology, NTNU, Norway a University of Piraeus, Greece b University of West Attica, Greece c Politecnico di Torino, Italy d Università di Cassino e del Lazio Meridionale, DiCeM, Italy e* National Technical University of Athens, Greece f r Ia iell *,e , t r r li f or egian niversity of Science and Technology, T , or ay a niversity of iraeus, reece b niversity of est Attica, reece c olitecnico di Torino, Italy d niversità di assino e del Lazio eridionale, i e , Italy e* ational Technical niversity of thens, reece 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 MedFract1 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 MedFract1 organizers Keywords: Preface, Fracture, Structural Integrity e t rs. lis e lse ier . . T is is a e access article er t e C - - lice se ( tt ://creati ec s. r /lice ses/ - c- / . /) Peer-re ie u er res si ilit f edFract1 organizers Key ords: Preface, Fracture, Structural Integrity 1. Preface The Greek Society of Experimental Mechanics of Materials (GSEMM) and the Italian Group on Fracture (IGF) are, respectively, the Greek and the Italian branches of the European Structural Integrity Society (ESIS). Being both countries embedded in the Mediterranean Sea, and thanks to the warm friendship between the members of GSEMM and IGF, at the end of 2019 the two associations decided to organize together the “1 st Mediterranean Conference on Fracture and Structural Integrity, MedFract1”, (Fig. 1) to be held in February 2020. The dates of the organization of this conference are quite important, as at the beginning of February 2020 the first problems with Covid-19 started to be evident in Europe, especially in Italy. The event was organized in the wonderful Athens, the cradle of European civilization. The event was really successful, and it attracted researchers from numerous countries, in addition to Greece and Italy (about 60 researchers . ref ce e ree ciet f eri e tal ec a ics f aterials ( ) a t e Italia r ract re (I ) are, res ecti el , t e ree a t e Italia ra c es f t e r ea tr ct ral I te rit ciet ( I ). ei t c tries e e e i t e e iterra ea ea, a t a s t t e ar frie s i et ee t e e ers f a I , at the end of t e t ass ciati s eci e t r a ize t et er t e “ st e iterra ea fere ce ract re a tr ct ral I te rit , e ract ”, ( i . ) t e el i e r ar . e ates f t e r a izati f t is c fere ce are ite i rta t, as at t e e i i f e r ar t e first r le s it i - starte t e e i e t i r e, es eciall i Ital . e e e t as r a ize i t e erf l t e s, t e cra le f r ea ci ilizati . e e e t as reall s ccessf l, a it attracte researc ers fr er s c tries, i a iti t reece a Ital (a t researc ers

* Corresponding author. Tel.: +39.07762993681 E-mail address: iacoviello@unicas.it * orresponding author. el.: 39.07762993681 - ail 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 MedFract1 organizers 2452-3216 e t rs. lis e lse ier . . is is a e access article er t e - - lice se ( tt ://creati ec s. r /lice ses/ - c- / . /) er res si ilit f e ract r a izers eer-re ie

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 MedFract1 organizers 10.1016/j.prostr.2020.06.001

Filippo Berto et al. / Procedia Structural Integrity 26 (2020) 1–2

2

2

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

from 15 different countries). The only problem was Covid-19. At the end of February 2020, in the northern Italy all the airports were closed and for many Italian researchers it was simply impossible to join the MedFract1. At the last minute, the Chairmen organized a “blended” conference, partially in presence and partially online. The event was integrally streamed in the IGF Facebook page (https://www.facebook.com/GruppoItalianoFrattura/) and two remote sessions were organized using a commercial platform and allowing participants to remotely join the conference. The experience was really interesting and it was among the first of the thousands of events that in the following months have been organized online. In addition, according to the IGF tradition, all the presentations were video-recorded and they are now available in the IGF YouTube channel (https://www.youtube.com/c/IGFTube). This volume collects the papers of many presentations, both in presence and remote. The number and the quality of the papers confirm that the fracture and structural integrity community is really active. The Editors would like to express their sincere thanks to the Authors who contributed to this volume and, also, to Dr Ermioni Pasiou for her assistance in the final formatting of the manuscript.

Fig. 1. MedFract1 banner.

Available online at www.sciencedirect.com Available online at www.sciencedirect.com Available online at www.sciencedirect.com

ScienceDirect

Procedia Structural Integrity 26 (2020) 336–347 Structural Integrity Procedia 00 (2019) 000–000 Structural Integrity Procedia 00 (2019) 000–000

