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Rate-dependent crack growth in adhesives
International Journal of Adhesion & Adhesives 23 (2003) 9–13
Rate-dependent crack growth in adhesives I. Modeling approach Chongchen Xu, Thomas Siegmund*, Karthik Ramani School of Mechanical Engineering, Purdue University, 1288 Mechanical Engineering Building, West Lafayette, IN 47907-1288, USA Accepted 17 September 2002
Abstract Rate effects in the failure behavior are investigated for adhesive bonds in which the source of time- dependence is the rate of the material separation in the fracture process zone. A rate-dependent cohesive zone model (CZM) is described and used to analyze rate effects during debonding. The model consists of a rate-independent CZM in parallel to a Maxwell element. General implications derived for the model with respect to rate effects in the failure behavior of adhesives are presented. The application of the model to the analysis of crack growth is described in a companion paper. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: D. Viscoelasticity; Adhesive; Cohesive zone model
1. Introduction It is the goal of the present study to investigate effects related to the rate-dependent failure behavior of polymeric adhesives. Failure of adhesive joints is investigated in a system where a rate-dependent material separation process zone exists only in the adhesive itself, and the adherends adjacent to the adhesive deform elastically without any additional rate effects. Part I of the study, as presented in this paper, is concerned with a model description of rate-dependent material separation. Part II of the study, as presented in a companion paper, describes the applications of the model to a crack growth in a thermoplastic adhesive. For adhesives, and specifically for those based on thermoplastic polymers, the presence of the extended fracture process zone does not allow the use of linear elastic fracture mechanics concepts, and the presence of an extended fracture process zone at the crack tip is to be accounted for. In addition to the properties of the adhesive itself, the processes in this zone are dependent on specimen geometry, the elastic properties of the adherend, and the applied loading rate. Thus, for the investigation of the structural integrity of bonded *Corresponding author. +1-765-494-9766; fax: +1-765-494-0539. E-mail address: [email protected] (T. Siegmund).
structures, a constitutive model that describes the fracture process zone at the crack tip and its dependence of the mechanical constraints is required. A cohesive zone model (CZM) is ideally suited for that purpose. In such a model the material separation characteristics of the adhesive are described by a relationship between the displacement jump across the crack surface, i.e. across the adhesive, and the associated traction. With an appropriate traction-separation law in the CZM, one can account for the loss of load carrying capacity, and finally for crack growth. A number of different approaches have been used in the past in order to incorporate rate effects into material separation models. One of the approaches undertaken is based on the idea that a viscoelastic constitutive equation is combined with a damage function. This approach was applied by Knauss and Losi [1] and Knauss [2] and was also described in overview by Schapery [3]. A model based on multiplication of a viscoelastic hereditary integral with a traction–separation function was introduced instead of a continuum damage function [4,5]. The hereditary integral used for description of the fracture process was taken to be identical to the one determined for the bulk polymer. No separate determination of the rate-dependence of the fracture process itself was undertaken in these models. In viscoplastic interface models [6,7], the material
0143-7496/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 7 4 9 6 ( 0 2 ) 0 0 0 6 2 - 3
10
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
separation in the fracture process zone is the sum of an elastic and viscoplastic contribution, and cohesive tractions are rate-dependent only if plastic opening exists. Also, a rate-dependent CZM based on creep constitutive equations was developed [8]. In CZMs developed on the basis of time-dependent damage models [7,9], the material separation rate determines the rate of damage evolution and finally the strength degradation. Rate-dependence was included also by the use of pre-definition rate-dependencies of the cohesive zone properties. Such models were used recently for dynamic fracture investigations where a complex functional dependence of the rate-dependent cohesive strength was introduced [10]. Another approach bases the rate-dependence on a general fluid model. This approach was established in [11] and was used [12] to describe fracture of polymers. Finally, rate-dependent CZMs were constructed by direct inclusion of the traction–separation law of the CZM into viscoelastic constitutive equations. A general model was presented in Refs. [13,14]. Simplifications based on the use of a CZM within a Kelvin element were presented in Refs. [15,16]. In the present paper, the goal is to demonstrate the use of a rate-dependent CZM in which both the rateindependent and rate-dependent material separation parameters can be determined with reasonable experimental and numerical analysis effort. The standard linear solid model (SLS) provides a starting point for this development. A single Maxwell element combined with a rate-independent CZM was used to construct the rate-dependent traction–separation constitutive relationship. While the underlying rate-independent CZM possesses two material parameters, i.e. a cohesive strength and a critical separation, the proposed model adds two additional parameters for the description of the rate dependency. A CZM based on the linear standard model can be expected to provide information only over a restricted range of loading rates. Nevertheless, it allows for a description of the main important observations in viscoelastic material behavior.
