Research Papers

J. Appl. Mech. 2017;84(7):071001-071001-8. doi:10.1115/1.4036475.

A solution to the problem of a hydraulic fracture driven by an incompressible Newtonian fluid at a constant injection rate in a permeable rock is presented in this paper. A set of governing equations are formed to obtain the fracture half-length, crack opening, and net fluid pressure. The solution is derived under the assumptions of plane strain, zero lag between fluid front and crack tip, followed by negligible fluid viscosity. The last assumption is related to a toughness-dominated fracture propagation regime therefore leading to a uniform fluid pressure along the crack surface. Early-time and late-time asymptotic solutions are obtained, which correspond to both regimes when the fluid contains within the crack and most of the injected fluid infiltrates into the rock, respectively. It is shown that these asymptotic solutions are in a simple form when the fracture propagation is dominated by the material toughness. The transient solution for the evolution from the early time to the late time is also obtained by a numerical method.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071002-071002-9. doi:10.1115/1.4036613.

Wrinkles are widely found in natural and engineering structures, ranging from skins to stretchable electronics. However, it is nontrivial to predict wrinkles, especially for complicated structures, such as multilayer or gradient structures. Here, we establish a symplectic analysis framework for the wrinkles and apply it to layered neo-Hookean structures. The symplectic structure enables us to accurately and efficiently solve the eigenvalue problems of wrinkles via the extended Wittrick–Williams (w–W) algorithm. The symplectic analysis is able to exactly predict wrinkles in bi- and triple-layer structures, compared with the benchmark results and finite element simulations. Our findings also shed light on the formation of hierarchical wrinkles

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071003-071003-9. doi:10.1115/1.4036696.

The nonaxisymmetric transverse free vibrations of radially inhomogeneous circular Mindlin plates with variable thickness are governed by three coupled differential equations with variable coefficients, which are quite difficult to solve analytically in general. In this paper, we discover that if the geometrical and material properties of the plates vary in generalized power form along the radial direction, then the complicated governing differential equations can be reduced into three uncoupled second-order ordinary differential equations which are very easy to solve analytically. Most strikingly, for a class of solid circular Mindlin plates with absolutely sharp edge, the natural frequencies can be expressed explicitly in terms of elementary functions, with the corresponding mode shapes given in terms of Jacobi polynomials. These analytical expressions can serve as benchmark solutions for various numerical methods.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071004-071004-10. doi:10.1115/1.4036672.

Bistable tape springs are ultrathin fiber-reinforced polymer composites, which could self-deploy through releasing stored strain energy. Strain energy relaxation is observed after long-term stowage of bistable tape springs due to viscoelastic effects and the tape springs might lose their self-deployment abilities. In order to mitigate the viscoelastic effects and thus ensure self-deployment, different tape springs were designed, manufactured, and tested. Deployment experiments show that a four-layer, [−45/0/90/45], plain weave glass fiber tape spring has a high capability to mitigate the strain energy relaxation effects to ensure self-deployment after long-term stowage in a coiled configuration. The two inner layers increase the deployment force and the outer layers are used to generate the bistability. The presented four-layer tape spring can self-deploy after more than six months of stowage at room temperature. A numerical model was used to assess the long-term stowage effects on the deployment capability of bistable tape springs. The experiments and modeling results show that the viscoelastic strain energy relaxation starts after only a few minutes after coiling. The relaxation shear stiffness decreases as the shear strain increases and is further reduced by strain energy relaxation when a constant shear strain is applied. The numerical model and experiments could be applied in design to predict the deployment force of other types of tape springs with viscoelastic and friction effects included.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071005-071005-5. doi:10.1115/1.4036776.

A thin film is clamped at the periphery to form a circular freestanding diaphragm before a uniform pressure, p, is applied to inflate it into a blister. The bulging membrane adheres to a rigid constraining plate with height, w0, from the nondeformed membrane. Increasing pressure expands the contact circle of radius, c. Depressurization causes shrinkage of the contact and “pull-off” or spontaneous detachment from the plate. Simultaneous measurement of (p, w0, c) allows one to determine the adhesion energy, γ. A solid mechanics model is constructed based on small strain and linear elasticity, which shows a characteristic loading–unloading hysteresis. The results are consistent with a large deformation model in the literature.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071006-071006-7. doi:10.1115/1.4036819.

