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Engineering Fracture Mechanics

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

Mode-I dynamic fracture initiation toughness using torsion load

Ali Fahem, Addis Kidane⁎, Michael A. SuttonDepartment of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, United States

A B S T R A C T

An experimental and numerical approach is proposed to determine the dynamic fracture initiation toughness of materials from a cylindricalspecimen with spiral surface crack subjected to dynamic torsional load using a torsional Hopkinson bar apparatus. The torsion load creates pre-dominantly tensile stress perpendicular to the spiral crack of the specimen, resulting in nominally Mode I conditions. The torque applied to thespecimen is measured by strain gages attached to the bar and the time at which the crack propagation initiated is measured using stereo imaging andstereo digital image correlation. Using the measured torque and the time of fracture as input, a commercial FE package, ABAQUS, is utilized toanalyze the spiral crack and numerically extract the dynamic fracture parameters. A 3D format of the dynamic interaction integral method is utilizedto calculate the three components of the applied dynamic stress intensity factors. The result demonstrates that the spiral crack-torsional loadingconfiguration indeed generates nominally Mode I conditions and can be used to measure the dynamic fracture initiation toughness. To demonstratethe proposed method, three aluminum alloys; Al 7050-T6, Al 2024-T3, and Al 6061-T6, were experimentally studied. The results are consistent,repeatable and in good agreement with literature data.

1. Introduction

Fracture mechanics has been a subject of interest in the engineering community for decades. During this period, fracture para-meters such as Stress Intensity Factors (SIFs), J-integral, Crack-Tip Opening Displacement (CTOD), Crack-Tip Opening Angle (CTOA)and the three-dimensional Crack Tip Displacement (CTD) have been developed and used to characterize the fracture properties ofmany engineering materials. Under quasi-static loading conditions fracture parameters typically are obtained experimentally byusing standard methods such as ASTM E399 for the Mode I stress intensity factor, KIc

static, E1820 for elastic-plastic toughness, JIcstatic,

and E1920 to evaluate CTOD [1–4]. These parameters are essential in the selection and judgment of materials for a particularengineering design application.

Conversely, in many critical engineering applications, components are subjected to sudden or high strain loading which couldresult in dynamic fracture. Quasi-static methods are insufficient to accurately determine the dynamic fracture parameters in materialsunder extreme conditions [5]. In light of this, investigators have been trying to develop experimental methods to determine thedynamic fracture initiation toughness of materials subjected to extreme loading conditions.

Currently, there are two methods (with some modifications) that have been widely used to estimate the dynamic fracturetoughness of materials: Charpy impact test with V-notched specimens, and the Hopkinson pressure bar apparatus. The Charpy impacttest is a standard method to determine the amount of energy absorbed by a material during fracture. One of the limitations of theCharpy test is that the fracture strength can be measured only at an intermediate loading rate (10–100/s). Also, there is no physicalrelationship between the hammer load and fracture parameters. In this case, empirical equations are used to estimate fractureparameters [6]. Due to its simplicity, despite its well-known theoretical weaknesses, the standard Charpy test is still popular in theindustry to characterize fracture toughness of materials at intermediate loading rates [7–9].

The split Hopkinson pressure bar (SHPB) is a widely utilized method to characterize the dynamic behavior of materials at high

https://doi.org/10.1016/j.engfracmech.2019.03.039Received 22 November 2018; Received in revised form 22 March 2019; Accepted 25 March 2019

⁎ Corresponding author at: 300 Main St, Columbia, SC 29208, United States.E-mail address: kidanea@cec.sc.edu (A. Kidane).

Engineering Fracture Mechanics 213 (2019) 53–71

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

T

strain rates, up to −10, 000 s 1 [10]. Though SHPBs are mainly employed to obtain the high strain rate constitutive response ofmaterials [11], they have been modified and used to investigate the dynamic fracture toughness of materials [6,12,13]. Typically, thecompression SHPB apparatus with three points bending and Brazilian disk specimens, have been used to obtain Mode I and mixedMode I/II fracture parameters, respectively. Due to the unavailability of a closed-form solution in the SHPB experiment, a quasi-staticequation oftentimes is used to extract the fracture parameters. In other words, to calculate the dynamic fracture toughness, re-searchers usually use the plane strain quasi-static fracture mechanics equation, Eq. (1), by replacing the static load (P) with dynamicload (P(t)) [14,15]. In this form, the technique can be used to estimate the fracture toughness of materials only if the time of fractureis sufficiently long to neglect inertia effects. To satisfy this condition and avoid transient effects, such experiments generally areperformed at a low impact speed, which limits the application of this method to low rate loading conditions [6,15].

= ⎛⎝

⎞⎠

× ⎧⎨⎩

⇒⇒

⎫⎬⎭

K t SBW

f aW

P StaticP t Dynamic( ) ( )I 3/2 (1)

where P is the load applied; a is the crack length, S is the span of the specimen, W is the width of the specimen and ( )f aW is the

geometry correction factor.Given the limitations noted above, an alternative method to measure the dynamic fracture toughness of materials is essential.On the other hand, Truss in 1984 and Sweeney in 1985 used a cylindrical specimen with small v-notch crack inclined at 45° to its

axis subjected the specimen to a pure torsion load to study the mode I fracture. Since pure torsion load produces principal tensilestress in a 45° plane, with a spiral notch at 45° the torsion load generates opening mode conditions and thus can be used to determinethe Mode I fracture toughness [16,17]. Similarly, a torsional specimen with a full spiral v-notch crack at 45° to its axis was used byWang and his group to determine the quasi-static fracture toughness of different materials, such as ceramics, metal, polymer, andconcrete [18–20].

