Stress Analysis of Functionally Graded Disk with Exponentially Varying Thickness using Iterative Method

In this paper, variable thickness disk made up of functionally graded material (FGM) under internal and external pressure is analyzed using a simple iteration technique. Thickness of FGM disk and the material property, namely, Young’s modulus are varying exponentially in radial direction. Poisson’s ratio is considered invariant for the material. Navier equation is used to formulate the problem in the differential equation form under plane stress condition. Displacement, stresses, and strains are obtained under the influence of material gradation and variable thickness. Three different material combinations are considered for the FGM disk. The mechanical response of disk obtained for different functionally graded material combinations are compared with the homogenous disk, and results are plotted graphically

A nonlinear rotation-free shell formulation with prestressing for vascular biomechanics

We implement a nonlinear rotation-free shell formulation capable of handling large deformations for applications in vascular biomechanics. The formulation employs a previously reported shell element that calculates both the membrane and bending behavior via displacement degrees of freedom for a triangular element. The thickness stretch is statically condensed to enforce vessel wall incompressibility via a plane stress condition. Consequently, the formulation allows incorporation of appropriate 3D constitutive material models. We also incorporate external tissue support conditions to model the effect of surrounding tissue. We present theoretical and variational details of the formulation and verify our implementation against axisymmetric results and literature data. We also adapt a previously reported prestress methodology to identify the unloaded configuration corresponding to the medically imaged in vivo vessel geometry. We verify the prestress methodology in an idealized bifurcation model and demonstrate the significance of including prestress. Lastly, we demonstrate the robustness of our formulation via its application to mouse-specific models of arterial mechanics using an experimentally informed four-fiber constitutive model.

Near Crack Line Elastic-Plastic Field for Mode I Cracks under Plane Stress Condition in Rectangular Coordinates

By using the crack line analysis method, this paper carries out an elastic-plastic analysis for mode I cracks under plane stress condition in an elastic perfectly plastic solid and obtains the general form of matching equations of the elastic stress field and the plastic stress field near the crack line in rectangular coordinate form. The analysis in rectangular coordinates in this paper avoids the conversion from rectangular coordinates into polar coordinates in the existing analysis and greatly simplifies the power series forms of the elastic stress field and plastic stress field near the crack line during the solving process. Furthermore, by focusing on a new problem, i.e., the center-cracked plate with finite width under unidirectional uniform tension, this paper obtains the elastic stress field, plastic stress field, and the length of the elastic-plastic boundary near the crack line by using the general form of the solution. When the dimensions of the plate tend to be infinite, the results of this paper are consistent with those obtained for an infinite plate with a mode I crack. Furthermore, the variation curves of the length of the elastic-plastic boundary are also delineated in different sized center-cracked plates, and the results are compared with those obtained under the small-scale yielding conditions. The solving process and the results in this paper abandon the small-scale yielding conditions completely. The method used in this paper not only makes the solving process simpler during the elastic-plastic analysis near the crack line but also enriches the crack line analysis method.

Mining-Induced Failure Criteria of Interactional Hard Roof Structures: A Case Study

Due to the additional abutment stress, interactional hard roof structures (IHRS) affect the normal operation of the coal production system in underground mining. The movement of IHRS may result in security problems, such as the failure of supporting body, large deformation, and even roof caving for nearby openings. According to the physical configuration and loading conditions of IHRS in a simple two-dimensional physical model under the plane stress condition, mining-induced failure criteria were proposed and validated by the mechanical behavior of IHRS in a mechanical analysis model. The results indicate that IHRS, consisting of three interactional parts—the lower key structure, the middle soft interlayer, and the upper key structure—are governed by the additional abutment stress induced by the longwall mining working face. The fracture of the upper key structure in IHRS can be explained as follows: Due to the crushing failure, lower key structure, and middle soft interlayer yield, the action force between the upper and lower key structures vanishes, resulting in fracture of the upper key structure in IHRS. In a field case, when additional abutment stress reaches 7.37 MPa, the energy of 2.35 × 105 J is generated by the fracture of the upper key structure in IHRS. Under the same geological and engineering conditions, the energy generated by IHRS is much larger than that generated by a single hard roof. The mining-induced failure criteria are successfully applied in a field case. The in-situ mechanical behavior of the openings nearby IHRS under the mining abutment stress can be clearly explained by the proposed criteria.

Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed-form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order [Formula: see text] in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending–curvature tensor, with coefficients depending on the initial curvature of the shell.

Edge Waves in an Initially Stressed Visco-Poroelastic Plate under Plane Stress Condition

The purpose of this paper is to investigate the propagation of edge waves in a homogeneous visco-poroelastic plate which is initially stressed in horizontal direction. The pertinent governing equations are derived and the frequency equation is obtained in the framework of Biot’s theory. Frequency and attenuation are computed as a function of wavenumber. For the numerical process, solids namely, sandstone saturated with kerosene, sandstone saturated with water is considered and the results are presented graphically.

A displacement-based approach to geometric instabilities of a film on a substrate

When a thin film adhered to a compliant substrate is growing, it will eventually buckle in order to release the compressive stresses accumulated within the film due to growth. Such geometric instabilities caused by compressive stresses prevail among all living systems in nature and their outcomes range from highly beneficial to destructive. Therefore, understanding compression induced instabilities is of crucial importance. Note that the origin of the “compression” need not necessarily be differential growth, as it may be due to pre-stretch or thermal expansion. A commonly accepted solution strategy for instabilities in bilayer structures dates back to the seminal work of Allen and employs the Airy stress functions. Owing to its reliance on a stress-based approach, the Allen solution is limited to linear two-dimensional problems and its success depends entirely on choosing an appropriate Airy function. The main objective of this contribution is to circumvent these limitations via a displacement-based approach formally suitable for three-dimensional problems, anisotropic materials, and even applicable to finite deformations. Furthermore, the Allen solution in its original form is valid for the plane-stress condition but often it is mistakenly compared with the numerical simulations corresponding to the plane-strain condition. We analyze the subtle difference between the solutions associated with the plane-strain and plane-stress conditions. Next, the analytical solution is compared against the computational results using the finite element method via eigenvalue analysis. Finally, it is briefly explained how the current approach can be utilized beyond the classical bilayer systems.