+ All Categories
Home > Documents > Engineering Fracture Mechanics - Salviato Lab

Engineering Fracture Mechanics - Salviato Lab

Date post: 20-Feb-2022
Category:
Upload: others
View: 2 times
Download: 0 times
Share this document with a friend
18
Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech Strength and cohesive behavior of thermoset polymers at the microscale: A size-eect study Yao Qiao, Marco Salviato William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195-2400, USA ARTICLE INFO Keywords: Notch Crack Size eect Micro Macro Residual stress ABSTRACT This study investigated, experimentally and numerically, the fracturing behavior of thermoset polymer structures featuring cracks and sharp u-notches. It is shown that, even for cases in which the sharpness of the notch would suggest otherwise, the failure behavior of cracked and pre- notched specimens is substantially dierent, the failure loads of the former conguration being about three times lower than the latter one. To capture this interesting behavior a two-scale cohesive model is proposed. The model is in excellent agreement with the experimental data and its predictions allow to conclude that (a) residual plastic stresses cannot explain the very high failure loads of notched structures; (b) the strength of the polymer at the microscale can be from six to ten times larger than the values measured from conventional tests whereas the fracture energy at the microscale can be about forty times lower; (c) the pre-notched specimens in- vestigated in this work failed when the stress at the tip reached the microscale strength whereas the cracked specimens failed when the energy release rate reached the total fracture energy of the material. The foregoing considerations are of utmost importance for the design of microelectronic devices or polymer matrix composites for which the main damage mechanisms are governed by the strength and cohesive behavior at the microscale. 1. Introduction Thermoset polymers nd extensive application across many engineering elds from e.g. automotive, aerospace and civil en- gineering to microelectronics [14]. Thermosets are also the material of choice for the manufacturing of advanced ber composites [5,6] although the demand for recyclability, high manufacturability, and damage tolerance is gradually shifting the focus to ther- moplastics [710]. Considering the several structural applications of thermosets, understanding the fracturing behavior of these materials is quin- tessential and has been the subject of extensive research in the past four decades [1117]. However, while signicant progress has been made in the characterization and modeling of crack initiation and propagation, far less attention has been devoted to the damaging and fracturing behavior of thermosets in the presence of sharp notches [1824]. This is an important issue considering that most of the thermoset structures inevitably feature sharp notches. Examples include microelectronic devices in which sharp geo- metrical features are ubiquitous or ber composites where the matrix is subjected to the stress concentration induced by the bers. Narisawa et al. [12] investigated the fracturing surface morphology of epoxy resin and showed that an internal crack may nucleate at the boundary between the plastic and elastic regions, thus generating a Fracture Process Zone that aects the fracturing behavior. A comprehensive analysis on the failure mechanism of epoxy resin was provided by Kinloch et al. [13], who proposed a https://doi.org/10.1016/j.engfracmech.2019.03.033 Received 27 December 2018; Received in revised form 21 March 2019; Accepted 22 March 2019 Corresponding author. E-mail address: [email protected] (M. Salviato). Engineering Fracture Mechanics 213 (2019) 100–117 Available online 26 March 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved. T
Transcript
Page 1: Engineering Fracture Mechanics - Salviato Lab

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

journal homepage: www.elsevier.com/locate/engfracmech

Strength and cohesive behavior of thermoset polymers at themicroscale: A size-effect study

Yao Qiao, Marco Salviato⁎

William E. Boeing Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195-2400, USA

A R T I C L E I N F O

Keywords:NotchCrackSize effectMicroMacroResidual stress

A B S T R A C T

This study investigated, experimentally and numerically, the fracturing behavior of thermosetpolymer structures featuring cracks and sharp u-notches. It is shown that, even for cases in whichthe sharpness of the notch would suggest otherwise, the failure behavior of cracked and pre-notched specimens is substantially different, the failure loads of the former configuration beingabout three times lower than the latter one. To capture this interesting behavior a two-scalecohesive model is proposed. The model is in excellent agreement with the experimental data andits predictions allow to conclude that (a) residual plastic stresses cannot explain the very highfailure loads of notched structures; (b) the strength of the polymer at the microscale can be fromsix to ten times larger than the values measured from conventional tests whereas the fractureenergy at the microscale can be about forty times lower; (c) the pre-notched specimens in-vestigated in this work failed when the stress at the tip reached the microscale strength whereasthe cracked specimens failed when the energy release rate reached the total fracture energy of thematerial. The foregoing considerations are of utmost importance for the design of microelectronicdevices or polymer matrix composites for which the main damage mechanisms are governed bythe strength and cohesive behavior at the microscale.

1. Introduction

Thermoset polymers find extensive application across many engineering fields from e.g. automotive, aerospace and civil en-gineering to microelectronics [1–4]. Thermosets are also the material of choice for the manufacturing of advanced fiber composites[5,6] although the demand for recyclability, high manufacturability, and damage tolerance is gradually shifting the focus to ther-moplastics [7–10].

Considering the several structural applications of thermosets, understanding the fracturing behavior of these materials is quin-tessential and has been the subject of extensive research in the past four decades [11–17]. However, while significant progress hasbeen made in the characterization and modeling of crack initiation and propagation, far less attention has been devoted to thedamaging and fracturing behavior of thermosets in the presence of sharp notches [18–24]. This is an important issue considering thatmost of the thermoset structures inevitably feature sharp notches. Examples include microelectronic devices in which sharp geo-metrical features are ubiquitous or fiber composites where the matrix is subjected to the stress concentration induced by the fibers.

Narisawa et al. [12] investigated the fracturing surface morphology of epoxy resin and showed that an internal crack maynucleate at the boundary between the plastic and elastic regions, thus generating a Fracture Process Zone that affects the fracturingbehavior. A comprehensive analysis on the failure mechanism of epoxy resin was provided by Kinloch et al. [13], who proposed a

https://doi.org/10.1016/j.engfracmech.2019.03.033Received 27 December 2018; Received in revised form 21 March 2019; Accepted 22 March 2019

⁎ Corresponding author.E-mail address: [email protected] (M. Salviato).

Engineering Fracture Mechanics 213 (2019) 100–117

Available online 26 March 20190013-7944/ © 2019 Elsevier Ltd. All rights reserved.

T

Page 2: Engineering Fracture Mechanics - Salviato Lab

quantitative model accounting for the blunting at the notch tip prior to the onset of crack initiation. Several potential tougheningmechanisms for polymer structures weakened by cracks and sharp notches were discussed by Argon et al. [15,16] whereas the effectof different pre-notching methods on the fracturing behavior of polymers was studied in [18–21]. In these contributions it wasproposed that residual plastic stresses induced by the manufacturing of the notch may lead to very high values of apparent fracturetoughness of the polymer.

According to the foregoing contributions, the failure behavior of cracked and pre-notched specimens is substantially different,even when the sharpness of the notch and Linear Elastic Fracture Mechanics (LEFM) would suggest otherwise. The present workattempts to clarify this difference leveraging computational cohesive fracture mechanics and size effect testing of both pre-crackedand pre-notched Single Edge Notch Bending (SENB) specimens. A two-scale cohesive law is proposed and shown to be in excellentagreement with the experimental data. Using the model, not only the notch mechanics of thermosets is clarified but also un-precedented insight on the strength and cohesive behavior of the polymer at the microscale is obtained. It is shown that the mi-croscale strength can be from six to ten times larger than the values estimated from macroscale tests. In contrast, the fracture energyat the microscale is estimated to be roughly forty times lower than the values obtained from traditional fracture tests.

The results of this work disprove the hypothesis, largely accepted in the literature [18–22], that residual plastic stresses generatedby the notching process are the cause of the different behavior of notched and cracked structures. More importantly, the two-scalecohesive model proposed in this study represents a first step towards the better understanding of the cohesive behavior of thermosetsat the microscale. This information is quintessential for the formulation of accurate computational models for microelectronic devicesor the damaging and fracturing behavior of the matrix in fiber composites. Further, the novel insight on the cohesive behavior canpave the way for the development of new nanomodification strategies targeting specifically the enhancement of the behavior at themicroscale [25–29].