More recently, the potential of the technique for studying the dynamic fracture properties of materials by using a torsionalHopkinson bar has been studied [21–25]. As the main advantage, torsional waves are non-dispersive, which allows the torsional waveto propagate along the bars without a change in its form. Due to this, in a torsional Hopkinson bar, strain gages can be placed at anyposition along the bar and reliable measurement can be measured. More importantly, the radial inertia does not affect the wavepropagation [26–29], which makes it ideal for measuring properties at low, intermediate and high strain rate loading while holdingthe equilibrium condition.

The objective of this work is to demonstrate the use of a cylindrical specimen with a spiral crack subjected to dynamic torsionalloading conditions to measure dynamic Mode I fracture properties of materials. To the authors’ knowledge, this is the first work usinga cylindrical specimen with spiral notch and a torsional Hopkinson bar to extract dynamic Mode I fracture properties for materials.The method takes advantages of the non-dispersive wave propagation properties of torsional waves and the negligible axial inertia ofa torsional wave at high strain rate. Dynamic fracture experiments are performed on cylindrical specimens with spiral grove using atorsional Hopkinson bar in conjunction with stereo digital image correlation. The torque related to fracture initiation and the time atwhich the crack propagation initiated are measured experimentally and finite element simulations are performed to extract fractureparameters based on the dynamic interaction integral method.

Nomenclature

List of symbols

J s( ) the energy release rate at point (s) correspondingto the weighted function q s( )k

J s¯ ( ) a dynamic weighted average of J-integral over thecrack front volume segment as shown in Fig. 7

V as illustrated in Fig. 7, the volume enclosed bysurfaces ±S S S S S, , , ,1 2 3 4

±S S, , , ,1 2 3 4 the crack face surfaces, an upper surface, anouter surface, an inner surface, and bottom surfacerespectively, of the volume domain shown in Fig. 7

sΓ( ) contour path around (s) point and perpendicularon the spiral crack front that swept along LΔ togenerate a volume integral domain (V)

qk the smooth continuous weight function (unity atthe surface close to the crack tip and vanish as theouter surface as shown in Fig. 7B)

ui displacementt timeσ ε;ij ij Cauchy stress tensor and strain tensors position along the crack frontρ material densityAq the projected area of the q-function

Table 1Properties of materials under quasi-static condition [30].

Aluminum alloy Densityρ g cc( / )

Modulus of elasticity E(GPa)

Poisson's ratioν( )

Yield stress(MPa)

Shear modulus(GPa)

Fracture toughnessK MPa m( )Ic

ASTM

2024-T3 2.78 73.10 0.33 324.00 28.00 32.00 (TL) B2106061-T6 2.70 68.90 0.33 276.00 26.00 29.00 (TL) B4297050-T6 2.81 71.70 0.33 503.00 26.90 27.50 (TL) B211

where TL is Orientation.

A. Fahem, et al. Engineering Fracture Mechanics 213 (2019) 53–71

54

2. Experimental approach

2.1. Material and specimen geometry

Spiral notched cylindrical torsion (SNT) specimens, with a spiral notch at 45° with respect to the longitudinal axis, are manu-factured from aluminum alloys 2024-T3, 6061-T6 and 7050-T6. These materials are common in aerospace and automobile appli-cations and are chosen to demonstrate the method. The as-received mechanical properties for all three materials are given in Table 1[30]. Several 3.15mm thick tubular cylinder specimens with a 45° spiral v-notch grove were prepared from the as-received solid bars(see Fig. 1). The spiral notch is machined on the outer surface of these specimens using a 4-axis lathe. The outer diameter, insidediameter and gage length of the specimens are 19.00mm, 12.70mm and 59.66mm, respectively. The notch depth is 2.00mm, andthe remaining ligament is 1.15mm. The accuracy of specimens dimensions was within±0.01mm. The average grain size forAluminum alloys 2024-T3, 6061-T6, and 7075-T6 used for the study is about 13.70 µm, 14.00 µm, and 31.70 µm respectively [31,32].The total number of grains across a 1000.00 μm thick ligament, in 2024-T3, 6061-T6, and 7075-T6 are 73, 72 and 32 respectively. Ingeneral, a representative volume element (RVE), for measuring the physical properties of materials, of 8–10 grains is sufficient andhence the as-manufactured specimen thickness and crack ligament are sufficient to extract continuum-level fracture parameters[33,34]. In addition, Knauss and Ravi-Chandra [35,36] have shown that, in brittle materials, the plane-stress and plane-strainconditions have very little influence on the Mode I dynamic fracture initiation value. Chao et al. [37,38] showed that the T-stress(higher-order term of William series) decreases as the loading rate increases. Thus, the effect of the three-dimensional stress state inthe vicinity of the crack tip during crack initiation at high loading rate is not significant.

2.2. Torsional Hopkinson bar (THB)

Fig. 2 illustrates a schematic of the experimental setup of THB used in this work. The full experimental setup, including the signalconditioning amplifier and the high-speed cameras, is shown in Fig. 3. Details of the THB and background are available in theliterature [26,27,39]. For the sake of completeness, a brief description is provided. The THB used in this work has a 240,000mm longincident bar and a 2300.00mm long transmitter bar. Both bars are 25.40mm in diameter and manufactured from high-strengthGrade 5 Titanium (ASTM B348). The bars are supported in Teflon rings in a horizontal plane and are free to rotate around theircentral axis. An internal hexagonal groove is manufactured at the end of the incident and transmitter bar. The spiral notch specimenis sandwiched between the two bars via the hexagonal joint. A thin layer of JB-Weld epoxy was applied around the hexagonalinterface to reduce slip between the specimen and the bars. The assembly provides a reliable and consistent connection that can beused to load the samples at high loading rate without slip.

During loading, a hydraulically driven rotary actuator, shown in Fig. 3, is used to apply and store shear strain in the 635.00mm

Fig. 1. Schematic of spiral V-notch torsion specimen (SNTS).

