Strain-rate effects on the tensile behavior of strain-hardening cementitious composites

Construction and Building Materials 52 (2014) 96–104

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Strain-rate effects on the tensile behavior of strain-hardening cementitious composites En-Hua Yang a,⇑, Victor C. Li b a b

Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singapore Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA

h i g h l i g h t s  Sources responsible for strain-rate effects of strain hardening cementitious composites (SHCC) were discovered.  Rate dependence in component phases, i.e. fiber, matrix, and interface, were experimentally determined.  A dynamic micromechanics-based strain hardening model was developed for SHCC component tailoring and ingredient selection.

a r t i c l e

i n f o

Article history: Received 6 February 2013 Received in revised form 23 October 2013 Accepted 5 November 2013 Available online 30 November 2013 Keywords: Strain-rate effects Strain-hardening cementitious composites SHCC Micromechanics

a b s t r a c t This paper investigated the strain-rate effects on the tensile properties of strain-hardening cementitious composite (SHCC) and explored the underlying micromechanical sources responsible for the rate dependence. Experimental studies were carried out to reveal rate dependence in component phases, i.e. fiber, matrix, and fiber/matrix interface. A dynamic micromechanical model relating material microstructure to SHCC tensile strain-hardening under high loading rates was developed. It was found fiber stiffness, fiber strength, matrix toughness and fiber/matrix interface chemical bond strength were loading rate sensitive and they increase with loading rates in a polyvinyl alcohol fiber-reinforced SHCC (PVA-SHCC) system. These changes in component properties result in the reduction of tensile strain capacity of PVA-SHCC as the strain-rate increases from 105 to 101 s1. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Terrorist attacks and natural hazards highlight the need for assuring human safety in civil infrastructure under extreme loading such as projectile impacts and bomb blasts. While concrete has served as an eminently successful construction material for years, reinforced concrete (R/C) infrastructure can be vulnerable under severe dynamic loading [1]. Many catastrophic failures of R/C structures subjected to impact or blast loading (IBL) were associated with the brittleness of concrete material in tension as suggested by Malvar and Ross [2]. Brittle failures, such as cracking, spalling, and fragmentation, of concrete were often observed in R/C structures when subjected to IBL [3], and can lead to severe loss of structural integrity. Apart from that, high speed spalling debris ejected from the backside of the structural elements can cause serious injury to personnel behind the structural elements. There is a need to enhance concrete ductility to enhance the safety of R/C infrastructure under IBL. ⇑ Corresponding author. Address: N1-01b-56, 50 Nanyang Avenue, Singapore 639798, Singapore. Tel.: +65 6790 5291; fax: +65 6791 0676. E-mail address: [email protected] (E.-H. Yang). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.11.013

Strain-hardening cementitious composites (SHCC), a new class of concrete material featuring high ductility and damage tolerance under tensile and shear loading, offers a potential solution to reducing R/C structure vulnerability under IBL. SHCC with tensile strain capacity in excess of 3% under quasi-static uniaxial tensile loading can be attained with only 2% fiber content by volume [4]. However, the success of SHCC as a ductile concrete material under IBL depends on its ability to retain tensile ductility at high strainrates [5], which requires a systematic investigation. Literatures on strain-rate effects of SHCC were not always consistent. For example, the authors reported the tensile strain capacity of a polyvinyl alcohol fiber-reinforced SHCC (PVA-SHCC) decreases while the tensile strength increases with increasing strain-rate from 105 to 101 s1 as shown in Fig. 1 [5]. Similar results were observed and reported by others [6,7]. This observation implies the brittleness of SHCC increases with loading rate which is unfavorable to the high energy absorption demand when R/SHCC structures are subjected to IBL. Maalej et al. [8], however, reported the tensile strain capacity of a hybrid steel and polyethylene fiber-reinforced SHCC shows negligible strain-rate effects when strain-rate increases from 106 to 101 s1. Boshoff and van Zijl [9] reported similar results with another version of SHCC. These

