Crack-Tip Stress Concentration and Structure Size in Nonlinear Structured Materials

2019

Phys. Rev. Materials

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Recently, it has been found experimentally that hydrated nacre exhibits a nonlinear mechanical response. While mechanical nonlinearity has been shown to be important in other biological structures, such as spider webs, the implications of mechanical nonlinearity in nacre have not been explored. Here, we show that the nonlinear mechanical response of nacre can be reproduced by an analytical model, which reflects a nacre-like layered structure, consisting of linear-elastic hard sheets glued together by weakly nonlinear-elastic soft sheets. We develop scaling analysis on this analytical model, and perform numerical simulations using a lattice model, which is a discrete counterpart of the analytical model. Unexpectedly, we find the weak nonlinearity in the soft component significantly contributes to enhancing toughness by redistributing the stress at a crack tip over a wider area. Beyond demonstrating a mechanism that explains the unusual properties of biological nacre, this study points to a general design principle for constructing tough composites using weak nonlinearity, and is useful as a guiding principle to develop artificial layered structures mimicking nacre. I. INTRODUCTIONNatural materials often exhibit remarkable hierarchical structures leading to outstanding mechanical characteristics [1-5], as observed in bone, spider silk [6], and the exoskeletons of crustaceans [7,8]. Nacre, which is found in many seashells and protects shellfish from their environment, is composed of soft and hard layers, and has been studied as a prototype material for several decades [9][10][11]. Researchers have been inspired by nacre's remarkable structure to develop new materials that demonstrate excellent mechanical performance. [12][13][14][15][16][17][18].There have been many studies discussing the mechanisms responsible for the toughness of nacre. Toughening mechanisms have been proposed based on a variety of experimental observations, such as (1) step-wise elongation [19], (2) micro-cracking and crack bridging [20], (3) thin compressive layers [21], (4) rough layer interfaces [22], (5) mineral bridges [23], and (6) wavy surface of the plates [24]. On the theoretical side, various approaches have been explored, including (1) elastic models [21] based on analytical solutions [25, 26] and on scaling arguments [27, 28], (2) viscoelastic

“…This is a generalized version of a nonlinear Griffith's failure formula that was proposed theoretically in Ref. [49] and confirmed experimentally in Ref. [50].…”

Section: Details Of Derivationsupporting

confidence: 64%

“…In fact, this second principle is numerically confirmed in Refs. [49,54,55]. The authors considered a line crack of size a in a two dimensional nonlinear network system whose mesh size is d, and showed that, as expected, the maximum stress always appears at the crack tips.…”

Section: Details Of Derivationmentioning

confidence: 76%

Toughening in a nacre-like soft-hard layered structure due to weak nonlinearity in the soft layer

Aoyanagi

2019

Phys. Rev. Materials

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“…where k and η are the spring constant and viscosity, respectively. Here, x i,j stands for the vertical position of the bead located originally at the lattice point (s = 1 and 3 correspond to shear and s = 2 and 4 correspond to stretch) [25]. 55 In order for a crack to propagate, every spring is broken when the force acting on it reaches the critical value f c .…”

Section: Simulation Modelmentioning

confidence: 99%

“…) with the natural length ℓ (s) of the springs with ℓ (1) = ℓ (3) = 0 and ℓ (2) = ℓ (4) = d (s = 1 and 3 correspond to shear and s = 2 and 4 correspond to stretch) [25].…”

Section: Simulation Modelmentioning

confidence: 99%

“…We study the crack propagation in a lattice model that incorporates a linear viscoelasticity in a simple manner. The use of lattice model is motivated 25 by the previous theories [9,10,11], in which the dynamics originating from the far-field linear viscoelastic contribution are fairly insensitive to near-crack treatments. As a result, we reproduce crack propagation with a constant velocity.…”

Section: Introductionmentioning

confidence: 99%

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Stationary crack propagation in a two-dimensional visco-elastic network model

Aoyanagi

2017

Polymer

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We investigate crack propagation in a simple two-dimensional visco-elastic model and find a scaling regime in the relation between the propagation velocity and energy release rate or fracture energy, together with lower and upper bounds of the scaling regime. On the basis of our result, the existence of the lower and upper bounds is expected to be universal or model-independent: the present simple simulation model provides generic insight into the physics of crack propagation, and the model will be a first step towards the development of a more refined coarse-grained model. Relatively abrupt changes of velocity are predicted near the lower and upper bounds for the scaling regime and the positions of the bounds could be good markers for the development of tough polymers, for which we provide simple views that could be useful as guiding principles for toughening polymer-based materials.

Fracture of Soft Foam Solids: Interplay of Visco- and Plasto-elasticity

Kashima

2014

ACS Macro Lett.

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Fracture mechanical properties of a very soft solidified foam of polyethylene with Young's modulus about 1 MPa are studied by changing stretching velocity in a wide range, by using sheets of the material in order to suppress finite-size effects. Unexpectedly, we find that the fracture can be described well by linear elastic fracture mechanics for a given fracture rate in the wide range. This allows a direct determination of velocity-dependent fracture energy of the soft foam. As a result, we find that the fracture energy is composed of a static plastoelastic component and another dynamic viscoelastic component, elucidating a simple physical interpretation of each component and giving guiding principles useful for practical applications to reinforce industrial polymer materials. Furthermore, we introduce a finite stress criterion for fracture that is similar in spirit to the cohesive zone model and, using our data, demonstrate that this stress criterion is consistent with the Griffith's energy balance. I n nature and in daily life, there is a variety of cellular structures or foam solids, ranging from familiar ones, such as cork, balsa, bread, coral, and apples, 1 to exotic remarkable ones, such as the stereom of echinoderms, 2 the skeleton of a certain sponge, 3 and the frustules of diatoms. 4 Cellular solids are generally lightweight smart materials like spider webs 5,6 and support our life, for example, daily by their insulating or shock-absorbing features. Active studies on foam solids to date have revealed, e.g., (1) simple fracture mechanical properties that are well-understood as a function of the volume fraction of solid ϕ 7−9 and (2) advantages of the porous structure for reinforcement. 10,11 However, experimental studies are limited to hard cellular solids with Young's modulus E larger than 3000 MPa, such as poly(methyl acrylate) (PMA), rigid polyurethane (PU), and cellular glass. In addition, any velocity dependences of fracture properties of foam solids have never been discussed in a systematic way, while such an issue has received considerable attention for adhesive interface, 12−14 flexible laminates, 15 viscoelastic solids, 16,17 and weakly cross-linked gels. 18,19 Recently, a very soft polyethylene foam with E around 1 MPa has been studied, and scaling laws different from those for hard porous materials were established. 20 Here, we study the fracture energy of similar soft foams with changing fracture rate in a wide range to surprisingly find that linear elastic fracture mechanics 21,22 works well for a fixed velocity in the wide range. The velocity-dependent fracture energy is shown to be composed of a static plastoelastic component and another dynamic viscoelastic component, with the latter scaling linearly with the rate. The velocity dependence originates from that of a yield stress that is introduced through the opening distance at crack tips ( Figure 1). Furthermore, we demonstrate that the Griffith criterion can be regarded as a stress criterion: we find with our data that the Griffith's ene...