www.elsevier.com / locate / procedia www.elsevier.com / locate / procedia

© 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 MedFract1 organizers Abstract Plane elasticity has allowed the development of the current knowledge on fracture mechanics since the dawn of this discipline. Linear elastic fracture mechanics (LEFM) has taken advantage of the possibility to obtain analytical closed-form solutions. Indeed, three dimensional e ff ects, related basically to the Poisson’s ratio and the plate thickness, have been mostly neglected for a long time. The proof of the existence of corner point singularities, i.e. stress singularity at the intersection of crack front with a free surface, has contributed to explain fracture phenomena, such as the bowed crack fronts and the behavior of stress intensity factors along the thickness of a cracked / notched body. Although fatigue crack growth is Mode I dominated and corner point singularities become insignificant, for the estimation of brittle failure of cracked / notched they may represent a substantial source of error, both in terms of magnitude of the stress field and failure location. In this paper we provide before a brief overview on 3D corner point singularities. By exploiting of 3D finite element (FE) modeling, to understand the theoretical basis of this phenomenon, numerical experiments on sharply-V-notched linear elastic plates under mode 1 loading are performed. A quantitative and qualitative description of the causes and e ff ects of 3D vertex singularities under mode 1 in relation with Poisson’s ratio is given, including negative values of this latter material parameter, never before investigated. Besides, by means of numerical simulations of the same notched plates subjected to mode 3 loading, observations on the possible connection between 3D corner point singularities and the out-of-plane shear stress distribution near the model surface are discussed. Finally, future perspectives on this research field are addressed. 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 / ) P er-review under respons bility of MedFract1 organizers. Keywords: Fracture Mechanics; 3D e ff ects; corner point singularities; stress singularities; stress intensity factors The 1 st Mediterranean Conference on Fracture and Structural Integrity, MedFract1 3D e ff ects on Fracture echanics: corner point singularities Marco Maurizi a, ∗ , Filippo Berto a a NTNU - Norwegian University of Science and Technology –Department of Mechanical Engineering, 7491 Trondheim, Norway Abstract Plane elasticity has allowed the development of the current knowledge on fracture mechanics since the dawn of this discipline. Linear elastic fracture mechanics (LEFM) has taken advantage of the possibility to obtain analytical closed-form solutions. Indeed, three dimensional e ff ects, related basically to the Poisson’s ratio and the plate thickness, have been mostly neglected for a long time. The proof of the existence of corner point singularities, i.e. stress singularity at the intersection of crack front with a free surface, has contributed to explain fracture phenomena, such as the bowed crack fronts and the behavior of stress intensity factors along the thickness of a cracked / notched body. Although fatigue crack growth is Mode I dominated and corner point singularities become insignificant, for the estimation of brittle failure of cracked / notched they may represent a substantial source of error, both in terms of magnitude of the stress field and failure location. In this paper we provide before a brief overview on 3D corner point singularities. By exploiting of 3D finite element (FE) modeling, to understand the theoretical basis of this phenomenon, numerical experiments on sharply-V-notched linear elastic plates under mode 1 loading are performed. A quantitative and qualitative description of the causes and e ff ects of 3D vertex singularities under mode 1 in relation with Poisson’s ratio is given, including negative values of this latter material parameter, never before investigated. Besides, by means of numerical simulations of the same notched plates subjected to mode 3 loading, observations on the possible connection between 3D corner point singularities and the out-of-plane shear stress distribution near the model surface are discussed. Finally, future perspectives on this research field are addressed. 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 under responsibility of MedFract1 organizers. Keywords: Fracture Mechanics; 3D e ff ects; corner point singularities; stress singularities; stress intensity factors The 1 st Mediterranean Conference on Fracture and Structural Integrity, MedFract1 3D e ff ects on Fracture Mechanics: corner point singularities Marco Maurizi a, ∗ , Filippo Berto a a NTNU - Norwegian University of Science and Technology –Department of Mechanical Engineering, 7491 Trondheim, Norway

Nomenclature Nomenclature

x , y , z Cartesian coordinates r , θ , z cylindrical / polar coordinates σ i j i , j − th stress component 2 h plate thickness β crack front intersection angle x , y , z Cartesian coordinates r , θ , z cylindrical / polar coordinates σ i j i , j − th stress component 2 h plate thickness β crack front intersection angle

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 MedFract1 organizers 10.1016/j.prostr.2020.06.043 ∗ 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 under responsibility of MedFract1 organizers. ∗ 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 under responsibility of MedFract1 organizers.

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

337

2

γ supplementary V-notch opening angle and crack surface intersection angle 2 α V-notch opening angle u displacement field, subscripts r , θ , z denote directions λ stress singularity, subscript indicating the mode of loading G shear modulus ν Poisson’s ratio f i j angular functions for stress field K i mode i (notch) stress intensity factor (N)SIF; superscript ∞ for far-field (N)SIF

1. Introduction

In a previous recent our paper Maurizi and Berto (2020), we reviewed the state of the art on three dimensional e ff ects on Fracture Mechanics. Specifically, the main numerical results regarding the in-plane and out-of-plane shear coupled modes were described and contextualized in the various analytical frameworks so far developed. Moreover, 3D corner point singularities were briefly introduced, without an in-depth-analysis. To the best of our knowledge, Benthem (1977, 1980) was the first author to analytically prove the existence of a particular state of stress at the vertex, i.e. at the intersection of the crack front with a surface (from here the term surface singularity ), of a quarter infinite crack in a half-space. The author solved an eigenvalue problem in the linear elastic fracture mechanics (LEFM) framework, assuming the existence of a state of stress as σ i j = r λ − 1 f i j ( λ, θ, Φ ) around the vertex of a crack in spherical coordinates ( r , θ , Φ ), whose infinite and enumerable eigenvalues and eigenfunctions were the singularity exponents λ − 1 and stress functions, respectively; obviously, the displacements will behave like r λ . Analogously to Benthem (1977, 1980), Bažant and Estenssoro (1979) based their analysis on the assumption of separation of variables on the stress and displacement functions, meaning that at di ff erent angles, identified by the couple ( θ, Φ ), the behavior is invariant with respect to r . From the strain energy, they derived a variational principle, proving how this latter, together with the eigenvalue problem, must be nonsymmetric, leading to complex eigenvalues. Their main finding was that to establish a numerical relation between the stress corner singularity (1 − λ ), the Poisson’s ratio ν , the crack surface angle γ and the crack front angle β (Fig. 1).

Fig. 1: (a) Crack front intersection angle. (b) Crack surface intersection angle.

