Int J Fract (2019) 215:77–89 https://doi.org/10.1007/s10704-018-00335-9
ORIGINAL PAPER
Finite strain theory of a Mode III crack in a rate dependent gel consisting of chemical and physical cross-links
Chung Yuen Hui · Jingyi Guo · Mincong Liu · Alan Zehnder
Received: 2 June 2018 / Accepted: 27 November 2018 / Published online: 3 January 2019 © Springer Media B.V 2019
Abstract In previous works we developed a three- 1 Introduction dimensional large deformation constitutive theory for a dual cross-link hydrogel gel with permanent and tran- A hydrogel is a network of polymer chains swollen sient bonds. This theory connects the breaking and in water. Due to the large water content, they can be reforming kinetics of the transient bonds to the non- highly compliant (with shear moduli ranging from a linear elasticity of the gel network. We have shown few Pa to 100 kPa), bio-compatible and exhibit low that this theory agrees well with experimental data friction, making them ideal candidates for many bio- from both tension and torsion tests. Here we study the engineering applications such as scaffolds for cells in mechanics of a Mode III or anti-plane shear crack for tissue engineering (Kuo and Ma 2001; Lee and Mooney this particular class of rate dependent solids. We first 2001), artificial cartilage (Kwon et al. 2014) and vehi- show that our constitutive model admits a non-trivial cles for drug delivery (Qiu and Park 2001). However, state of anti-plane shear deformation. We then establish the high water content of these gels also limits their use a correspondence principle for the special case where as structural materials since they have very low resis- the chains in the network obey Gaussian statistics. For tance to fracture. There have been rapid advances in this special case there is a one to one analogy between this field since 2003, when Gong et al. (2003) synthe- our model and standard linear viscoelasticity. We also sized a highly extensible double network (DN) hydro- study the asymptotic behavior of the time dependent gel with fracture toughness ∼103 J/m2, comparable to crack tip field, and show that for a wide class of net- that of synthetic rubber. This gel consists of two inter- work energy density functions, the spatial singularities penetrating networks: the first network is tightly cross- of these fields are identical to a hyper-elastic, cracked linked and swollen while the second network is loosely body with the same but undamaged networks. cross-linked and highly extensible (Gong et al. 2003; Webber et al. 2007). When subjected to mechanical Keywords Self-healing hydrogel · Mode III crack · loading, the first network bears most of the load and Asymptotics · Rate dependence · Correspondence undergoes progressive damage and the second exten- principle sible network prevents the formation and growth of macroscopic cracks (Brown 2007; Gong 2010; Naka- jima et al. 2013; Wang and Hong 2011). More recently, C. Y. Hui (B) · J. Guo · M. Liu · A. Zehnder several research groups (Henderson et al. 2010; Lin Field of Theoretical and Applied Mechanics, Department et al. 2010; Mayumi et al. 2013; Narita et al. 2013; Sun of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14583, USA et al. 2012, 2013) have found that highly stretchable e-mail: [email protected] 123 78 C. Y. Hui et al. and tough hydrogels can also be made by replacing the simple case of Mode III or anti-plane shear fracture. covalent bonds in the stiff network by temporary physi- The kinematic simplicity of Mode III fracture allows cal bonds. Since physical bonds can reform after break- us to reduce the mathematical complexities associated age, upon unloading and resting, these DN gels can par- with plane strain or plane stress cracks and to gain tially or fully recover to their original state depending insight into the mechanics of fracture in more com- on the resting period. Due to the kinetics of the physical plex geometries. For example, Rice studied the stresses bonds, these self-healing gels exhibit time dependent due to sharp notches in elastic–plastic solids loaded in behavior similar to nonlinear viscoelastic solids (Lin anti-plane shear using the hodograph technique (Rice et al. 2010; Mayumi et al. 2013; Narita et al. 2013; Sun 1967), Chitaley and McClintock (1971) studied the et al. 2012, 2013). plastic deformation of a steady state growing crack There are many open questions