E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104

component rate dependence, so that its application was limited to static loading. Under higher loading rates, inertial effect and material component (matrix and fiber bridging properties) rate dependence shall be accounted for in energy balance considerations. The dynamic energy release rate for a crack propagating at velocity V was defined [15] as

9 8 7

Tensile Stress (MPa)

97

6

 Z  1 @ui @ui @ui dy  rij nj W þ qV 2 ds C!0 C 2 @x @x @x

5

GðCÞ ¼ lim 4 3 2 1 0 0

1

2

3

4

Tensile Strain (%) Fig. 1. Rate dependence of tensile stress–strain curve of PVA-SHCC (after [5]).

experimental results revealed that SHCC tensile properties may or may not exhibit rate dependence and the magnitude and the tendency of rate dependence, if it exists, will likely depend on the microstructure and material composition. For example, it has been reported that the pull-out resistance and probability of fiber rupture increased with the increase of the pull-out rate [10,11]. In addition, it was found that changes in temperature can lead to pronounced changes in the SHCC performance as well, likely due to changes in the properties of the individual components of the system [12]. These observations on the tensile strain-hardening behavior of SHCC under higher loading rates underscore a need to study the sources of strain-rate effects in SHCC. In this paper, a dynamic micromechanics-based strain-hardening model was proposed first. This analytical model linked microscopic constituent properties to macroscopic SHCC tensile strain-hardening behavior under high loading rates. Experimental studies were carried out to discover rate dependence in SHCC component phases, i.e. fiber, matrix, and fiber/matrix interface. Through the measured rate dependent constituent properties combined with the dynamic strainhardening model, sources of strain-rate effects on the tensile behavior of SHCC were discovered. 2. Dynamic strain-hardening model for randomly oriented discontinuous fiber-reinforced brittle matrix composites The theoretical foundation behind the multiple cracking phenomenon of SHCC under static loading was first explored by Aveston et al. [13] whose work was later extended by Marshall and Cox [14]. These investigations focused on brittle matrix composites reinforced with continuous and aligned fibers with relatively simple bridging laws. Emphasis was placed on the conditions for steady-state matrix cracking – extension of bridged crack length with flat crack surfaces so that the bridging elements remain intact after the passage of a crack. The fully bridged flat crack with limited width allows load transfer from the bridging fibers back into the matrix to activate additional flaw sites into new microcracks. Especially transparent in Marshall and Cox, the steady-state crack analysis employed the concept of energy balance between external work, crack tip energy absorption through matrix breakdown (matrix toughness), and crack flank energy absorption through fiber/ matrix interface debonding and sliding. The steady-state flat crack model did not capture inertia effects nor did it incorporate material

ð1Þ

where W is the strain energy density, q is the density, ui is the displacement component, rij is the stress component, and ni is the unit normal vector. Eq. (1) is in general not path independent except for the special case when the crack propagates in a steady-state mode (Appendix A). For a static crack (V = 0), Eq.(1) reduces to the J-integral of Rice [16] which is identical to the strain energy release rate for elastic behavior. For a steady-state crack propagating against a matrix toughness of Gtip and crack face bridging traction rB(d) under remote steady-state tensile stress rss and steady-state crack opening dss, it can be shown (Appendix B) that

Gtip ¼ rss dss 

Z

dss

0

rB ðdÞdd  G0b

ð2Þ

Specifically, since G0b is limited by its maximum value attained when rss = r0 when dss = d0, as shown in Fig. 2, the condition for steady-state flat crack propagation mode under dynamic condition can be stated as

Gtip < r0 d0 

Z

dss

rB ðdÞdd  C

ð3Þ

0

where C is the maximum complimentary energy as defined above. The crack face bridging traction rB(d), which can be viewed as the constitutive law of fiber bridging behavior, was derived by using analytic tools of fracture mechanics, micromechanics, and probabilistic theory [17]. In particular, the energetics of tunnel crack propagation along an interface was used to quantify the debonding process and the bridging force of a fiber with given embedment length. Probabilistics was introduced to describe the randomness of fiber location and orientation with respect to a crack plane. The random orientation of fiber also necessitated the accounting of the mechanics of interaction between an inclined fiber and the matrix crack. As a result, rB(d) was expressible as a function of ten micromechanics parameters, including fiber properties and fiber/ matrix interfacial properties. An additional condition for strainhardening was that the matrix cracking strength rfc must not exceed the fiber peak bridging strength r0.

