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Liquid blister trapped between an elastic sheet and a rigid substrate. The blister is driven by a steadily moving blade, kinematically constrained to enforce a constant gap w 0 at the receding back end of the blister. A negative pressure −σ 0 acts in the lag cavity behind the separation front.
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... key result concerns the dependence of the scaled force on the two numbers controlling the solution of the moving liquid blister. Figure 1 illustrates a liquid blister forced to advance between a thin elastic sheet and a rigid substrate by the action of a rigid frictionless blade moving at constant velocity v, while enforcing a constant gap w 0 at the blister back end. The elastic layer of thickness h is characterised by flexural rigidity D = Eh 3 /12(1 − ν 2 ), where E is Young's modulus and ν is Poisson's ratio. ...
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... This load would move horizontally at a constant level and may occur by movement of crossed pipelines. 18 Arch-like structures under a sliding load may be found some applications in the liquid-¯lled blister, 19,20 or bubbles in elastic¯lm. The elastic layer is protruded and delaminated from the substate. ...
This paper investigates the behavior of a pre-stressed arch under a sliding load, where the initial configuration can be obtained from the buckling of a straight column. The shape of the pre-stressed arch can be varied by increasing the end shortening. Subsequently, a sliding load is applied at a certain height level. The orientation of the applied load maintains the right angle with respect to the tangential line of the arch. By moving the load horizontally, the behavior of the arch can be explored. The governing di®erential equations of the problem can be obtained by equilibrium equations, constitutive relation, and nonlinear geometric expressions. The exact closed-form solutions can be derived in terms of elliptic integrals of the¯rst and second kinds. In this problem, the arch can be divided into two segments where each segment is a part of a buckled hinged-hinged column mounted on an inclined support. The shooting method is employed to solve the numerical solutions for comparing with the elliptic integral method. The stability of the pre-stressed arch is evaluated from vibration analysis, where the shooting method is again utilized for solving the natural frequencies in terms of a square function. A simple experiment is set up to explore the equilibrium shapes. Poly-carbonate sheet is utilized as the pre-stressed arch. From the results, it is found that the results obtained from elliptic integral method are in excellent agreement with those obtained from shooting method. The equilibrium shapes from the theoretical results can also compare with those from the experiment. The pre-stressed arch can lose its stability and snap into an upside-down (inverted) con¯guration depending on the ratio of rise to span-length and loading height. The instability of the arch is not only detected during the pushing of the sliding load but a pulling load can also cause unstable behavior of the arch.
... This load would move horizontally at a constant level and may occur by movement of crossed pipelines. 18 Arch-like structures under a sliding load may be found some applications in the liquid-¯lled blister, 19,20 or bubbles in elastic¯lm. The elastic layer is protruded and delaminated from the substate. ...
This paper investigates the behavior of a pre-stressed arch under a sliding load, where the initial configuration can be obtained from the buckling of a straight column. The shape of the pre-stressed arch can be varied by increasing the end shortening. Subsequently, a sliding load is applied at a certain height level. The orientation of the applied load maintains the right angle with respect to the tangential line of the arch. By moving the load horizontally, the behavior of the arch can be explored. The governing differential equations of the problem can be obtained by equilibrium equations, constitutive relation, and nonlinear geometric expressions. The exact closed-form solutions can be derived in terms of elliptic integrals of the first and second kinds. In this problem, the arch can be divided into two segments where each segment is a part of a buckled hinged-hinged column mounted on an inclined support. The shooting method is employed to solve the numerical solutions for comparing with the elliptic integral method. The stability of the pre-stressed arch is evaluated from vibration analysis, where the shooting method is again utilized for solving the natural frequencies in terms of a square function. A simple experiment is set up to explore the equilibrium shapes. Poly-carbonate sheet is utilized as the pre-stressed arch. From the results, it is found that the results obtained from elliptic integral method are in excellent agreement with those obtained from shooting method. The equilibrium shapes from the theoretical results can also compare with those from the experiment. The pre-stressed arch can lose its stability and snap into an upside-down (inverted) configuration depending on the ratio of rise to span-length and loading height. The instability of the arch is not only detected during the pushing of the sliding load but a pulling load can also cause unstable behavior of the arch.