on the rate depen- under anti-plane shear. More recently, Long and Hui dent mechanical behavior of these gels. Until our studied the effect of finite chain extensibility on the recent works (Long et al. 2014, 2015) there have stress and strains near the tip of Mode III crack (Long been no quantitative constitutive descriptions for the and Hui 2015). large strain time dependent behavior of these gels. In Many previous works use a linearized version of these works (Long et al. 2014, 2015) we studied the anti-plane shear which assumes small deformation. As mechanical behavior of a poly(vinylalcohol) (PVA) noted by the seminal work of Knowles (1977), only dual crosslinked gel with PVA chemically crosslinked a certain class of hyper-elastic incompressible solids by glutaraldehyde and physically crosslinked by Borax admits a non-trivial state of anti-plane shear defor- ions (Mayumi et al. 2013; Narita et al. 2013). Although mation. Since our constitutive model is nonlinear vis- this gel has lower resistance to fracture in comparison to coelastic, it is not clear that it admits a non-trivial the polyampholyte gel developed by Sun et al. (2013), it state of anti-plane shear deformation. The proof that it has a well-defined simple chemical structure with only indeed does and a brief review of our constitutive model one type of physical crosslink between PVAchains. As are given in Sects. 2 and 3 of this paper. In Sect. 4,we a result, in linear rheology tests, the loss modulus has studied several special cases which lends insight to the a well-defined relaxation time. In addition, the dynam- time dependent deformation and stress fields in these ics of bond breaking and healing are found experimen- gels. Summary and discussion is given in Sect. 4. tally to be nearly independent of strain over a wide range of strain amplitude, which is typically not the case for more complex gels. From this perspective it 2 Review of constitutive model serves a model system to study the interplay between bond kinetics and macroscopic mechanical behavior We briefly review our constitutive model. Details can of the hydrogel having transient crosslinks. In several be found in our previous works (Guo et al. 2016; Long papers (Guo et al. 2016; Long et al. 2014, 2015), we et al. 2014). Macroscopically the gel is assumed to have shown that a three-dimensional constitutive model be isotropic and incompressible. It consists of two which combines the finite strain elasticity of the net- independent networks: the chemical crosslinks form a works with the kinetics of bond breaking and reattach- permanent elastic network and the physical crosslinks ment can accurately predict the behavior of uni-axial form a transient elastic network that can break and reat- tension and torsion tests with complex loading histo- tach with rates independent of the stress or strain acting ries. on it. We assume the chemical cross-links do not break. Although there have been some experimental stud- Further, when a polymer chain in the transient network ies on the fracture behavior of these self-healing gels breaks, it instantaneously releases the strain energy it (Mayumi et al. 2016), we are not aware of any quan- carries. Also, when a broken chain reattaches at time titative analyses of their fracture mechanics. This is τ, it has zero strain energy at that time, even though the because these gels are often subjected to very large transient network is under stress. The deformation of deformation; the nonlinearity due to finite strain kine- temporary chains reconnected at time τ is described by matics and material behavior as well as history depen- the deformation gradient tensor Fτ→t . The superscript dence make quantitative analysis extremely difficult. τ → t indicates that the temporary chain experiences Our aim is to provide such an analysis, albeit for the the deformation history from its birth (reattachment) 123 Finite strain theory of a Mode III crack 79