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portion of the incident bar located between the rotary actuator and the clamp system. The stored shear strain is suddenly released bybreaking a brittle notched bolt installed in the clamping mechanism. During this time, half of the stored shear strain propagatestowards the specimen through the incident bar, and half of the stored strain is released towards the clamp. Typical dynamicallypropagated and released strain signals are shown in Fig. 4. When the incident wave reached the specimen, some of the waves willtransmit to the output bar through the specimen, and the rest will reflect back to the incident bar. The incident, transmitted andreflected shear strain data is acquired using strain gauges attached to the bars. Two two-element 90-degree Rosette (MMF003193)strain gages are attached diametrically opposite each other, to minimize the bending effect, on both bars.

Fig. 2. Schematic of torsional Hopkinson bar (THP) apparatus (Dimensions in mm).

Fig. 3. The experimental setup used in this work: (A) high-speed cameras, (B) specimen, (C) signal condition and amplifier, (D) input bar, (E)clamping system, (F) store bar and (G) twist angle measurement.

Fig. 4. Typical shear strain plot; stored, dynamic propagated and static released.

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2.3. Experimental data analysis

Classical torsion theory and one-dimensional wave analysis are used to calculate the torque applied to the specimen, T t( )s . Theincident torque, T t( )i , is obtained as shown in Eq. (2):

= ×T t GD π γ t( )16

( )i I

3

(2)

where G is the shear modulus of the bar; D is the bar diameter and γ t( )I is the incident wave.As shown in Fig. 5, T t( )1 is the torque at the input bar-specimen interface, and T t( )2 is the torque at the output bar-specimen

interface and can be written as shown in Eqs. (3) and (4) [40];

= +T t GD π γ t γ t( )16

[ ( ) ( )]I R13

(3)

=T t GD π γ t( )16

[ ( )]T23

(4)

where γ t γ t γ t( ), ( ), ( )I R T are incident, reflected and transmitted shear strain, respectively.The complete input bar, specimen, and output bar assembly are modeled with appropriate boundary conditions. More details

regarding the model are provided later in the in a finite element solution section.

2.4. High-speed imaging and stereo-digital image correlation

Full-field measurements of the specimen surface around the edge of the crack face were obtained using stereo digital imagecorrelation (Stereo DIC or 3D-DIC). A typical speckle pattern around the crack edges with corresponding gray‐scale histograms isshown in Fig. 6A. The gray‐scale intensity depicted in Fig. 6B shows a bell-shaped distribution of the intensity pattern without havingsaturated pixels; such a distribution is suitable for DIC measurements [34]. Two high-speed Photron SAX-2 cameras with two sets ofTokina 100mm lenses are used to record the surface deformation around the spiral notch edges at a rate of 200,000 frames persecond with a resolution of 256× 152 pixels2 (8.11 pixel/mm). The images are processed using VIC-3D, a commercial digital imagecorrelation software developed and distributed by Correlated Solutions, Inc. The correlation and calibration parameters for the StereoDIC system depicted in Fig. 6c are shown in Tables 2a and 2b respectively. Typical torque-time relationships, measured and cal-culated based on displacement fields on the specimen from the 3D DIC measurements, are plotted in Fig. 6C. The two measurementsagree very well, with a maximum difference of less than 2.2%. The full field displacement u v and w, were also used to measurethe opening displacement at the crack edges and to estimate the time at which fracture initiated.

3. Numerical solution methodology

An outline of the numerical method used to extract the fracture parameters is shown in this section. The interaction integralmethod used in this work is first discussed. Then, the extraction of the stress intensity factor from the interaction integral method isexplained. Finally, the complete finite element model is presented.

3.1. Interaction integral for 3-D spiral crack

The J-integral method used in this work was first developed as a measure of energy release rate for non-linear materials near thecrack tip by Rice [41]. For dynamic condition, the J-Integral formulation of a non-growing crack is extended by adding the kineticenergy density T( ) to the strain energy density W( ), as shown in Eq. (5.1) [42–44].

∫ ⎜ ⎟= ⎛⎝

+ − ∂∂

⎞⎠→

J Lim W T n σ n ux

d( ) Γij ji

Γ 0 Γ 11 (5.1)

Fig. 5. Schematic representation of the specimen and the application of the torsional load.

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Fig. 6. Specimen geometry and typical speckle pattern and the corresponding grayscale value.

Table 2aVIC-3D stereo-DIC correlation parameters.

Image parameters Values

Subset size (pixels × pixels) 25.00× 25.00Subset spacing (pixels) 5.00Average speckle size (Pixel× Pixel) 5.00× 5.00Interpolation Optimized 8-tapGrid calibration dot spacing 5.00mmCalibration score 0.02Strain filter size and type 9.00 (Lagrange)Software Vic-3DStereo angle ≅ 14(Degree)Field of view (FOV) 21.00mm

Table 2bCalibration system parameters obtained of the stereo cameras setup used.

Parameter Camera 0 Camera 1 Relative position (Tx y z α β γ, , , , , )

Result SD* Result SD* Parameter Result SD*

Center (x) (pixels) 0496.79 02.036 0499.19 02.074 =Tx 167.05 (mm) 0.013Center (y) (pixels) 0511.16 03.715 0516.47 03.846 =Ty 01.90 (mm) 0.001Focal length (x) 5633.15 15.839 5628.16 15.858 =Tz 17.41 (mm) 0.384Focal length (y) 5633.17 15.878 5628.54 15.863 =Tα 00.07 (deg.) 0.000Skew (deg.) 0000.17 00.018 0000.02 00.018 =Tβ 13.00 (deg.) 0.000Kappa 1 0000.12 00.000 0000.13 00.000 =Tγ 00.71 (deg.) 0.000

SD* (Standard deviation).