Fig. 2. A typical rB(d) curve for tensile strain-hardening composite. Hatched area represents maximum complimentary energy C. Shaded area represents crack tip toughness Gtip.

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rfc < r0

ð4Þ

where r0 was determined from rB(d) and rfc is a function of matrix toughness, pre-existing internal flaw size, and crack face bridging traction rB(d). A cohesive traction model with two interface types was used to calculate the equilibrium ambient load r1 for a given crack length, from which the matrix cracking strength can be determined. As shown in Fig. 3a, a two dimensional plate with 160 mm in width and 100 mm in length is modeled using the finite element method. Due to x- and y-axis symmetries, only a quarter of the plate is modeled. The boundary conditions are fixed in the y direction for the bottom line and fixed in the x direction for the left sideline. The plate is subject to an ambient uniaxial tensile load r1. The matrix element is assumed to act linear-elastically. Interface elements are used to simulate the developing crack and are assigned the userdefined cohesive traction law as material property. Except for those elements representing a pre-existing crack, they are assumed to be linear elastic until a predefined tensile strength ft is reached. After that, the cohesive behavior follows the imposed cohesive traction law. Specifically, two sets of interface elements are deliberately employed as shown in Fig. 3a. A traction-free interface (Interface 1) with 2.5 mm in length was used to simulate an inherent defect site of 5 mm, the largest flaw commonly found in

PVA-SHCC matrix [18]. A total cohesive traction as a function of half crack opening (Fig. 3b) was assigned to the interface (Interface 2) with 77.5 mm in length. The total cohesive traction is the superposition of matrix retention and fiber bridging. The simulations were performed using a commercial FEM program, DIANA, that allows implementation of user-defined cohesive traction properties in the interface element. Details of the model can be found in Yang and Li [19]. While the energy criterion Eq.(3) governs the crack propagation mode, the strength-based criterion represented by Eq.(4) controls the initiation of cracks. Satisfaction of both Eqs.(3) and (4) is necessary to achieve ductile strain hardening behavior; otherwise, normal tension-softening FRC behavior results. It is important to recognize here that the material parameters in Eqs.(2) and (4) can be rate dependent. Specifically, matrix toughness Gtip and crack face bridging traction rB(d) can be loading rate sensitive. Rate dependence of rB(d) may result from the rate dependent fiber properties and/or fiber/matrix interfacial properties. Rate dependence of Gtip, C, rfc, and/or r0 may result in a violation of the inequality sign in Eqs.(3) and (4) under higher loading rates even when this inequality is satisfied under quasistatic loading condition. This implies that a ductile SHCC under quasi-static loading may become brittle under higher loading rates.

matrix elements

interface elements

Interface 2

Interface 1

(a) 7 6

σ (MPa)

5 4 3 Total cohesive traction Matrix retention Fiber bridging

2 1 0 0

0.02

0.04

0.06

half crack opening, δ/2 (mm)

(b) Fig. 3. (a) Cohesive traction model with two interface types, and (b) total cohesive traction as a summation of fiber bridging and matrix retention.

99

E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104 3. Experimental program To trace the sources of strain-rate effects in SHCC under tension, investigations were conducted in revealing the rate dependence in component phases of the polyvinyl alcohol fiber-reinforced SHCC (Fig. 1) reported by the authors in previous studies [5]. Single fiber uniaxial tensile tests were conducted to examine the rate dependence of fiber properties. A single fiber was cast into the SHCC matrix at the two ends with a long embedment length to prevent fiber pullout. The sample was then tested under uniaxial tension by UTM at different loading rates. Each data point was the average of at least five tests. Rate dependence of matrix toughness was measured through three point bending toughness tests [20]. Each data point was the average of at least three tests. Single fiber pullout tests [21] as shown in Fig. 4 were conducted to reveal the rate dependence in the fiber/matrix interfacial properties, including interface chemical bond strength Gd, interface frictional bond strength s0, fiber strength reduction coefficient f0 , and slip hardening coefficient b. Each data point was the average of at least eight tests. A servohydraulic material test machine was used in the displacement control mode for all three tests. The loading rate tested in component phases ranged from 103 to 10 mm/s which corresponds to strain-rate of 105 to 101 s1 tested in the uniaxial tensile tests of SHCC shown in Fig. 1.

