Notch effect in highly deformable material sheets

Thin-Walled Structures 105 (2016) 90–100

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Notch effect in highly deformable material sheets Roberto Brighenti n, Andrea Spagnoli, Andrea Carpinteri, Federico Artoni Dept. of Civil-Environmental Engineering and Architecture, University of Parma, Viale delle Scienze 181/A - 43124 Parma, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 5 October 2015 Received in revised form 12 February 2016 Accepted 29 March 2016 Available online 14 April 2016

Defect tolerance is usually understood as the ability of a material to withstand an external load in the presence of a geometrical flaw. The case of a defect represented by a notch (i.e. a geometric discontinuity with finite curvature radius) can be described by the so-called stress (strain) concentration factor at the notch root and by the stress (strain) gradient in the vicinity of the notch root which is a preferential site for crack nucleation and propagation. Under static loads and within the elastic regime, notch effect in traditional structural materials is simply governed by the initial notch geometry. On the other hand, in highly deformable materials, such as soft matters (biological tissues, colloids, polymers, gels, foams, etc.), notch effect must be evaluated by considering large strain values arising around the notch, responsible for its blunting. In addition, when notches are contained in non-confined plate-like components, a sort of augmented notch blunting might occur as a consequence of local flexural instability of the material plate in the compressed zones. In the present paper, an experimental and theoretical study is discussed for a silicon sheet with different levels of notch severity. It is shown as the safety against notch concentration effect (that can reduce up to about 40–70%) can be greater than that in low deformable materials, and so notch blunting and plate's localized instability are beneficial contributions, leading to an increase of the element’s tensile strength. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Soft materials Stress concentrators Notch blunting Defect tolerance

1. Introduction As is well-known, any geometric discontinuity contained in a structural element is responsible for a stress concentration effect [1–6]. In practical applications, such flaws cannot be avoided as in the case of holes, threaded, fillets, smooth geometric variations, etc. Furthermore, different types of defects are frequently originated during the production process or under service life. The main effect of a geometric discontinuity can be recognized to be a local stress raising with the appearance of a noticeable stress gradient in the neighboring region surrounding the socalled notch root. Even if the stress raiser – conversely to the cases of sharp notches or cracks – does not produce any stress singularity, its presence must carefully be evaluated in the structural safety assessment. The highly stressed regions are worth of attention because of the easily appearance, and sometimes propagation, of crack-like defects, especially in the presence of timevarying loading [7–10]. Several authors have examined the problem of cracks and stress concentrators [2,11–15]. In particular, it is worth mentioning the study by Creager [4] who generalized the problem of a stress concentrator on the basis of fracture mechanics concepts by n

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

http://dx.doi.org/10.1016/j.tws.2016.03.030 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

accounting for the finite value of the notch radius. A careful study of the notch effect in stressed materials is of great importance for their safety assessment under in-service conditions. As far as ductile materials are concerned, very high stress levels (theoretically unlimited in perfectly elastic materials with sharp cracks, corresponding to zero curvature notch root) are responsible for a localized plasticity around the notch root with a beneficial effect on the stress concentration, as a so-called crack blunting takes place leading to a notch with a finite curvature radius. A similar effect can also occur in soft matters, i.e. in highly deformable materials, where the highly strained regions significantly deform, leading to a relevant change in notch geometry, usually corresponding to a mitigation of the notch effect on the strength of the flawed structural components [16].

2. A brief overview on the mechanics of soft materials Many common materials – such as cleaning products, paints, plastics, biological tissues – are complex at the microscopic scale, and are often made of several kinds of highly organized molecules or particles, which are held together by weak electrostatic forces that justify the term ‘soft’ to define their main mechanical quality. Polymers, soft colloids and biological materials are the most common soft materials.

