Energetics of cracks and defects in soft materials: The role of surface stress
Introduction
In elastic materials, the driving configurational energetic force for crack propagation is the energy release rate J, defined as the change in the sum of the elastic strain energy stored in the crack specimen and the potential energy of the loading system per unit area of crack growth. The resistance to fracture is usually characterized by the fracture toughness which is defined as the critical energy release rate for a pre-existing crack to grow. While the fracture toughness is a material property, the energy release rate J depends on the manner of loading as well as the specimen geometry. Usually, in experiments, one chooses a specimen with a pre-existing crack, computes the energy release rate J as a function of load and geometry, and determines by noting that crack growth occurs when J reaches . Eshelby [1] was the first to determine the configurational energetic forces on defects in linear elastic solids, in terms of the J-integral. By considering cracks as defects, Rice [2] and Cherepanov [3] independently showed that the J-integral is the energy release rate for cracks in linear elastic solids. Knowles and Sternberg [4] and Eshelby [5] extended the J integral to hyperelastic cracked solids subjected to large deformation. In the large strain formulation, the components of the energetic force acting on a defect are where C is a smooth simple closed curve in the reference configuration which encloses the defect at the material point (Fig. 1(a)), is the arc length, is the strain energy density, are the Cartesian components of the unit outward normal vector on C, are the Cartesian components of the displacement, and are the Cartesian components of the first Piola–Kirchhoff stress. In (1a), , where denote the in-plane material coordinates in the reference configuration. If the defect is a straight crack on the axis, then the energetic force is the energy release rate and is given by the J-integral where is a smooth simple curve in the reference configuration which surrounds the crack tip, starting from the lower flat crack face and ending on the upper flat crack face, as shown in (Fig. 1b). In this work, Greek indices range from 1 to 2 and summation convention over repeated indices is used. Unless stated otherwise, the positive direction of path integrals is counterclockwise.
Research on the fracture mechanics of soft materials dates to the 1950s when a number of seminal works [6], [7], [8], [9] quantitatively characterized the fracture behaviors of rubbers. For example, Rivlin and Thomas [6] conducted several fracture experiments on natural rubbers and proposed an expression for the energy release rate J under large deformation. Recent advances in soft materials, especially the ability to make highly compliant and tough hydrogels, have rekindled interest in the study of fracture behavior of elastic, stretchable materials [10], [11], [12], [13]. These soft and yet tough materials have great potential in applications such as clinical devices [14], [15], tissue engineering [16], [17], [18], and soft robotics [19], [20], [21]. However, the role of surface stress in determining the energy release rate is often overlooked.
Surface stress effects are felt over a characteristic length scale, the elastocapillary length, defined as , where E is the Young modulus of the solid and the magnitude of the surface stress. For simple solids, surface stress is isotropic and constant. Its value lies in a relatively narrow range, on the order of tens of mN/m for soft solids such as elastomers and gels and up to a few N/m for metals and ceramics [22], [23], [24]. However, for the same range of materials, elastic modulus varies over seven orders of magnitude. For metals and ceramics, the value of elastocapillary length is small. For soft solids such as elastomers and gels with elastic modulus in the kPa to MPa range, the corresponding value of elastocapillary length is on the order of tens of nanometers to hundreds of microns or larger. Thus, for soft materials, surface stress is far more likely to play a significant and sometimes dominant role in surface mechanical phenomena. For instance, surface stress can flatten sharp features by smoothing corners and undulations [25], [26], [27], [28]; drive instabilities [29]; stiffen fluid–solid composites [30]; invalidate the classical contact and adhesive mechanism by the Hertz and Johnson–Kendall–Roberts (JKR) theories [31], [32], [33], [34], [35], [36]; violate the classical Young Equation [27], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]; and alter the flow field in porous media [49], [50], [51].
Many recent experiments and theories have also demonstrated that surface stresses can significantly alter the local deformation and stress fields near stress concentrators and hence alter the energy release rate [52], [53], [54], [55], [56]. However, most of these analyses assume small strains and linear elastic behavior. One of the goals of this work is to find an expression for the energy release rate for cracks and defects that accounts for surface stress, under arbitrarily large deformation. We consider plane strain deformation where the out-of-plane displacement is identically zero, and the in-plane displacements depend only on the in-plane material coordinates . Hence the defects are line elastic singularities, e.g. a center of dilation in 2D. Here line singularity refers to our assumption of plane strain where the singular fields are uniform in the out-of-plane directions. Note that a traction free crack or an edge dislocation are not line defects because the displacement field is discontinuous on the crack faces or the glide plane. Hence configurational energetic forces acting on cracks or dislocations (which can be viewed as surface defects) can be different from those on line defects.
The plan of this paper is as follows. Section 2 contains background materials for finite deformation and surface stress for a hyperelastic body subjected to plane strain deformation. Although this information represents a special case of a general surface theory, see for example, Steigmann and Ogden [57] Gao et al. [58], Gurtin and Murdoch [59], Green [60] and Liu et al. [61], it is given in some detail here for the sake of clarity. Section 3 shows how the J integral given by (1a,b) is modified to account for surface stress. We also give some simple examples to highlight the new formulation in Section 4.
Section snippets
Finite deformation theory of plane strain in hyperelastic solids with surface stress
We consider hyperelastic solids that are isotropic and homogeneous. The bulk deformation gradient tensor is where is the position of a material point after deformation, and are the Cartesian basis vectors in the reference and current configurations, respectively, and is the standard tensor product. We assume the bulk deformation gradient is continuously differentiable except at the defect or the crack tip. Since the elastostatics equations governing the interior
Energetic force on a line defect with surface stress
The expression of the energetic force on a line defect accounting for surface stresses is still given by (1a). This can be explained by observing that the effect of surface stress occurs in the boundary condition (9) and that integral defined by (1a) is path independent, that is, it has the same value for all simple closed paths C enclosing the defect inside the bulk solid. Indeed, even though surface stresses changes the stress and deformation fields, the balance laws governing bulk
Energetic force on a plane strain crack
For a planar traction-free crack subjected to external loading, the crack face separates, so the undeformed crack faces must be considered as a part of the boundary. Without loss of generality, we assume that the undeformed crack lies in the direction. Since the energy release rate has to do with translation of the crack, the only physical meaningful component is which we shall denote by J. Because surface stress causes traction discontinuity across the deformed crack faces, the energy
Summary and discussion
The key result of this paper is that an expression (modified J-integral) for the energy release rate that considers surface stress effect is developed. Compared to the previous research [52], [53], [54], [55], [56] assuming small strains and linear elastic behaviors, our theoretical work is valid for large deformation and complex surface mechanical behaviors (e.g., surface stiffening). Further, the modified J-integral given in (18a) is path-dependent. To our best knowledge, there is no
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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