ð2Þ
where z is 16e/9, e is exp (1). The hat over the cohesive zone quantities indicates rate-independent parameters. The current location of the crack tip is defined by un>dc. The rate-independent cohesive zone material parameters are the cohesive strength, s# max ; and the cohesive length, dc : The cohesive energy for the rate# is given by G # ¼ 9=16 s# max dc : The independent case, G; SLS, Fig. 1, is the basis for the development of the present rate-dependent CZM. This model is commonly used to characterize the behavior of solid polymer components [18, 19]. The SLS model consists of a spring in parallel to a Maxwell element that consists of a second spring and a dashpot in series. The uniaxial constitutive relationship between stress, a, and strain, e, given by the SLS model is ds de Z þ E2 s ¼ ZðE1 þ E2 Þ þ E1 E2 e; ð3Þ dt dt where Z is the viscosity, E1 and E2 are moduli, and t is time. This combination of elements provides an equilibrium modulus, E1, characteristic of low loading rates, and a glassy modulus, E1+E2, for high loading rates. On the basis of the SLS model, Eq. (3), a standard
σ Tn
E2 k2
E1
dTˆn dun
η η CZ
2. The standard linear CZM The rate-independent traction–separation relationship used in this investigation was an exponential traction–separation law as described in Ref. [17]. Only normal material separation was considered. The constitutive equation relating the rate-independent normal crack surface traction, T# n ; to the normal material separation, un, are un un # Tn ¼ s# max ze exp z ðun rdc Þ; dc dc
T# n ¼ 0 ðun > dc Þ;
σ Tn
ð1Þ Fig. 1. The standard linear material model.
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
linear CZM (SL-CZM) can be established to characterize the rate dependence of fracture. The SL-CZM as used in the present investigation, Fig. 1, is based on the following developments:
Z
dt
þ k2 T n ¼ Z
CZ
# dTn dun þ k2 T# n : þ k2 dun dt
Parametric studies were conducted to explore the traction–separation response of the SL-CZM. In the first group of results, the model response to various loading speeds was investigated. One of the main characterizing parameters in the SL-CZM is the ratio between the secondary stiffness of the model, k2, and the initial stiffness, k1 ¼ s# max ze=dc ; of the rate-independent CZM, where e and z were identified in Section 2.
= 100 = 10
un
Tn / ˆ max
=1
c
0.8 un
0.6
= 0.1
c
un
0.4
=0.01
c
un
0.2
= 0.001
c
0 0
0.2
0.4
0.6
un /
(a)
0.8
1
c
1.2
/ ˆ max
1.1
1.15
ˆ max
1
0.9
0.8
0.001
(b)
3. Results
1.2
un
1
ð4Þ
By combining the material parameters in the SL-CZM, a characteristic time constant, t ¼ ZCZ =k2 ; and a reference separation rate, d’ c ¼ k2 dc =ZCZ ; are introduced. The definition of the crack tip by the condition un=dc was retained for the rate-dependent CZM model, and the traction was set to zero when this condition was fulfilled. Furthermore, the critical separation, dc ; at which loss of load carrying capacity in the cohesive zone occurs due to craze breakdown, was taken to be constant and independent of rate. This assumption is based on experimental accounts reported in studies on craze growth and fracture [20–23]. Similar assumptions were used in the rate-dependent CZM models of Landis et al. and Liechte and Wu [6, 16]. Due to the dependence of the traction–separation law on rate, the cohesive energy is no longer given from cohesive length and strength and has now to be determined by the use of numerical integration.