R. Brighenti et al. / Thin-Walled Structures 105 (2016) 90–100

Nomenclature Principal axes of the elliptical notch (maximum and minimum, respectively). Width of the transversal compressed region of the c notched plate. Elastic tensor of the material. C Young modulus of the material. E Green-Lagrange strain tensor in the reference system E t , y at the notch root. Ett ≡ Exx, Eyy, Ety Green-Lagrange strain tensor components in the reference system t , y at the notch root. Green-Lagrange strain tensor in the rotated reference E′ system ξ, η at the notch root. E11, E22, E12 Green-Lagrange strain tensor components in the rotated reference system ξ, η at the notch root. Deformation gradient and its determinant, F, J respectively. Stress-Intensity Factor (SIF) of the notch-equivalent KI crack. Stress concentration factor for a finite notched plate. Kt Stress concentration factor for a notched plate taking Kt* into account flexural plate instability leading to additional notch blunting. Stress concentration factor for an infinite notched Kt

2a, 2b

Highly-deformable materials, sometimes also termed ‘soft’ materials [17,18], are of fundamental and practical relevance in modern mechanics applications. Liquid crystals polymers, viscoelastic fluids [19], emulsions, colloidal matters, elastomers, foams, gels [20,21], synthetic hydrogels, sylicons, fibrous materials [22], biological tissues (muscles, blood, cartilage, etc.), just to mention a few, are becoming important in a wide range of technical applications (such as in adhesives, paints, lubricants, rubbers, sealants, packaging materials) and scientific fields (nanotechnologies and biotechnology). Soft materials have become well-known after the awarding of the nobel prize to P.G. de Gennes [23] for his studies on the modeling of complex soft materials within a simple thermodynamic framework. Soft matters are usually organized into mesoscopic physical structures that are much larger than the microscopic scale (atomic and molecular scale), and much smaller than the macroscopic scale of the material used for structural components. Their overall mechanical properties depend on physicalchemical causes, the weak interactions of these different scales, and the high number of internal degrees of freedoms of the elementary constituents. A proper balance between entropic and enthalpy contributions to the free energy and their sensitivity to external conditions are responsible for their softness at the macroscopic scale and capability to show metastable states. Very weak chemical actions can induce a heavy change of the material’s mechanical properties. Typically soft materials have mechanical properties falling within the following ranges: elastic modulus 0.1–1.5 MPa, tensile and compressive strengths 1–10 MPa and 20–60 MPa, respectively, maximum tensile and compression strains 1000–2000% and 90– 95%, respectively, fracture energy 100–1000 J/m2. Soft materials usually show a mechanical behavior characterized by complex constitutive laws [24], large strain effects, unusual fracture [25–28] and fatigue [29]. Generally speaking, their

Kt, ε r, θ

91

plate. strain concentration factor. Cylindrical co-ordinates of a generic point with respect to the crack tip of the equivalent crack. Second Piola-Kirchhoff stress tensor.

S Stt , Syy, Sty Second Piola-Kirchhoff stress tensor components in the reference system t , y at the notch root. Deviatoric stress tensor. t Remote tensile strain in the y direction. εy,0 εx x ′, εyy , εxy ′ Strain tensor components in the x′ − y reference system. ν Poisson’s coefficient of the material. Curvature radius at the notch root. ρ Initial curvature radius at the notch root in the unρ0 deformed material. Incremented curvature radius at the notch root due to ρ′ a variation of the remote stress σy,0. Curvature radius at the notch root accounting for ρ* flexural plate instability leading to additional notch blunting. σ Stress tensor. Remote tensile stress in the y direction. σy,0 σx ′, σy, σz , .... Stress tensor components in the x′ − y reference system.

behavior is not fully understood and is worth of deeper investigations. For instance, a so-called delay fracture has been observed in polymeric gels [28], that is, under a given load level, fracture takes place after a period of time from the load application, due to the evolution of the local stress field around existing defects; such an evolution is consequent to visco-elastic creep coupled with solvent migration in the matter.