at τ to the current time t. The total strain energy is ∞ ( ) ≡¯γ φ (τ/ ) τ equal to the sum of the strain energy carried by each n t ∞ B tB d t −α chain in both networks and is given by an energy den- 2 B −α sity function. Although not necessary, we assume the tB t 1 B =¯γ∞ 1 + (αB − 1) (1c) strain energy densities of the undamaged networks are 2 − αB tB the same; denoted by W, a function only of the strain Note that n(t) is a decaying function of time since 2 > T invariant I1 ≡ tr F F , where F is the deformation αB > 1. gradient. We also assume the breaking and reattach- To gain physical insight, consider the special case ing of the physical chains reaches a dynamic equilib- of uniaxial tension; (1a) reduces to rium soon after the gel is synthesized. This steady state 1 P = 2 [ρ + n(t)] W (I ) λ(t) − assumption implies that the breaking and reattachment 11 1 H(0,t) λ2(t) rates are equal and independent of time. These rates are t t − τ + γ¯∞ φ ( ) denoted by γ¯∞. 2 B W I1 H(τ,t) tB Without loss of generality, we assume loading com- 0 λ(t) λ(τ) mences at t = 0 and prior to that no mechanical load- × − dτ (1d) λ2(τ) λ2( ) ing has ever been applied. With these assumptions, the t λ nominal stress tensor P is related to the deformation where P11 is the nominal stress and is the stretch ratio gradient Fτ→t by in the loading direction, and 2 −T I (t) = λ2 (t) + , =− 0→t + ( ) + ρ ( ) 1 λ (t) P p F 2 [n t ] W I1 I =H(0,t) 1 (τ, ) = λ ( ) /λ (τ) 2 t − τ H t [ t ] 0→t t × + γ¯∞ φ ( ) F 2 B W I1 I =H(τ,t) + 2λ (τ) /λ (t) ,τ
(kPa) where σ
10 w,α ≡ ∂w/∂xα, (2d–f) w ( , ,τ) = w ( , ) − w ( ,τ) 5 x t x t x and ω33 = 1 − w,1 (x,τ)w,1 −w,2 (x,τ)w,2 Nominal Stress 0 Also, -5 1 1.05 1.1 1.15 1.2 1.25 1.3 Stretch λ → T → H (x, 0, t) ≡ tr F0 t F0 t (2g)
Fig. 1 Experimental results and model prediction of the stress- = + w, ( , )w, ( , ) stretch behavior of one PVA dual-crosslink hydrogel sample 3 α x t α x t in constant rate uniaxial tension tests of different loading and unloading rates. The solid curves are experimental results and the dashed curves are model predictions. The material parame- ( ,τ, ) = τ→t T τ→t ters in our model are: μρ = 3.8091 kPa, μγ¯∞ = 18.68 kPa/s, H x t tr F F (2h) αB = 1.6137, tB = 0.6606 s = 3 + w,α (x, t,τ) w,α (x, t,τ)
Substituting (2b–h) into (1a), the nominal stress com- cal region in its undeformed configuration is said to be ponents are: deformed to a state of anti-plane shear if the displace- ment of each material point is parallel to the generators P α = 2 [n(t) + ρ] W (H ) w,α 3 0 of the cylinder (which we took to be in the direction of t t − τ + γ¯ φ ( ) w, τ the unit vector E ) and is independent of x , the out of 2 ∞ B W H α d (3a) 3 3 0 tB plane coordinate.” We identify a material point in the undeformed reference configuration by the 2D vector P33 =−p + 2 [n(t) + ρ] W (H0) t t−τ ≡ α α α = , φB x x E 1 2(2a)+ γ¯∞ tB 2 0 W (H)(−w,1 w,1 −w,2 w,2 +1) dτ Here {E1, E2, E3} are orthonormal vectors which form (3b) a right handed basis. In the following, we used the P = P =−p + 2 [n(t) + ρ] W (H ) 11 22 0 t summation convention of summing over repeated t − τ + 2γ¯∞ φ W (H) dτ (3c) indices. Also, we assume quasi-static deformation in B t 0 B the absence of body forces. t t − τ Pα3 = pw,α (x, t) − 2γ¯∞ φB By definition, the only non-vanishing displacement 0 tB w ( , , ) is x1 x2 t in the out of plane direction. A straight W (H) w,α (x,τ) dτ (3d) forward but tedious calculation using the definition of P12 = P21 = 0 (3e) −1 deformation gradient and Fτ→t = F0→τ F0→t shows that the matrices representing the tensors F0→t Since the quantity inside the curly bracket in (3b)is
τ→ →τ −T and F t F0 with respect to the Cartesian basis independent of x3, the equilibrium equation in the out , + , = { , , } of plane direction P3α α P33 3 0 reduces to E1 E2 E3 are: p, = P α,α = 2 [n(t) + ρ] W (H ) w,α ,α 3 3 0 ⎡ ⎤ t − τ t 100 + 2γ¯∞ φ W (H) w,α ,α dτ 0→t ⎣ ⎦ B F = 010, (2b,c) 0 tB w,1 (x, t)w,2 (x, t) 1 (4) 123 Finite strain theory of a Mode III crack 81
Since the RHS of (4) is independent of x , p must be a c(x, t) = f (t)w(x, t) + 2 [n(t) + ρ] W (H ) 3 0 t linear function of x3, that is, t − τ + 2γ¯∞ φB W (H) dτ + d(t) p = φ(x, t)x3 + c(x, t) (5a) 0 tB (11) where where d(t) is an arbitrary function of time. φ(x, t) = 2 [n(t) + ρ] W (H ) w,α ,α 0 f (t) d(t) t To determine the functions and ,weuse t − τ + 2γ¯∞ φB W (H) w,α ,α dτ, the fact that the crack faces are traction free for all 0 tB times. Here when we assume the crack is straight and (5b) occupies a closed interval C on x2 = 0, the traction and c(x, t) is an undetermined function. To determine free condition is: c(x, t), in-plane equilibrium requires: = = = ∀ ∈ P21 P22 P23 0 tx1 C (12) , + , =− , + ( ) + ρ ( ) P11 1 P13 3 p 1 2 [n t ] W H0 By (3e), P21 = 0 is automatically satisfied. Using (3d), = t (10) and (11), the condition P22 0 on the