un
c
Considering equilibrium and continuity, the constitutive equation for the SL-CZM model is: CZ dTn
1.4
c
max
(a) The displacement jump, un, across the cohesive zone and the crack surface traction, Tn, were used to replace strain, e, and stress, s, in the SLS, respectively; (b) The modulus, E1, in the SLS is replaced by stiffness of the cohesive zone in the rate-independent case, # dT=du n ; as given in Eq. (1); (c) A secondary cohesive zone stiffness, k2, with a unit of force per relative displacement per area (spring coefficient per area), and the cohesive zone viscosity, ZCZ ; with a unit of force per velocity per area (dashpot coefficient per area) were used to replace the modulus, E2 ; and the dashpot, Z; in the SLS, respectively.
11
0.01
0.1
1
un /
10
100
c
Fig 2. (a) Traction–separation response for k2=0.05k1, at different normalized material separation rates, (b) Dependence of the cohesive strength on the normalized separation rate.
Traction–separation curves for the SL-CZM for two parameter sets, k2=0.05k1 and k2 ¼ k1 ; are shown in Figs. 2a and 3a, respectively, for a set of normalized separation rates, u’ n =d’ c : The dependence of the maximum tractions, i.e. the cohesive strength in the rate dependent case, smax ; on the normalized separation rate are depicted in Figs. 2b and 3b. In analogy to the SLS, the behavior of the SL-CZM can be divided into three regimes. For low normalized separation rates, u’ n =d’ c o0:1; the SL-CZM behaves in the same way as the original rate-independent CZM. With increasing values of the normalized separation rate, the traction– separation response is shifted towards larger values of traction. For large normalized separation rates, u’ n =d’ c > 10; an upper limit of the traction–separation response is reached and the SL-CZM becomes a rate-independent CZM model in parallel to the spring E2. In this limit case, the cohesive traction can be obtained by the following equation: un un Tn ¼ s# max ze exp z ð5Þ þ k 2 un : dc dc At low rates, the rate-dependent cohesive strength, smax ; remains equal to the rate-independent cohesive strength, s# max ; and the viscous properties have little impact on the material separation behavior. When the normalized
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
12 14 12
un
10
un
= 100
Tn / ˆ max
c
= 10
c
un
8
=1
c
6 un
4
un
= 0.1
c
= 0.01
c
2
un
= 0.001
c
0
0
0.2
0.4
(a)
un /
c
un /
c
0.6
0.8
1
1
10
100
14 12
max
/ ˆ max
10
13.2 ˆ max
8 6 4 2 0 0.001
(b)
0.01
0.1
transition in the failure behavior of polymers as the glass transition temperature is approached from the elevated temperature side. Implications of this qualitative change in characteristics of the Tn-un response on crack stability were discussed in [2]. Failure properties of adhesives are known to be dependent on the polymer molecular weight or on the content of reinforcements in the adhesive. An investigation of the effects of variations in the magnitude of the rate-independent cohesive zone parameters, s# max anddc ; on the cohesive energy in the rate-dependent case provides an understanding of the impact of such modifications of the adhesive on its rate-dependent failure properties. Fig. 4a depicts the dependence of the cohesive energy, G; as normalized by its rate-indepen# on variations in the critical separation, dc. dent limit, G; Variations in critical separation, dc, are normalized to a reference cohesive zone possessing dref c : Fig. 4a demonstrates that a reduction in the critical separation—under all other parameters remaining constant—leads to a reduced dependence of the cohesive energy, i.e. the fracture toughness, on separation rate, u= ’ d’ c : This observation is independent of the applied loading rate. Fig. 4b demonstrates that an increase in the cohesive
Fig 3. (a) Traction–separation response for k2=k1, at different normalized material separation rates, (b) Dependence of the cohesive strength on the normalized separation rate.