2. Materials and methods

2.1. Material preparation

Following [30,31], the thermoset polymer used in this work was composed of an EPIKOTE™ Resin MGS™ and an EPIKOTE™curing Agent MGS™ RIMH 134-RIMH 137 (Hexion [32]) combined in a 100:32 ratio by weight.

The epoxy and hardener were mixed for 10min and degassed for 20min in a vacuum trap using a Vacmobile mobile vacuumsystem [33] in order to remove any air bubbles. After degassing, the mixture was poured into silicone molds made of RTV siliconefrom TAP Plastics [34] to create geometrically-scaled specimens with consistent sizes. Finally, the resin was allowed to cure at roomtemperature for approximately 48 h and then post-cured in an oven for 4 h at 60 °C.

2.2. Specimen preparation

Several previous investigations on Compact Tension (CT) and Single Edge Notch Bending (SENB) specimens made of thermosetpolymers have shown that the method used to create the notch has a significant influence on the failure behavior and the ultimateload [18–22]. In particular, it has been shown that specimens with pre-cracks created by tapping may exhibit values of the fracturetoughness from 5 to 20 times lower than the ones estimated by testing pre-notched specimens, regardless of the way the pre-notch ismade (e.g. insertion of a Teflon sheet or micro-sawing). This discrepancy cannot be explained by Linear Elastic Fracture Mechanics(LEFM) since, considering the very small ratios between the notch tip radius and the notch depth, this theory predicts the same failureload in case of the pre-notch or the pre-crack [35–38].

To clarify the foregoing differences and to provide an objective investigation of the fracture properties of the polymer both pre-cracked and pre-notched Single Edge Notch Bending (SENB) specimens were investigated in this work. The pre-notched specimenswere created by means of a 0.3 mm wide diamond coated saw leading to a 0.4 mm wide notch as illustrated in Fig. 1 which shows amagnification of the notch tip by means of Scanning Electron Microscopy (SEM). The pre-cracked specimens were created through atwo-stage process. The first step consisted in creating a notch about one quarter of the specimen width by means of the 0.3 mm widediamond coated saw. Then, tapping by means of a sharp razor blade followed to create the last portion of the crack.

2.2.1. Pre-cracked specimensThe design of the Single Edge Notch Bending (SENB) specimens was based on ASTM D5045-99 [39]. In order to study the scaling

of the fracturing behavior of pre-cracked specimens, geometrically-scaled specimens of three different sizes were prepared as illu-strated in Fig. 2. The dimensions, scaled as 1:2:4, were ×10 36 mm, ×20 72 mm, and ×40 144 mm, respectively. The various cracklengths of the specimens were approximately in the range 0.35D to 0.55D, where D is the width of the specimen. This aspect was veryimportant to guarantee a proper geometrical scaling. The scaling did not involve the thickness, t, which was kept about 12mm for allthe investigated sizes.

2.2.2. Pre-notched specimensAs illustrated in Fig. 3, geometrically scaled specimens of four different sizes were prepared. The dimensions, scaled as 1: 2: 4: 8,

were ×10 36 mm, ×20 72 mm, ×40 144 mm and ×80 288 mm, respectively while the width of the notch was kept the same for all theinvestigated sizes. Accordingly, the ratio between the depth and the radius of the notch, a b/0 , was 25, 50, 100, and 200 respectively.The notch length was always half of the width of the specimen. The thickness, t, was kept about 12mm for all the investigated sizes.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

101

Page 3: Engineering Fracture Mechanics - Salviato Lab

2.3. Fracture testing

Three-point bending tests were performed using a closed-loop, electro-actuated 5585H Instron machine. To minimize the effectsof the strain rate on the tests, the applied displacement rate was scaled with the specimen size to provide the same nominal strain rate∊̇N , the nominal strain being defined as ∊ = PL tD E3 /(2 )N

2 . In the definition of nominal strain, P is the load, L is the span between thetwo supports, and E is the Young’s modulus.

3. Experimental results

The load-displacement curves of the three-point bending tests are plotted in Fig. 4. As can be noted, for both pre-cracked and pre-notched specimens, the mechanical behavior is linear up to the peak load which is followed by unstable crack propagation. This is anindication of pronounced brittle behavior for all the investigated specimens. Further, as the figure shows, the stiffness of the spe-cimens is not affected by the sharpness of the notch which is not surprising given the significantly large aspect ratios. In contrast, the

Fig. 1. Notch tip geometry from Scanning Electron Microscopy (SEM).

Fig. 2. Geometry of the pre-cracked specimens. Units: mm.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

102

Page 4: Engineering Fracture Mechanics - Salviato Lab

peak load of the pre-notched specimens is approximately 3 times higher that the one of the pre-cracked specimens for all theinvestigated sizes. This difference, which agrees with previous investigations [18–22], will be discussed in the following sections.

The crack or notch length, peak load, and nominal strength =σ P L tD3 /2Nc c2 for all the geometrically scaled specimens tested in

this work are tabulated in Table 1. In the definition of nominal strength, Pc is the critical load and other quantities have the meaningas discussed in previous sections.

4. Analysis and discussion

4.1. Estimation of the mode I fracture energy by LEFM

In recent works [30,31,40,41], the effect of the Fracture Process Zone (FPZ) size on the fracturing behavior of thermoset na-nocomposites was investigated leveraging size effect testing and analysis. It was shown that, Linear Elastic Fracture Mechanics(LEFM) provides a very accurate description of the fracture scaling in epoxy for typical laboratory-scaled, pre-cracked specimens assuggested by the ASTM D5045-99 [39]. This confirms that, for the pure epoxy and sufficiently large specimens, the FPZ size has anegligible effect. On the other hand, several investigations [13,35–38,42] showed that, when the notch radius is sufficiently smallerthan Irwin’s characteristic length, the notch has the same effects on the ultimate failure load as a crack of the same length. In thisstudy, the notch radius b= 0.2 mm is significantly smaller than the Irwin’s characteristic length = ≈∗l E G f/ 0.75ch F t

2 mm where=E 2263 MPa, =ν 0.35 and =f 51.6t MPa [30].In light of the foregoing considerations, both the pre-cracked and pre-notched specimens were analyzed by means of LEFM, using

the equations for a cracked specimen [39]:

=∗

G ασ D

Eg α( ) ( )F

Nc0

2

0 (1)

where =α a D/0 0 =normalized initial crack length, =σ PL tD3 /2N2 =nominal stress, =∗E E for plane stress and = −∗E E ν/(1 )2 for

plane strain, ν is Poisson’s ratio, and g α( )0 =dimensionless energy release rate which can be easily calculated following the pro-cedure described in [30,31,40,43–48]. The mode I fracture energy calculated by means of Eq. (1) is shown in Fig. 5 for the differentspecimen sizes and notch types. Not surprisingly, the fracture energy calculated by LEFM is not affected by the specimen size. This is aconfirmation that, for all the specimens investigated in this work, the nonlinear damage in the Fracture Process Zone (FPZ) did notaffect the structural behavior significantly and linear theories such as LEFM can be used to provide a first estimate of the fractureproperties.

On the other hand, the results showed in Fig. 5 indicate a very significant effect of the type of notch, the fracture energy of the pre-

Fig. 3. Geometry of the pre-notched specimens. Units: mm.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

103

Page 5: Engineering Fracture Mechanics - Salviato Lab

notched specimens being approximately 10 times higher than the one of the pre-cracked specimens. This result, abundantly con-firmed in the literature [18–22], cannot be explained by LEFM since, according to this theory, the notched specimens investigated inthis work should fail at the same load as the cracked ones and the fracture energy should be a material property not affected by anygeometrical feature.

In [18–22], the higher apparent fracture energy was ascribed to the emergence of residual, plastic stresses during the pre-notchingprocess. However, a direct validation of this statement was never provided. The following section focuses on this particular aspectand shows that these hypothetical residual stresses should be unrealistically high to justify the difference in fracture energy reportedin the present work and in the literature [18–22].