Expressing the energy flux which flows into the crack front edge by means of the Rice’s J-integral, they deduced that the condition of propagating crack is Re ( λ ) = 1 / 2, which is the classical 2D stress singularity. However, from the numerical relationship between the angles in Fig. 1 and λ , these authors have also shown that the crack front edge angle ( β ) of a propagating crack must be oblique, whose amplitude depends on the crack surface angle ( γ ). Therefore, the shape of a propagating crack front is adjusted such that the classical square root singularity is held, intersecting the surface at a critical angle β c . This latter was empirically quantified by Pook (1994), linking it to the Poisson’s ratio for every mode of fracture, assuming λ = 1 / 2. Numerical and experimental studies, such as that of Heyder et al. (2005) confirmed this continuous process of adjusting β to keep constant the stress singularity during a crack propagation. Nonetheless, for the fracture assessment of brittle materials knowing the behavior of the stress singularity and (notch)

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

338

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

3

stress intensity factor (SIF or NSIF), i.e. the stress field, along the crack / notch front edge, including the corner point, is needed; specifically, the failure location can change as these parameters vary. Assuming the values of the angles in Fig. 1 be equal to 90 ◦ , a few researchers, after Benthem (1977, 1980) and Bažant and Estenssoro (1979), have studied the 3D corner point singularities for stationary cracks. Shivakumar and Raju (1990) performed simulations by using 3D finite element models of middle-crack tension and bend specimens under mode 1 loading, studying the variation of the stress singularity along the crack front. These authors have shown that the near-tip stress field component can be expressed as σ yy = C ( θ, z ) r − 1 / 2 + D ( θ, Φ ) r λ 1 C − 1 , being sum of two components. The first one represents the classical stress field in polar coordinates ( r , θ , z ), vanishing at the corner point, whereas the second component identifies the vertex singularity, which dominates the field at the surface, and it goes to zero away from a region called boundary layer. This latter has a thickness depending on the Poisson’s ratio, reaching the maximum value at about 5 % of the specimen thickness for ν = 0 . 45. Despite the small region size of influence, the corner point singularity is able to modify the (N)SIF along the crack / notch front, moving therefore the failure location. With the goal of determining the e ff ect of the 3D vertex singularity and the Poisson’s ratio (despite they are strictly related to each other) on the plasticity-induced fatigue crack closure, De Matos and Nowell (2008) studied the stress singularity and SIF change along the crack front of a cracked plate under mode 1 through a log-log linear regression analysis of the near-tip stress and displacement fields and by means of the J-integral, obtained by three-dimensional finite element simulations. Although e ff ort has been put on the identification of the e ff ects of 3D corner point singularities on stationary cracks, a lack of studies on their inherent causes, extending the concept to sharp V-notches (highlighted only by Bažant and Estenssoro (1979); Kotousov (2010))), seen as the theoretical generalization of cracks Williams (1952), is evident. In the present paper, we address the problem of determining quantitatively and qualitatively the causes of 3D corner point singularities on sharp V-notches loaded in mode 1 in the framework of LEFM, through three-dimensional accurate finite element simulations. Moreover, Poisson’s ratio values from -0.3 to 0.45 are considered, investigating the e ff ect of a negative ν compared to the positive counterpart. A clear representation of the 3D vertex singularity in terms of notch stress intensity factor is hence provided. Finally, a few observations on the e ff ect of the surface singularity on sharp V-notches under anti-plane loading are given, discussing previous explanations provided by some authors for cracks. To make the work clear and repeatable, we report in Fig. 2 the adopted notation for the stress field near the sharp V-notch tip and the relative geometrical features.

Fig. 2: (a) Notation for the stress field close to the V-notch and corner point in cylindrical coordinates ( r , θ , z ). Cartesian local coordinate system ( x , y , z ) centered at the V-notch tip on the mid-plane. (b) Geometrical features of the sharp V-notch.

2. Numerical experiments

In the theoretical framework of LEFM, to understand the causes and e ff ects of 3D corner point singularities on the stress state in the region close to the intersection between the sharp V-notch front and the surface (Fig. 3b), numerical experiments were performed by exploiting of three-dimensional accurate finite element models. Regarding to Fig. 3, in a generic body for which the hypotheses of LEFM hold, i.e. small scale plasticity in the process region, the 3D stress state region, whose size was quantified (Harding et al. (2010); Nakamura and Parks (1989) to be ∼ h , being 2 h the plate thickness, only for crack / notch under in-plane shear loading, is assumed to be included in a K-dominance

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

339

4

zone, where the conditions of plane stress are completely recovered at su ffi cient distance ( ∼ 2 h ); this hypothesis can be later verified by analyzing the results.

Fig. 3: (a) Representation of the various regions with the correspondent stress state near a crack / notch tip. (b) 3D finite element model of a sharp V-notch surrounded by a cylindrical region. The local Cartesian coordinate system ( x , y , z ) is centered at the notch tip on the mid-plane, as reported specifically in Fig. 2a.

As shown in Fig. 3b, a 3D FE model of a cylindrical portion of a sharply-V-notched plate of thickness (2 h ) 20 mm and radius R = h was built by using the commercial FE software Abaqus / Standard, through Python language. The model consisted of 15-node wedge elements (C3D15 in Abaqus) around the V-notch tip, surrounded by 20-node hexahedral (brick) elements (C3D20 in Abaqus) along the radial direction, whose biggest element size was 2.1 mm on the outer boundary of the cylinder, progressively reduced towards the notch tip down to 0.0001 mm (the nearest node at 0.00005 mm from the notch tip). The supplementary V-notch angle γ (Fig. 2b) was divided in 16 parts, whereas the plate thickness was meshed with weighted divisions from the mid-plane, whose first element size was h / 10, i.e. 1 mm, toward the external surface, with the smallest element size being 0.01 mm. This mesh resulted in ∼ 120000 elements and ∼ 500000 nodes, slightly varying with the angle 2 α (Fig. 2b). Due to the symmetry of the problem with respect to the mid-plane under Mode 1, only one half of the cylinder was modeled, as evident in Fig. 3b. Consequently, symmetric displacement boundary conditions were applied on the nodes of the mid-plane ( z = 0), constraining the displacements along z (coordinate system in Fig. 2a). Moreover, assuming that the conditions of plane stress are attained at distance equal to half of the thickness from the notch tip, a plane stress far-field displacement (Eq. (1a)-(1b)) under mode 1 was imposed to the outer periphery of the cylinder. This displacement field, i.e. the components u r and u θ (coordinate system Fig. 2a), can be obtained as a function of the asymptotic plane stress NSIF ( K ∞ 1 ) under mode 1 by simply integrating for separation of variables the constitutive equation (Lamè Equation) in terms of displacements, and knowing the expression of the first singular term of the asymptotic expansion of the stress field around a V-notch (see Lazzarin and Tovo (1998)), as follows:

f rr ( θ, λ 1 , χ 1 ) − ν f θθ ( θ, λ 1 , χ 1 ) λ 1

r λ 1 K ∞ 1

1 E √ 2 π 1 E √ 2 π

u r ( r , θ ) =

(1a)

(1 + λ 1 ) + χ 1 (1 − λ 1 ) r λ 1 K ∞ 1 (1 + λ 1 ) + χ 1 (1 − λ 1 ) 1 + ν

f θθ ( θ, λ 1 , χ 1 ) d θ − ν +

f rr ( θ, λ 1 , χ 1 ) d θ , (1b)

λ 1

1 λ 1

θ

θ

u θ ( r , θ ) =

0

0

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

340

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

5

where E is the Young’s modulus, λ 1 is the eigenvalue of the 2D stress singularity under mode 1, obtained from sin ( λ 1 2 γ ) + λ 1 sin (2 γ ) = 0 (Williams (1952)), χ 1 = − sin [(1 − λ 1 ) γ ] / sin [(1 + λ 1 ) γ ] (Lazzarin and Tovo (1998)), whereas f rr ( θ, λ 1 , χ 1 ) and f θθ ( θ, λ 1 , χ 1 ) are elementary trigonometric functions, which appear in the expressions for the stress components σ rr and σ θθ (Lazzarin and Tovo (1998)), respectively, reported for the sake of clarity in the Ap pendix (Eq. (A.1)-(A.2)). By imposing unitary plane stress NSIF ( K ∞ 1 = 1 MPa mm 1 − λ 1 ) in Eq. (1a)-(1b), numerical simulations (experiments) under mode 1 were performed varying the Poisson’s ratio in the range -0.3 to 0.45, assumed to be the direct cause of vertex singularities, for 30 ◦ and 60 ◦ of V-notch opening angle 2 α . Besides, a Young’s modulus of 200 GPa was arbitrarily adopted, without loss of generality. To provide some observations on the e ff ects of the 3D corner point singularity on the out-of-plane shear stress com ponent σ yz in the proximity of V-notches / cracks under anti-plane loading, a few numerical simulations were carried out by exploiting of the FE model (Fig. 3b), setting material constant and geometrical parameters to E = 200 GPa, ν = 0 . 3, and 2 α = 45 ◦ , respectively. By virtue of the anti-symmetry of the problem, the displacements u x and u y of the nodes at the mid-plane were fully constrained. Moreover, analogously to Eq. (1a)-(1b), the plane stress asymptotic anti-plane displacement field, reported in Eq. (2a)-(2c), was applied to the outer boundary of the cylindrical plate:

2 K ∞ 3 G √ 2 π

r λ 3 sin ( λ

3 θ )

(2a)

u z ( r , θ ) =

u r = 0 u θ = 0 ,

(2b) (2c)

where u z is the displacement component along z (coordinate system Fig. 2a), K ∞ 3 is the far-field mode 3 NSIF, imposed to be equal to 1 MPa mm 1 − λ 3 , while G is the shear modulus, which can be expressed due to the isotropy as G = E / 2(1 + ν ). The eigenvalue λ 3 for a sharp V-notch under anti-plane loading can be found as solution of the simple equation cos ( λ 3 γ ) = 0 (see Berto et al. (2013)). Besides, two orders of magnitude, i.e. 2 and 20 mm, for the plate thickness were tested to investigate possible e ff ects on the out-of-plane shear stress component distribution on the plate surface. Hypothesizing that the i j -th stress component, σ yy for mode 1 or σ yz for mode 3, around the notch front in a 3D problem can be expressed in cylindrical coordinates (Fig. 2a) as follows:

1 √ 2 π

K ( z , 2 α, ν ) r λ ( z , 2 α,ν ) − 1 f

σ i j ( r , θ, z ) =

i j ( θ, λ, 2 α ) ,

(3)

where the NSIF K ( z , 2 α, ν ) depends on the geometry, i.e. notch opening angle 2 α , on the type of loading and boundary conditions, and it is assumed it varies along the notch front ( z ) and with the Poisson’s ratio. The power-law exponent is hypothesized to change with z , 2 α and ν . The unknown function f i j performs the same role of the angular functions of Eq. (1a)-(1b). The singularity exponent, i.e. λ − 1, and the NSIF, i.e. K are hence su ffi cient to characterize the stress field, which is influenced near the corner point by the vertex singularity. The stress components, i.e. σ yy for mode 1 or σ yz for mode 3, were extracted along the bisector line from the notch tip up to a distance equal to R / 10, for di ff erent z -coordinates, i.e. planes z = constant. By using a log-log regression analysis, under the hypothesis of Eq. (3), λ − 1 was determined as the slope of the line log( σ i j ) versus log( r ), whereas the intensity factor by the following classic definition:

K ( z , ν ) = √ 2 π lim r → 0

1 − λ ,

σ i j r

(4)

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

341

6

performing the average of the obtained values along the distance from the notch tip r , in the range 0 to R / 10. The stress component to use in Eq. (4) depends on the mode of loading: yy -component for mode 1, and yz -component for anti-plane mode. It is worth noting that the stress components were extrapolated from integration points to nodes along the bisector line, giving rise to further numerical approximations. The results were therefore confirmed by performing the log-log regression also on the displacement field, which behaves like r λ and it is directly extracted at the nodes. This latter procedure resulted to be useful at the plate surface, where the corner point singularity influence might create numerical errors, as mentioned.