3. Stress concentration near blunt notches in highly deformable materials The stress concentration factor is defined as the ratio between the maximum stress and the nominal stress, the latter measured with reference to a uniformly stressed section of the structural component (related to either its gross or net cross section):

Kt = σmax/σnom

(1)

Creager and Paris [30] showed that the stress field around a blunt notch has locally the same characteristics as those of an equivalent crack (Fig. 1), and can substantially be described through the same parameters used in linear elastic fracture mechanics framework (LEFM). In other words, the problem of a blunt notch can be solved by the knowledge of the Stress-Intensity Factor (SIF) of an equivalent crack and by taking into account the notch root radius. By adopting the reference system shown in Fig. 1, Creager and Paris [30] determined the stress field around the notch root by observing that it is well described by the singular stress field of a properly shifted crack with the same notch centerline axis. In particular, for a notched structural component under a remote uniform stress acting parallel to the y-axis, σnom = σy,0 , the plane stress tensor components are given by the following expressions:

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Fig. 1. Creager's reference system and the equivalent crack at ρ/2 from the notch root. Fig. 3. Scheme of the notch profile modification due to large strains occurring in the material during an increment dσy,0 of the applied remote stress.

KI ⎛ ρ ⎞⎟ ⎜1 − , 2r ⎠ 2π r ⎝

σx = ′ σz = 0

(plane stress),

σy = σy,0 +

KI ⎛ ρ ⎟⎞ ⎜1 + , 2r ⎠ 2π r ⎝

τx

σz = ν(σx ′ + σy) (plane strain)

′y

= 0, (3)

The parameter KI can be related to the stress concentration factor Kt in a notched infinite plate considering the stress component σy at the notch root, i.e. at r = ρ /2 and θ = 0º :

Kt (ρ) =

σy, max σy,0

=1+

σy(ρ /2, 0º) σy,0

=1+

KI

⎛ ⎜

σy,0 2πr ⎝

2KI 1 D ⋅ =1+ σy,0 πρ ρ

1+

ρ ⎞⎟ 2r ⎠

r=

ρ 2

(4)

Eq. (4) indicates that the sole relevant parameter for the definition of the maximum stress and of the near root stress profile is the notch radius. This approach can thus be adopted for cases involving notches of different shapes, once their local curvature radius at the root is known. As a relevant example, let us consider the case of a plate with an elliptical notch with semi-axes a, b (Fig. 2); the curvature at the root of such an elliptical notch and the corresponding stress concentration factor in the case of an infinite plate can be written as follows [4]:

Fig. 2. Geometric parameters of an elliptically notched plate.

KI ⎛ ρ KI θ⎛ θ 3θ ⎞ 3θ ⎞ ⎜ ⎟+ ⎟ cos cos ⎜ 1 − sin sin 2 ⎠ 2⎝ 2 2 ⎠ 2πr ⎝ 2r 2πr KI ⎛ ρ KI θ⎛ θ 3θ ⎞ 3θ ⎞ ⎜ ⎟+ ⎟ σy(r , θ ) = σy,0 + cos cos ⎜ 1 + sin sin 2 ⎠ 2⎝ 2 2 ⎠ 2πr ⎝ 2r 2πr KI ⎛ ρ KI θ θ 3θ ⎞ 3θ ⎜ ⎟+ τx y(r , θ ) = − sin sin cos cos ′ 2 ⎠ 2 2 2 2πr ⎝ 2r 2πr

=

ρ = b2 /a,

σx ′(r , θ ) = −

(2)

where KI = σy,0Y πa is the SIF of the equivalent crack having its tip placed at ρ/2 from the notch root (where Y , a are the geometric factor and the reference crack size, respectively). Along the x′-axis (horizontal line identified by θ = 0º ) with origin O at the equivalent crack tip position, the stresses can be expressed as follows (Fig. 1):

Kt = 1 + 2a/b = 1 + 2 a/ρ

(5)

Dealing with soft matters such as gels, sylicon, rubbers, it is generally observed that they exhibit a nearly incompressible behavior, i.e. the deformation of the material is only slightly sensitive to the volumetric stress and the only effective stress state is the deviatoric one, which is defined as follows at the notch root:

t=σ−

tr σ I, 3

with

(6a)

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93

Fig. 4. Strain concentration factor evolution with the applied remote strain for specimen (a) No.1, (b) No.2, (c) No.3. Table 1 Geometric parameters and notch root stress concentration factors for the Sylgards specimens used in the experimental tests. Specimen No.