4.2. Residual stresses

To check the possibility that the higher apparent fracture energy of pre-notched specimens is related to the presence of residualstresses as generally accepted in the literature, a simple analysis can be conducted within the framework of the Linear Elastic FractureMechanics (LEFM). Taking advantage the superposition principle, the total mode I Stress Intensity Factor (SIF) can be calculatedsumming the effects of the applied load P and the residual stress distribution ahead of the crack tip:

+ =K K KI I I total(1) (2) , (2)

where, as illustrated in Fig. 6, KI (1) refers to the SIF associated to the concentrated load on the middle top of the specimen without theeffect of residual stresses while KI (2) refers only to compressive stresses acting on the equivalent Fracture Process Zone of length cf[30,31,40,43–48]. The length cf is proportional to the Irwin’s characteristic length lch and it is defined as an additional equivalentcrack length required to capture the nonlinear effects of the FPZ.

The stress intensity factor KI (1) can be expressed as ∗E G for a plane strain condition whereas KI (2) has the following expression[49]:

Fig. 4. Experimental load-displacement curves for pre-cracked and pre-notched specimens of different sizes.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

104

Page 6: Engineering Fracture Mechanics - Salviato Lab

∫= −− −

+K σ x

πa

fx2 ( ) ( , )

(1 ) [1 ( ) ]dI a

a c θx

aaD

aD

xa

(2)0

3/2 2 1/2f

0

0 00

00 (3)

where σ x( )θ is the magnitude of the compressive stresses applied on the equivalent FPZ,= + − +f g g g g( , ) ( ) ( ) ( ) ( ) ( )x

aaD

aD

xa

aD

xa

aD

xa

aD1 2

23

340

0 00

00

00

0 is a dimensionless geometry function, and x is the distance from the bottomsurface of the SENB specimen as shown in Fig. 6. The value of the constants gi with = …i 1 4 can be found in [49]. At incipient crack

Table 1Max load and nominal strength of pre-notched and pre-cracked specimens at different sizes.

Specimen type Specimen width (mm) Crack or notch length (mm) Max load (N) Nominal strength (MPa)

Pre-notched D=10 5.00 647.01 28.12Pre-notched D=10 5.00 543.07 23.60Pre-notched D=10 5.00 613.01 26.65Pre-notched D=10 5.00 654.10 28.40Pre-notched D=10 5.00 561.12 24.39Pre-notched D=10 5.00 537.22 23.34Pre-notched D=10 5.00 553.40 24.03Pre-notched D=10 5.00 630.18 27.38Pre-notched D=20 10.00 800.66 17.49Pre-notched D=20 10.00 758.70 16.57Pre-notched D=20 10.00 742.83 16.22Pre-notched D=20 10.00 779.50 17.02Pre-notched D=20 10.00 693.15 15.14Pre-notched D=20 10.00 731.95 15.99Pre-notched D=20 10.00 814.50 17.79Pre-notched D=20 10.00 774.82 16.92Pre-notched D=40 20.00 1100.05 12.04Pre-notched D=40 20.00 1077.89 11.80Pre-notched D=40 20.00 1108.72 12.14Pre-notched D=80 40.00 1580.11 8.66Pre-notched D=80 40.00 1542.20 8.45Pre-notched D=80 40.00 1653.11 9.06Pre-notched D=80 40.00 1532.03 8.40Pre-notched D=80 40.00 1455.30 7.97Pre-notched D=80 40.00 1755.01 9.62

Pre-cracked D=10 5.03 169.33 7.73Pre-cracked D=10 4.96 164.62 7.47Pre-cracked D=10 4.38 201.01 9.48Pre-cracked D=20 7.63 289.27 7.91Pre-cracked D=20 9.26 306.58 6.65Pre-cracked D=20 9.28 325.62 6.61Pre-cracked D=40 16.46 455.31 5.37Pre-cracked D=40 17.27 385.44 4.93Pre-cracked D=40 16.88 422.30 5.12

Fig. 5. Fracture energy estimated from LEFM for pre-cracked and pre-notched specimens of different sizes.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

105

Page 7: Engineering Fracture Mechanics - Salviato Lab

onset, the stress intensity factor KI total, is ought to be equal to the fracture toughness of the pre-cracked specimens measured from theexperiments. Accordingly, with the assumption that ≈ ≈ ≈a D c l/ 0.5, 0.5 0.4f ch0 mm [44,45], and the residual stresses are uniformlydistributed, the magnitude can be estimated. As illustrated in Fig. 7, the residual stress for the pre-notched specimens investigated inthis work is higher than 100MPa which seems unrealistically high to be created by sawing during the pre-notching process. It isworth mentioning that Eq. (3) is obtained by superimposing the Stress Intensity Factors (SIFs) induced by a uniform distribution ofinfinitesimal forces, =dP σ dxθ , acting on the equivalent crack faces, ∈ +x a a c[ , ]f0 0 .

The choice of the equivalent crack length cf ≈ 0.5 lch is based on the assumption that, in first approximation, the traction-separation law can be considered as a linear function. This hypothesis was validated by previous size effect tests on cracked SingleEdge Notch Bending Specimens (SENB) made of the same thermoset polymer investigated in this work [30,31]. In fact, although thebest match of the experimental data was achieved using a bi-linear cohesive law, it was shown that a linear traction-separation couldbe used to provide a course approximation of the fracturing behavior. Once it is agreed that the cohesive law can be approximated bya linear function, a parametric study on the relation between the Irwin’s characteristic length and the effective FPZ size is notnecessary. In fact, it should be noted that the effective FPZ size cf is always equal to 0.44 lch provided that the cohesive law is linear,regardless of the mechanical properties of the material or the geometry of the specimen. This was proven via an extensive com-putational study by Cusatis and Schauffert [45].

In the foregoing analysis, the Fracture Process Zone (FPZ) at the notch tip was assumed to be fully developed and the residualstresses were considered to act on the effective crack length. Although it is very unlikely that due to the sawing process the FPZpropagates sub-critically to form a longer crack, an additional computational study was conducted to investigate the effects of the sizeof the FPZ on the magnitude of the residual stresses required to match the experiments. For the analysis, a FE model of the SENBspecimen of width =D 20 mm was constructed. The mesh was created using 4-node iso-parametric quadrilateral elements for a totalnumber of degrees of freedom of 35,452. Thanks to the fine mesh, more than 100 elements were located along the FPZ. The crack was

Fig. 6. Schematic description of the boundary conditions used for the estimation of the residual stresses by the superposition principle.

90

95

100

105

110

115

120

0 20 40 60 80 100 120 140Specimen Size [mm]

ssertS laudiseR[M

Pa]

Crack FPZ

Residual Stress

Eq. (2): (1) + (2) = ,

Fig. 7. The residual stress estimated from equivalent linear elastic fracture mechanics, Eq. (2).

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

106

Page 8: Engineering Fracture Mechanics - Salviato Lab

simulated via the cohesive interaction algorithm available in ABAQUS/Implicit with a linear traction-separation law featuring astrength of 55MPa and a fracture energy of 0.8 N/mm. These values were obtained from previous tests on cracked specimens [30,31].An uniform pressure representing the residual stresses was applied on the left and right faces with cohesive interaction, from theinitial crack tip to a distance l from the tip. A schematic of the geometry of the problem and related boundary conditions is providedin Fig. 8. For every length l, the residual stresses were adjusted so that the peak load applied to the specimen matched the experi-mental value from the tests. This way, a more thorough analysis of the effects of the residual stresses on the development of thecohesive zone ahead of the notch was possible. Fig. 8 shows the evolution of the residual stresses as a function of the extent of theplastic stress region normalized by Irwin’s characteristic length lch. As can be noted, the relation is highly nonlinear, the residualstresses increasing significantly with decreasing values of l. It is remarkable that, for a plastic stress region close to 0.4 lch, the residualstresses required to match the experiments are in the order of 500MPa. On the other hand, for plastic regions exceeding a length of0.5 lch the stresses decrease at a very low rate, reaching about 90MPa for a plastic region as large as the Irwin’s characteristic length.For the residual stresses to be within a realistic range, the plastic zone size should be more than 30% larger than Irwin’s characteristiclength. This is considered by the authors as a very unlikely event.