3. Results and Discussion

After mesh convergence tests on the FE model of Fig. 3b, the numerical simulations under mode 1 allowed to characterize the near-notch-front stress field and to understand the mechanisms behind the causes of the 3D corner point singularity and its e ff ect on brittle failure of materials subjected to in-plane tensile loading. The simulations of the plate (Fig. 3b) subjected to anti-plane loading (mode 3) were performed to observe the vertex singularity e ff ect on the out-of-plane shear stress component.

3.1. In-plane Mode 1 loading

In Fig. 4a the generic stress component σ yy as a function of the notch front distance r , along distinct planes z / h = constant , is shown; whereas, in Fig. 4b the same variable is plot along the notch front at three di ff erent radii.

Fig. 4: Stress component on a sharply-notched plate with thickness 2 h = 20 mm, notch opening angle 2 α = 60 ◦ and Poisson’s ratio ν = 0 . 3, under mode 1. (a) Stress component σ y y along the bisector line vs. distance from the notch tip, for di ff erent z -planes. (b) Stress component σ y y along the notch front at three distinct radii. Unambiguously the stress σ yy intensity drops down as the free surface is approached, in agreement with what reported by Kotousov (2010) for the NSIF, with higher gradient as the notch front distance decreases (Fig. 4b). In Fig. 4a a slight bending of the line log( σ yy ) vs. log( r ) at the free surface occurs, probably due to the numerical approximation before mentioned. The stress singularity, i.e. the power-law exponent 1 − λ 1 of Eq. (3), for the opening angles 2 α = 30 ◦ and 60 ◦ , is plotted along the notch front ( z / h ), highlighting the e ff ect of the Poisson’s ratio, in Fig. 5. The range 0 . 8 < z / h < 1, in the proximity of the free surface, in Fig. 5 was selected to highlight the singularity behaviour due to the corner point, while the upper value of the Poisson’s ratio range was limited to 0.3 for the opening angle of 30 ◦ (extended plot in Fig. A.9 for 2 α = 60 ◦ ) due to numerical "instabilities" of the computed stress near the free surface due probably to the stronger e ff ect of the vertex singularity (which reflects on the extrapolations from the integration points of the FE model), symptom of a powerful Poisson’s ratio influence. The asymptotic trend of 1 − λ 1 towards the 2D eigenvalue solution (Williams (1952)) for z / h < 0 . 8 is clearly visible (values down to z / h = 0 not shown for clarity) for both the notch opening angles, recovering at the mid-plane the 2D singularity of the stress

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

342

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

7

Fig. 5: Stress singularity under mode 1 along the notch front, varying the notch opening angle and the Poisson’s ratio. (a) Stress singularity 1 − λ 1 vs. z / h , notch opening angle 2 α = 30 ◦ . (b) Stress singularity 1 − λ 1 vs. z / h , notch opening angle 2 α = 60 ◦ ; extended plot including nu = 0 . 45 is reported in Fig. A.9. (a) and (b) show a zoom in the range 0 . 8 < z / h < 1.

field corresponding to the applied (plane stress) displacement field of Eq. (1aa-b), hence confirming the correctness of the assumption R = h . The first notable trend corresponds to ν = 0; in fact, for both the opening angles the stress singularity remains constant along the notch front, corresponding to the 2D stress solution (Williams (1952)). As the Poisson’s ratio becomes positive, 1 − λ 1 starts increasing (relative di ff erence greater than 0.1 %) at about z / h = 0 . 95 − 0 . 90 for ν = 0 . 1 and 0.3, respectively, i.e. at ∼ 2 . 5 − 5 % of the plate thickness (2 h ) of distance from the free surface, reaching a maximum just before ( < 1 % of the plate thickness) the free surface and falling to values smaller than the far-field singularity at z / h = 1. This e ff ect, i.e. maximum and minimum in the proximity of the free surface, is amplified as the Poisson’s ratio increases, as also evident from Fig. A.9, in agreement with previous results for cracks by Shivakumar and Raju (1990); De Matos and Nowell (2008). The region where the stress singularity at the crack front deviates from the far-field 2D solution and oscillates was called boundary layer zone by Shivakumar and Raju (1990), and it represents a small area close to the free surface, whose size (between 0 and 5 % of the plate thickness) depends on ν , where Eq. (3), applied to cracks and with λ 1 not being function of z , might loose its meaning, and a sum of two singularities, i.e. the asymptotic and vertex singularity, considered instead. The mentioned authors proved that inside such region the separation of variables assumption fails, meaning that the stress singularity and the displacement power-law exponent, computed by the log-log regression analysis of σ yy and u y , respectively, depend on the angle θ . However, at the inner and outer boundary of this region, i.e. at the beginning of the deviation from the far-field value and at the free surface, the single power-law equation was demonstrated (Shivakumar and Raju (1990); De Matos and Nowell (2008)) to be suitable to describe the near-crack-front stress field. Despite the evidence that shows the limit of a single power-law model for cracks, a clear proof of the suggested model (sum of two singularities) is still not available. Extending the reasoning also to sharp v-notches, object of the present work, and for lack of other information, we limited our analysis to assuming that Eq. (3) as well as the separation of variables is valid at the extreme of the boundary layer, as done by Shivakumar and Raju (1990) and De Matos and Nowell (2008); further investigations are left for future works. On the other hand, merely as a speculative approach, it could be argued that Eq. (3) may be corrected by including the dependence of λ 1 on θ , without having anyway a physical proof of the validity. The stress singularity seems to have an opposite trend for negative values of Poisson’s ratio. Indeed, 1 − λ 1 has a minimum just before the free surface, where it suddenly goes up. A weaker change of the singularity at the free surface with respect to the far-field solution, compared to that corresponding to the positive Poisson’s ratio counterpart (e.g. ν = − 0 . 3 and ν = 0 . 3), appears to be evident for both the opening angles. Fig. 6a can help to deeply understand this latter consideration and to highlight the 3D corner point singularity dependence on the Poisson’s ratio for sharp V-notches. The vertex singularity 1 − λ 1 c was obtained from the slope of the near-notch-front displacement component u y at the free surface in the log( u y )-log( r ) space (see Fig. A.10), evaluated at θ = γ (Fig. 2b for notation). Since the displace-