= 25
ref c
u ref c
Γ/Γˆ
1.5
= 2.5 u ref c
1
= 0.25
u = 0.025 ref c
0.5
0 0
0.5
1
(a)
1.5
2
ref c
c/
5 4
Γ/Γˆ
separation rate is in an intermediate range, the cohesive strength is dependent on the normalized separation rate, and increases from the lower rate-independent limit to the upper rate-dependent limit in the type of transition well-known from the modulus rate dependence of a viscoelastic solid. The value of smax can be obtained from the condition dTn =dun ¼ 0: In general, there is no closed form solution from which to obtain smax in the intermediate range of normalized separation rates, and numerical results must be used to determine smax : A closed form solution for the cohesive strength can be provided for the case that the maximum traction is reached at un=dc, then smax ¼ s# max zexpð1 zÞ þ k2 dc While the increase in smax in dependence of the normalized loading rate is rather small for k2=0.05k1, namely smax ¼ 1:15 s# max ; a considerable increase occurs for k2=k1, namely smax ¼ 13:2 s# max : It is important to notice that for large values of k2/k1 the SL-CZM predicts a tendency of changes in the characteristics of the traction–separation behavior. The initial rate-independent traction–separation law described in Eq. (1) possesses a distinct softening behavior. However, this softening part of the Tnun response is lost as k2/k1 increases. Assuming that the ratio of k2/k1 increases as temperature decreases, the model well represents the
u
2
un
3
=1
c
un
2
= 0.1
c
1
un
= 0.01
c
0 0
(b)
1
2
3
4
5
6
ˆ ′max / ˆ max
Fig 4. Effects of the cohesive zone material parameters on the cohesive energy in the SL-CZM. (a) Influence of critical separation on cohesive energy for various material separation rates, (b) Influence of rateindependent cohesive strength on cohesive energy for various material separation rates.
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
strength, s# max ; again all other parameters remaining constant, also leads to a reduction in the ratedependence of the fracture performance of the adhesive. These predictions obtained from the present SL-CZM are consistent with general trends observed in fracture experiments on polymeric materials. Below glass the transition temperature, polymers are in a brittle state and will possess high s# max and small dc. In this state they commonly exhibit only limited rate-dependence in fracture behavior. On the other hand, polymers tested at temperatures above their glass transition temperatures can be described by small values of s# max and large values of dc. For this combination of cohesive zone parameters the model predicts a considerable dependence of the cohesive energy on separation rate, in line with experimental accounts [24].
4. Conclusions A standard linear CZM (SL-CZM) was developed to account for rate effects in the failure behavior of polymeric adhesives. The model presented is developed on the basis of the standard linear solid model (SLS). It predicts that the fracture behavior can be divided into three different regimes based on the separation rate. For the SLS, the equivalent modulus becomes constant if the separation rate is either low or high relative to the reference rate. At intermediate rates, the equivalent modulus depends on the separation rate. For the SLCZM model developed here the cohesive strength and subsequently the cohesive energy show similar behavior based on material separation rate in the process zone. However, while the SLS always obeys the Boltzman superposition principle, the SL-CZM is history-dependent. The SL-CZM captures important features in ratedependent fracture of polymeric materials. It provides a representation of the general correlations between the rate dependency and the testing temperature relative to the glass transition temperature. An application of the model to a study of crack growth in a thermoplastic
13
adhesive is presented in the companion paper that appears next.
Acknowledgements Early portions of this work were supported in part by the NSF CAREER Award (DMI 9501646) to K. Ramani. The authors would also like to thank Bemis Associates Inc. for providing adhesive for this study.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Knauss WG, Losi G. J Appl Mech 1993;60(4):793. Knauss WG. J Eng Mater Technol 1993;115(7):262. Schapery RA. Int J Fract 1999;97(1–4):33. Rahul-Kumar P, Jagota A, Bennison SJ, Saigal S, Muralidhar S. Acta Mater 1999;47(15–16):4161. Allen DH, Searcy CR. Mech Mater 2001;33(3):177. Landis CM, Pardoen T, Hutchinson JW. Mech Mater 2000;32(11):663. Corigliano A, Ricci M. Int J Solids Struct 2001;38(4):547. Fager LO, Bassani JL. J Eng Mater Technol 1993;115(3):252. Chaboche JL, Feyel F, Monerie Y. Int J Solids Struct 2001;38(18):3127. Lee Y, Prakash V. Int J Solids Struct 1999;36(22):3293. Glennie EB. J Mech Phys Solids 1971;19:255–72. Xu DB, Hui CY, Kramer E. J Appl Phys 1992;75(8):3294. Ba$zant ZP, Li YN. Int J Fract 1997;86(3):247. Li YN, Ba$zant ZP. Int J Fract 1997;86(3):267. Costanzo F, Walton JR. Int J Eng Sci 1997;35(12/13):1085. Liechti KM, Wu JD. J Mech Phys Solids 2001;49(5):1039. Needleman A. Int J Fract 1990;42(1):21. Ferry JD. Viscoelastic properties of polymers. New York: Wiley, 1980. Osswald TA. Materials science of polymers for engineers. Cincinnati: Hanser/Gardner, 1996. Kramer EJ, Berger LL. Adv Polym Sci 1990;91/92:1–68. . W. Adv Polym Sci 1983;59:105–68. Doll Pandya KC, Williams JG. Plast Rubber Compos 2000;29(9):439. Pandya KC, Williams JG. Polym Eng Sci 2000;40(8):1765. Clegg DW, Collyer AA. The structure and properties of polymeric materials. London: Institute of Materials, 1993.