crack face t − τ + 2γ¯∞ φB W (H) dτ ,1 is 0 tB + p, w, = 0 (6a) − f (t)x − f (t)w(x, t) − d(t) = , 3 1 3 0 (13) P22,2 +P23,3 =−p,2 + 2 [n(t) + ρ] W (H0) for all times and all x3, these imply t t − τ f (t) = d (t) = 0(14) + 2γ¯∞ φ W (H) dτ , B t 2 0 B Substituting (14)into(11) and using (10), p is + p,3 w,2 = 0 (6b) p = c(x, t) = 2 [n(t) + ρ] W (H0) Equation (6a,b) can be rewritten as t t − τ + γ¯ φ ( ) τ 2 ∞ B W H d (15) p − 2 [n(t) + ρ] W (H0) 0 tB t A special case is when the chains obey Gaussian statis- t − τ + 2γ¯∞ φB W (H) dτ tics, for this case W is neo-Hookean, i.e., 0 tB , α μ μ − , w, = W = (I − 3) ⇒ W = (16) p 3 α 0(7)2 1 2 Substituting (5a)into(7)gives: where μ is the small strain shear modulus. Direct eval- uation of the integral in (15)using(1b) and (16) results , = ( ) + ρ ( ) p α 2 [n t ] W H0 in t t − τ p = μ [n(t = 0) + ρ] . (17) + 2γ¯∞ φ W (H) dτ B t 0 B ,α For this special case p is a constant, independent of + φ( , )w, x t α (8) deformation and time. Finally, (15) and (3a, 3d)imply = where H0 = H (x, 0, t) and we hide the arguments in that P3α Pα3 and this ensures the true stress is sym- H to simplify notation. Since the RHS of (8) is indepen- metric. The true stress tensor can be computed using 0→t T dent of x3, p,α must be independent of x3; substituting τ = P F and is (5a)into(8) shows that τ = τ = τ = τ = , φ,α ( , ) = ⇒ φ( , ) = ( ) , 11 12 22 21 0 x t 0 x t f t (9) τ α = P α = 2 [n(t) + ρ] W (H ) w,α ( ) 3 3 0 for some unknown f t . Equation (5a) can now be t t−τ (18a–c) + 2γ¯∞ φB W (H) w,α dτ, written as 0 tB τ33 = P31w,1 +P32w,2 +P33 p = f (t)x3 + c(x, t) (10)
Substituting (10)into(8) with φ(x, t) = f (t) and inte- The nominal stress P33 in (3b) can be simplified using grating gives: (15) and is 123 82 C. Y. Hui et al. t t − τ P33 =−2γ¯∞ φB W (H) (18d) 0 tB [w,1 w,1 +w,2 w,2] dτ
Note, in contrast to linearized anti-plane theory, neither of the out of plane stress component P33 nor τ33 are zero. Although this stress is completely determined by Fig. 2 The undeformed specimen consists of an infinite strip of the shear stresses and the displacement gradient, they height 2h with a semi-infinite crack. The strip is infinite in the x1 and x3 (out of plane) direction and is loaded by imposing equal have higher singularity at the crack tip than the out of and opposite out of plane displacement in the upper and lower plane shear stresses because the displacement gradi- faces ents are also singular there. Like the T stress in plane fracture, τ is parallel to the crack plane and in hyper- 33 ation test, the boundary of the specimen is subjected elastic crack problems, it has no effect on the energy to a fixed displacement instantaneously. For example, release rate. However, the T stress is known to affect in the strip specimen shown in Fig. 2, this boundary the stability of crack path (Cotterell and Rice 1980). condition is Unlike the T stress, τ33 is highly singular and therefore + in theory, could affect crack path in a significant way. If w(x1, x2 =±h, t ≥ 0 ) =±w0, (20) this hypothesis is correct, then the normal to the crack where w0 is the imposed out of plane displacement. We plane should rotate about the x1 axis in Fig. 2.How- seek a solution wE (x) in which the displacement in the ever, we are not aware of any literature that discuss this cracked body is independent of time, i.e., effect. We end this section by noting that (9) and (14)imply w(x1, x2, t) = wE (x1, x2) (21) that the partial differential equation (PDE) governing This assumption implies that w in the integral in (19) the displacement w is: is identically zero for all times. Thus, the governing t − τ equation for the displacement is: t 2 [n(t) + ρ] W (H0) w,α ,α +2γ¯∞ φB t ( ) w,α ,α = 0 B W H0 0(22) W (H) w,α ,α dτ = 0 (19) where H0 = 3+w,α (x)w,α (x). It is interesting to note Note that (19)isnot the general equation for anti-plane that (22) is the governing equation of anti-plane defor- shear deformation since it uses the traction free bound- mation for a hyper-elastic solid with strain energy den- ary condition on the crack faces to determine f and d. sity function W (I1). Knowles (1977) and Rice (1967) have shown that the nonlinear PDE (22) can be trans- formed to a linear PDE using the Holograph transform. 