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

343

8

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

Fig. 6: (a) 3D corner point stress singularity vs. Poisson’s ratio ( ν ) for notch opening angles of 30 ◦ and 60 ◦ , and 24 ◦ and 96 ◦ extracted from the work of Bažant and Estenssoro (1979), indicated as [Previous work] in the plot. (b) Size of non-uniform longitudinal strain ( ε zz ) in terms of % of the plate thickness (2 / h ). The inset shows the strain component ε zz around the corner point for 2 α = 60 ◦ and ν = 0 . 3, highlighting the region aroud the corner point where the non-uniformity occurs.

ments are directly computed at the nodes, their magnitude is not a ff ected by the extrapolation from the integration points, as for the stresses, whose values at the free surface resulted to be abundantly distorted. Therefore, the slope of u y at the free surface as determined is the 3D corner point singularity. This latter strongly depends on the Poisson’s ratio, especially for positive values. In fact, as this material parameter decreases, down to ν = − 0 . 3, the gradient of 1 − λ 1 c does so. To have a benchmark, values extracted from the work of Bažant and Estenssoro (1979) are plotted in Fig. 6a. Despite di ff erent values of notch angle are available in literature, Fig. 6a shows that the singularity monoton ically decreases for rising values of Poisson’s ratio, in a similar fashion, also in agreement with the results obtained by Bažant and Estenssoro (1979); Benthem (1980) for cracks under mode 1, collected by Pook (1994). Moreover, looking for example at the data for ν = 0 . 3, the nonlinear increment of the singularity as the angle changes stands out. Catastrophic brittle failure of materials caused by stress raisers, such as notches, occurs suddenly when a certain ex ternal load limit is reached, provoking a crack initiation and rapid propagation. Therefore, neglecting the conditions of crack propagation, which necessarily force one to taking into account the e ff ects of the intersection angles (Fig. 1), the only parameter involved in 3D corner point singularities is the Poisson’s ratio. This latter material constant can hence be considered the cause behind the mechanism of vertex singularity, at least under mode 1. The inset in Fig. 6b features the contour plot of the strain component ε zz around the corner point, which unambiguously produces a non-uniform region in terms of lateral contraction (in z -direction, Fig. 2a) of the plate, undermining the hypothe sis of generalized plane strain (Lazzarin and Zappalorto (2012)), almost always adopted to formulate 3D analytical closed solutions for elastic problems. The depth of such region along the notch front was deduced by identifying the z -coordinate at the notch tip for which the function ε zz ( z ) started changing, compared to the uniform far-field value (at the mid-plane). The estimation of the relative non-uniform zone size with respect to the plate thickness, displayed in Fig. 6b, steeply rises for positive values of ν , remaining nearly 0 % in the negative range of Poisson’s ratio, leading to an asymmetric behavior, also confirmed by the singularity at the vertex (Fig. 6). The region of non uniformity can be considered as the boundary layer previously defined; in fact, the order of magnitude for positive Poisson’s ratio resembles that obtained by Shivakumar and Raju (1990); De Matos and Nowell (2008). It is further worth noting that at the higher opening angle, 60 ◦ , corresponds a greater region size, especially at ν = 0 . 45, and at ν = 0 the non uniformity disappears, confirming the Poisson’s ratio-leading nature of 3D corner point singularity around V-notches under mode 1. Finally, the normalized NSIF ( K I / K ∞ I ), whose critical value activates the brittle fracture, is featured in Fig. 7. Whilst inside the boundary layer region the validity of the single power-law is still unclear, computing the NSIF by means of Eq. (4) provides interesting information. The trend of σ yy for ν = 0 . 3 in Fig. 4b demonstrates a continuous decreasing in the proximity of the free surface for every radius r ; the opposite occurs for negative values (not reported here for brevity). Therefore, if at the extremes of the boundary layer region the model represented by Eq. (3) is

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

344

9

Fig. 7: Normalized notch stress intensity factor under mode 1 along the notch front, varying the notch opening angle and the Poisson’s ratio. (a) Normalized notch stress intensity factor K I / K ∞ I vs. z / h , notch opening angle 2 α = 30 ◦ . (b) Normalized notch stress intensity factor K I / K ∞ I vs. z / h , notch opening angle 2 α = 60 ◦ .

