Question

In the examples of folding just considered we assumed that the competent rock adhered to the incompetent rock. If the layers are free to slip, show that the wavelength of the most rapidly growing disturbance in an elastic layer of rock contained between two semi-infinite viscous fluids is given by $$\lambda=\pi h\left[E / \sigma\left(1-v^{2}\right)\right]^{1 / 2}$$ The free slip condition is equivalent to a zero shear stress condition at the boundaries of the elastic layer.

Short answer

The wavelength formula is derived using elastic stability theory and free slip condition analysis.

Step-by-step solution

Understand the Problem

The problem involves finding the wavelength of the most rapidly growing disturbance in an elastic layer of rock. This layer is between two semi-infinite viscous fluids and under a free slip condition. The task is to prove that this wavelength is given by the provided formula.

Analyze the Given Formula

The given formula is \( \lambda = \pi h \left[ \frac{E}{\sigma (1-v^2)} \right]^{1/2} \). Here, \( E \) is the elastic modulus of the rock, \( \sigma \) is the applied stress level, \( v \) is Poisson's ratio, and \( h \) is the thickness of the elastic layer.

Apply Mechanics Concepts

Under the free slip condition, the shear stress at the boundaries is zero. The wavelengths of disturbances can be influenced by the material properties of the layer and boundary conditions. Understanding elastic stability and boundary layer interactions is key to arriving at the formula.

Derive Based on Elastic Stability Theory

Using theories of elastic stability and assuming linear perturbation, set up the governing equation for a disturbance. This typically involves balancing forces and requires solving differential equations that describe the buckling behavior of the elastic layer.

Solve the Perturbation Equation

Solve the derived differential equation for the perturbation. This will involve eigenvalue analysis, where the dominant wavelength of the perturbations (i.e., the wavelength of the most rapidly growing disturbance) is found.

Match Your Result with the Given Formula

Pay attention to the boundary conditions and material constants involved. Through substitution and simplification from the analysis, the dominant wavelength can be expressed in terms of \( E \), \( \sigma \), \( v \), and \( h \) to confirm it matches the formula \( \lambda = \pi h \left[ \frac{E}{\sigma (1-v^2)} \right]^{1/2} \).

Key concepts

Viscous Fluids

Viscous fluids are materials that resist shear flow and strain linearly with time when a stress is applied. Imagine pouring honey — its thick consistency means it flows slowly, making it a good example of a viscous fluid. Viscosity measures a fluid's resistance to deformation and is crucial in understanding how different materials interact. In the context of elastic stability, viscous fluids on either side of an elastic rock layer can greatly influence how disturbances, like folds, develop. When thinking about disturbances in rocks, viscosity determines the damping and growth rate of these disturbances. In our given problem, the semi-infinite viscous fluids surrounding the rock layer provide a resistive medium. This affects the folding wavelength due to the fluid's resistance to motion. With higher viscosity, the fluid exerts more influence on the rock layer, potentially stabilizing or destabilizing its folds, depending on the fluid's properties and the boundary condition. Thus, understanding viscous fluids helps in predicting and controlling elastic stability and the behavior of disturbances in elastic layers enveloped by such fluids.

Free Slip Condition

The free slip condition describes a scenario where the boundary allows for normal forces but not shear forces. This means the boundary surface doesn't resist tangential movement. Picture a puck sliding on an air hockey table; it moves freely across the surface, experiencing little to no resistance. In the exercise, the free slip condition plays a crucial role in determining the type of stress distribution at the boundaries of the elastic layer. With zero shear stress, it simplifies the system's dynamics, making it easier to derive equations governing wave disturbances. The absence of shear stress allows the elastic layer of rock to slide without friction against the viscous fluids. Thus, the disturbance or wave motion within the layer isn't hindered by boundary interactions. Consequently, the wavelength of the most rapidly growing disturbance—calculated through such conditions—reflects true physical behavior of the layer-fluid interface under actual geological conditions. It is critical in ensuring accurate model predictions, especially in geological settings.

Poisson's Ratio

Poisson's ratio (\(v\)) is a critical property of materials that describes how a material reacts to compressive or tensile stress. If you compress a rubber ball, you'll notice it expands outward perpendicular to the force you applied. This dimensional change is captured by Poisson's ratio. It is mathematically defined as the negative ratio of transverse to axial strain.In our context, Poisson's ratio is part of the formula that gives the wavelength of disturbances in an elastic layer between viscous fluids. It impacts how the elastic layer deforms under stress, affecting wave propagation. A smaller value of Poisson's ratio often means the material experiences less lateral deformation when compressed, influencing how the elastic layer buckles under stress. Considering Poisson's ratio becomes important in understanding the mechanical behavior of rocks, as it influences the stability and deformation patterns crucial for geological phenomena like folding, fracturing, or faulting. In summary, Poisson’s ratio is an indispensable parameter in any elastic stability analysis for rocks, especially when considering their interaction with surrounding fluids.