To shed more light on the possible effects of residual stresses on the fracturing behavior, additional tests on pre-notched speci-mens were conducted. However, this time, a sharp razor blade was pre-inserted into the specimens during the manufacturing processand eventually removed after the curing of the epoxy resin to create the notch. Thanks to this procedure, notch tip radii similar to the

Fig. 8. The evolution of residual stresses with the extent of the plastic region, l, through cohesive zone modeling (D=20mm).

Fig. 9. (a) experimental load-displacement curves for pre-cracked and pre-notched specimens, and specimens with the notch made by a pre-insertedrazor blade; (b) silicone molds for the manufacturing of the specimens with the pre-inserted razor blade; (c) typical specimens with the pre-insertedrazor blade right after curing. Note that the blade was removed before the tests.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

107

Page 9: Engineering Fracture Mechanics - Salviato Lab

ones of the sawed specimens were obtained without the possible emergence of plastic residual stresses.Fig. 9 compares the load displacement curves for specimens with pre-cracks made by tapping, with pre-notches made by sawing,

and with pre-notches made by pre-inserting a razor blade. As can be noted, all the notched specimens exhibit a significantly largerpeak load compared to cracked specimens. More importantly, the mechanical behavior of specimens with notches made by pre-inserting a razor blade is basically identical to the one reported for the case of sawed notches. Considering that the former cannotfeature any residual stress, it is concluded that plastic residual stresses are not likely the dominant reason of the higher apparentfracture energy of pre-notched specimens. To prove this statement, we took advantage of the transparency of epoxy and investigatedthe region close to the notch tip via Transmission Optical Microscopy (TOM) with cross-polarized light. In this technology, cross-polarized light is transmitted through the specimen enabling the visualization of permanent deformation by the analysis of the fringepatterns. This is now an established approach to visualize crazing and plastic deformation at the crack tip in thermoset polymers (see,among others, [50–52]). Fig. 10a–c summarize the results of the analysis for cracked and notched specimens. These Figures showthree cases: (a) a pre-cracked specimen before applying any load, (b) a pre-notched specimen before applying any load, and (c) a pre-notched specimen after applying a load equal to 80% of the failure load and unloading. Micrographs taken using bright field andcross-polarized light are shown for comparison. As can be noted from Fig. 10d, the pre-cracked specimens exhibited a damage zoneprior the application of any mechanical load. This damage took the form of micro-crazing as the typical fringes in the insert show. Asimilar morphology was reported for other thermoset systems in [50–52].

Not surprisingly, the micrographs on the pre-crack tip confirmed that tapping indeed leads to the formation of an FPZ before theapplication of any mechanical load. What was unexpected is that, as clearly shown in Fig. 10e, the notch tip of the pre-notchedspecimens did not exhibit significant permanent deformations. To further corroborate this observation, the specimens were loaded upto 80% of the failure load and unloaded. Then, TOM pictures of the notch tip were taken for comparison. As can be noted fromFig. 10f, in this case the stress concentration at the notch indeed led to damage as confirmed by the very visible fringes ahead of thetip. This provides further evidence that if a region of pre-existing plastic residual stresses exists, either it is significantly smaller thanthe notch width or the residual stresses are almost negligible. In this context, it is not surprising that the specimens featuring notchesmade by pre-inserting a razor blade exhibited the same failure loads as the one made by sawing. In both cases the plastic residualstresses were negligible.

A possible explanation of the true cause is presented next leveraging cohesive fracture mechanics.

4.3. Cohesive zone modeling of thermoset polymers

In this section a two-scale cohesive zone model is proposed to capture the fracturing behavior of both pre-cracked and pre-notched specimens seamlessly. The main observations that led to the development of the model are described next along with thecomparison with the experimental data.

Fig. 10. Transmission Optical Microscopy (TOM) micrographs of specimens under: (a–c) bright field and (d–f) cross-polarized light. Note thatfigures (a–e) are the specimens before applying any load whereas figures (c and f) are the specimens after applying a load equal to 80% of the failureload and unloading.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

108

Page 10: Engineering Fracture Mechanics - Salviato Lab

4.3.1. Bi-linear cohesive law for large crack opening displacementsIn previous investigations on pre-cracked, geometrically-scaled specimens made of thermoset resin [31,40], it was found that the

cohesive behavior of this material is best described by a bi-linear law. This is evident from Fig. 11a–b showing that the bi-linearcohesive law enables the matching of the nominal strength σNc with errors always lower than 7% for all the specimen sizes in-vestigated. In contrast, the errors are in the order of 30% in case of a linear cohesive law (see Fig. 11c–d).

As shown in Fig. 12b, the bi-linear cohesive law can be described through four parameters: (a) tensile strength ft, (b) initialfracture energy, Gf

b, which represents the area under the initial segment of the bi-linear cohesive law; (c) total fracture energy, GFb,

which is the total area under the bi-linear cohesive law; (d) change-of-slope stress, σk, which is the value of stress at the intersection ofthe initial and tail segment. By matching the size effect data on the structural strength of the pre-cracked specimens, the cohesiveparameters of the bi-linear cohesive law are tabulated in Table 2.

However, notwithstanding the excellent agreement with the data on pre-cracked specimens, it is impossible to match the ex-perimental results on the pre-notched specimens by means of the foregoing cohesive law. A relatively good agreement is possible onlyby increasing the total fracture energy to =G 7.2F

b N/mm, the value corresponding to the energy estimated by LEFM. However, thisvalue would lead to a significant over-prediction of the structural strength of the pre-cracked specimens. Considering that the co-hesive law should be a material property, the discrepancy between the results for pre-cracked and pre-notched specimens must beexplained using the same cohesive law. A possible solution is presented next.

4.3.2. Two-scale cohesive lawIn a recent publication [42], Di Luzio and Cusatis investigated the fracturing behavior of structures featuring blunt notches and

characterized by materials exhibiting a linear cohesive law. In their numerical study they found that when the notch tip radius isapproximately equal to the Irwin’s characteristic length = ∗l E G f/ch f t

2 the failure occurs when the maximum stress at the notchapproximately reaches the material strength, i.e. =σ f k/Nc t with k =stress concentration factor. On the other hand, for sufficientlysharp notches, the fracture is driven by the formation of a Fracture Process Zone (FPZ) in front of the notch which ultimatelypropagates and leads to the final failure.

Based on the foregoing considerations, a cohesive zone model should be able to capture the behavior of notched and crackedspecimens seamlessly provided that some particular features are added to the cohesive law. In this work, it is assumed that thecohesive behavior can be described by a two-scale cohesive law as represented in Fig. 12a. As shown in the figure, ft

μ represents the

Fig. 11. Load-displacement curves vs. cohesive zone model featuring a linear and bi-linear cohesive law for geometrically-scaled, pre-crackedspecimens of different sizes.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

109

Page 11: Engineering Fracture Mechanics - Salviato Lab

initial strength whereas Gfμ is the initial fracture energy or, in other words, the area under the first linear branch of the curve. The rest

of the curve is identical to the cohesive law identified by fracture tests on geometrically-scaled SENB specimens [40,41] and ischaracterized by ft = macroscopic strength, σk =stress at the third change of slope, Gf =initial macroscopic fracture energy (areaAOBDE), and the total fracture energy GF (area AOBCDE).

The proposed cohesive law must differ from the linear one investigated in [42] to capture the peculiar cohesive behavior ofthermoset polymers. Until the crack opening displacements are in the sub-micron regime, the resulting cohesive stresses areequivalent to the ones predicted by a linear cohesive law of strength ft

μ and total energy Gfμ. For larger opening displacements, the

cohesive law becomes equivalent to the bi-linear law that provided an excellent agreement with the experimental data on pre-crackedspecimens. Since the first portion of the cohesive law describes the cohesive stresses at the microscale while the rest of the cohesivecurve captures the behavior for larger displacements, the proposed cohesive law is characterized by two very distinct length scales.For this reason, in this and in future contributions the model will be referred to as a two-scale cohesive model.

In this work, it is assumed that for sub-micron crack opening displacements the cohesive strength ftμ is about 300MPa and the

cohesive stresses decrease linearly and very steeply with increasing crack openings. This captures the fact that, due to statistical sizeeffect, the microscopic strength of the polymer can be several times higher than the one measured from macroscopic tests [53–55]. Atthe same time, the steep initial part of the cohesive curve, leading to an initial fracture energy of only about 2.5% of the total one,captures the lower energy dissipation occurring at the sub-micron scale.