W (mm)

2a (mm)

2b (mm)

t (mm)

2a/W

a/b

1 2 3 4

117 117 117 117

40 40 40 40

10 5 2 o 0.1

2.0 2.0 2.0 3.2

0.342 0.342 0.342 0.342

4 8 20 4 400

ρ (mm)

Y

Kt (ρ)

Kt (ρ)

1.250 0.3125 0.0050 –

 1.06  1.06  1.06  1.06

9.40 17.80 43.00 –

12.43 24.67 61.73 –

Fig. 5. Qualitative view of the (a) undeformed and (b, c) increasing deformed notched plate for the specimen No.1.

⎡0 0 ⎢ 2KI σ = ⎢ 0 σy,0 + ⎢ πρ ⎢ ⎣0 0 ⎡ ⎛ ⎢ − 1 ⎜ σy,0 + ⎢ 3⎝ ⎢ ⎢ t=⎢ 0 ⎢ ⎢ ⎢ 0 ⎢⎣

0⎤ ⎥ 0 ⎥, ⎥ ⎥ 0⎦ 2KI ⎞ ⎟ πρ ⎠

⎤ ⎥ ⎥ ⎥ ⎛ ⎞ 2KI ⎥ 2 ⎜ σy,0 + ⎟ 0 ⎥ 3⎝ πρ ⎠ ⎥ 2KI ⎞⎥ 1⎛ ⎟⎥ 0 − ⎜ σy,0 + 3⎝ πρ ⎠⎥⎦ 0

notch root, deformation is characterized by both stretching and distortion. The Green-Lagrange strain tensor E at the notch root (with components Ett , Eyy, Ety ) in the reference system t − y (with origin at the notch root point o, Fig. 3) is related to the corresponding tensor E′ (with components E11, E22, E12) in the local reference system ξ, η through the well-known relationship [31]:

0

E′ = NT EN,

E11 = c 2Ett + s 2Eyy,

E22 = s 2Ett + c 2Eyy,

E12 = csEtt − csEyy with (6b)

In the case under study, the only non-zero stress at the notch root is the vertical one, σy , so that an infinitesimal material element, having square shape and edges initially parallel to the x-y reference system axes, shows deformation corresponding to simple stretch in the vertical and in the horizontal directions (approximately by maintaining unchanged its initial volume) without any distortion. On the other hand, by considering an infinitesimal element with edges inclined by an angle α with respect to the x-axis, i.e. locally parallel to the notch profile in the vicinity of the

c = cos α,

(7a)

s = sin α

As is explained in the following, the components E11, E22, E12 of the Green-Lagrange strain tensor E′ are needed to determine the current notch profile. This can be obtained from the Green-Lagrange strain tensor components in the reference system t − y , that can be found through the Hooke’s expressions in the plane stress condition:

Ett = − ν Syy/E,

Eyy = Syy/E,

Ety = 0

(7b)

where the components Stt , Syy, Sty of the second Piola-Kirchhoff stress tensor S can be evaluated by the relationship [31]:

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Fig. 6. Strain maps obtained from DIC analyses on Sylgards specimens No.1: strains Eyy for an applied remote strain equal to (a) εy,0 = 0.087, (b) 0.140 and (c) 0.260 ; (d, e, f) corresponding maps for the strains Ett .

S = J F −1σF −T

(8a)

F being the deformation gradient ( Fij = ∂ϑi/∂θj , with ϑ1 = t , ϑ2 = y the coordinates of the deformed current configuration and θ1 = T , θ2 = Y the coordinates in the undeformed reference configuration), and J being its determinant. The components of the true stress tensor σ can be determined through Eqs. (2) and (3). Note that we indicate the coordinate in the undeformed configuration with the upper case letters, whereas the coordinates in the deformed one are shown with lower case letters. Since the described problem is highly nonlinear, the above relationships need to be evaluated in an incremental form, i.e.:

dE′ = NT dEN ,

dE = C−1dS = C−1J F −1dσF −T

(8b)

C being the elastic tensor of the material. By considering the displacement field at the notch root (Fig. 3), a generic point is characterized by the following coordinates in the deformed configuration:

ϑ1 = t = ( X cos γ − Y sin γ ) 1 + 2E22 , ϑ2 = y = ( X sin γ + Y cos γ ) 1 + 2E22