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where

∫=W σ ε dεε

ij ij ij0

ij

(5.2)

= ∂∂

∂∂

T ρ ut

ut

12

i i(5.3)

For a 3-D curve (like spiral crack), the divergence theorem was applied to Eq. (5) to convert it from the line integral to area andvolume integral.

As shown in Fig. 7, the segment of volume integral domain at a specific point on the crack front is extended from point a to point cthrough the volume center point b. The general solution of J-integral of the volume segment on a spiral crack front is calculatedfollowing previous studies [42,45–48],

∫= = + +−J J s q s ds J J J¯ [ ( ) ( )] ¯ ¯ ¯S S S

St( ) 1 2 3a c

a

c

(6.1)

Fig. 7. 3-D Schematic of a Partition of Spiral Crack, Pointwise Volume Integral Domain, and q-Function.

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where

∫ ⎜ ⎟= ⎛⎝

∂∂

∂∂

−∂∂

⎞⎠

J σ ux

qx

Wqx

dV¯V ij

i

k

k

j

k

k1

(6.2)

∫ ⎜ ⎟= − ⎛⎝

∂∂

− ∂∂ ∂

⎞⎠

→J Wx

σ ux x

q dV Thermal strains effect¯ ( )V k

iji

j kk2

2

(6.3)

∫ ⎜ ⎟= − ⎛⎝

∂∂

− ∂ ∂∂ ∂

+ ∂ ∂∂ ∂ ∂

⎞⎠

→J Tqx

ρ u ut x

q ρ u ut t x

q dV Dynamic loading effects¯ ( )V

k

k

i i

kk

i i

kk3

2

2

2

(6.4)

and

∂∂

→Tqx

Flux of the Kinetic energy the direction of the crack propagation: ink

k (6.4.1)

∂ ∂∂ ∂

→ρ u ut x

q Represents the material acceleration:i i

kk

2

2 (6.4.2)

∂ ∂∂ ∂ ∂

→ρ u ut t x

q Identified the spatial gradient of the velocities:i i

kk

2

(6.4.3)

Then, the mean value of the J-integral at point b (the middle of the volume segment) can be written as,

∫∫

= = −J sJ s q ds

q dsJA

( )[ ¯ ( ) ] ¯

ba

ct

ac

t

a c

q (7)

For 3D elasto-dynamic problem, neglecting thermal effects, the J-integral can be written as,

∫= = + = ⎛⎝

− − + − ⎞⎠

−J J J J σ u σ ε ρ u u ρ u u u ρ u u u q dV¯ ¯ ¯ ¯ 12

12

a cact

V ij i ij ij i i i i i i i i j1 3 ,1 ,1 ,1 , (8)

On the basis of the dynamic J-integral formula, an auxiliary load field was added to the spiral’s crack front. The auxiliary J-integral, J aux, can be written as,

∫= ⎛⎝

− − + − ⎞⎠

J σ u σ ε ρ u u ρ u u u ρ u u u q dV¯ 12

12

auxV ij

auxiaux

ijaux

ijaux

iaux

iaux

iaux

iaux

iaux

iaux

iaux

iaux

j,1 ,1 ,1 , (9)

The auxiliary loading field Eq. (9) was added to the actual field load Eq. (8), thus the total J-integral around crack front can bewritten as,

∫= ⎡

⎣⎢

+ + − + + − + + +

+ + + − + + +⎤

⎦⎥J

σ σ u u σ σ ε ε ρ u u u u

ρ u u u u u u ρ u u u u u uq dV¯ ( )( ) ( )( ) ( )( )

( )( )( ) ( )( )( )Sup

Vij ij

auxi i

auxij ij

auxij ij

auxi i

auxi i

aux

i iaux

i iaux

i iaux

i iaux

i iaux

i iaux j

,1 ,112

12

,1 ,1 ,1 ,1,

(10)

Now, according to the definition, the dynamic interaction integral JIntrecan be written as [49],

= − −J J J J¯ ¯ ¯ ¯InterSup aux

. (11)

Now, substitute Eqs. (8)–(10) into Eq. (11), yield to Eq. (12).

∫=

⎡

⎣

⎢⎢⎢⎢⎢⎢

+ + + − − − − −

− − − + + + + + −

+ + + + − + + − +

− + + − +

⎤

⎦

⎥⎥⎥⎥⎥⎥

J

σ u σ u σ u σ u σ ε σ ε σ ε σ ε

ρu u ρu u ρu u ρu u ρ u u u u u u u u u u

ρ u u u u u u u u u u σ u σ ε ρu u ρ u u u

ρ u u u σ u σ ε ρu u ρ u u u ρ u u u

q dV¯ ( ) ( )

( ) ( )

Inter V

ij i ij iaux

ijaux

i ijaux

iaux

ij ij ij ijaux

ijaux

ij ijaux

ijaux

i i i iaux

iaux

i iaux

iaux

i i i iaux

iaux

i iaux

iaux

i iaux

i i i iaux

iaux

i iaux

iaux

i iaux

ij i ij ij i i i i i

i i i ijaux

iaux

ijaux

ijaux

iaux

iaux

iaux

iaux

iaux

iaux

iaux

iaux

j

,1 ,1 ,1 ,112

12

12

12

12

12

12

12 ,1 ,1

,1 ,1 ,112

12 ,1

,1 ,112

12 ,1 ,1

,

(12)

Furthermore Eq. (12) can be simplified by assuming the auxiliary velocity is zero, =u 0iaux

i [48]. Also, for a linear elastic system,× = ×u u u u u u i i i i i i,1 ,1. Thus, the interaction integral does not depend on material velocity. Thus, Eq. (12) can be simplified and

written as Eq. (13).

