Damage, self-healing, and hysteresis in spider silks

Abstract

In this article, we propose a microstructure-based continuum model to describe the material behavior of spider silks. We suppose that the material is composed of a soft fraction with entropic elasticity and a hard, damageable fraction. The hard fraction models the presence of stiffer, crystal-rich, oriented regions and accounts for the effect of softening induced by the breaking of hydrogen bonds. To describe the observed presence of crystals with different size, composition, and orientation, this hard fraction is modeled as a distribution of materials with variable properties. The soft fraction describes the remaining regions of amorphous material and is here modeled as a wormlike chain. During stretching, we consider the effect of bond-breaking as a transition from the hard- to the soft-material phase. As we demonstrate, a crucial effect of bond-breaking that accompanies the softening of the material is an increase in contour length associated with chains unraveling. The model describes also the self-healing properties of the material by assuming partial bond reconnection upon unloading. Despite its simplicity, the proposed mechanical system reproduces the main experimental effects observed in cyclic loading of spider silks. Moreover, our approach is amenable to two- or three-dimensional extensions and may prove to be a useful tool in the field of microstructure optimization for bioinspired materials.

Figure 1

Schematic behavior of a single protein macromolecule. (A) Scheme of a typical force-displacement diagram (bold lines) under a WLC assumption, using different equilibrium branches with different contour lengths (a–e). (B) Schematic of the microstructure configuration of the equilibrium branches in A.

Figure 2

Constitutive assumptions. (A) Stress-strain relation of the hard phase. (B–D) Distribution of the hard and soft phases in the probability space during loading (B), unloading (C), and reloading (D).

Figure 3

Mechanical behavior under a variable soft fraction. (A) Primary loading path for three systems (a–c) characterized by different assumptions of the soft fraction. (B) Depiction of the experimental behavior when the soft fraction is varied, as described by Guinea et al. (6) and reproduced by our model (A). The constitutive parameters are listed beneath the figure. Prestrain induces an alignment in the soft fraction, with increasing stiffness.

Figure 4

Mechanical behavior when the hard and soft fraction are both varied. (A) Primary loading path for two systems characterized by two different assumptions of the crystalline fraction. The constitutive parameters are listed below the figure. (B) Depiction of the experimental behavior reported by Denny (1) and reproduced by our model (A), where the different curves correspond to frame (a) and viscid (b) silk, characterized by a large and small crystalline fraction, respectively.

Figure 5

Unloading behavior. The system is unloaded at ɛ_M = 0.18. The constitutive parameters are listed below the figure.

Figure 6

Cyclic loading. The system is unloaded at ɛ_M = 0.18. The constitutive parameters are listed below the figure.

Figure 7

Cyclic loading for frame and viscid silks. (Upper) Model behavior. For the frame silk, we unloaded at ɛ_M = 0.18, whereas for the viscid silk we unloaded at ɛ_M = 1. The constitutive parameters are listed below the figure. (Lower) Experimental behavior of frame and viscid silk, reproduced from Denny (1).