The value of about 300MPa is estimated from the failure loads of the pre-notched specimens. In fact, since the initial fractureenergy Gf

μ is only a fraction of the total energy dissipated and the initial strength ftμ is significantly larger than in macroscopic tests,

the Irwin’s characteristic length = ∗l E G f/( )chμ

tμ 2 related to the initial formation of the FPZ is significantly smaller than the width of

the notches investigated in this work. Accordingly, following [45] the fracturing behavior of the notched specimens must depart fromthe one of the cracked specimens and the nominal stress at failure is determined by the elastic limit condition =σ f k/Nc t

μ . It is worthmentioning here that, thanks to the foregoing considerations, the initial strength can be determined very precisely. Small variationson the value proposed in this work would lead to significant changes on the predicted structural strength.

It is worth mentioning here that the microscale strength estimated in this work is significantly larger than the values obtainedfrom typical macroscopic tests. To the best the authors’ knowledge, a direct in-situ measurement of the polymer strength has neverbeen accomplished or reported in the open literature. This is due to the evident challenges deriving from conducting a tensile test on amicro thermoset specimen. However, some researchers did try to estimate the strength by means of indirect methods. In 2007,Hobbiebrunken et al. [53] proposed an experimental approach for the investigation of the microscale strength in epoxy polymers.

Fig. 12. (a) Schematic of the two-scale cohesive law proposed in this study describing the cohesive stresses associated to micro to macro crackopening displacements; (b) Bi-linear portion describing the fracturing for large crack opening displacements.

Table 2Calibrated cohesive parameters for linear, bi-linear and two-scale cohesive laws.

Cohesive Law GF (N/mm) GFb (N/mm) GF

l (N/mm) Gf (N/mm) Gfb (N/mm) ft (MPa) ft

μ (MPa) σk (MPa)

Two-scale 0.8 / / 0.645 / 55 300 5Bi-linear / 0.78 / / 0.625 55 / 5Linear / / 0.78 / / 55 / /

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

110

Page 12: Engineering Fracture Mechanics - Salviato Lab

With the inherent assumption that the source of size effect on the strength is mainly statistical-energetic (Type I), they manufacturedepoxy fibers featuring diameters from 22 to 50 μm and various gauge volumes and they tested them under tensile loading. Theirresults showed a significant size effect, the tensile strength reaching 166MPa for a gauge volume of × −1.5 10 3 mm3 compared to astrength of 92MPa estimated from tensile tests on traditional dog bone specimens (gauge volume 66mm3). Although the gaugevolume for the microscale tests was still not small enough, the authors concluded from the reported size effect that the inherentstrength of epoxy at the microscale can be close to its theoretical value, estimated as 275MPa in their work. Although the thermosetsystem investigated in Hobbiebrunken et al. [53] is not the same as the one characterized in this study, their strength is very close tothe value estimated in the present experimental and computational investigation.

Another indirect approach was proposed in 2016 by Zike et al. [54] who investigated the strain deformation close to the tip of ablunt notch in Double Cantilever Beam (DCB) specimens made of epoxy. The 2mm-deep specimens had a length of approximately70mm, a width of 10mm and featured a 0.7mm wide notch. The specimens were speckled for Digital Image Correlation (DIC) andtested in an Environmental Scanning Electron Microscope (ESEM). Leveraging the strain field measured from DIC and the loadcaptured by the load-cell, the authors were able to estimate the nonlinear stress-strain response of the polymer by estimating the J-integral from DIC for different subcritical load levels. They concluded that the strength of the polymer they investigated must bewithin 220–300MPa, a range very close to the one estimated in the present study. Although it can be argued that the specimens werenot micrometric, it should be noted that the estimate of the constitutive behavior was conducted using in situ strains. The agreementbetween Zike’s and the results of this work, using two completely different approaches further corroborates the possibility of asignificantly larger strength at the microscale.

Having clarified the reason for the very high strength at the microscale, another question needs to be answered: why do thecohesive stresses of the initial portion of the cohesive curve decrease so steeply? The reason is that, if the initial slope of the cohesivecurve were milder, it would not be possible to capture the size effect data on pre-cracked specimens. In the presence of a crack or asufficiently sharp notch, the stress intensity at the tip at incipient failure would always lead to elastic stresses that are greater than themicroscale strength. Accordingly, since the cohesive stresses decrease very quickly in the initial part of the cohesive curve, a cohesiveFPZ can emerge and develop up to the second portion of the two-scale cohesive curve. The cohesive law becomes equivalent to the bi-linear law of strength ft and total fracture energy GF

b since the initial energy Gfμ is only about 2.5% of GF or, in other words,

≈ ≈G G 0.80F Fb N/mm. Thanks to the foregoing considerations it is possible to estimate the initial slope of the two-scale cohesive

model very precisely by testing notched specimens of various notch tip radii and matching the experimental data by means of thetwo-scale cohesive model. Such a comprehensive experimental campaign is beyond the scope of the present work and will be thesubject of future contributions. In this study, the initial fracture energy was calibrated against one notch tip radius only, leading toGf

μ =0.02 N/mm.Finally, one last question remains: why was not the foregoing cohesive behavior found before from tests on pre-cracked speci-

mens? The answer lies in the particular shape of the cohesive curve. Since the initial fracture energy is only a negligible portion of thetotal fracture energy and, in laboratory-scale cracked specimens the cohesive stresses always overcome the initial portion of thecohesive law, it is not possible to characterize ft

μ and Gfμ unless micro-metric specimens are tested. In fact, any tests on large

specimens would provide an estimate of GF which takes approximately the same value as GFb. It is worth mentioning here that micro-

tests are very challenging and are generally affected by significant uncertainties. However, based on the results of this work, a validalternative is to test both cracked and pre-notched specimens and use the results on the latter configuration to characterize ft

μ.To verify the foregoing assumptions, both pre-cracked and pre-notched specimens were simulated in ABAQUS/Explicit 2017. The

models combined 4-node bi-linear plain strain quadrilateral elements (CPE4R) with a linear elastic isotropic behavior and 4-nodetwo-dimensional cohesive elements (COH2D4) with the traction-separation law shown in Fig. 13 to model the crack. To guarantee anaccurate characterization of the cohesive behavior, 20 elements were assigned in the microscale FPZ leading to 600 elements for thetotal FPZ. The width of the cohesive band was 0.1 μm and the stable time increment was about −10 9 s. The parameters of the two-scalecohesive model that provided the best matching with the experimental data are given in Table 2.

Fig. 13 shows a comparison between the experimental load-displacement curves and simulations by the two-scale cohesive model.As can be noted the model successfully matches the experimental curves of both the pre-notched and pre-cracked specimens ofdifferent sizes. The most remarkable aspect related to these results is that the excellent matching was possible leveraging the samecohesive law which can now be treated as a material property.

It is interesting to plot the structural strength σNc as a function of the structure size D in double-logarithmic scale. As can be notedfrom Fig. 14a, the computational model is in excellent agreement with the experimental data for all the sizes and types of notch.

In case of cracked specimens the radius at the crack tip is extremely small, approximately zero. For sufficiently large specimens,the FPZ size is negligible compared to the structure size and, as predicted by LEFM, the structural strength scales with −D 1/2. Fordecreasing sizes, the fraction of the structure occupied by the nonlinear FPZ becomes larger and larger thus affecting the fracturingbehavior. The structural strength departs from the values predicted by the LEFM which is a linear theory and, inherently, ignores thecohesive stresses in the FPZ. For sufficiently small geometrically-scaled structures the nominal strength tends to the plastic limit.