Rate-dependent crack growth in adhesives I. Modeling approach Chongchen Xu, Thomas Siegmund*, Karthik Ramani School of Mechanical Engineering, Purdue University, 1288 Mechanical Engineering Building, West Lafayette, IN 47907-1288, USA Accepted 17 September 2002
Abstract Rate effects in the failure behavior are investigated for adhesive bonds in which the source of time- dependence is the rate of the material separation in the fracture process zone. A rate-dependent cohesive zone model (CZM) is described and used to analyze rate effects during debonding. The model consists of a rate-independent CZM in parallel to a Maxwell element. General implications derived for the model with respect to rate effects in the failure behavior of adhesives are presented. The application of the model to the analysis of crack growth is described in a companion paper. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: D. Viscoelasticity; Adhesive; Cohesive zone model
1. Introduction It is the goal of the present study to investigate effects related to the rate-dependent failure behavior of polymeric adhesives. Failure of adhesive joints is investigated in a system where a rate-dependent material separation process zone exists only in the adhesive itself, and the adherends adjacent to the adhesive deform elastically without any additional rate effects. Part I of the study, as presented in this paper, is concerned with a model description of rate-dependent material separation. Part II of the study, as presented in a companion paper, describes the applications of the model to a crack growth in a thermoplastic adhesive. For adhesives, and specifically for those based on thermoplastic polymers, the presence of the extended fracture process zone does not allow the use of linear elastic fracture mechanics concepts, and the presence of an extended fracture process zone at the crack tip is to be accounted for. In addition to the properties of the adhesive itself, the processes in this zone are dependent on specimen geometry, the elastic properties of the adherend, and the applied loading rate. Thus, for the investigation of the structural integrity of bonded *Corresponding author. +1-765-494-9766; fax: +1-765-494-0539. E-mail address: [email protected] (T. Siegmund).
structures, a constitutive model that describes the fracture process zone at the crack tip and its dependence of the mechanical constraints is required. A cohesive zone model (CZM) is ideally suited for that purpose. In such a model the material separation characteristics of the adhesive are described by a relationship between the displacement jump across the crack surface, i.e. across the adhesive, and the associated traction. With an appropriate traction-separation law in the CZM, one can account for the loss of load carrying capacity, and finally for crack growth. A number of different approaches have been used in the past in order to incorporate rate effects into material separation models. One of the approaches undertaken is based on the idea that a viscoelastic constitutive equation is combined with a damage function. This approach was applied by Knauss and Losi [1] and Knauss [2] and was also described in overview by Schapery [3]. A model based on multiplication of a viscoelastic hereditary integral with a traction–separation function was introduced instead of a continuum damage function [4,5]. The hereditary integral used for description of the fracture process was taken to be identical to the one determined for the bulk polymer. No separate determination of the rate-dependence of the fracture process itself was undertaken in these models. In viscoplastic interface models [6,7], the material
0143-7496/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 7 4 9 6 ( 0 2 ) 0 0 0 6 2 - 3
10
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