0.5mm

Mix design of SHCC matrix in the single fiber pullout test as well as the matrix toughness test can be found in Table 1. It comprised of standard mortar components, including type I Portland cement (C), water (W), fly ash (FA, class F), sand (S, 0.11 mm nominal grain size), and superplasticizer (SP). Fibers used in the single fiber uniaxial tensile tests and the single fiber pullout tests were polyvinyl alcohol (PVA) fibers produced by Kuraray Co. Mechanical properties of PVA fibers were summarized in Table 2. The mix design and the fiber used in component phases were identical to the SHCC shown in Fig. 1.

4. Results and discussion 4.1. Rate dependence in component phases Figs. 5–7 show the rate dependence of fiber, matrix, and fiber/ matrix interfacial properties, respectively. As can be seen, fiber stiffness Ef, in situ fiber strength rfu, i.e. fiber strength obtained when embedded in cement matrix, toughness Gtip and interface chemical bond strength Gd are loading rate sensitive. The general trend shows that these parameters increase with loading rates. Interface frictional bond s0, slip-hardening coefficient b, and fiber strength reduction coefficient f0 , on the other hand, show negligible rate dependence over the tested loading rates. Rate dependent component properties as shown in Figs. 5–7 can potentially alter SHCC tensile strain-hardening. For example, increase in matrix toughness Gtip as a result of higher loading rate may cause a violation of the energy criterion Eq.(3) and therefore introducing a negative impact on SHCC tensile ductility. Parametric studies were conducted in the following section to reveal impacts

5mm

10mm

15

Fig. 4. Geometry for a single fiber pullout specimen.

10

Material

C

W

FA

S

SP

SHCC matrix

1.0

0.53

1.2

0.8

0.03

Gtip (J/m 2)

Table 1 Mix proportions of SHCC matrix by weight.

5

Table 2 Mechanical properties of PVA fibers. Diameter (lm)

Tensile strength (MPa)

Tensile modulus (GPa)

PVA

40

1600

42*

0 0.0001

0.01

1

Test Speed (mm/s)

Initial tangent modulus.

Fig. 6. Rate dependence of matrix toughness, Gtip.

(a)

45

(b) 1400

40

1200

35

20 15

(MPa)

25

800

fu

1000

30

E f (GPa)

*

Fiber type

600 400

10 200

5

0

0 0.0001

0.01

1

Test Speed (mm/s)

100

0.0001

0.01

1

100

Test Speed (mm/s)

Fig. 5. Rate dependence of fiber properties: (a) fiber stiffness Ef (secant modulus) and (b) in situ fiber strength rfu.

100

100

E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104

(a) 8

(b) 8

6

6

5

5

7

0 (MPa)

Gd (J/m2)

7

4

4

3

3

2

2

1

1

0 0.0001

0.01

1

100

0 0.0001

Test Speed (mm/s)

In-situfiber strength σfu (MPa)

1.5

β

1

100

(d) 1200

(c) 2.0

1.0

0.5

0.0 0.0001

0.01

Test Speed (mm/s)

0.01

1

100

1100 1000 900 800 700 600 500

Test Speed (mm/s)

(e)

0.6 0.5 0.4 0.3 0.2 0.1 0

Fig. 7. Rate dependence of fiber/matrix interfacial properties: (a) interface chemical bond strength Gd, (b) interface frictional bond strength s0, (c) slip hardening coefficient b, (d) fiber strength reduction as a function of pullout angle, and (e) fiber strength reduction coefficient f0 (i.e. coefficient of the power of exponential function in (d)).