correct, together with the previous observation, the estimate of the normalized NSIF can be considered su ffi ciently representative of the real trend. For ν = 0, the NSIF is constant along the notch front, as expected, assuming a value slightly smaller than K ∞ I due to the asymptotic character of the plane stress region, hypothesized to be completely developed at r = h (to save computational time); nonetheless, the behavior and the relative change of K I as a function of ν were una ff ected. The typical increase of the NSIF over the 2D value toward the mid-plane (Pook (2013)), i.e. for z / h < 0 . 5, is clear, especially for ν = − 0 . 3, 0 . 3 and 0 . 45. It can further be noted that for positive values of Poisson’s ratio, the intensity drops near the free surface, whereas for ν = − 0 . 1 and − 0 . 3 it rises. Such observations could explain for example why the crack initiation of sharply notched plates, with typical positive Poisson’s ratio, loaded in mode 1 is located around the mid-plane, where the hypothesis of plane elasticity represents a good approximation, neglecting the slight increment of the 3D solution. The opposite trend exhibited at negative values of ν might reveal a new phenomenon: auxetic notched materials (Greaves et al. (2011)) subjected to in-plane tensile loading might fail locally to the free surface, where the NSIF tends apparently to infinity or simply to a maximum. Future experimental confirmations are anyway needed. The thickness was considered a constant parameter. However, being the relation between 3D corner point singularities and plate thickness still unclear (Kotousov (2010)), a deeper research focused only on the e ff ect of the thickness may be necessary in the future. Additionally, it must be noted that the normalized NSIF in Fig. 7 is not formally adimensional close to the corner point. In fact, although K ∞ I has physical dimensions [ F ][ L ] − 2 [ L ] 1 − λ ∞ 1 , the corresponding ones for K I are [ F ][ L ] − 2 [ L ] 1 − λ 1 , where [ F ] and [ L ] represents the dimensions of force and length, respectively. The dimensional contribution is provided by the di ff erence between λ 1 , the 3D stress singularity at the coordinate z , and λ ∞ 1 , the asymptotic plane stress singularity. Recent works, such as Pook et al. (2015) and Berto et al. (2017), have dealt with an apparent paradox related to the 3D stress field around cracks loaded under mode 3. They observed that the stress intensity factor as well as the out-of-plane shear stress ( τ yz , Fig. 2a for coordinate system) at the free surface of cracked plates and discs is not zero, as the boundary conditions, i.e. external surface loads equal to zero, impose. In fact, evaluating the SIF along the crack front at fixed distance from it, the SIF seems tending to zero near the free surface; however, its distribution at the free surface as a function of r suggests an increase as the crack front is approached. This contradiction is in part corroborated by Bažant and Estenssoro (1979), who predicted the mode 3 SIF ( K 3 ) to tend to infinity in the proximity of the free surface. In Fig. 8a the out-of-plane shear stress component ( τ yz ) as a function of the distance from the notch front in a log-log plot is featured for a set of planes at z / h = constant (codified by scale of colors). 3.2. Out-of-plane Mode 3 loading

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

345

10

Fig. 8: Out-of-plane shear stress ( τ yz ) distributions in the proximity of sharp-V-notches with opening angle 2 α = 45 ◦ loaded under mode 3, as function of the distance from the notch front ( r ). (a) Distribution along di ff erent planes z / h = constant in the range 0 to 1. Plate thickness equal to 20 mm. (b) Distribution along a 1 MPa shear-loaded surface. Plate thickness equal to 2 mm.

As obtained by Pook et al. (2015); Berto et al. (2017) for cracks, for sharp V-notches an analogous e ff ect on the τ yz distribution close to the free surface was found. As the free surface is approached, the stress component distribution is deflected from the straight line, i.e. the single power-law (as Eq. (3)). Besides, the distance from the notch front at which the deviation occurs is smaller as the distance from the free surface is reduced. We postulate that this e ff ect is caused by the 3D vertex singularity, in agreement with the results of the previous studies. However, further confirmations are needed. The core of the apparent paradox lies on the boundary conditions at the model surface, which is usually traction-free. However, this latter hypothesis, adopted by Benthem (1977); Bažant and Estenssoro (1979); Benthem (1980) in the analytical framework of self-equilibrating states of stress (eigenvalue problem), is not necessary so that the surface singularity can a ff ect the out-of-plane shear stress distribution, as shown in Fig. 8b. It was indeed obtained as a result of a numerical simulation, in which a shear surface force field along the y − direction of intensity equal to 1 MPa was applied to the model surface. It is hence proved that the 3D vertex singularity still influences the anti-plane shear stress in a region close to the notch front, even if the surface is not traction-free. The present work was conducted to understand the generative mechanism of 3D vertex singularities and their e ff ects on sharply-V-notched plates loaded under mode 1 as well as to highlight the presence of the apparent paradox on the out-of-plane shear stress component under mode 3. What is already known for cracks, was also underlined for sharp V-notches subjected to in-plane tensile loading, highlighting that the Poisson’s ratio is the fundamental parameter which causes a non uniform transversal strain region around the corner point, therefore confuting the usual assumption of generalized plane strain in the current available 3D analytical frameworks. These latter approaches cannot in fact predict neither the stress intensity factor variation (only a linear change of K 3 ) along the crack / notch front nor the 3D vertex singularity, which have been proven in this work to be strictly related to each other for sharply V-notched plates under mode 1. Therefore, a new analytical breakthrough for 3D cracked / notched plates seems to be needed. A new e ff ect on auxetic materials was further observed: numerical simulations of sharply-V-notched plates with negative values of Poisson’s ratio loaded under mode 1 revealed an opposite e ff ect both on the stress singularity and on the notch stress intensity factor. This latter was seen to increase in the proximity of the free surface, predicting a possible change in failure location for brittle auxetic materials compared to the classic ones. Despite auxetic materials are not nowadays widespread, inconceivable future applications, where brittle fracture represents the main mechanism of failure, might take advantage of the understanding previously developed. 4. Conclusions

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

346

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

11

In light of the results of the present paper and the previous studies, a greater future e ff ort in developing new analytical and empirical solutions, and further experimental verifications is hence justified.