separation in the fracture process zone is the sum of an elastic and viscoplastic contribution, and cohesive tractions are rate-dependent only if plastic opening exists. Also, a rate-dependent CZM based on creep constitutive equations was developed [8]. In CZMs developed on the basis of time-dependent damage models [7,9], the material separation rate determines the rate of damage evolution and finally the strength degradation. Rate-dependence was included also by the use of pre-definition rate-dependencies of the cohesive zone properties. Such models were used recently for dynamic fracture investigations where a complex functional dependence of the rate-dependent cohesive strength was introduced [10]. Another approach bases the rate-dependence on a general fluid model. This approach was established in [11] and was used [12] to describe fracture of polymers. Finally, rate-dependent CZMs were constructed by direct inclusion of the traction–separation law of the CZM into viscoelastic constitutive equations. A general model was presented in Refs. [13,14]. Simplifications based on the use of a CZM within a Kelvin element were presented in Refs. [15,16]. In the present paper, the goal is to demonstrate the use of a rate-dependent CZM in which both the rateindependent and rate-dependent material separation parameters can be determined with reasonable experimental and numerical analysis effort. The standard linear solid model (SLS) provides a starting point for this development. A single Maxwell element combined with a rate-independent CZM was used to construct the rate-dependent traction–separation constitutive relationship. While the underlying rate-independent CZM possesses two material parameters, i.e. a cohesive strength and a critical separation, the proposed model adds two additional parameters for the description of the rate dependency. A CZM based on the linear standard model can be expected to provide information only over a restricted range of loading rates. Nevertheless, it allows for a description of the main important observations in viscoelastic material behavior.
ð2Þ
where z is 16e/9, e is exp (1). The hat over the cohesive zone quantities indicates rate-independent parameters. The current location of the crack tip is defined by un>dc. The rate-independent cohesive zone material parameters are the cohesive strength, s# max ; and the cohesive length, dc : The cohesive energy for the rate# is given by G # ¼ 9=16 s# max dc : The independent case, G; SLS, Fig. 1, is the basis for the development of the present rate-dependent CZM. This model is commonly used to characterize the behavior of solid polymer components [18, 19]. The SLS model consists of a spring in parallel to a Maxwell element that consists of a second spring and a dashpot in series. The uniaxial constitutive relationship between stress, a, and strain, e, given by the SLS model is ds de Z þ E2 s ¼ ZðE1 þ E2 Þ þ E1 E2 e; ð3Þ dt dt where Z is the viscosity, E1 and E2 are moduli, and t is time. This combination of elements provides an equilibrium modulus, E1, characteristic of low loading rates, and a glassy modulus, E1+E2, for high loading rates. On the basis of the SLS model, Eq. (3), a standard
σ Tn
E2 k2
E1
dTˆn dun
η η CZ
2. The standard linear CZM The rate-independent traction–separation relationship used in this investigation was an exponential traction–separation law as described in Ref. [17]. Only normal material separation was considered. The constitutive equation relating the rate-independent normal crack surface traction, T# n ; to the normal material separation, un, are un un # Tn ¼ s# max ze exp z ðun rdc Þ; dc dc
T# n ¼ 0 ðun > dc Þ;
σ Tn
ð1Þ Fig. 1. The standard linear material model.
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
linear CZM (SL-CZM) can be established to characterize the rate dependence of fracture. The SL-CZM as used in the present investigation, Fig. 1, is based on the following developments:
Z
dt
þ k2 T n ¼ Z
CZ
# dTn dun þ k2 T# n : þ k2 dun dt
Parametric studies were conducted to explore the traction–separation response of the SL-CZM. In the first group of results, the model response to various loading speeds was investigated. One of the main characterizing parameters in the SL-CZM is the ratio between the secondary stiffness of the model, k2, and the initial stiffness, k1 ¼ s# max ze=dc ; of the rate-independent CZM, where e and z were identified in Section 2.