4 Results This fact allows us to establish the following correspon- dence principle for relaxation crack problems. That is, In the following, we assume the specimen geometry in a relaxation test, the displacement of the cracked and the manner of loading are consistent with anti- body is independent of time and is identical to displace- plane shear deformation. Although it is possible to ment wE of an identical hyper-elastic cracked body consider multiple cracks, we assume a single trac- with the same energy density function W. However, tion free stationary crack occupying the interval C = the true stresses are time dependent and decay with { ≤ ≤ , = } a x1 b x2 0 . We allow a to be infinite, for time according to: example, a =−∞, b = 0, corresponds to a semi- τ = = ( ) + ρ + w , w , infinite crack. An example is shown in Fig. 2. 3α P3α 2 [n t ] W 3 E β E β wE ,α ≡ [n(t) + ρ] τE3α (23a) τ = [n(t) + ρ][τ w , +τ w , ] (23b) 4.1 Relaxation test 33 E31 E 1 E32 E 2 where the subscript E in displacement and stress are Relaxation test is commonly used to study the mechani- used to indicate that these fields belong to the hyper- cal behavior of rate dependent solids. In an ideal relax- elastic solid. Thus, the stresses in the cracked body 123 Finite strain theory of a Mode III crack 83 behaves like a hyper-elastic body where the shear stress tensor are proportional to 1/r. Also, as noted modulus is multiplied by a decaying function of time. earlier, τE33 τE3α as r → 0. Since all Mode III Since n(t →∞) = 0, the cracked body relaxes to a crack tip fields has the same functional form given by hyper-elastic cracked body with reduced modulus μρ (25a–25c), K can be interpreted as an intensity fac- at long time. Physically, this stress relaxation corre- tor. However, K cannot be determined by a local anal- sponds to the breaking of physical bonds that are loaded ysis since it depends on specimen geometry and the at t = 0+, the integral term vanishes since physical manner of loading. Knowles (1977) has shown that for bonds that are healed do not carry any load. GNH cracked bodies loaded under anti-plane shear, K As a concrete example, assume that the energy den- is related to the path independent J integral by sity function is given by 1−N B fN A2nK2N = J, μ B N πμ ( ) = + ( − ) − , W I1 1 I1 3 1 (24) N 2B N 4N 4 fN = (26) where μ is the small strain shear modulus, N > 1/2isa N (2N − 1)2N−1 2N 2 − 2N + 1 > strain hardening parameter and B 0 is a dimension- An example where J can be obtained by elementary = less material constant. The special case of N 1 cor- means is the strip sample shown in Fig. 2. Assuming responds to a neo-Hookean solid; for this reason, this that the upper and lower edges of the strip are sub- model is called the “Generalized neo-Hookean (GNH)” jected to a uniform displacement ± w0, translational solid. invariance implies that The asymptotic displacement wE and the true tress J = 2hW∞ (27) field τE3α near the crack tip for a GNH solid was obtained by Knowles (1977). With respect to a polar where W∞ is the strain energy density at x1 =∞given coordinate system (r,θ) in the reference configuration, by they are μ w 2 N = + B 0 − m 1 W∞ 1 1 (28) wE ∼ Kr v(θ) m = 1 − θ ∈ [−π, π] 2B N h 2N (25a) Substituting (28)into(27) and using (26) relates the
− B N−1 −( −( / )) intensity factor K to the applied displacement w , i.e., τ α ∼ μK 2N 1 τˆα (θ) r 1 1 2N , 0 E3 N τˆ (θ) = ˆ N−1 (θ) − 1 v (θ) (θ) −N N α h 1 cα (25b) hB fN B w0 2 2N K 2N = 1 + − 1 (29a) +˙v (θ) εβαcβ (θ) π N h − B N 1 2N ˆ N −1 For very large deformations where w0/h 1, (29a) τE33 ∼ μK h (θ) r (25c) N is well approximated by where v˙ (θ) = dv/dθ, εβα is the 2D alternator where 1/2N hfN 1 w0 ε = ε = 0,ε =−ε = 1 and K = √ (29b) 11 22 12 21 π N h c (θ) = cos θ, c (θ) = sin θ (25d) − 1 2⎡ ⎤ Since the local stresses is proportional to K 2N 1 [see 2 (1 − 1/N)2 cos2 (θ/2) (25b, 25b)], (29b) implies that for large N the local v (θ) = sin (θ/2) ⎣ 1 − ⎦ 1 + ω (θ, N) stresses increase very rapidly with the imposed far field w / N−1 strain 0 h. [ω (θ, N) + (1 − 1/N) cos θ] 2N (25e) For the special case of N = 1, the stresses are given ω (θ, N) = 1 − (1 − 1/N)2 sin2 θ (25f) by 2 μK (2N − 1) − / −1/2 hˆ (θ) = [(1 − 1/N) cos θ + ω (θ, N)] 1 N τE31 ∼− r sin (θ/2) (30a) 4N 3 2 μ (25g) K −1/2 τE ∼ r cos (θ/2) (30b) 32 2 Equations (25b, 25c) show that the stress field near the − τ ∼ μK 2r 1/4 (30c) crack tip increases with strain hardening. Indeed, for E33 w where K = √2 0 . very large N, all non-vanishing components of the true πh 123 84 C. Y. Hui et al.