∫= ⎡⎣ + − − ⎤⎦J σ u σ u σ ε σ ε q dVInter V ij iaux

ijaux

i ij ijaux

ijaux

ij j,1 ,112

12 ,

(13)

As shown in Eq. (13), the kinetic energy term is eliminated from the dynamic interaction integral relation. It is also worth tomention that, the dynamic J-integral is available in Abaqus-implicit as the form of Eq. (13) and neglects the kinetic energy effect [50].Vargas and Dodds [42–44], shows that for most impact responses, the inertia components of the J-integral contributes less than 0.1%of the total J and can be neglected from the analysis. In the case of torsional loading, the inertial effect is very minimal [26–29], and

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60

Eq. (13) can be used safely. It is worth to mention that the interaction integral method built based on J-integral theory and works in asmall-scale-yielding(SSY) condition [51]. In general, Eq. (13) can be written in three different fracture modes that depend on theauxiliary loading field as shown in Eq. (14), where α represents the modes;

∫ ∫= + − +J t σ t u t σ t u t q dV σ t ε t σ t ε t q dV¯ ( ) ( ( )( ( )) ( ( )) ( )) 12

( ( )( ( )) ( ( )) ( ))Interα

V ij iaux α

ijaux α

i j V ij ijaux α

ijaux α

ij j. ,1 ,1 , , (14)

Similar to Eq. (7), the result of Eq. (14) is justified along a 3-D segment by using a weighted function q s( ) as,

∫∫

= =J b tJ s q ds

q dsno sum on α I II and III¯ ( , )

[ ¯ ( ) ]( , , )Intre

α ac α

t

t (15)

where =J b t J b t J b t J b t¯ ( , ) [ ¯ ( , ), ¯ ( , ), ¯ ( , )]Interα

InterI

InterII

InterIII T

. . . . .The J b t¯ ( , )Inter

α. is the interaction integral of a unit virtual advance of a finite crack front segment for a specific mode at a specific

point as a function of time. The discretized form of interaction integral for a three-dimensional domain is used in a finite elementsolution. The stresses, strains, and displacement were calculated with a standard Gauss quadrature procedure and all the integrationpoint in each element inside the volume domain were assembled as shown in Eq. (16).

∑ ∑= ⎡⎣

⎛⎝

+ − − ⎞⎠

⎤⎦

J σ t u t σ t u t σ t ε t σ t ε t q J w¯ ( )( ( )) ( ( )) ( ) 12

( )( ( )) 12

( ( )) ( ) detInterV

elements

element

G Q P

ij iaux α

ijaux α

i ij ijaux α

ijaux α

ij i p p.

. ,

,1 ,1 ,(16)

In this caseG Q P. . is a Gaussian quadrature integration point at each element, wp is respective weight function at each integrationpoint, [......]p are evaluated at Gauss points [52], and Jdet is determinant of Jacobian for 3D coordinates. The FE commercial software“ABAQUS Standard Dynamic-Implicit 2017” was used to solve Eq. (16). Additional details for the numerical solution method areavailable in open literature [47,52,53].

3.2. Extraction of stress intensity factors

In the case of isotropic linear elastic materials and infinitesimal deformation, the stress intensity factors are related to the cor-responding J-integral as shown in Eq. (17) [49,50,54].

= −Jπ

K B K18

· ·T 1(17)

where

K= K K K[ , , ]I II III T : Stress intensity factor vector components (opening mode (Mode-I), in-plane shear mode (Mode-II), and out of plane shear mode (Mode-III) respectively).

J= J J J[ , , ]I II III Tint int int : J-Integral components related to three modes of fracture.

B= Energy Factors[ ]: A second-order tensor, depends on the directions and elastic properties of the material. It called the pre-logarithmic energy factortensor [54].

The J-integral define in Eq. (17) is a general mixed mode relationship representing energy release rate on a crack. The integralinteraction method was used again to separate the J-integral into the corresponding SIFs due to the different fracture modes. Thismethod was introduced by Asaro and Shih [49,54].

The general equation Eq. (17), can be further expanded as Eq. (18),

= + + + + +− − − − − −Jπ

K B K K B K K B K K B K K B K K B K18

[ 2 2 2 ]I I II II III III I II I III II III111

221

331

121

131

231

(18)

The individual parameters can be obtained from the above relation following the procedure explained below. The procedure toobtain Mode-I is discussed in detail.

Eq. (18) can be rearranged by collecting like terms as,

= + + +− − −Jπ

K B K K B K K B K terms not include K18

[( 2 2 ) ]I I I II I III I111

121

131

(19)

Following a similar procedure, the J-integral for an auxiliary Mode-I JauxI

. can be written as,

= −Jπ

κ B κ18aux

II I. 11

1(20)

Now, superposing the auxiliary field Eq. (20), onto the actual fields Eq. (19), the total field of J-integral can be written as,

= + + + + + + +− − −Jπ

K κ B K κ K κ B K K κ B K terms not include K18

[(( ) ( ) 2( ) 2( ) ) ]totalI

I I I I I I II I I III I111

121

131

(21)

The interaction integral for Mode-I can be written as Eq. (22) [47,49].

= − − = + + − − + + −− − − −J J J Jπ

K κ B K κ K B K κ B κ K κ B K18

(( ) ( ) 2( ) ( ) .....)InterI

totalI

auxI

I I I I I I I I I I II. . 111

111

111

121

(22)

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For homogeneous, isotropic and linear elastic materials and infinitesimal deformation, Bαβ is a diagonal matrix, and for planestrain condition can be written as [54],

= =−

=+

= = =B B Eπ υ

and B Eπ υ

and B B B8 (1 )

,8 (1 )

, 011 22 2 33 12 13 23 (23)

Substituting Eq. (23) into Eq. (22), the final relation for Mode I can be written as,

= + + + − − =−

−JB

πK K κ K κ κ K κ

πκ B K

8( ) 1

4InterI

I I I I I I I I I I.11

12 2 2 2

111

(24)

A similar procedure can be used for Mode II and Mode III and the J-integral for each mode can be obtained as,

= =−J sπ

κ B K no sum on α I II and III( ) 14

( , , )Interα

α αβ β.1

(25)

where κα is auxiliary stress intensity factors, and it can be assumed unity. Thus, in dynamic case, Eq. (25) can be rewritten as,

= =K t π B J t no sum on α I II III( ) 4 ( ) ( , , )β αβαint (26)

And the corresponding stress-intensity factor as a function of J-integral can be written as,

∑=− =

K t Eυ

J t( )10(1 )

( )Ii

IntreI

21

5

.

