Nonlinear viscoelasticity and generalized failure criterion for polymer gels Bavand Keshavarz,1, ∗ Thibaut Divoux,2, 3 S´ebastienManneville,4 and Gareth H. McKinley1 1Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 2Centre de Recherche Paul Pascal, CNRS UPR 8641 - 115 avenue Schweitzer, 33600 Pessac, France 3MultiScale Material Science for Energy and Environment, UMI 3466, CNRS-MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA 4Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France (Dated: January 6, 2017) Polymer gels behave as soft viscoelastic solids and exhibit a generic nonlinear mechanical response characterized by pronounced stiffening prior to irreversible failure, most often through macroscopic fractures. Here, we aim at capturing the latter scenario for a protein gel using a nonlinear integral constitutive equation built upon (i) the linear viscoelastic response of the gel, here well described by a power-law relaxation modulus, and (ii) the nonlinear viscoelastic properties of the gel, encoded into a \damping function". Such formalism predicts quantitatively the gel mechanical response to a shear start-up experiment, up to the onset of macroscopic failure. Moreover, as the gel failure involves the irreversible growth of macroscopic cracks, we couple the latter stress response with Bailey's durability criterion for brittle solids in order to predict the critical values of the stress σc and strain γc at the failure point, and how they scale with the applied shear rate. The excellent agreement between theory and experiments suggests that the crack growth in this soft viscoelastic gel is a Markovian process, and that Baileys' criterion extends well beyond hard materials such as metals, glasses, or minerals. PACS numbers: 62.20.mj, 83.80.Kn, 82.35.Pq, 83.10.Gr Introduction.- Polymer gels find ubiquitous applica- ing the Boltzmann superposition principle, the damping tions in material science, from biological tissues to man- function is used to construct a time-strain separable con- ufactured goods, among which food stuffs and medical stitutive equation of K-BKZ (Kaye{Bernstein-Kearsley- products are the most widespread [1{3]. These materi- Zapas) form [14, 15] that predicts the gel mechanical als commonly feature a porous microstructure filled with response to steady-shear experiments. This approach water, which results in solid-like viscoelastic mechani- robustly captures the strain-stiffening of the gel during cal properties. While soft polymer gels share common start up of steady shear tests up to the appearance of features with hard materials, including delayed failure a stress maximum that is accompanied by the onset of [4, 5], crack propagation [6, 7] or work-hardening [8], the first macroscopic crack. Moreover, in order to link their porous microstructure also confers upon them re- the nonlinear viscoelastic response of the gel to its subse- markable nonlinear viscoelastic properties. Indeed, such quent brittle-like rupture, we adopt the Bailey criterion, soft solids strongly stiffen upon increasing deformation, which describes the gel failure as arising from accumu- which stems from the inherent nonlinear elastic behavior lation of irreversible damage [17, 18]. The combination of the polymer chains composing the gel network [9{12]. of the stress response predicted by the K-BKZ consti- Polymer gels hence endure large strains to failure and tutive formulation with the Bailey criterion allows us to dissipate substantial mechanical work, leading to very predict accurately the scaling of the critical stress and tough hydrogels and elastomers [13]. However, to date strain at failure with variations in the magnitude of the no quantitative link has been made between the nonlin- applied shear rate. Our results extend Bailey's criterion ear viscoelasticity of polymer gels and the failure that is to viscoelastic soft solids and provide a unified consistent arXiv:1607.08300v2 [cond-mat.soft] 5 Jan 2017 subsequently observed as the strain-loading is increased framework to describe the failure of protein gels under beyond the initial stiffening regime. various shear loading histories. In the present Letter, we apply the concept of a strain Experimental.