(8c)

The case of E22 = 0 corresponds to a rigid body rotation, while γ = 0, E22 ≠ 0 to a stretch in the η direction. In general, the above expressions locally represent a rotation combined with extension (for E22 > 0) of the material. In the case of highly deformable materials, large strains must be considered so that the edge of the deformed element is not tangent to the initial notch profile (note that the angle β is such that A is a point in the neibourhood of the notch root), and such strains can be used to determine the new notch shape, characterized by an increased curvature radius ρ′ at its root (Fig. 3). By analyzing a small square element of edge length equal to h, and evaluating its deformed configuration, the curvature radius increment against the applied remote stress variation dσy,0 can be expressed through the following differential relationship:

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95

Fig. 7. Strain maps obtained from DIC analyses on Sylgards specimens No.2: strains Eyy for an applied remote strain equal to (a) εy,0 = 0.077, (b) 0.144 and (c) 0.262; (d, e, f) corresponding maps for the strains Ett .

dρ ≅

h⋅ 1 + 2dE22 2 cos(β + dγ )

=

with respect to the current curvature value:

h

(

cos β + 2 c⋅s

D ⋅ E ρ ⋅(1

)

⋅

+ ν )dσy,0

K (ρ)⋅D⋅(c 2 − ν⋅s 2) 1+2 t dσy,0 E⋅2 ρ

ρ=

(9)

where the Green-Lagrange strain tensor components have been used and the angle variation γ can be expressed as γ = arcsin⎡⎣2E12/ 1 + 2E11 ⋅ 1 + 2E22 ⎤⎦. Eq. (9) can be obtained by considering the increase of the curvature radius under the generic remote stress variation dσy,0

(

ρ + dρ =

{ h + ⎡⎣ h⋅

where dE22 =

)

} 2 cos(β 1+ dE

1 + 2dE22 ⎤⎦ ⋅

Kt (ρ)⋅D 2 (c − ν⋅s 2)⋅dσy,0, E ρ

12 )

dE12 =

,

Kt (ρ)⋅2⋅c⋅s⋅D (1 + ν ) dσy,0 E ρ (10)

h h ≅ 2 cos(β ) 2 cos(β + γ )

(11)

In the last expressions, the shear strain E12 is assumed to take small values in comparison with β , where β identifies the current inclination of the small reference square element (Fig. 3). The differential of the true stress concentration factor Kt can explicitly be obtained as follows:

dKt = −

D 2ρ3/2

dρ = −

D⋅h 2ρ3/2

K (ρ) ⋅ D ⋅ (c 2 − ν ⋅ s 2) 1+2 t dσ y,0

⋅

E⋅2 ρ

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ D ⋅(1 + ν )dσ y,0 ⎟ cos⎜ β + 2 c⋅s ⋅ E ρ  ⎟ ⎜⎜ ⎟ ⎝ ⎠ γ

with cos(β + γ ) = cos β cos γ − sin β sin γ =

h cos γ − 2ρ

1−

h2 4ρ 2

sin γ

(12)

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R. Brighenti et al. / Thin-Walled Structures 105 (2016) 90–100

Fig. 8. Strain maps obtained from DIC analyses on Sylgards specimens No. 3: strains Eyy for an applied remote strain equal to (a) εy,0 = 0.076 , (b) 0.142 and (c) 0.181; (d, e, f) corresponding maps for the strains Ett .

Such a differential presents negative values, indicating a decrease of the stress concentration factor with increasing the applied remote stress. As a first attempt to apply the above model describing the attenuation of stress concentration factor, some two dimensional geometrically non-linear Finite Element (FE) analyses are performed. The strain concentration factor (that can trivially be obtained from the stress concentration factor) is determined at the notch root as a function of the remote applied strain. A plane stress FE model with 8-node elements is developed; given the two axes of symmetry, a quarter of the notched plate is considered. An isotropic linear elastic constitutive law is assumed for the material (the value of the elastic constants are reported in the subsequent section). A geometrically non linear analysis is carried out under load control (symmetry boundary conditions are imposed on the nodes along the x and y axes, while a constant tensile traction along the y direction is applied to the upper edge of the plate). In Fig. 4, the comparison between the results by the theoretical model and those by non-linear FE analyses are shown for different geometries of the notched plate (see Table 1).