Fig. 8. Finite element model of a cylindrical spiral specimen along with torsional Hopkinson bar.

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∑=− =

K t Eυ

J t( )10(1 )

( )IIi

IntreII

21

5

.(27)

∑=+ =

K t Eυ

J t( )10(1 )

( )IIIi

IntreIII

1

5

.

where JInterα

. are evaluated numerically from Eq. (15). The finite element model was generated to calculate the stress intensity factor ateach point (in the middle of the volume segment) along the spiral’s crack front line. At each plane, as shown in Fig. 8, five differentvolume segments around the crack front were generated to extract the fracture parameters. Since the J-integral is path-independent,the mean value at each plane is used as a final value to calculate the stress intensity factor at each middle point along the crack front.

3.3. Finite element analysis

A numerical method is performed to calculate the dynamic stress intensity factor following the dynamic interaction methodpresented in Eq. (27). As the torsional load is uniform along the spiral length, modeling a quarter section of the specimen is sufficient[55]. A commercial finite element software ABAQUS/ Standard Dynamic implicit (Direct-Integration Dynamic Procedure), was usedto model a quarter revolution spiral crack specimen and the input and the output bar assembly [50]. An outline of the finite elementsimulation is shown in Fig. 8. First, a solid circular bar with a 25.40mm diameter was extruded for a total length of 3555.00mm;which includes 1220.00mm input bar, 2300.00 mm output bar, and 35.00mm specimen. Then, a cut revolves tool was used togenerate the final shape of the specimen. Finally, a shell revolve was used to make a 15.00mm pitch (quarter spiral revolution) spiralseam crack as shown in Fig. 8. The model was divided into a number of planar to obtain the structure element in the whole model. Torefine the elements around the crack tip, a small circle was created at the crack tip and extended along the crack front line, generatinga small cylindrical solid concentrated at the crack front. The middle area of the solid cylinder was divided into 52 slices, about 6.9°each element, which generated a robust and refined Spider mesh around the crack tip as shown in Fig. 8. The gage length is dividedinto 40 planers as shown in Fig. 8. The planer thickness, which represents the volume segment thickness = −L s s(Δ )a c is 250.00 µm.The model was built with a 3D solid structure quadratic hexahedral C3D20R element, and at the crack front the elements collapsedinto a wedge element.

The incident torque measured experimentally was used as input to the finite element model. The boundary conditions are appliedin three steps. First, one end of the bar was fixed in three dimensions (r θ and z, , ). Second, the torsional impulse load was appliedon the other end as a moment load. Finally, the crack tip and crack faces were fixed with respect to Z, Rx, and Ry [23,55]. Thedynamic J-integral is calculated by using keyword *CONTOUR INTEGRAL subroutine program of a standard ABAQUS dynamic-implicit. The dynamic stress intensity factor was calculated at each node on the crack front. With the assumption of linear elasticfracture mechanics and SSY condition, isotropic linear elastic constitutive model, =σ D ε( )ij ijkl kl , where σij is Cauchy stress tensor, Dijklis a fourth order elastic tensor, and εkl is a total elastic strain tensor, was used in the finite element model [50].

As a benchmark, the quasi-static model of a quarter spiral crack was developed, and the fracture toughness of different materialswas calculated and compared with the existing value in the literature. The benchmark verification was done at the quasi-staticcondition to make sure the procedure and the boundary condition are appropriate. As shown in Table 3, the finite element model is inexcellent agreement with available data in the literature [18].

4. Result and discussion

A typical incident, transmitted, and reflected wave signals measured, in this experiment, are shown in Fig. 9. The shear wavetravels in the Titanium-G5 bar at a velocity of 3152.00m/s. Once the incident wave reached the specimen at 1100.00 μs, part of theincident wave was transmitted to the output bar through the SNT specimen, and the rest of incident was reflected from the interfacebetween the input bar and the specimen. The reflected wave signal has two local maxima as shown in Fig. 9. The first maxima, point(1), could be associated with the reflection of the wave at the interface due to materials and geometries different (impedancemismatch). The second maxima reflection point (2) is believed to be associated with a crack initiation in the specimen. The noticeabledrop of transmitted waves at the same time with a rapid increase of the reflected wave can evidently show the crack propagationinitiation instance. However, the exact time at which crack propagation initiated is challenging to specify based on only the wavesignals, and high-speed imaging is used in this work as discussed later.

Table 3Benchmark verifications of the FE model.

Materials K MPa m( )Ic Difference %

FE model Ref. [18] ASTM

Al. 7475-T6 47.60 47.30 48.30 ≈ 1.45Steel A302B 54.20 54.90 55.20 ≈ 1.81Ceramic 02.00 02.10 02.12 ≈ 5.66

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4.1. Dynamic stress-state equilibrium verification

For reliable dynamic fracture initiation toughness and valid Hopkinson torsional experimental results, one of the fundamentalassumptions that must be held during the test is stress equilibrium at two sides of the specimen (incident-specimen, and specimen-transmitted interfaces). The torques applied on both sides of the spiral crack specimen, T1 and T2 are shown in Fig. 10. The equi-librium time is about 10 μs and after equilibrium time the two torques are in a good agreement, indicating that the specimen wassubjected to a pure torsional load.