Traction between adsorbed islands and the substrate is commonly seen in both living and material systems: deposited material gathers into islands at the early stage of polycrystalline film deposition and generates stress due to lattice mismatch, cells exert cellular traction to extracellular matrix to probe their surrounding microenvironment in vivo, and so on. The traction between these islands and the substrate can result in perceivable macroscopic deformation in the substrate and may be measurable if the substrate is a cantilever beam. However, currently broadly used Stoney equation is incapable of handling such boundary condition. In this paper, we give the closed-form expression on the resulted curvature in substrate beams by distributed tractions. Such a relationship could be employed to monitor the stress evolution during thin film deposition, to quantify the stress level of cell traction as cells adhere to cantilever beams, and other related mechanical systems like charging–discharging induced stress in island-patterned electrode films. Moreover, we found that follower traction induced by an array of islands could lead to negative curvature. It shields light on the early stage compressive stress during polycrystalline film deposition.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071007-071007-5. doi:10.1115/1.4036825.

Building upon the previous work for the failure of quasi-isotropic fiber composite laminates, the much more difficult and more important general case of orthotropic laminates is now taken up. The full and complete failure criterion is derived for fiber dominated, general 0, 90, ±45-deg laminated materials, with the relative volume fractions to be specified for each direction. Quasi-isotropy is a special case of the orthotropic formalism, and the general orthotropic results are just as rigorous as the previous specialized quasi-isotropic results. The orthotropic failure criterion results are of direct and immediate relevance in composite materials applications.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071008-071008-17. doi:10.1115/1.4036821.

For the development of a new family of implicit higher-order time integration algorithms, mixed formulations that include three time-dependent variables (i.e., the displacement, velocity, and acceleration vectors) are developed. Equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables in the mixed formulations, and the time finite element method and the modified weighted-residual method are applied to the velocity–displacement and velocity–acceleration relations of the mixed formulations. Weight parameters are introduced and optimized to achieve preferable attributes of the time integration algorithms. Specific problems of structural dynamics are used in the numerical examples to discuss some fundamental limitations of the well-known second-order accurate algorithms as well as to demonstrate advantages of using the developed higher-order algorithms.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(7):071009-071009-13. doi:10.1115/1.4036822.

For the development of a new family of higher-order time integration algorithms for structural dynamics problems, the displacement vector is approximated over a typical time interval using the pth-degree Hermite interpolation functions in time. The residual vector is defined by substituting the approximated displacement vector into the equation of structural dynamics. The modified weighted-residual method is applied to the residual vector. The weight parameters are used to restate the integral forms of the weighted-residual statements in algebraic forms, and then, these parameters are optimized by using the single-degree-of-freedom problem and its exact solution to achieve improved accuracy and unconditional stability. As a result of the pth-degree Hermite approximation of the displacement vector, pth-order (for dissipative cases) and (p + 1)st-order (for the nondissipative case) accurate algorithms with dissipation control capabilities are obtained. Numerical examples are used to illustrate performances of the newly developed algorithms.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Appl. Mech. 2017;84(7):074501-074501-6. doi:10.1115/1.4036820.

In a companion paper,ff2 we have obtained the closed-form solutions to the stress and strain fields of a two-dimensional Eshelby inclusion. The current work is concerned with the complementary formulation of the displacement. All the formulae are derived in explicit closed-form, based on the degenerate case of a three-dimensional (3D) ellipsoidal inclusion. A benchmark example is provided to validate the present analytical solutions. In conjunction with our previous study, a complete elasticity solution to the classical elliptic cylindrical inclusion is hence documented in Cartesian coordinates for the convenience of engineering applications.

Topics: Tensors , Displacement
Commentary by Dr. Valentin Fuster

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