Appendix A.

The angular functions of Eq. 1a can be found in Lazzarin and Tovo (1998) as:

f rr ( θ, λ 1 , χ 1 ) = (3 − λ 1 ) cos [(1 − λ 1 ) θ ] − χ 1 (1 − λ 1 ) cos [(1 + λ 1 ) θ ] f θθ ( θ, λ 1 , χ 1 ) = (1 + λ 1 ) cos [(1 − λ 1 ) θ ] + χ 1 (1 − λ 1 ) cos [(1 + λ 1 ) θ ] .

(A.1) (A.2)

Since the integrals of Eq. A.1-A.2, appearing in Eq. 1b, can be trivially computed, they are not reported here for brevity. In Fig. A.9 the stress singularity 1 − λ 1 along the notch front for the angle 2 α = 60 ◦ is reported, including also the values for ν = 0 . 45.

Fig. A.9: Stress singularity 1 − λ 1 vs. z / h , notch opening angle 2 α = 60 ◦ , including the values for ν = 0 . 45. The displacement component u y along the V-notch flank, i.e. θ = γ ( γ in Fig. 2b), for both the opening angles is reported in Fig. A.10.

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

347

12

Fig. A.10: Displacement component u y vs. distance from the notch front r in a double logarithmic space, along the V-notch flank on the free surface, for di ff erent values of Poisson’s ratio. (a) Notch opening angle 2 α = 30 ◦ . (b) Notch opening angle 2 α = 60 ◦ .

References

Bažant, Z.P., Estenssoro, L.F., 1979. Surface singularity and crack propagation. International Journal of Solids and Structures 15, 405–426. Benthem, J., 1977. State of stress at the vertex of a quarter-infinite crack in a half-space. International Journal of Solids and Structures 13, 479–492. Benthem, J., 1980. The quarter-infinite crack in a half space; alternative and additional solutions. International Journal of Solids and Structures 16, 119–130. Berto, F., Kotousov, A., Lazzarin, P., Pegorin, F., 2013. On a coupled mode at sharp notches subjected to anti-plane loading. European Journal of Mechanics, A / Solids 38, 70–78. URL: http://dx.doi.org/10.1016/j.euromechsol.2012.09.007 , doi: 10.1016/j.euromechsol. 2012.09.007 . Berto, F., Pook, L.P., Campagnolo, A., 2017. Corner point singularities under in-plane and out-of-plane loading: A review of recent results. Engineering Solid Mechanics 5, 167–176. doi: 10.5267/j.esm.2017.6.002 . De Matos, P., Nowell, D., 2008. The influence of the poisson’s ratio and corner point singularities in three-dimensional plasticity-induced fatigue crack closure: A numerical study. International Journal of Fatigue 30, 1930–1943. Greaves, G.N., Greer, A., Lakes, R.S., Rouxel, T., 2011. Poisson’s ratio and modern materials. Nature materials 10, 823–837. Harding, S., Kotousov, A., Lazzarin, P., Berto, F., 2010. Transverse singular e ff ects in V-shaped notches stressed in mode II. International Journal of Fracture 164, 1–14. doi: 10.1007/s10704-010-9449-x . Heyder, M., Kolk, K., Kuhn, G., 2005. Numerical and experimental investigations of the influence of corner singularities on 3d fatigue crack propagation. Engineering Fracture Mechanics 72, 2095–2105. Kotousov, A., 2010. E ff ect of plate thickness on stress state at sharp notches and the strength paradox of thick plates. International Journal of Solids and Structures 47, 1916–1923. Lazzarin, P., Tovo, R., 1998. A notch intensity factor approach to the stress analysis of welds. Fatigue and Fracture of Engineering Materials and Structures 21, 1089–1103. doi: 10.1046/j.1460-2695.1998.00097.x . Lazzarin, P., Zappalorto, M., 2012. A three-dimensional stress field solution for pointed and sharply radiused V-notches in plates of finite thickness. Fatigue and Fracture of Engineering Materials and Structures 35, 1105–1119. doi: 10.1111/j.1460-2695.2012.01698.x . Maurizi, M., Berto, F., 2020. 3d e ff ects on fracture mechanics. Procedia Structural Integrity, Submitted . Nakamura, T., Parks, D.M., 1989. Antisymmetrical 3-D stress field near the crack front of a thin elastic plate. International Journal of Solids and Structures 25, 1411–1426. URL: http://dx.doi.org/10.1016/0020-7683(89)90109-1 , doi: 10.1016/0020-7683(89)90109-1 . Pook, L., 1994. Some implications of corner point singularities. Engineering Fracture Mechanics 48, 367–378. Pook, L.P., 2013. A 50-year retrospective review of three-dimensional e ff ects at cracks and sharp notches. doi: 10.1111/ffe.12074 . Pook, L.P., Campagnolo, A., Berto, F., Lazzarin, P., 2015. Coupled fracture mode of a cracked plate under anti-plane loading. Engineering Fracture Mechanics 134, 391–403. URL: http://dx.doi.org/10.1016/j.engfracmech.2014.12.021 , doi: 10.1016/j.engfracmech.2014. 12.021 . Shivakumar, K., Raju, I., 1990. Treatment of singularities in cracked bodies. International journal of fracture 45, 159–178. Williams, M., 1952. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of applied mechanics 19, 526–528.