For the cases shown in Fig. 14a for pre-notched specimens, all the geometrical features of the structure are geometrically-scaledexcept for the thickness and notch tip radius, b. Accordingly, the aspect ratio of the notch a b/0 increases along with the structure size,D. As can be noted, for sufficiently large specimens with the investigated aspect ratios, the structural strength coincides with theelastic limit. For decreasing sizes with the same aspect ratios, the specimens tend to be similar to the cracked specimens andeventually converge to the performance of cracked specimens when the specimen size is sufficiently small. It is worth mentioninghere that the elastic limit can also be obtained without using any cohesive models. As shown in Fig. 14b, the smooth curve represents

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

111

Page 13: Engineering Fracture Mechanics - Salviato Lab

the elastic limit calculated numerically in ABAQUS/Implicit for various specimen sizes whereas the diamond symbols represent theelastic limit for all the investigated specimen sizes obtained from the experiments. The computational model featured 8-node bi-quadratic plane strain quadrilateral elements (CPS8) with a linear elastic isotropic behavior. The smallest mesh size at the notch tip

Fig. 13. Cohesive zone modeling results: (a) calibrated cohesive law; (b) load-displacement curves vs. two-scale Cohesive Zone Model (CZM)simulations for both pre-cracked and pre-notched specimens of different sizes.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

112

Page 14: Engineering Fracture Mechanics - Salviato Lab

was about 0.1μm leading to 20 elements in the microscopic FPZ and 600 elements for the total FPZ. In the expression of elastic limit=ψ b D f k b D( , ) / ( , )t

μ , the stress concentration factor is a function of D since the notch tip radius is not geometrically scaled. As can benoted in Fig. 14b, the function ψ b D( , ) agrees very well with the experimental data. This confirms that, for the notched specimensinvestigated in this work, all the failures happened by approximately reaching the elastic limit of the structure.

On the other hand, it is also interesting to have a comparison between the simulations by the two-scale cohesive model and thegeneralized cohesive crack size effect curves proposed by [42]. As shown in Fig. 14b, the generalized CCSECs are plotted by using theinitial fracture energy for pre-notched specimens while both initial and total fracture energy are used for pre-cracked specimens. Ascan be noted, for the smallest aspect ratio investigated (a b/0 =25), the simulation results exactly follow the generalized CCSEC. Thisindicates that, for decreasing the specimen size, the cohesive stress only develops into the first portion of the two-scale cohesive lawsince the generalized CCSEC is based on the assumption of a linear cohesive law and the initial fracture energy is used for thesecurves. However, for increasing aspect ratios, the simulation results start to depart from the generalized CCSEC. This is not surprisingsince the cohesive stress in these cases develops into the remaining portion of the two-scale cohesive law. For sufficiently large aspectratios ( = ∞a b/0 ), the simulation results are located between the two asymptotes, the generalized CCSEC by using initial fractureenergy and total fracture energy. These two asymptotes successfully capture the simulation results for sufficiently small and largespecimens. This indicates that the cohesive stress completely develops for sufficiently large specimens thus the shape of the cohesivelaw does not affect the results. However, for decreasing the specimen size, the cohesive stress develops only partially and the shape ofthe cohesive law plays an important role as shown in the figure.

Finally, it is interesting to compare the cohesive stresses at the condition of incipient failure for the case of the specimensweakened by cracks and notches of equal depth considered in this work. As shown in Fig. 15a and b, the cohesive stresses in these twocases differ significantly, in accordance with the observations that motivated the proposed two-scale cohesive model. For notches, theFPZ is developed only partially, with the minimum cohesive stress being significantly larger than ft. This means that, in agreementwith the theory, only the initial part of the two-scale cohesive model governs the behavior of the notched structures investigated in

Fig. 14. Size effect on the notched and cracked specimens investigated in this work: (a) nominal strength predicted by the two-scale cohesive modelas a function of the specimen size. Note the excellent agreement with the experimental data for both cracked and notched specimens; (b) comparisonon the elastic limit between simulations without any cohesive model and experimental data for notched specimens; (c) comparison between the two-scale cohesive model and the Generalized Cohesive Crack Size Effect Curves using the initial and total fracture energy [42].

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

113

Page 15: Engineering Fracture Mechanics - Salviato Lab

this work. In contrast, in case of pre-cracked specimens, the FPZ is far more developed and the minimum cohesive stress at the tip isgenerally much lower than ft. This means that the second part of the cohesive law is entered and the cohesive behavior is equivalentto a bi-linear cohesive law of total fracture energy ≈G GF

bF . All the foregoing observations confirm firmly the validity of the two-scale

cohesive model and the following information provides a significant insight on the main damage mechanisms associated with thetwo-scale cohesive model.

In a recent publication, Yang and Qu [55] used a course grained molecular dynamics model to investigate the tensile behavior ofthermoset polymers. In contrast to other MD simulations, their approach enabled the analysis of one of the largest volumes eversimulated by MD (85×85×85 μm). Thanks to this, it was possible to capture phenomena such as void formation and coalescence

Fig. 15. FE models and cohesive stresses at the condition of incipient failure for (a) pre-cracked specimen and (b) pre-notched specimen.

Fig. 16. Two-scale cohesive law with related main damage mechanisms. The pictures are the MD simulation results of the deformation of athermoset polymer. Pictures reproduced from Yang and Qu 2014 [55].

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

114

Page 16: Engineering Fracture Mechanics - Salviato Lab

which occur at a length scale that is typically one or two orders of magnitude larger than the length of a polymer chain. The result ofthe simulation is shown in Fig. 16a–e which were re-adapted from [55] and show the typical deformation occurring during a tensiletest. In the figures, the volumes colored in yellow represent regions of high-densely packed polymer chains. In grey are the volumes oflow density featuring voids and particles with very low coordination numbers. According to [55] the material reaches its yieldstrength of 298MPa at about 7% strain. As reported by [55], this happens by micro-crazing and stretching of the densely packedregions with cavity nucleation and coalescence in the remaining portion of the domain (note that a similar mechanism was observedexperimentally in [50–52]). The opening displacement required to stretch and re-align the short polymer chains in the denselypacked regions are low, in the order of several nanometers. After that, cavity nucleation and coalescence as shown in Fig. 16b–e leadsto a sudden drop of the tensile stress. As shown in Fig. 16b–e, cavities grow both longitudinally and laterally. The scenario presentedby the foregoing MD analysis is in perfect agreement with the two-scale cohesive law proposed in this study. In fact, the first steepdecrease of the cohesive stresses in the proposed traction-separation law captures the sudden drop of the stress due to the cavityinitiation and coalescence occurring at sub-micron opening displacements. The second branch of the cohesive curve captures the laterstages of the fracturing process occurring at significantly lower stresses by re-alignment of the polymer chains in the less denselypacked regions. This happens with a very mild decrease of the cohesive stresses.

Furthermore, based on the two-scale model proposed in this work, some considerations on the applicability of LEFM for a courseestimate of the failure load can be made. In case of pre-cracked specimens, LEFM can only be applied when the FPZ size is negligiblecompared to the structure size. If this condition is met for micro-metric structures, then the fracture energy to be used in thecalculations is Gf

μ. This means that to predict e.g. the onset of a microcrack in a polymer matrix composite by LEFM, one has to firstverify that the micro Irwin’s characteristic length lch

μ is significantly smaller than the smallest geometrical features (e.g. fiber diameter,inter-fiber distance etc). On the other hand, if the condition of negligible FPZ size is met for structures of size ≫ =D l E G f* /ch F t

2, thetotal fracture energy GF should be used in the calculations.

In case of pre-notched specimens, LEFM can only be applied when the notch tip radius is extremely small, smaller than the microIrwin’s characteristic length. If this condition is met, then the same observations on the applicability of LEFM to pre-crackedstructures hold.

Last but not least, it is worth mentioning that thermosets are inherently less viscous compared to other polymers such as e.g.thermoplastics thanks to the very short polymer chains and strong covalent bonds between particles. Yet, as shown for concrete andother quasibrittle materials [56,57], thermosets do exhibit strain rate dependence and increase of brittleness as the loading ratebecomes slower. To reduce the foregoing effects on the size effect data, the applied displacement was scaled with the specimen size soto keep the same nominal strain rate as mentioned in Section 2.3. While this approach limited strain-rate-dependent shifts of the sizeeffect curve, it could not prevent the fact that the two-scale cohesive model we are proposing holds for the given strain-rate appliedduring the tests. Although the authors think that the traction-separation law would change significantly only for extremely largerates, a study on the rate effect would be needed to validate this statement. This is beyond the scope of this work and will be subject offuture investigations.