- We consider two acid-induced protein damping function, traditionally used for polymeric liq- gels with substantially different mechanical properties: uids and rubber-like materials [16], to quantify the non- the first one shows pronounced strain-hardening, while linear viscoelastic response of a prototypical protein gel. the second does not. They are prepared by dissolving The form of the damping function is constructed exper- caseinate powder (Firmenich) at 4% wt. (resp. 8% wt.) imentally through a series of independent stress relax- in deionized water under gentle mixing at 600 rpm and ation tests that allow us to probe large deformations T = 35◦C. Homogeneous gelation is induced by dissolv- while injecting very little energy into the gel, hence lim- ing 1% wt. (resp. 8% wt.) glucono-δ-lactone (GDL, Fir- iting as much as possible any plastic damage. Follow- menich) in the protein solution [19, 20]. While still liquid, 2 4 10 0.4 0.3 α 0.2 3 0.1 −3 −2 −1 0 1 10 10 10 10γ 10 10 0 0 10 3 2 10 ) 2.5 0 )[Pa] γ 2 ,t ( ) 0 0 γ 1 h ( 1.5 γ −1 h ( 10 10 1 G 0.5 0 γ(t) 3 10 γ0 0 −3 −2 −1 0 1 10 10 10γ 10 10 1 −2 0 −1 t 10 −3 −2 −1 0 1 10 −1 0 1 2 3 4 10 10 10 10 10 10 10 10 10 10 10 γ0 t[s] FIG. 2: (color online) Strain damping function h(γ0) of a 4% FIG. 1: (color online) Nonlinear relaxation function wt. casein gel as defined in the text. Same color code as in G(γ0; t) = σ(t)/γ0 vs time t determined by step strain tests, Fig. 1. The solid black line is the best fit function by power each one performed on a freshly prepared 4% wt. casein gel. 2 series in γ0 as proposed in [27] which captures the stiffening Colors from blue to red represent strain values ranging from behavior, but does not account for the softening part of the γ0 = 0:002 to γ0 = 5. The black line is the best power-law gel response at strains larger than 50%. The red continuous fit of the data in the linear region (γ0 ≤ 0:01). Inset: Stress line is the best fit function h~(γ) of the data (see text). Inset: relaxation exponent α extracted from the power-law fit of the same data plotted in semilogarithmic scales. data shown in the main graph with the same color code. The horizontal line is the average exponent α = 0:18 ± 0:01. V α = 266 5 Pa.s [22, 23]. For γ0 & 0:01, the stress re- laxation still± exhibits a power-law decrease in time, with the protein solution is poured into the gap of a cylindri- the same exponent α, after t & 0:1 s but the magnitude cal Couette shear cell connected to a strain-controlled of the stress at a given time first stiffens and then softens rheometer (ARES, TA Instruments, Delaware) [21]. In as γ0 is increased. situ gelation is achieved within 12 hours after which ei- Since α is insensitive to the strain amplitude, we can ther a step strain or a constant shear rate is imposed on use the concept of strain-time separability [24] to quan- the sample while the resulting stress response is moni- tify the strain dependence of the stress relaxation re- tored. In both cases, images of the gel deformation are sponse by computing the damping function [16], defined recorded simultaneously to rheology in order to monitor as h(γ ) = G(γ ; t)=G(t) where ::: denotes the time 0 h 0 it h it the nucleation and growth of cracks. average for 1 t 1000 s for each of the step-strain ≤ ≤ Damping function.- To first characterize the viscoelas- experiments. The resulting damping function reported tic properties of the 4% wt. casein gel, we perform a in Fig. 2 thus fully characterizes the strain dependence series of step strain tests. Each experiment is performed of the viscoelastic response in the material. The gel dis- on a freshly prepared gel and consists of two successive plays a linear response (i.e. h = 1) up to γ0 = 0:1, strain steps. The first step is applied within the linear whereas for intermediate strain amplitudes, the gel ex- deformation regime and the stress relaxation is followed hibits a pronounced strain-stiffening, that is character- over the next 4000 s and serves as a reference for the ized by a maximum value h 2:2 reached at γ = 0:5. ' comparison of two independent experiments. This is fol- Finally, for even larger strains, the material softens due lowed by a second step at a strain amplitude chosen be- to network rupture and the damping function decreases −3 tween 10 γ0 5 and the stress is monitored again abruptly as a power-law function of the imposed strain for 4000 s to≤ measure≤ the gel viscoelastic response. The with an exponent of 3. Yet, we emphasize that for ∼ − stress relaxation functions G(γ0; t) = σ(t)/γ0 associated all step strain tests, the gel remains visually intact even with the second step of strain are reported in Fig. 1. at strain amplitudes as large as γ 5 [25]. 0 ' At low applied strains (γ0 . 0:01), the magnitude of As proposed in [26, 27], the strain hardening portion the viscoelastic stress scales linearly with the imposed of the damping function is captured in a power series ex- ? strain and the relaxation modulus exhibits a remarkable pansion of h (γ0) [28], with the fractal dimension db of power-law decrease over four decades of time, which is the stress-bearing network backbone as the only fitting well modeled by a spring-pot (or fractional viscoelastic parameter.