= 100 = 10
un
Tn / ˆ max
=1
c
0.8 un
0.6
= 0.1
c
un
0.4
=0.01
c
un
0.2
= 0.001
c
0 0
0.2
0.4
0.6
un /
(a)
0.8
1
c
1.2
/ ˆ max
1.1
1.15
ˆ max
1
0.9
0.8
0.001
(b)
3. Results
1.2
un
1
ð4Þ
By combining the material parameters in the SL-CZM, a characteristic time constant, t ¼ ZCZ =k2 ; and a reference separation rate, d’ c ¼ k2 dc =ZCZ ; are introduced. The definition of the crack tip by the condition un=dc was retained for the rate-dependent CZM model, and the traction was set to zero when this condition was fulfilled. Furthermore, the critical separation, dc ; at which loss of load carrying capacity in the cohesive zone occurs due to craze breakdown, was taken to be constant and independent of rate. This assumption is based on experimental accounts reported in studies on craze growth and fracture [20–23]. Similar assumptions were used in the rate-dependent CZM models of Landis et al. and Liechte and Wu [6, 16]. Due to the dependence of the traction–separation law on rate, the cohesive energy is no longer given from cohesive length and strength and has now to be determined by the use of numerical integration.
un
c
Considering equilibrium and continuity, the constitutive equation for the SL-CZM model is: CZ dTn
1.4
c
max
(a) The displacement jump, un, across the cohesive zone and the crack surface traction, Tn, were used to replace strain, e, and stress, s, in the SLS, respectively; (b) The modulus, E1, in the SLS is replaced by stiffness of the cohesive zone in the rate-independent case, # dT=du n ; as given in Eq. (1); (c) A secondary cohesive zone stiffness, k2, with a unit of force per relative displacement per area (spring coefficient per area), and the cohesive zone viscosity, ZCZ ; with a unit of force per velocity per area (dashpot coefficient per area) were used to replace the modulus, E2 ; and the dashpot, Z; in the SLS, respectively.
11
0.01
0.1
1
un /
10
100
c
Fig 2. (a) Traction–separation response for k2=0.05k1, at different normalized material separation rates, (b) Dependence of the cohesive strength on the normalized separation rate.
Traction–separation curves for the SL-CZM for two parameter sets, k2=0.05k1 and k2 ¼ k1 ; are shown in Figs. 2a and 3a, respectively, for a set of normalized separation rates, u’ n =d’ c : The dependence of the maximum tractions, i.e. the cohesive strength in the rate dependent case, smax ; on the normalized separation rate are depicted in Figs. 2b and 3b. In analogy to the SLS, the behavior of the SL-CZM can be divided into three regimes. For low normalized separation rates, u’ n =d’ c o0:1; the SL-CZM behaves in the same way as the original rate-independent CZM. With increasing values of the normalized separation rate, the traction– separation response is shifted towards larger values of traction. For large normalized separation rates, u’ n =d’ c > 10; an upper limit of the traction–separation response is reached and the SL-CZM becomes a rate-independent CZM model in parallel to the spring E2. In this limit case, the cohesive traction can be obtained by the following equation: un un Tn ¼ s# max ze exp z ð5Þ þ k 2 un : dc dc At low rates, the rate-dependent cohesive strength, smax ; remains equal to the rate-independent cohesive strength, s# max ; and the viscous properties have little impact on the material separation behavior. When the normalized
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
12 14 12
un
10
un
= 100
Tn / ˆ max
c
= 10
c
un
8
=1
c
6 un
4
un
= 0.1
c
= 0.01
c
2
un
= 0.001
c
0
0
0.2
0.4
(a)
un /
c
un /
c
0.6
0.8
1
1
10
100
14 12
max
/ ˆ max
10
13.2 ˆ max
8 6 4 2 0 0.001
(b)
0.01
0.1
transition in the failure behavior of polymers as the glass transition temperature is approached from the elevated temperature side. Implications of this qualitative change in characteristics of the Tn-un response on crack stability were discussed in [2]. Failure properties of adhesives are known to be dependent on the polymer molecular weight or on the content of reinforcements in the adhesive. An investigation of the effects of variations in the magnitude of the rate-independent cohesive zone parameters, s# max anddc ; on the cohesive energy in the rate-dependent case provides an understanding of the impact of such modifications of the adhesive on its rate-dependent failure properties. Fig. 4a depicts the dependence of the cohesive energy, G; as normalized by its rate-indepen# on variations in the critical separation, dc. dent limit, G; Variations in critical separation, dc, are normalized to a reference cohesive zone possessing dref c : Fig. 4a demonstrates that a reduction in the critical separation—under all other parameters remaining constant—leads to a reduced dependence of the cohesive energy, i.e. the fracture toughness, on separation rate, u= ’ d’ c : This observation is independent of the applied loading rate. Fig. 4b demonstrates that an increase in the cohesive
Fig 3. (a) Traction–separation response for k2=k1, at different normalized material separation rates, (b) Dependence of the cohesive strength on the normalized separation rate.