Numerous crack propagation criteria have been employed in numerical investigations of hydraulic fracturing. Some criteria rely on linear elastic fracture mechanics, which assume the presence of an inverse-square-root stress singularity and an elliptical crack profile. However, it has been observed that hydraulic fracturing exhibits a weaker stress singularity, which is governed by a dimensionless parameter known as apparent fracture toughness. This parameter represents the energy required to fracture the solid compared to the energy needed to overcome fluid friction. Abnormal stress singularity was found for the viscosity dominated regime where the apparent fracture toughness approaches zero from the positive side. Variations in results can occur when different criteria are employed, primarily due to the breakdown of the inverse-square-root stress singularity. In this paper, a mathematical model utilizing a straight slit crack was employed to investigate these discrepancies. Four commonly used criteria were implemented in the same mathematical model, and high-precision methods based on modified Chebyshev polynomials and implicit algorithms were employed for calculations. The transient crack dimensions and wellbore pressure were found to diverge for these criteria. Notably, the CTOD, CDM, and FPZ criteria were found to be challenging to apply in the viscosity-dominated regime where the apparent fracture toughness was less than 1.0. The underlying mechanisms were discussed based on crack tip asymptotics and crack opening behavior. This study provides valuable insights for selecting appropriate crack propagation criteria in hydraulic fracturing.
One investigates the post-shut-in growth of a plane-strain hydraulic fracture in an impermeable medium while accounting for the possible presence of a fluid lag. After the stop of fluid injection, the fracture may present three distinct propagation patterns: an immediate arrest, a temporary arrest with delayed propagation, and a continuous fracture growth. These three patterns are all followed by a final fracture arrest yet the fracture behaviour prior to that results from the interplay between the dimensionless toughness $$\mathcal {K}_m$$ K m , the shut-in time $$t_s/t_{om}$$ t s / t om , and the propagation time $$t/t_s$$ t / t s . $$\mathcal {K}_m$$ K m characterizes the energy dissipation ratio between fracture surface creation and viscous fluid flow under constant rate injection. $$t_s$$ t s and $$t_{om}$$ t om represent respectively the timescale of shut-in and the coalescence of the fluid and fracture fronts. The immediate arrest occurs when the fracture toughness dominates the fracture growth at the stop of injection ( $$\mathcal {K}_m \gtrapprox 4.3$$ K m ⪆ 4.3 ). It may also occur upon an early shut-in at low dimensionless toughness associated with an overshoot of fracture extension and a significant fluid lag. For intermediate values of $$\mathcal {K}_m$$ K m and $$t_s/t_{om}$$ t s / t om , the fracture may experience a temporary arrest followed by a restart of fracture propagation. The period of the temporary arrest becomes shorter with higher dimensionless toughness and later shut-in until it drops to zero. The fracture behaviour after shut-in then transitions from temporary arrest to continuous propagation. These propagation patterns result in different evolution of fracture dimensions which possibly explains the various emplacement scaling relations reported in magmatic dikes.
This paper describes the dynamics of a liquid blister forced to advance between a thin elastic sheet and a rigid substrate, by the dual action of a piston and a flat frictionless sleeve at the receding end. Compared to the removal of a viscous blister by sliding a frictionless blade (Wang and Detournay, 2021), the present problem is a steady-state one due to the absence of bleeding at the back end. We seek to obtain a travelling-wave solution, in particular the dependence of the external driving force on the blister velocity and other parameters characterizing this problem, such as the fluid viscosity, the elastic properties of the sheet, and the interface toughness. The peculiarity of this problem lays in the Eshelbian (rather than Newtonian) nature of the horizontal driving force applied by the sleeve on the elastic sheet. The Eshelbian nature of this horizontal force is then discussed and alternative expressions of this force are derived from both variational and energy balance considerations. Scaling of the governing equations indicates that the solution depends on three numbers: namely, dimensionless toughness K, residual gap W and length γf of the fluid-filled part of the blister, a proxy for the volume of the fluid. We use the method of matched asymptotic expansions to predict the horizontal force on a long blister in both the viscosity- and toughness-dominated asymptotic regimes in the back end boundary layer. The numerical solution of the finite blister is then compared with the asymptotic solutions. The key result concerns the dependence of the scaled horizontal force on the three numbers controlling the solution of the moving liquid blister.