5. Conclusions

This study investigated the fracturing behavior of thermoset polymer structures featuring cracks and sharp u-notches. Based onthe results obtained in this work, the following conclusions can be elaborated:

1. the fracturing behavior of thermoset polymer structures featuring cracks is distinctively different than the one of structuresweakened by u-notches. Although this result is not surprising for blunt notches with large tip radii, it was not expected for thenotches investigated in this work which featured a tip radius significantly smaller than the Irwin’s characteristic length of thematerial;

2. for the cases studied in this work, the direct use of LEFM to estimate the mode-I fracture energy of the material provides valuesfor the notched specimens that are about 10 times larger than the ones estimated from cracked specimens (7.2 N/mm for notchedspecimens vs 0.78 N/mm for cracked specimens);

3. in contrast to what has been proposed in the literature, plastic residual stresses induced by the manufacturing of the notch cannotbe the dominant cause of such a high apparent fracture toughness. In fact, the residual stresses that would be required to justifysuch a high toughness are larger than 100MPa and it seems unlikely that they can be generated by the sawing procedure. Thisstatement is supported by additional tests on specimens with notches created by pre-inserting a sharp razor blade. Even if in suchcase plastic stresses cannot be induced, the failure loads were similar to those of the other notched specimens. A further con-firmation for this statement is provided by the results from Transimission Optical Microspocy (TOM) with cross-polarized lightshowing almost negligible residual stresses due to the pre-notching process.

4. to capture the behavior of both the cracked and notched specimens, a two-scale cohesive model is proposed. For sub-micron crackopening displacements, the cohesive strength is about 300MPa and the cohesive stresses decrease linearly and very steeply withincreasing crack openings. When the cohesive stresses reach the strength estimated from typical laboratory-scale specimens(about 55MPa) and the opening displacements are beyond the sub-micron range, the slope of the cohesive law becomes milder.Finally, the cohesive law ends with another linear branch starting when the stresses reach 5MPa. This last part of the cohesivelaw is characterized by a very slow decay of the cohesive stresses;

5. the proposed two-scale cohesive model captures the fact that, due to energetic-statistical size effect [58–62], the microscopicstrength of the polymer can be several times higher than the one measured from macroscopic tests [53–55]. At the same time, the

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

115

Page 17: Engineering Fracture Mechanics - Salviato Lab

steep initial part of the cohesive curve, leading to an initial fracture energy of only about 2.5% of the total one, captures the lowerenergy dissipation occurring at the sub-micron scale;

6. thanks to the foregoing model the different behavior of polymer structures featuring cracks and sharp u-notches can be easilyexplained. Since the initial fracture energy Gf

μ is only a fraction of the total energy dissipated and the initial strength ftμ is

significantly larger than in macroscopic tests, the Irwin’s characteristic length = =∗l E G f/( ) 5.4chμ

tμ 2 mμ related to the initial

formation of the FPZ is significantly smaller than the width of the notches investigated in this work. Accordingly, the fracturingbehavior of the notched specimens must depart from the one of cracked specimens and the nominal stress at failure is determinedby the elastic limit condition [42], i.e. =σ f k/Nc t

μ with =k stress concentration factor. On the other hand, in laboratory-scaledcracked specimens the cohesive stresses in the FPZ at incipient failure always reach the second portion of the cohesive curve. Insuch a case, the Irwin’s characteristic length lch describing the FPZ is significantly larger since it depends on the total fractureenergy ≈G G40F f

μ and a significantly lower strength ≈f f /6t tμ . Accordingly, the failure behavior of macroscopic cracked spe-

cimens depends only on the total fracture energy and macroscopic strength;7. the higher fracture energy of pre-notched specimens has not meaning since the investigated specimens fail at the elastic limit.

This indicates that LEFM cannot be applied to the calculation of the fracture energy for pre-notched specimens in general, unlessthe width is sufficiently lower than the Irwin’s characteristic length associated to the initial part of the traction-separation law;

8. since the initial fracture energy is only a negligible portion of the total fracture energy and, in laboratory-scale cracked specimensthe cohesive stresses always overcome the initial portion of the cohesive law, it is not possible to characterize ft

μ and Gfμ unless

micro-metric specimens are tested. These tests are very challenging and are generally affected by significant uncertainties. Basedon the results of this work, a valid alternative is to test both cracked and pre-notched specimens and use the results on the latterconfiguration to characterize ft

μ;9. the foregoing considerations are supported by the excellent agreement between the model predictions and experimental data on

geometrically-scaled Single Edge Notch Bending (SENB) specimens. As can be noted from Figs. 13 and 14, the two-scale cohesivemodel is able to capture not only the load-displacement curves but also the related size effect on structural strength for bothcracked and pre-notched specimens;

10. the foregoing results are of utmost importance for the design of microelectronic devices or polymer matrix composites. In fact, inthese systems, the main damage mechanisms belong to the sub-micron scale and the fracturing behavior is dominated by theinitial portion of the cohesive law since the opening displacements are not large enough to reach the second portion of the curve.The two-scale model proposed in this work and the related testing protocol, represent a first step towards the accurate yet simplesimulation of thermoset polymers at the microscale.

Acknowledgments

Marco Salviato acknowledges the financial support from the Haythornthwaite Foundation through the ASME HaythornthwaiteYoung Investigator Award and from the National Science Foundation (NSF) through grant CMMI 1624513. This work was alsopartially supported by the Joint Center For Aerospace Technology Innovation through the grant titled “Design and Development ofNon-Conventional, Damage Tolerant, and Recyclable Structures Based on Discontinuous Fiber Composites”.

References

[1] lukaszewicz DHJA. Automotive composite structures for crashworthiness. In: Elmarakbi A, editor. Advanced composite materials for automotive applictions:structural integrity and crashworthiness. Wiley; 2013. p. 100–27.

[2] Rana S, Fangueiro R. Advanced composites materials for aerospace engineering. United Kindom: Woodhead Publishing; 2016.[3] Song D, Gupta RK. The use of thermosets in the building and construction industry. In: Guo QP, editor. Thermosets: structure, properties, and applications.

Elsevier; 2012. p. 165–88.[4] Maier G. Polymers for microelectronics. Mater Today 2001;4:22–33.[5] Stenzenberger HD. Recent developments of thermosetting polymers for advanced composites. Compos Struct 1993;24:219–31.[6] Pascault, Williams RJJ. Overview of thermosets: present and future. In: Guo QP, editor. Thermosets: structure, properties, and applications. Elsevier; 2018. p.

3–34.[7] Takahashi J, Wan Y. Tensile and compressive properties of chopped carbon fiber tapes reinforced thermoplastics with different fiber lengths and molding

pressures. Compos Part A 2016;87:271–81.[8] Blok LG, Longana ML, Yu H, Woods BKS. An investigation into 3D printing of fibre reinforced thermoplastic composites. Addit Manuf 2018;22:176–86.[9] Yao SS, Jin FL, Rhee KY, Hui D, Park SJ. Recent advances in carbon-fiber-reinforced thermoplastic composites: a review. Compos Part B 2018;142:241–50.[10] Biron M. Future prospects for thermoplastics and thermoplastic composites. In: Biron M, editor. Thermoplastics and thermoplastic composites. Plastics Design

Library; 2018. p. 1083–126.[11] Knauss WG. Mechanics of polymer fracture. Appl Mech Rev 1973;26:1–17.[12] Narisawa I, Murayama T, Ogawa H. Internal fracture of notched epoxy resins. Polymer 1982;23:291–4.[13] Kinloch AJ, Williams JG. Crack blunting mechanisms in polymers. J Mater Sci 1980;15:987–96.[14] Dear JP, Williams JG. Simple method of determining the fracture-resistance for rapidly propagating cracks in polymers. J Mater Sci 1993;28:259–64.[15] Argon AS, Cohen RE, Mower TM. Mechanisms of toughening brittle polymers. Mater Sci Eng 1994;176:79–90.[16] Argon AS, Cohen RE. Toughenability of polymers. Polymer 2003;44:6013–32.[17] Kwon HJ, Jar PY. Fracture toughness of polymers in shear mode. Polymer 2005;46:12480–92.[18] Xiao K, Ye L, Kwok YS. Effects of precracking methods on fracture behaviour of an Araldite-F epoxy and its rubber-modified systems. J Mater Sci