4. Experimental tests In order to verify the expected behavior of thin soft notched sheets under tension, some experimental tests have been performed. In particular, pre-notched sheets made by Sylgards polymer have been tested, and experimental results have been elaborated with Digital Image Correlation (DIC) technique by using the NCORR software [32,33]. Sylgards elastomer is a commonly used silicon polymer (obtained from the crosslinking of two components, a vinyl-terminated polydimethylsiloxane matrix and an hydride terminated siloxane curing agent, with a Pd-catalyzed hydrosilylation) characterized by low Young modulus (equal to about 0.84 MPa) and Poisson’s ratio equal to about 0.36–0.38. The geometric characteristics of the specimens are shown in Table 1. Three notches with different values of the root radius have been prepared and one cracked specimen has been used in order to simulate the asymptotic case of notch with zero curvature radius. Note that the symbol Kt (ρ) indicates the stress concentration factor – that represents the ratio between the maximum notch root stress and the corresponding remote applied one – for a notched infinite

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97

Fig. 9. Strain maps obtained from DIC analyses on Sylgards specimens No.4: strains Eyy for an applied remote strain equal to (a) εy,0 = 0.057, (b) 0.089 and (c) 0.118; (d, e, f) corresponding maps for the strains Ett .

plate, and Kt (ρ) is the corresponding value for the analogous notched plate with a finite width. The initial (undeformed) and the generic stretched shapes of specimen No.1 (see Table 1) at two levels of remote applied stress are shown in Fig. 5. The material is able to comply with very high deformations, leading to a severe defect re-modeling characterized by an evident blunting. In Figs. 6–9, the strain patterns obtained from the DIC analysis are shown for increasing levels of the remote stress. The stress concentration phenomenon seems to be correctly predicted by such a contactless measurement technique. A transversally compressed region ( Ett ≡ Exx < 0) arises in the plate due to the contraction effect normal to the loading direction. Such a compressed region is responsible for a very particular behavior of the plate, and contributes to the further reduction of the stress concentration factor as is discussed in the following. In Figs. 6 and 7, the large strain pattern arising in the examined notched specimens under various remote strain values is displayed. In particular, the strain Eyy parallel to the loading direction and the strain Exx transversal to such a direction are shown. It is worth noting that, for the cracked case No. 4, the stress or

strain concentration factor cannot be defined, while a stress intensity should more properly be introduced. However, experiments provide a finite strain value at the crack tip region due to the limited resolution used in DIC analysis. This is not a limitation of the experimental measurements if very soft materials are concerned, because they spontaneously tend to smooth out peak stress [16] thanks to the crack shape evolution to a notch-like profile occurring under loads. It can be remarked that the experimental tests highlight a transversal compression stress state just in front and behind the elliptical hole. Due to the slenderness and low elastic modulus of the plate, such compressed zones (which can be observed in Figs 6–9) tend to buckle very easily, and become immediately ineffective in the load bearing mechanism of the tensioned plate (Fig. 10). This phenomenon noticeably contributes to the notch blunting, i.e. the initial elliptical notch behaves as if it were much more rounded at its extremities, leading to a significant decrement of the stress (and strain) concentration factor when the applied tensile load increases.

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R. Brighenti et al. / Thin-Walled Structures 105 (2016) 90–100

along the line x = 0, lying on the y-axis), are shown together with the applied remote stress (acting on the upper edge y = H ), where the notch side ( y = 0, − a ≤ x ≤ a ) is stress free. By considering the portion of the plate lying in the half-plane y > 0 as a beam having height H and span equal to 2s ( s is the distance between the centre of the elliptical notch and the resultant R of the stress distribution along the ligament, σy(x, 0)), supported by two finite-width columns having width approximately equal to s − a , the stress field can be evaluated by the Fourier series expansion of the stress function Φ(η, x ) [35] (a new reference system (η, x ), having origin in O′, is shown in Fig. 11). By writing the stress function in the form: ∞

∑ Φn(η)⋅ cos αnx

Φ(η, x) =

(13)

n= 0 2nπ

where αn = s , then the well-known field equation ∇2∇2Φ(η, x )=0 (which governs the plane elastic equilibrium problem in the absence of body forces) becomes an ordinary fourth-order differential equation [35,36]. The stresses can be determined by deriving the resulting stress function as follows: σx = Φ , xx , σy = Φ , ηη τx, y = − Φ , ηx , and the following expressions can be obtained: ∞

Fig. 10. Tensioned notched plate: transversally qualitative compressed zone around the elliptical notch and definition of the new blunted notch root radius ρ*.

