An Analytical Elastic Solution for Right-Angle Trapezoidal Opening in Steeply Inclined Coal Seam

In the process of underground mining, steeply inclined rocks or coal seams are often encountered, forming the openings of right-angle trapezoid. According to the geological conditions of a mining project in China, an analytical elastic solution of stress and displacement around right-angle trapezoidal opening in a homogeneous, isotropic, and linear elastic geomaterial is presented, which is based on the evaluation of the conformal mapping representation by an appropriate numerical calculation and the complex potential functions. The different results from other shaped openings are shown as follows. In a right-angle trapezoidal opening, the maximum displacements of roof falling occur on the low side, while the most horizontal displacements on the low side are around the roof and the most horizontal displacements on the high side are around the middle of the high side in this opening. These results are also compared with the numerical calculations in FLAC software, illustrating that the solution may be easily applied to rock mechanics or rock engineering for understanding the deformation of floor heave and roof falling down. The solution is also suitable for optimum design of bolt supporting in a right-angle trapezoidal opening, which is different from the traditional concept of symmetrical bolt supporting. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of an opening without symmetrical axis.

Elastic Solution of a Polygon-Shaped Inclusion With a Polynomial Eigenstrain

This paper presents the Eshelby’s tensor of a polygonal inclusion with a polynomial eigenstrain, which can provide an elastic solution to an arbitrary, convex inclusion with a continuously distributed eigenstrain by the Taylor series approximation. The Eshelby’s tensor for plane strain problem is derived from the fundamental solution of isotropic Green’s function with the Hadmard regularization, which is composed of the integrals of the derivatives of the harmonic and biharmonic potentials over the source domain. Using the Green’s theorem, they are converted to two line (contour) integrals over the polygonal cross section. This paper evaluates them by direct analytical integrals. Following Mura’s work, this paper formulates the method to derive linear, quadratic, and higher order of the Eshelby’s tensor in the polynomial form for arbitrary, convex polygonal shapes of inclusions. Numerical case studies were performed to verify the analytic results with the original Eshelby’s solution for a uniform eigenstrain in an ellipsoidal domain. It is of significance to consider higher order terms of eigenstrain for the polygon-shape inclusion problem because the eigenstrain distribution is generally non-uniform when Eshelby’s equivalent inclusion method is used. The stress disturbance due to a triangle particle in an infinite domain is demonstrated by comparison with the results of the finite element method (FEM). The present solution paves the way to accurately simulate the particle-particle, partial-boundary interactions of polygon-shape particles.

CREEP OF INHOMOGENEOUS POLYMER CYLINDRICAL SHELL DURING HEATING

The article deals with the problem of creep of a polymer thick-walled cylindrical shell under the action of an uneven temperature field. The calculation is based on the nonlinear Maxwell – Gurevich equation, which is widely used in the calculations of polymer structures. To solve a quasi-stationary nonlinear problem, a "layer-by-layer" method is used, where a layer means a certain time of the process. Such methods for solving creep problems are also called step, sequential loading, incremental, etc. The problem is solved in an axisymmetric formulation under conditions of plane deformation. The use of the Maxwell – Gurevich differential equation allows solving rheology problems when exposed to a timevarying temperature field. The change in the stress state with time and its significant difference from the elastic solution are shown.

Effect of liquid elasticity on the behaviour of high-speed focused jets

We investigate the effect of highly contrasting non-Newtonian liquid properties on the formation of liquid jets with a focused shape. By using two nozzle-free ejection techniques, mechanically impact- and laser-induced, fast jets of a highly elastic (sodium polyacrylate) and weakly elastic (xanthan-gum) diluted polymer solutions are generated. A unique high-speed effect is encountered at the jet ejection onset of the highly elastic solution. Its jet-tip speed is on average 1.4 times faster in comparison to a Newtonian (glycerin/water) and the weakly elastic liquids. We explain this effect occurring as a result of the high viscoelasticity of the sodium polyacrylate solution. Additionally, a ‘bungee jumper’ jet behaviour (Morrison and Harlen in Rheol Acta 49(6):619–632, 2010) is observed in a regime of high speed ($$10<V_j<40$$ 10 < V j < 40 m/s) and high viscosity ($$\mu >20$$ μ > 20 mPa s) not previously examined. We additionally characterise the viscoelastic non-breakup jet limit using the Bazilevskii et al. (Fluid Dyn 40(3):376–392, 2005) ejection criterion. Herein, the extensional rheological parameters are measured implementing a novel DoS-CaBER technique (Dinic et al. in Lab Chip 17(3):460–473, 2017). Our findings may influence results of inkjet printing technologies and recent nozzle-free ejection systems for ejecting liquids with non-Newtonian properties. Graphical abstract

Antiplane point load solution for a linear slot with rounded ends

An exact linear elastic solution is derived for a pair of opposing point loads, which act in the middle of a straight slot having rounded ends. This problem is analogous to the mode III crack problem where two concentrated forces act in opposite directions to open the surfaces of a finite-length crack. The corresponding path independent J integral for this slot problem is also determined.