1998;33:2831–6.[19] Cayard MS, Bradley WL. The effect of various precracking techniques on the fracture toughness of plastics. General papers, International Conference of Fracture

ICF7 (Houston. USA);1989. p. 2713–23.[20] Salazar A, Patel Y, Williams JG. Influence of crack sharpness on the fracture toughness of epoxy resin. In: 13th international conference on fracture.[21] Souza JM, Yoshimura HN, Peres FM, Schön CG. Effect of sample pre-cracking method and notch geometry in plane strain fracture toughness tests as applied to a

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

116

Page 18: Engineering Fracture Mechanics - Salviato Lab

PMMA resin. Polym Test 2012;31:834–40.[22] Kuppusamy N, Tomlinson RA. Repeatable pre-cracking preparation for fracture testing of polymeric materials. Eng Fract Mech 2016;152:81–7.[23] Zappalorto M, Salviato M, Quaresimin M. Mixed mode (I+ II) fracture toughness of polymer nanoclay nanocomposites. Compos Sci Technol 2013;111:50–64.[24] Zappalorto M, Salviato M, Pontefisso A, Quaresimin M. Notch effect in clay-modified epoxy: a new perspective on nanocomposite properties. Compos Interface

2013;20:405–19.[25] Zappalorto M, Salviato M, Quaresimin M. Influence of the interphase zone on the nanoparticle debonding stress. Compos Sci Technol 2011;72:49–55.[26] Zappalorto M, Salviato M, Quaresimin M. A multiscale model to describe nanocomposite fracture toughness enhancement by the plastic yielding of nanovoids.

Compos Sci Technol 2012;72:1683–91.[27] Salviato M, Zappalorto M, Quaresimin M. Plastic shear bands and fracture toughness improvements of nanoparticle filled polymers: a multiscale analytical

model. Compos Part A 2013;48:144–52.[28] Salviato M, Zappalorto M, Quaresimin M. The effect of surface stresses on the critical debonding stress around nanoparticles. Int J Fract 2011;172:97–103.[29] Quaresimin M, Salviato M, Zappalorto M. A multi-scale and multi-mechanism approach for the fracture toughness assessment of polymer nanocomposites.

Compos Sci Technol 2014;91:16–21.[30] Mefford CH, Qiao Y, Salviato M. Failure and scaling of graphene nanocomposites. Compos Struct 2017;176:961–72.[31] Qiao Y, Salviato M. Study of the fracturing behavior of thermoset polymer nanocomposites via cohesive zone modeling. Compos Struct 2019. https://doi.org/10.

1016/j.compstruct.2019.03.092.[32] Hexion, Bellevue, USA http://hexion.com.[33] Vacmobiles, Auckland, New Zealand https://www.vacmobiles.com/.[34] TAP Plastics, Seattle, USA http://www.tapplastics.com.[35] Goḿez FJ, Elices M, Berto F, Lazzarin P. A generalized notch stress intensity factor of U-notched components loaded under mixed mode. Eng Fract Mech

2008;75(16):4819–33.[36] Gómez FJ, Guinea GV, Elices M. Failure criteria for linear elastic materials with U-notches. Int J Fract 2006;143:99–113.[37] Atrzori B, Lazzarin P, Meneghetti G. Fracture mechanics and notch sensitivity. Fatigue Fract Eng M 2003;26:257–67.[38] Atrzori B, Lazzarin P. Notch sensitivity and defect sensitivity under fatigue loading: two sides of the same medal. Int J Fract 2001;107:1–8.[39] ASTM D5045-99 - Standard Test Methods for Plane-Strain Fracture Toughness and Stain Energy Release Rate of Plastic Materials; 1999.[40] Qiao Y, Salviato M. A cohesive zone modeling study on the fracturing behavior of thermoset polymer nanocomposites. In: Proceedings to 33rd Annual Technical

Conference, 18th US-Japan Conference on Composite Materials ASTM D30.[41] Guo K, Qiao Y, Salviato M. Scaling of fatigue crack growth in pristine epoxy. In: Proceedings to 33rd Annual Technical Conference, 18th US-Japan Conference on

Composite Materials ASTM D30.[42] Di Luzio G, Cusatis G. Cohesive crack analysis of size effect for samples with blunt notches and generalized size effect curve for quasi-brittle materials. Eng Fract

Mech 2018;204:15–28.[43] Ko S, Tuttle ME, Yang J. and Salviato M, Characterization and computational modeling of the fracturing behavior in discontinuous fiber composite structures.

Proceedings to 33rd Annual Technical Conference, 18th US-Japan Conference on Composite Materials ASTM D30.[44] Deleo AA, Salviato M. Computational study for size effect in composites and nanocomposites. In: Proceedings to 33rd annual technical conference, 18th US-

Japan conference on composite materials ASTM D30.[45] Cusatis G, Schauffert EA. Cohesive crack analysis of size effect. Eng Fract Mech 2009;76:2163–73.[46] Bažant ZP, Daniel IM, Li Z. Size effect and fracture characteristics of composite laminates. J Eng Mater-T ASME 1996;118:317–24.[47] Bažant ZP, Planas J. Fracture and size effect in concrete and other quasi-brittle materials, Boca Raton; 1998.[48] Salviato M, Kirane K, Ashari SE, Bažant ZP. Experimental and numerical investigation of intra-laminar energy dissipation and size effect in two-dimensional

textile composites. Compos Sci Technol 2016;135:67–75.[49] Tada H, Paris PC, Irwin GR. The stress analysis of cracks handbook, third ed., New York; 2000.[50] Morgan RJ, O’Neal JE. The microscopic failure processes and their relation to the structure and their relation to the structure of amine-cured bisphenol-A-

diglycidyl ether epoxies. J Mater Sci 1977;12:1966–80.[51] Sue HJ, Garcia-Meitin EI, Yang PC, Bishop MT. Crazing in high-performance thermoset resins. J Mater Sci Lett 1993;12:1463–6.[52] Sue HJ, Bertram JL, Garcia-Meitin EI, Puckett PM. Characterization and promotion of dilatation bands in toughenable thermosetting resins. J Polym Sci

1995;33:2003–17.[53] Hobbiebrunken T, Fiedler B, Hojo M, Tanaka M. Experimental determination of the true epoxy resin strength using micro-scaled specimens. Compos Part A

2007;38:814–8.[54] Zike S, Sorensen BF, Mikkelsen LP. Experimental determination of the micro-scale strength and stress-strain relation of an epoxy resin. Mater Des 2016;98:47–60.[55] Yang SR, Qu JM. Coarse-grained molecular dynamics simulations of the tensile behavior of a thermosetting polymer. Phys Rev E 2014;90.[56] Bažant ZP, Gettu R. Rate effects and load relaxation in static fracture of concrete. Aci Mater J 1992;89:456–68.[57] Di Luzio G. Numerical model for time-dependent fracturing of concrete. J Eng Mech 2009;135:632–40.[58] Salviato M, Bažant ZP. The asymptotic stochastic strength of bundles of elements exhibiting general stress-strain laws. Probabilist Eng Mech 2014;36:1–7.[59] Salviato M, Kirane K, Bažant ZP. Statistical distribution and size effect of residual strength of quasibrittle materials after a period of constant load. J Mech Phys

Solids 2014;64:440–54.[60] Le JL, Bažant ZP, Bažant MZ. Unified nano-mechanics based probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack growth, lifetime

and scaling. J Mech Phys Solids 2011;59:1291–321.[61] Bažant ZP, Le JL, Bažant MZ. Scaling of strength and lifetime probability distributions of quasibrittle structures based on atomistic fracture mechanics. Proc Nat

Acad Sci 2009;106(28):11484–9.[62] Xu Z, Le JL. A first passage based model for probabilistic fracture of polycrystalline silicon MEMS structures. J Mech Phys Solids 2017;99:225–41.

Y. Qiao and M. Salviato Engineering Fracture Mechanics 213 (2019) 100–117

117


Recommended