of rate dependence of individual component on SHCC tensile strain-hardening. Specifically, influence of rate dependent fiber stiffness Ef, fiber strength rfu, and interface chemical bond strength Gd on fiber bridging relation rB(d) and matrix cracking strength rfc were calculated based on the models described in Section 2. 4.2. Effects of component rate dependence on fiber bridging and matrix cracking strength – parametric study Fig. 8a shows the effects of rate dependent fiber stiffness Ef on fiber bridging properties. As can be seen, the stiffness of fiber bridging increases while the peak bridging strength r0 is maintained with increased fiber modulus as a result of higher loading rates. A smaller maximum complimentary energy C was observed with increased fiber stiffness which can cause a violation of the energy criterion Eq.(3). Higher fiber strength rfu, on the contrary, increases the peak bridging strength resulting in a higher maximum complimentary energy C, as shown in Fig. 8b, which is favorable to SHCC strain hardening. As for the interface chemical bond strength, higher Gd results

in a smaller maximum complimentary energy C which again can cause a violation of the energy criterion Eq.(3) as shown in Fig. 8c. Fig. 9 illustrates calculation results of matrix cracking strength rfc. As can be seen, stable crack growth occurred until peak load was reached, after which the crack grew unstably. Matrix cracking strength is defined as the ambient stress when crack propagation becomes unstable, which is the peak load in the curve. As shown in Fig. 9a and c, higher fiber stiffness Ef and interface chemical bond strength Gd increase matrix cracking strength which can cause a violation of the strength criterion Eq.(4). Higher fiber strength rfu does not affect matrix cracking strength as shown in Fig. 9b. Based on the parametric study, it can be concluded that rate dependence in increasing fiber stiffness, interface chemical bond, and matrix toughness creates negative impact on SHCC tensile strain-hardening and should be avoided in designing SHCC to sustain dynamic loading. On the other hand, rate dependence in increasing fiber strength has positive impact on SHCC tensile strength and tensile ductility and should be encouraged in future design of impact resistant SHCC.

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E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104

(a)

(b)

6

8 7

5

2

(MPa)

(MPa) B

3

5

B

6

4

3

4

2 1

1

0

0 0

0.05

0.1

0

0.15

0.05

(c)

0.1

0.15

(mm)

(mm) 6

4 3

B

(MPa)

5

2 1

0 0

0.05

0.1

0.15

(mm) Fig. 8. Fiber bridging as a function of rate dependent (a) fiber stiffness Ef, (b) fiber strength rfu, and (c) interface chemical bond strength Gd.

(b)

(a)

(c)

Fig. 9. Matrix cracking strength as a function of rate dependent (a) fiber stiffness Ef, (b) fiber strength rfu, and (c) interface chemical bond strength Gd.

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E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104

(a) 7

(b) 6

6

5 4

4

(MPa)

(MPa)

5

3

3

2

2

1

1

0

0

0

0.02

0.04

0.06

0.08

0.1

0

(mm)

5

10

15

Half crack length (mm)

Fig. 10. Rate dependence in fiber bridging and matrix cracking strength of PVA-SHCC.

4.3. Effects of rate dependent component properties on SHCC tensile strain-hardening Overall effects of rate dependent component properties on fiber bridging, matrix cracking strength, and tensile strain-hardening of the PVA-SHCC were evaluated in this section. Fig. 10 shows the effects of loading rates on fiber bridging and matrix cracking strength by taking into account all rate dependent component properties into the model. As can be seen in Fig. 10a, the fiber bridging curve rB(d) moved up with increasing loading rates. A 20% increase in the peak bridging strength r0 was observed at the highest testing rate. This observation suggests that the increase in SHCC tensile strength at higher loading rates as depicted in Fig. 1 is attributed to the rate dependent fiber bridging properties, in which peak bridging strength increases with increasing loading rate. A higher maximum fiber bridging strength r0 is favorable to the satisfaction of the strength criterion Eq.(4) at higher loading rates. On the other hand, a 15% reduction in the maximum complementary energy C was observed which suggests a potential violation of the energy criterion Eq.(3) at higher loading rates. As for the matrix cracking strength (Fig. 10b), a 40% increase in rfc was observed as the loading rate increased from 0.001 mm/s to 10 mm/s, which can cause a violation of the strength criterion Eq. (4). Pseudo strain hardening (PSH) indices (C/Gtip and r0/rfc) as a function of loading rate were calculated based on information obtained from Figs. 6 and 10. A 50% reduction in the PSH energy (C/ Gtip) index and a 10% reduction in the PSH strength (r0/rfc) index were observed when the loading rate increases from 0.001 to 10 mm/s for PVA-SHCC. Materials with lower values of PSH indices should have less chance of saturated multiple cracking and often leads to small tensile ductility [22]. As a result, PVA-SHCC showed strong strain-rate dependence, the tensile strain capacity decreases while the tensile strength increases with increasing strain-rate from 105 to 101 s1 as shown in Fig. 1 (see Fig. 11).