As is well known, the out of plane shear stresses known that such equations has a unique solution (Kress τE3α are identical to the linearized small strain anti- 2014) which in this case is the trivial solution where plane theory. For this case K is related to the ModeIII √ w,αα = 0 (33) stress intensity factor K III by K III = π/2μK .A key difference between the linear and large deformation Thus, the displacement field satisfies the 2D Laplace theory is that τE33 = 0 in the linearized theory whereas equation, just as in the linearized small strain anti-plane this is the dominant singularity in the nonlinear theory. shear theory. For the out of plane shear stresses, (18b) reduces to
4.2 A special case: neo-Hookean τ3α = P3α = μ [n0 + ρ] w,α (x, t) − μγ¯∞ t We are not aware of any analytical technique to obtain t − τ the general solution of (19) with arbitrary energy den- φB w,α (x,τ) dτ (34a) tB sity function (see Sect. 4.3 for more details). How- 0 ever, the special case of Gaussian chains where W whereas is neo-Hookean can be solved exactly. For this case, τ = μ + ρ |∇w ( , )|2 W = μ/2 and (19) reduces to 33 [n0 ] x t −2μγ¯∞w,α (x, t) t t t − τ t − τ ( ) + ρ w, +¯γ φ φ w,α (x,τ) dτ [n t ] αα ∞ B B t tB B 0 0 t (w,α) ,α dτ = 0 (31a) − τ t 2 + μγ¯∞ φB |∇w (x,τ)| dτ (34b) Equation (31a) can be simplified by noting that the inte- tB 0 gral term is Correspondence principle t t − τ Equation (31a) can be written in a more suggestive γ¯ φ (w, ) , τ =¯γ w, ∞ B α α d ∞ αα form by defining a time dependent modulus tB 0 μ (t) ≡ μ [ρ + n(t)] (35) t R t − τ (x, t) φB dτ −¯γ∞ Integrating by parts and using (35) and (1c) shows that tB (31a) can be rewritten in the form of a linear viscoelastic 0 t model − τ t τ = = μ ( ) w, ( , = ) φB w,αα (x,τ) dτ 3α P3α R t α x t 0 tB t 0 ∂w,α (x,τ) = w, ( , ) − ( ) −¯γ + μ (t − τ) dτ (36) αα x t [n0 n t ] ∞ R ∂τ t 0 t − τ φ w, ( ,τ) τ B αα x d (31b) where μR (t) is interpreted as the shear relaxation func- tB 0 tion. Equation (36) implies that the usual correspon- dence principle of linear viscoelasticity applies to our wherewehaveused(1c) and n0 ≡ n(t = 0) . Substi- tuting (31b)into(31a), the governing PDE becomes: model. Two simple examples are: (i) if traction boundary conditions are prescribed, then (n0 + ρ) w,αα (x, t) −¯γ∞ t the stresses in the cracked body are the same as t − τ the stresses in an identically elastic cracked body φ w, ( ,τ) τ = B αα x d 0(32)τ ) tB ( E3α subjected to the same traction (these stresses 0 are independent of the shear modulus), and the dis- Equation (32) is a linear Volterra integral equation of placement field can be obtained by inversion of (36) the second kind with a continuous kernel. It is well using Laplace transform, that is, 123 Finite strain theory of a Mode III crack 85
w,α = C (t) τE3α (x, 0) t ∂τ α (x,τ) + C (t − τ) E3 dτ ∂τ (37a) 0 where C (t) is the creep function and is related to μR (t) by the convolution identity t
μR (t − τ) C (τ) dτ = t. (37b) 0 (ii) If a displacement boundary condition is prescribed on the cracked body (except on the traction free crack faces), then the displacement field is the same as the displacement of an identical neo-Hookean cracked body (wE ), which is independent of mod- ulus, and the stresses can be found using (36). One can now use the large library of elastic crack solu- tions to solve a gel fracture problem. For more general boundary conditions, the solution can be obtained using Laplace transform. Fig. 3 An infinite block with a center crack 2a subject to remote The correspondence principle implies that the dis- cyclic displacement in Mode III placement and shear stresses near the crack tip must have the form: The shear stresses are 2K (t) w ∼ III r/2π sin (θ/2) r → 0 (38a) t μ ε∞ ∂ sin ωτ τ32 + iτ31 = μR (t − τ) dτ 1 − a2/z2 ∂τ 0 τ K (t) − sin (θ/2) (42) 31 ∼ √III (38b,c) τ32 2πr cos (θ/2) For very long times it is easy to show that the shear stresses approach a steady state given by where K III is the Mode III stress intensity factor. An example is a center crack of length 2a in an ε τ + τ → τ ss + τ ss = ∞ infinite block subjected to remote cyclic displacement 32 i 31 32 i 31 1 − a2/z2 of the form : μ˜ (ω) ω +˜μ (ω) ω R sin t R cos t (43a) w (x1, x2 =±∞, t) = ε∞x2 sin ωt (39) The geometry is shown in Fig. 3. By the correspondence where principle, the displacement field everywhere is ∞ w (x , x , t) = (ε∞ sin ωt) μ˜ (ω) = μρ + μω (η) ωη η 1 2 R n sin d 2 2 Im z − a where z = x1 + ix2 0 tB tB (40) = μρ + μγ¯∞ − μγ¯∞ √ 2 − αB αB − 1 ∞ where i = −1 . The crack opening displacement is ω 1 tB −α cos x (1 + x) 1 B dx, ± 2 2 α − w |x1| ≤ a, x2 = 0 , t = (γ∞ sin ωt) a − x 0 B 1 (41) (43b) 123 86 C. Y. Hui et al. ∞ N−1 − μ B tB N 1χ, , ∼ μ˜ (ω) = μω n (η) ωηdη = μγ¯∞ K I1 α α R cos α − 2 N B 1 0 2N−1 N−1 ∞ K χ,β χ,β χ,α ,α ∼ 0 (48) ωtB 1 ( + ) 1−α sin x 1 x B dx (43c) → αB − 1 to leading order as r 0. Note the homogeneity of 0 are the storage and loss modulus respectively, and the W ensures that this result is valid irrespective of the μ˜ (ω) μ˜ (ω) value of K . Hence, by (46), the leading behavior of (2d) second expressions of R and R are obtained by integration by parts and substituting the variable and (2f) at the crack tip is: x ≡ (αB − 1) η/tB. The long time modulus is given 2 μρ H (x → 0, t) ≈ k (t) χ,β χ,β , (49a,b) by . Because of dissipation caused by the breaking 0 2 and reattaching of physical cross-links, the stresses are H (x → 0,τ,t) ≈ [k(t) − k(τ)] χ,β χ,β not in phase with the applied displacement. 2N−2 N−1 W (H0) ∼ k (t) χ,β χ,β , (49c,d)