4.2. Dynamic fracture initiation time determination (tf )

In order to calculate the fracture initiation toughness accurately, identifying the fracture initiation time is critical. Stereo- digitalimage correlation data was used to provide more quantitative information on the crack initiation’s time. The data acquisition (DAQ)device was also used to synchronize the wave signals with the corresponding images so that the load, time and location of the crackinitiation can be easily identified.

Fig. 11 show the in-plane (v) displacements of two points across the crack front line on the surface of the specimen. At 200,000frame/s the camera provides images at an inter-frame time of 5 microseconds, which may not be fast enough to capture the onset ofthe crack. However, it is clear that both displacements have a distinct feature at about 210 μs. The corresponding time in the incident-reflected signals is shown in Fig. 9. It indicates that the fracture is initiated at the second maxima in the reflected signal discussedearlier, which also matches with the image at which the crack propagation becomes visible. In all three materials tested, the crackinitiation time tf is higher than the rise time to, >t t( )f o . Also, fracture initiation develops at a constant strain rate and in the rangewhere the dynamic equilibrium is valid as shown in the previous section.

4.3. Dynamic fracture initiation toughness K( )Id

As discussed earlier, using the incident torque, Eq. (2), Fig. 12, and initiation fracture time as an input, the dynamic initiationfracture toughness KId is determined numerically using dynamic energy release rate theory. Though the interest is on the openingmode, for completeness the three modes of dynamic fracture intensity factor K t K t and K t( ), ( ), ( )I II III are calculated. The stressintensity factors K t K t and K t( ), ( ), ( )I II III at a time of fracture initiation along the crack front line are plotted as shown in Fig. 13.As shown in the figure, the opening mode (Mode I) is at least one order magnitude higher than the other two modes. As expected,

Fig. 9. Typical incident, transmitted and reflected shear waves (Al. 2024-T3).

Fig. 10. Typical incident and transmitted torques (Al. 6061-T6).

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Fig. 11. Typical full field displacement at the crack edge of Al. 2024-T3.

Fig. 12. Typical incident and effective torque waves.

Fig. 13. FE result of dynamic stress intensity factors of Al. 7075-T6. =t t( )f

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Mode II is almost zero and Mode III is within the range of the numerical error.As shown in Fig. 13, the Mode I stress intensity factor value is maximum and almost constant in the first quarter of the specimen,

≅5.625–11.25mm from the loading edge. In the high-speed images, it was observed that this area is the region at which the crack isinitiated. Fig. 14 shows a full field displacement of Al. 6061-T6 at two different time scales. Two points perpendicular to the crack tipon both sides of the crack edges were chosen to estimate the crack edges displacement (CED) and crack mouth opening displacement(CMOD) and to evaluate the fracture initiation time as shown in Fig. 14. The displacement components values at the upper edge(black point) of the specimen are denoted as 0 U V W( , , )0 0 0 , and the displacement components values at the lower edge (red point) aredenoted as 1 U V W( , , )1 1 1 . The crack mouth opening displacement (CMOD) can be obtained from the two points, as shown in Eq.(28.1)–(28.3) [56–58]:

= −CMOD t ECD t ECD t( ) ( ) ( )0 1 (28.1)

= + +ECD t U t V t W t( ) ( ) ( ) ( )0 02

02

02 (28.2)

= + +ECD t U t V t W t( ) ( ) ( ) ( )1 12

12

12 (28.3)

A typical full field shear strain around the crack edge of Al 2024-T3 and Al 7075-T6 obtained from the 3D-DIC analysis are shownin Figs. 15 and 16, respectively. A maximum shear strain of ∼ 0.25% and ∼ 0.30% is observed in Al 2024-T3 and Al 7075-T6, re-spectively. Note, Figs. 15 and 16 shows an in-situ image of the specimen immediately after the fracture initiation >t t( )f . Fig. 17shows a typical final fractured spiral specimen. The figure shows that the crack is initiated in the middle section of the crack front.This is a significant behavior of spiral crack subjected to pure torsion, the crack initiated in the middle of the gage length far from theedge. From the figure, it is also evident that, the maximum opening displacement is developed at the middle location of the gagelength. Thus, the numerical result data of dynamic stress intensity factors at the middle nodes, Fig. 13, are extracted, averaged, andplotted as a function of time up to the fracture initiation time tf for each material.

The dynamic stress intensity factors of Al. 6061-T6 for Mode I, and III are shown in Fig. 18A. As shown in the figure, the Mode Iappear to be developed immediately after the loading wave reached the crack, however Mode III seems to have a delay about 20 µs.Also, the Mode I was constantly increasing until the initiation, however the Mode III is almost constant. It is clear that the Mode Ifracture is the driving factor. Mode II is expected to be zero, and in our analysis the value was within the numerical error and notpresent here to avoid confusion. For comparison, the stress intensity factor at crack initiation for Al. 6061-T6 and Al. 2024-T3 ispresented below (see Fig. 18).

==

K t MPa mK t MPa m

( ) 37.80( ) 3.90

I

III

Note that the total J-integral is 18.67 KJ/m2 and the weight of stress intensity factor of each mode comparing to the total energyrelease rate was evaluated as presented below.

→→

K JK J

93.30%5.80%

I total

III total

Fig. 14. Typical full field displacement data of Al. 6061-T6 measured using 3D-DIC.

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As shown above, the Mode-I controls the value of total energy around the crack tip and plays a great role in the crack propagationinitiation. Hence, the Mode-I stress intensity factor at the fracture time can be considered as the dynamic fracture initiation toughnessof the materials tested in this work.