(a)

5. Conclusions This paper investigated the strain-rate effects on the tensile properties of SHCC and explored the underlying micromechanical sources responsible for the rate dependence. The strain-rate effects of SHCC were traced to the rate dependence in the component phases, specifically the fiber, the matrix, and the fiber/matrix interface phase. A dynamic micromechanics-based strain hardening model was developed to gain insights into the source of the rate dependence in SHCC. The model provides guidance for ingredients selection and component tailoring for impact resistant SHCC design as shown in Yang and Li [5], in which deliberate tailoring of SHCC for impact resistance through microstructure control involving fiber, matrix and fiber/matrix interface was demonstrated. SHCCs with tensile strain capacity in excess of 3% at all strain rates spanning fours orders of magnitude from 105 to 101 s1 were developed. Drop weight tower tests on panels and beams confirmed that the material ductility of impact resistant SHCC transformed into enhancements of load and energy dissipation of these structural elements. Although the strain-rate range investigated in this paper was limited to between 105 and 101 s1 which corresponds to quasistatic to low speed impact loading rates [23], the methodology and approach of revealing the strain-rate effects on the tensile behavior of SHCC proposed in this study can be further extended to higher strain rates. Main conclusions drawn from this study include:  Fiber stiffness Ef, fiber strength rfu, matrix toughness Gtip and fiber/matrix interface chemical bond strength Gd were loading rate sensitive and they increase with loading rates in the PVA-SHCC system. Fiber/matrix interface frictional bond s0, slip-hardening coefficient b, and fiber strength reduction factor f0 , on the other hand, showed negligible rate dependence over the tested loading rates.

(b)

Fig. 11. PSH indices of PVA-SHCC as a function of loading rate.

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E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104

 Increase in fiber stiffness, interface chemical bond strength, and matrix toughness as a result of higher loading rate creates negative impact on SHCC tensile strain hardening and should be avoided in designing SHCC to sustain dynamic loading. On the other hand, rate dependence in increasing fiber strength has positive impact on SHCC tensile strength and tensile ductility and should be encouraged in future design of impact resistant SHCC.  Overall rate dependent component properties of PVA-SHCC resulted in lower PSH indices. As a result, PVA-SHCC showed strong strain-rate dependence, the tensile strain capacity decreases while the tensile strength increases with increasing strain-rate from 105 to 101 s1.

where x–y is a coordinate system attached to and moving with the crack tip at speed V. To calculate G for a propagating crack in an arbitrary contour, the displacement field within the region of A–A* should be known. Special case: Under conditions of steady-state crack propagation, the area integral vanishes and G is indeed path-independent.

G¼

Z

ðW þ TÞdy  rij nj

C

@ui ds @x

The steady-state crack propagation here means that the mechanical fields are time invariant in a reference frame traveling with the crack tip in the x-direction at a uniform speed, V [12]. Appendix B. Dynamic steady-state cracking criterion

Acknowledgements Partial support of this research by the National Science Foundation and by the University of Michigan is gratefully acknowledged. Appendix A. Dynamic energy release rate

C !0

Z  C

ðW þ TÞn1  rij nj

 @ui ds @x

C ðW

i þ TÞdy  rij nj @u ds @x

C ðW

i þ 12 q V 2 @u @x

R

¼ ¼

R R

r22 du2 C1 rss du2 C1

hþdss 2

¼ rss u2 j2h 2

¼ rss2dss

1 q u_ i u_ i 2

Path C2:

If another path, C, is chosen,

n2 = 0

Z 

 @ui ds GðCÞ ¼ lim ðW þ TÞn  r n 1 ij j C !0 @x C " # ) Z @u @ 2 ui € i i  qu_ i dA þ q u @x @t@x A—Aast 

GðCÞ  GðC Þ ¼ lim 

C !0

Z AA

"

@u @ 2 ui qu€i i  qu_ i @x @t@x

@ui Þdy @x

i  rij nj @u ds @x

Path C1: dy = 0, ds = dx n1 = 0, n2 = 1 r11 = r12 = r21 = 0, r22 = rss

GC1

is not a path-independent integral, where W is the strain energy density and T is the kinetic energy density, respectively.

T¼

¼

R

GC1 þ GC2 þ GC3 þ GC4 þ GC5 þ GC6 þ GC7 ¼ Gtip

For a propagating crack in an elastic body, the dynamic energy release rate, G, can be defined as the rate of mechanical energy flow out of the body and into the crack tip per unit crack advance. It is found that

GðC Þ ¼ lim 

G ¼

#

dA–0    Path dependency

r11 = r12 = r21 = 0,

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E.-H. Yang, V.C. Li / Construction and Building Materials 52 (2014) 96–104

@ui ¼0 @x Z GC2 ¼ ðW þ TÞdy C2

W¼

where

GC2 ¼

GC7 ¼ Re 0

rij deij ¼

i T ¼ 12 qV 2 @u @x

Z

Wdy ¼

C2

r2ss 2E

@ui @x

R e22 0

r22 de22 ¼ 2E

¼0

dss

yjh2þdss ¼  2

Z

r22 du2 ¼

Z

C3

2

r2ss h 4E

Z

rB ðu2 Þdu2 ¼

C3

r22 du2 ¼

Z

C4

Z

0

dss 2

rB ðu2 Þdu2

rB ðu2 Þdu2 ¼

Z

d2ss

rB ðu2 Þdu2

0

C4

Path C5: n2 = 0 r11 = r12 = r21 = 0,

@ui ¼0 @x Z GC5 ¼ ðW þ TÞdy C5

W¼

where

GC5 ¼

Re 0

rij deij ¼

i T ¼ 12 qV 2 @u @x

Z C5

Wdy ¼

r2ss 2E

@ui @x

0

2

r22 de22 ¼ r2Ess

¼0

hdss 2

yj2dss

R e22

¼

2

r2ss h 4E

Path C6: dy = 0, ds = dx n1 = 0, n2 = 1 r11 = r12 = r21 = 0, r22 = rss

GC6 ¼

Z

C6

rss du2 ¼ rss u2 j2hdss ¼

Path C7: n2 = 0 r11 = r12 = r21 = 0,

@ui ¼0 @x Z GC7 ¼ ðW þ TÞdy C7

h

2

2

0

rij deij ¼

i T ¼ 12 qV 2 @u @x

Z

Wdy ¼

C7

r2ss 2E

@ui @x

R e22 0

2

r22 de22 ¼ r2Ess

¼0

h

r2ss h

2

2E

yj2h ¼

Gtip

¼ GC1 þ GC2 þ GC3 þ GC4 þ GC5 þ GC6 þ GC7  2  R R dss 0 rss h ¼ rss2dss þ  4E þ dss rB ðu2 Þdu2 þ 02 rB ðu2 Þdu2 2  2   2  rss h rss h þ  4E þ rss2dss þ 2E Rd ¼ rss dss  0 ss rB ðdÞdd  G0b

References

Path C4: dy = 0, ds = dx n1 = 0, n2 = 1 r11 = r12 = r21 = 0, r22 = rB(d)

GC4 ¼

Re

r2ss

Path C3: dy = 0, ds = dx n1 = 0, n2 = 1 r11 = r12 = r21 = 0, r22 = rB(d)

GC3 ¼

W¼

where

rss dss 2

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