2N−2 2 2 4.3 Time dependent asymptotic crack tip fields W (H) ∼ k (t) − k (τ) χ,β χ,β
The difficulty with finding exact analytical solution Equations (48) and (49) imply that with (19) is that W (H ) w,α ,α ∼ 0 and 0 W (H) = W [3 + w,α (x,τ,t) w,α (x,τ,t)] W (H) w,α ,α ∼ 0asr → 0(50) (44) This means that (19) is satisfied to leading order as is a nonlinear function which depends on the history r → 0, so the asymptotic displacement as r → 0must of the displacement. Because of this feature, the holo- be given by (46). Of course, the function k (t) cannot graph transform does not work. Nevertheless, it is pos- be determined by local analysis, it depends on loading sible to obtain the asymptotic crack tip behavior for a history and geometry. wide class of energy density functions. For concrete- The near tip stresses are determined by substituting ness, we assume that the elasticity of the networks obey (46) into (18b,c) and keeping only the leading order the GNH model. Knowles (1977) has shown that the terms, this results in displacement field near the crack tip has the form N−1 B −(1−(1/2N)) m 1 τ3α ∼ μ Klocal (t/tB) r τˆα (θ) wE = Kr v (θ) , r → 0 m = 1 − (45) 2N N where K is a loading parameter and v (θ) is given by (51a) 2N−1 (25e). In our case we must allow the loading parameter Klocal (t/tB) = [n(t) + ρ][k(t)] K to be time dependent, which we indicate explicitly t − τ ( ) t 2N−1 by k t , that is, we assume the crack tip displacement +¯γ∞ φB [k(t) − k(τ)] dτ (51b) has the form: tB 0 w = ( ) mv (θ) ≡ ( )χ ( ,θ) , → k t r k t r r 0 (46) The function Klocal (t/tB) can be regarded as a local where to simplify notation we denote χ (r,θ) ≡ intensity factor since it carries all the loading informa- m r v (θ). Note that the GNH is locally homogeneous, tion via k(t). The asymptotic behavior of τ3α is deter- in the sense that mined by (46), (51a, 51b) and (18c). μ N−1 →∞ ≈ B N−1 > Matched asymptotics: small scale hardening W [I1 ] I1 N 1 (47) 2 N In our fracture experiments using plane stress edge = The case of N 1 corresponds to the neo-Hookean crack specimens, we find that the local crack open- strain energy density function and was already consid- ing displacements agree surprising well with the pre- m (θ) ered above. The fact that Kr v is the local solution diction of (1a)usingtheneo-Hookean energy density of the elasticity problem means that function (Guo et al. 2018). This suggests that strain N−1 μ B hardening must be confined to a region SH near the W (I ) wE ,α ,α ∼ 1 2 N crack tip that is very small in comparison with the 123 Finite strain theory of a Mode III crack 87
Fig. 4 Schematics of the small scale hardening
region of dominance of the stress field predicted using → k (t) r mv (θ) r → 0(54) the neo-Hookean model, which we denote by (see K Equation (54)impliesthat Fig. 4). Assuming that this is the case for a Mode III crack, we can link the near tip strain hardening behav- r D ,θ,t/tB , γ¯BtB, B/N,ρ ior predicted by (51a, 51b) to the applied loading on a K 2 (t)/μ2 III cracked body using a Small Scale Hardening (SSH) m r approach – a variation of the Small Scale Yielding ∼ ψ (t) v (θ) (55) 2 ( )/μ2 approach in Elastic–Plastic Fracture (Zehnder 2012). K III t In this approach, we use a boundary layer formulation where ψ is some unknown function of time and is inde- where the physical crack is replaced by a semi-infinite 2 ( )/μ2 pendent of K III t . Comparison of (54) and (55) traction free crack and the actual boundary conditions shows that replaced by the neo-Hookean singular displacement at 1−m K 2 (t) large distances in comparison with SH, as schemat- k (t) = III ψ (t) (56) ically shown in Fig. 4. In light of (34a), the far field μ2 boundary condition are: Substituting (56)into(51b) and using 1 − m = 1/2N 2K (t) w → III r/ π (θ/ ) r →∞ gives: μ 2 sin 2 as (52) ⎡ where K III (t) is the time dependent stress intensity ⎢ 2N−1 K (t/t ) = ⎣ [n(t) + ρ][ψ (t)] +¯γ∞ factor for the associated viscoelastic crack problem local B (with the neo-Hookean work function). ⎤ t Equation (19) and dimensional analysis imply that − τ t 2N−1 ⎦ the displacements must have the form: φB [ψ(t) − ψ(τ)] dτ tB 2 ( ) 0 K III t r w = D ,θ,t/tB, γ¯BtB, B/N,ρ 2N−1 μ2 K 2 (t)/μ2 K (t) N III III (57) (53) μ where D is a dimensionless function of its dimension- Equation (57) connects the remote Mode III stress ( ) less arguments. Equation (46) implies that intensity factor K III t to the local stress intensity factor Klocal. Unfortunately, the dimensionless func- 2 K (t) r ψ (t) III D ,θ,t/t , γ¯ t , B/N,ρ tion can be determined only by solving (19) with μ2 2 ( )/μ2 B B B K III t the boundary condition (52). Finally, although we have 123 88 C. Y. Hui et al. used the GNH energy density function in deriving the crack tip is still to be determined. An interesting ques- asymptotic fields and (57), there is nothing special tion is whether the local correspondence principle holds about this model. For example, the same method can be for cracks loaded under plane stress or plane strain con- applied to any energy density function as long as it is a ditions. We believe that it is possible to show that for function of I1 and W is locally homogeneous, that is: plane stress or plane strain stationary crack problems, β the asymptotic behavior of the integral term in Eq. (1a) W [I →∞] ≈ I , for some positive power β>0 1 1 as the crack tip is approached is either subdominant (58) or has the same order of singularity as corresponding the nonlinear elastic solution. If this is true, the spatial variation of the near tip stress fields in our gel should 5 Summary and discussion be identical to a hyper-elastic solid in which the chains obey the same work function. We are currently working In this work, we studied the near tip stress and strain to prove this hypothesis. fields of a Mode III crack in rate dependent hydro- Our analysis has several limitations. It assumes the gels. We establish a global correspondence principle crack is stationary and we have not proposed a local for the special case where the undamaged network failure mechanism for the initiation of crack growth. obeys Gaussian statistics. We showed that the local This failure mechanism is still being studied. Here we time dependent asymptotic stress and strain fields have note that the constitutive model cannot be applied to the same spatial dependence as the asymptotic stress study failure, since we have assumed that the chains and strain field of a hyper-elastic solid, as long as the between chemical crosslinks cannot break. To study energy density function is a function of I1 only and crack growth, we must relax this assumption to allow W is locally homogeneous. This result can be viewed for chain breakage. Furthermore, it is reasonable to as a local correspondence principle for the crack tip expect that crack growth can change the asymptotic fields. We show that the near tip stress and strain fields behavior of the stress and strain fields near the crack tip, are completely determined by a time dependent stress as is well known in elastic–plastic and elastic-power- intensity factor Klocal (t). The local time dependent law creeping materials (Chitaley and McClintock 1971; damage process near the crack tip is governed by the Hui and Riedel 1981). However, in slow crack growth, stress history. If we assume the region of local dam- it is likely that these asymptotic fields for growing age is sufficiently confined so that there exists a region cracks are highly concentrated near the crack tip, and near the crack tip where the stress field is completely as a result, the asymptotic fields of the stationary crack determined by the local stress intensity factor Klocal (t), may still control the crack growth process. then the local strains and damage would have to be con- A different limitation of our analysis is that it is based trolled by the history of Klocal (t). Hence Klocal (t) can on a constitutive model which is tested on a particular be used as a history dependent loading parameter to hydrogel ((PVA) dual crosslinked gel) and hence may determine when a pre-existing crack initiates growth. not be applicable to other hydrogels with both chem- Note that since the material is rate dependent, whether ical and transient bonds. However, our recent work a crack will grow or not depends on the entire time his- (to be published) shows that, with some modification, tory Klocal (t),0≤ t and not its current value alone. If our constitutive model can be used to characterize the the network energy density function is neo-Hookean, mechanical properties of much tougher systems such as then the local stress intensity factor can be related to the polyampholyte hydrogel developed by Ihsan et al. the remote loading history using classical theory of (2016). linear viscoelasticity via the correspondence principle, as shown in Sect. 4.2. For networks that do not obey Acknowledgements The authors acknowledge support by the Gaussian statistics, it is necessary to use the numerical National Science Foundation under Grant No. CMMI -1537087. method to determine the dependence of Klocal (t) on the ( ) loading history. Finally, this idea of using Klocal t as References a criterion for crack initiation in principle should apply to plane stress and plane strain cracks, except in these Brown HR (2007) A model of the fracture of double network cases the asymptotic behavior of the stress field near the gels. Macromolecules 40:3815–3818 123 Finite strain theory of a Mode III crack 89
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