= 25
ref c
u ref c
Γ/Γˆ
1.5
= 2.5 u ref c
1
= 0.25
u = 0.025 ref c
0.5
0 0
0.5
1
(a)
1.5
2
ref c
c/
5 4
Γ/Γˆ
separation rate is in an intermediate range, the cohesive strength is dependent on the normalized separation rate, and increases from the lower rate-independent limit to the upper rate-dependent limit in the type of transition well-known from the modulus rate dependence of a viscoelastic solid. The value of smax can be obtained from the condition dTn =dun ¼ 0: In general, there is no closed form solution from which to obtain smax in the intermediate range of normalized separation rates, and numerical results must be used to determine smax : A closed form solution for the cohesive strength can be provided for the case that the maximum traction is reached at un=dc, then smax ¼ s# max zexpð1 zÞ þ k2 dc While the increase in smax in dependence of the normalized loading rate is rather small for k2=0.05k1, namely smax ¼ 1:15 s# max ; a considerable increase occurs for k2=k1, namely smax ¼ 13:2 s# max : It is important to notice that for large values of k2/k1 the SL-CZM predicts a tendency of changes in the characteristics of the traction–separation behavior. The initial rate-independent traction–separation law described in Eq. (1) possesses a distinct softening behavior. However, this softening part of the Tnun response is lost as k2/k1 increases. Assuming that the ratio of k2/k1 increases as temperature decreases, the model well represents the
u
2
un
3
=1
c
un
2
= 0.1
c
1
un
= 0.01
c
0 0
(b)
1
2
3
4
5
6
ˆ ′max / ˆ max
Fig 4. Effects of the cohesive zone material parameters on the cohesive energy in the SL-CZM. (a) Influence of critical separation on cohesive energy for various material separation rates, (b) Influence of rateindependent cohesive strength on cohesive energy for various material separation rates.
C. Xu et al. / International Journal of Adhesion & Adhesives 23 (2003) 9–13
strength, s# max ; again all other parameters remaining constant, also leads to a reduction in the ratedependence of the fracture performance of the adhesive. These predictions obtained from the present SL-CZM are consistent with general trends observed in fracture experiments on polymeric materials. Below glass the transition temperature, polymers are in a brittle state and will possess high s# max and small dc. In this state they commonly exhibit only limited rate-dependence in fracture behavior. On the other hand, polymers tested at temperatures above their glass transition temperatures can be described by small values of s# max and large values of dc. For this combination of cohesive zone parameters the model predicts a considerable dependence of the cohesive energy on separation rate, in line with experimental accounts [24].
4. Conclusions A standard linear CZM (SL-CZM) was developed to account for rate effects in the failure behavior of polymeric adhesives. The model presented is developed on the basis of the standard linear solid model (SLS). It predicts that the fracture behavior can be divided into three different regimes based on the separation rate. For the SLS, the equivalent modulus becomes constant if the separation rate is either low or high relative to the reference rate. At intermediate rates, the equivalent modulus depends on the separation rate. For the SLCZM model developed here the cohesive strength and subsequently the cohesive energy show similar behavior based on material separation rate in the process zone. However, while the SLS always obeys the Boltzman superposition principle, the SL-CZM is history-dependent. The SL-CZM captures important features in ratedependent fracture of polymeric materials. It provides a representation of the general correlations between the rate dependency and the testing temperature relative to the glass transition temperature. An application of the model to a study of crack growth in a thermoplastic
13
adhesive is presented in the companion paper that appears next.
Acknowledgements Early portions of this work were supported in part by the NSF CAREER Award (DMI 9501646) to K. Ramani. The authors would also like to thank Bemis Associates Inc. for providing adhesive for this study.
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