σx(η, x) = 2A 0 −

∑ ⎡⎣ A n cosh αnη′ + Bnαnη′ cosh αnη′ n= 1

+ Cn sinh αnη′ + Dnαnx sinh αnη′⎤⎦⋅ cos αnx ∞

σy(η, x) =

∑ ⎡⎣ ( A n + 2Dn)cosh αnη + Bnαnη cosh αnη n= 1

+ ( 2Bn + Cn)sinh αnη + Dnαnη cosh αnη⎤⎦⋅ cos αnx ∞

τxy(η, x) =

∑ ⎡⎣ ( A n + Dn)sinh αnη n= 1

+ Bnαnη cosh αnη + ( Bn + Cn)cosh αnη + Dnαnξ cosh αnη⎤⎦⋅ sin αnx

(14)

where η = y − H/2, η′ = H /2 − y and the coefficients An , Bn, Cn, Dn are expressed by: An = − ( an + an )

sinh(α nH /2) + (α nH /2)⋅ cosh(α nH /2) sinh(α nH ) + α nH

Bn = ( an − an )

cosh(α nH /2) sinh(α nH ) − α nH

Cn = − ( an − an )

cosh(α nH /2) + (α nH /2)⋅ sinh(α nH /2) sinh(α nH ) + α nH

D n = ( an + an )

sinh(α nH /2) sinh(α nH ) + α nH

with an = 0, an = −

Fig. 11. Scheme of the stress distribution along the x and y line in a notched plate.

5. Notch blunting assessment in soft plates 5.1. Approximate stress field evaluation in the notched plate Since transversal buckling in the compressed regions of cracked plates under tension is a localized phenomenon [34], the size of such an ineffective zone can approximately be computed through simple considerations on the embedded compressed portion of the tensioned plate. Owing to the two symmetry axes (x- and y-axis), only a quarter of the tensioned plate can be examined. In Fig. 11, the qualitative stress distributions acting on such a portion of the plate are displayed: the external applied stress σy,0 (acting along the edge y = H ), the stress σy(x ) (acting along the ligament, identified by

y = 0, a ≤ x ≤ W /2, lying on the x-axis) and the stress σx(y ) (acting

2σy,0t ⋅ s (−1)n ⋅ n ⋅ π⋅c

sin

(15)

nπcw s

( ), n = 0, .. , ∞.

In Fig. 12, the dimensionless stress σx(x, y = 0) /σy,0 , acting along vertical lines parallel to the y-axis of symmetry, computed by analytical expressions (Eq. (14)), are displayed for different values of the plate aspect ratio W /H (equivalently identified by the ratio H /a if the notch main size a is kept constant). As can be observed, the stress field in the compressed region, evaluated along the y-axis, is approximately independent of the plate aspect ratio H /a . The extension of the compressed region, evaluated along vertical lines, is identified by the point corresponding to the change of the stress sign. It must also be noted that analytical results do not depend upon the Poisson’s ratio (see Eq. (14)). The vertical extension c (see Fig. 11) of the compressive stress region is indicated with respect to the notch main size a , that is, the parameter c /a is here adopted. It can be observed that such an extension varies from c /a = 0.73 (along the line placed at x/a = 0), up to c /a = 0.45 (along the line placed at x/a = 0.75). 5.2. Concentration factor assessment in soft notched plates The estimated transversally compressed region is shown in Fig. 13 where its approximation with an elliptical profile (dashed line) is also displayed. Neglecting the sheet’s compressed region

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Fig. 12. Stress distribution along the y-axis according to Eq. (14) for (a) x/a = 0.00 , (b) x/a = 0.25, (c) x/a = 0.50 and (d) x/a = 0.75.