Very similar behavior was seen in the dynamic stress intensity factor for Al 2024-T3 specimen as shown in Fig. 18B. As expected,the Mode I component is predominantly the driving feature for the fracture of the specimen. The dynamic initiation fracturetoughness, the stress intensity factor at the time of initiation, for Al 2024-T3 specimen is ≅ ∓K m38.20 0.1 MPaId . This value isslightly higher than the quasi-static value of m30.20 MPa . Furthermore, the fracture initiation time of Au. 2024-T3 is longer than Al.6061-T6, but it was difficult to understand the reason since the specimen geometry is slightly different. Consistently, similar resultsare observed in Al. 7075-T6, and without repeating the process only the final value is presented in Table 4.

For comparison purpose, the loading rate was calculated by dividing the fracture initiation toughness to the corresponding

Fig. 15. Typical full field shear strain of Al 2024-T3 measured by 3D-DIC.

Fig. 16. Typical full field shear strain of Al 7075-T6 measured by 3D-DIC.

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Fig. 17. Fracture initiation in Al 2024-T3.

Fig. 18. Typical FE result of dynamic stress intensity factors of (A) Al. 6061-T6, (B) Al. 2024-T3.

Table 4Dynamic fracture initiation toughness and loading rates.

Materials tf µs. K Pa m GI /s K MPa mIdDyna. K MPa mIc

Static KIcDyna

KIcStatic

.

.

Al. 2024-T3* ≈ ±210 5 181.90 38.20 ± 1.1 30.20 (T-L) ≈ 1.23Al. 6061-T6 ≈ ±076 5 494.10 37.80 ± 1.1 29.00 (T-L) ≈ 1.31Al. 7075-T6 ≈ ±120 5 333.00 40.20 ± 1.1 27.50 (T-L) ≈ 1.45

* Al. 2024-T3 properties from [61].

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initiation time =K K t( / )I Id f . The loading rate for the three of the specimens is found to be between⩽ ⩽m K m180 GPa /Sec. 494 GPa /Sec.I This range is higher than 100 GPam1/2/s, which is a minimum limit to define a dynamic

fracture [6]. The summary of the dynamic crack initiation toughness for all the materials considered is shown in Table 4. As shown inthe table, the dynamic initiation fracture toughness is higher than the quasi-static value. Furthermore, the dynamic initiationtoughness values obtained are in a good agreement with the literature values and matches well with the general understanding thatthe dynamic fracture toughness is at least 40% higher than the quasi-static fracture toughness [59,60].

A comparison of results based on the proposed torsional load on the spiral crack specimen with other methods, such as the 3-pointbending test on edge crack samples is shown in Fig. 19. As shown in Fig. 19, the result of a dynamic initiation toughness of the threeAluminum alloys from the spiral specimen, are plotted along with the corresponding results from the 3 points bending test [61–63].Unfortunately, the loading rates were not the same and a direct comparison was not possible. However, in general, the dynamicfracture toughness is in the same order of magnitude with a slight difference. We believe, the inertia component is significant in thecase of a direct loading condition in the 3-point bending experiment, which may be the reason for the difference observed.

4.4. Effect of crack tip temperature on the fracture parameters

It is well understood that during crack propagation, heat will be generated near the crack tip. The amount of heat produceddepends on the material property and loading conditions. The temperature generated during fracture propagation can be estimatedbased on the amount of plastic deformation and its change rate. Rice and Levy [64], showed that the effect of crack tip temperatureon the fracture properties of stationary crack at high strain loading is not significant. In metals for stationary crack under dynamicloading condition, the temperature usually remains below 100 °c [64]. However, in a propagating crack, the change in temperaturecan be huge, and the effect of the temperature should be incorporated. In this study, the crack propagation was not considered, onlyuntil crack initiation is considered, and hence the effect of temperature can be neglected.

5. Conclusion

A new approach to estimate the dynamic fracture initiation toughness of materials without inertia effect is proposed. A cylindricaltubular specimen with a spiral crack at 45° on the surface is used to study the dynamic fracture toughness of materials. To de-monstrate the method, three Aluminum alloys, 2024-T3, 6061-T6, and 7075-T6, are tested at room temperature. The specimens weresubjected to dynamic torsional loading using a torsional Hopkinson bar apparatus. The incident strain signal is measured, and thetorque applied to the specimen is analyzed using one-dimension wave theory. The time at which the crack propagation initiated ismeasured using stereo-digital image correlation. Using the torque measured and the time of crack initiation as input, a three-dimension full-size model is developed in ABAQUS to extract the fracture parameters. The 3D dynamic interaction integral method isused to calculate the stress intensity factors numerically based on the energy release rate theory. The Mode I dynamic fracturetoughness of all Aluminum alloys subjected to a loading rate between ⩽ ⩽m s K m s180 GPa / . 494 GPa / .I is found to be higher thanthe quasi-static value ≅ ∓i e K K( . . (1.4 0.15) )Ic

DynaIcStatic. . In addition, the following summary can be stated about the proposed method.

• Due to the advantage of the torsional wave being non-dispersive and the axial inertia is negligible, the proposed method is ideal toinvestigate the dynamic fracture toughness of materials at high loading rate.

• Due to its unique geometry and loading condition, the proposed method can be adapted to any material and size, by which itavoids the limitation of the plane strain condition required in other standard methods.

• The method can further extend to mixed mode loading by changing the angle of the spiral crack

• Since the spiral crack configuration does not have a closed form solution or analytical relation, the interaction integral formulawas used to calculate the stress intensity factor numerically.

Fig. 19. Comparison of the dynamic initiation fracture toughness of aluminum alloys based on the proposed spiral crack with other methods.

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Acknowledgment

Mr. Ali Fahem was financially supported by the Ministry of Higher Education and Scientific Research, University of Al-Qadisiyah,Iraq and is greatly acknowledged. The support of the University of South Carolina and Mr. Edward Walton, Chief Financial Officer, toprovide significant matching funds for development of a multiaxial loading laboratory is deeply appreciated.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfracmech.2019.03.039.

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