Strength and toughness of biocomposites consisting of soft and hard elements: A few fundamental models
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Introduction In nature, there are strong and tough biological materials that often possess remarkable hierarchical structures.1–4 Examples of such natural materials include nacre, the exoskeleton of crustaceans, and spider webs. Using these examples, we emphasize the role of simple models for structures, which yield physical insights and often reveal useful scaling laws. A scaling law gives an important physical quantity as a product of the powers of other physical parameters, as in Equations 1 and 2 given later in the text. Scaling laws are useful as guiding principles for various applications such as industrial development of reinforced materials, although they are mathematically exact only in the limit in which a number of physical parameters, X, Y, . . ., are much larger or smaller than certain values. Such a limit is often expressed as a set of inequalities, such as X >> X0 and Y > Es
STRENGTH AND TOUGHNESS OF BIOCOMPOSITES CONSISTING OF SOFT AND HARD ELEMENTS
and dh >> ds. The elastic energy of this simple model can be constructed in the limit of small ε under the existence of a line crack perpendicular to the layers, where ε is defined as ε = (Es/Eh)(d/ds) with d = dh + ds. Reasonable values of the parameters are dh = 0.5 µm, ds = dh/20, Eh = 50 GPa, and Es = 1 MPa (the soft layers are like gels34); for this set of parameters, ε is significantly small (ε ∼ 10−4). When the sample is stretched in the y direction, on the basis of the elastic energy in the small ε limit, the dominant component of deformation field ui and stress field σij are shown to be uy and σyy (termed σy in the following), with the other components being negligibly small in comparison. In addition, the dominant component of the deformation field uy was shown to be governed by an anisotropic Laplace equation.24 The equation for uy can be analytically solved under boundary conditions. For example, the conditions that describe a line crack of length 2a propagating in the r direction, as illustrated in Figure 1b,24,35 are specified in the following way: The stress σy is zero at the surface of the crack (i.e., at y = 0 and −a < r < a), the deformation uy is zero at y = 0 from symmetry, except for the region −a < r < a in which the crack is located. The fixed grip condition is specified by the requirements that the deformation uy is set to uy = u0 and −u0 at the top and bottom edges (i.e., y = L/2 and −L/2 where L is the height of the sample as defined in Figure 1b). With these boundary conditions, a complete analytical solution for uy is obtained via a special conformal mapping augmented by a transformation for avoiding the singularity that appears on the y = 0 axis. The analytical solution of the stress field σy is obtained from that of uy. Near the crack tip (|r| 0,
(2)
and
where L is the sample height and σ0 is a characteristic size of the stress at the top and bottom edges (at y = L/2 and −L/2). The above scaling laws predict that, compared with a monolith of aragonite, stress concentration is reduced by the factor ε1/4 and deformation is enhance
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