Fig. 13. Transversally compressed region behind the elliptical notch: position of the points (indicated with X) identifying the change of the sign of the stress σx along the vertical lines x/a = 0.00 , x/a = 0.25, x/a = 0.50 and x/a = 0.75 and interpolating ellipse (dashed line).

provides a larger elliptical notch (with semi-axes 2a and 2b*), characterized by the notch root radius ρ*, with ρ* > ρ , that can be assumed to represent the effective notch to be considered in the plate in order to determine its real stress field. Such an effect must be superposed to the large deformation effects discussed above, leading to a further reduction of the stress (or strain) notch concentration factor.

Fig. 14. Strain concentration factor variation vs the remote applied strain: DIC results and theoretical model assessment.

In Fig. 14, the strain concentration factor Kt, ε , defined as the ratio between the maximum strain ( εy, max ) at the notch root and the remotely applied strain ( εy,0 = σy,0/E ), Kt , ε = εy, max /εy,0 , against the remote strain is displayed according to the experimental tests and the present non-linear theoretical model with a notch root

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characterized by the radius ρ*. In Fig. 14, the experimentally determined strain concentration factors, obtained through the DIC technique for the specimens No. 1, 2, 3 and 4 vs the remote applied strain εy,0 are shown with symbols, while the corresponding trend of the strain concentration factor obtained by integrating Eq. (12) (indicated with Kt*) for the enlarged elliptical notch with radius ρ*, is shown by the continuous line (in the figure's legend the strain concentration factors obtained through the stress field given by Eq. (3) are indicated for the effective and the enlarged notch). It can be observed that, for relatively high values of the remote applied strain, different initial notch shapes show very similar trend in term of strain concentration factor, confirming that the notch blunting in soft materials is very significant, leading to a strain concentration factor reduction up to the order of about 40–70%. It can be concluded that soft matters are few sensitive to stress raisers, especially when thin elements are considered, thanks to the transversal buckling occurring in the compressed region of the notched tensioned plates and to the notch blunting due to high deformation at the notch root. The decreasing concentration factor trend against εy,0 according to the experiments is similar to the estimation of the simplified theoretical model, that systematically provides an overestimation of such a parameter. The case of sharp notches (cross symbols in Fig. 14) must carefully be considered, since the concentration factor cannot theoretically be defined. However, it can be noted that the experimentally determined Kt, ε values quickly decrease tending to those of the initially rounded notches. The safety assessment must also take into account the possibility of fracture failure of the cracked body because of the crack extension under the critical load. This observation is of particular relevance in the safety assessment of soft structural components used in practical applications such as in biomechanics, soft tissue behavior, foams applications, etc. More detailed studies conducted at the micro-scale level should be carried out to deeply understand the local blunting mechanism and, possibly, to control it in order to get advanced structural elements having the possibility to adjust their in-service notch sensitivities to desired values under in service loading.

6. Conclusions In the present paper, the problem of notch shape evolution in highly deformable materials, such as soft matters (biological tissues, colloids, polymers, gels, foams, etc.), has been addressed by both theoretical and experimental approach. In such materials, the notch effect must be evaluated by considering large strain values arising around the notch, responsible for the notch root blunting. Moreover, when notches are contained in non-confined thin platelike components, a further notch blunting might occur as a consequence of local instability of the material in the compressed zones arising transversally to the load direction. Experimental and theoretical results on this aspect have been presented for a silicon sheet with different values of notch severity. A simple nonlinear model has been proposed for the assessment of the notch stress concentration factor in soft materials that surprisingly can withstand very high remote stresses thanks to the favorable notch profile evolution and local buckling effect under load. These results allow us to properly evaluate the safety level of notched soft structural components that nowadays find numerous and advanced applications. It has been shown as the notch safety increases significantly in soft thin sheets (the strain concentration factor can reduce up to about 40-70%), and so notch blunting and

plate's localized instability are beneficial contributions that naturally arise in such structural components, leading to a relevant increment of their tensile strength.

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