Question

An unstressed surface is covered with sediments with a density of \(2500 \mathrm{~kg} \mathrm{~m}^{-3}\) to a depth of \(5 \mathrm{~km}\). If the surface is laterally constrained and has a Poisson's ratio of \(0.25,\) what are the three components of stress at the original surface?

Short answer

The stress components are: \( \sigma_z = 122.625 \, \mathrm{MPa} \), \( \sigma_x = \sigma_y = 40.875 \, \mathrm{MPa} \).

Step-by-step solution

Identifying the stress components

When a surface is loaded with sediments, the stress components at the original surface due to vertical loading are the vertical stress (\( \sigma_z \)) and the horizontal stresses (\( \sigma_x \) and \( \sigma_y \)). Since it's isotropic, \( \sigma_x = \sigma_y \), and these can be calculated using Poisson's ratio and the vertical stress.

Calculate vertical stress

The vertical stress, \( \sigma_z \), at the original surface can be calculated using the weight of the sediment: \( \sigma_z = \rho \cdot g \cdot h \), where \( \rho \) is the density of the sediments (\(2500 \mathrm{~kg/m^3}\)), \( g \) is the acceleration due to gravity (approximately \(9.81 \mathrm{~m/s^2}\)), and \( h \) is the depth of the sediment (\(5000 \mathrm{~m}\)).\[\sigma_z = 2500 \times 9.81 \times 5000 = 122,625,000 \ \, \mathrm{Pa} = 122.625 \, \mathrm{MPa}\]

Calculate horizontal stresses

The horizontal stresses \( \sigma_x \) and \( \sigma_y \) can be calculated using Poisson's ratio (\( u = 0.25 \)) and the formula:\[\sigma_x = \sigma_y = u \cdot \sigma_z/(1 - u)\]Substitute the values:\[\sigma_x = \sigma_y = 0.25 \times \frac{122.625}{1-0.25} = 40.875 \, \mathrm{MPa}\]

Finalize the stress components

Summarizing the calculated stress components: the vertical stress at the original surface is \(122.625 \, \mathrm{MPa}\), and both horizontal stresses (\( \sigma_x \) and \( \sigma_y \)) are \(40.875 \, \mathrm{MPa}\).

Key concepts

Stress components

In geomechanics, understanding stress components is essential for analyzing how forces affect subsurface materials. When a surface undergoes loading, it experiences different types of stress, specifically vertical and horizontal stresses. For an isotropic medium, the surface stress components are uniform, simplifying calculations.

• The vertical stress (\( \sigma_z \)) is the stress applied perpendicular to the surface, often due to the weight of overlying materials.
• Horizontal stresses (\( \sigma_x \) and \( \sigma_y \)) act parallel to the plane, commonly influenced by external constraints and material properties such as Poisson's ratio.
• In lateral constrained systems, where horizontal deformations are prevented, \( \sigma_x \) and \( \sigma_y \) are mainly derived from properties like Poisson's ratio and the magnitude of vertical stress.

These components help us assess and predict subsurface stability under varying conditions.

Vertical stress calculation

The vertical stress calculation is vital for determining the load exerted by sediments or other materials above a certain point in the Earth's subsurface. It is simply the weight of the material column above the point of interest. By using the formula:\[\sigma_z = \rho \cdot g \cdot h \]where \( \rho \) is the density of the sediments, \( g \) is the acceleration due to gravity, and \( h \) is the depth,you can calculate it easily.

• Density (\( \rho \)) represents the mass of the material per unit volume.
• The gravitational constant (\( g \approx 9.81 \ \mathrm{m/s^2} \)) determines the force per unit mass due to Earth's gravity.
• Depth (\( h \)) accounts for the thickness of the sediment layer above the surface.

For example, in our case, with a density of \(2500 \, \mathrm{kg/m^3}\) and a depth of \(5000 \, \mathrm{m}\), the vertical stress is \(122.625 \, \mathrm{MPa}\).Calculated vertical stress provides crucial insights into potential pressures exerted on subsurface structures.

Poisson's ratio

Poisson's ratio is a fundamental property in geomechanics, describing the relationship between longitudinal expansion and lateral contraction of a material when under stress. In simpler terms, it measures how much a material will become thinner in width when it is stretched or grows in length.

• A typical Poisson's ratio, denoted as \( u \), usually ranges between 0 and 0.5 for most materials.
• Materials with \( u = 0.25 \), like in the exercise, will produce equal horizontal stresses when the surface is constrained laterally.

A higher Poisson's ratio indicates a greater tendency for the material's width to contract when it is expanded longitudinally. Understanding Poisson's ratio helps us better predict and calculate the resultant stress states, particularly the induced horizontal stresses, when examining sediment-loaded surfaces or other stressed earth systems.

Sediment load

Sediment load refers to the weight and pressure exerted by accumulated sediments on a subsurface layer. Sediments can vary widely in density depending on the type of sedimentary material present, and this directly impacts the subsurface stress calculations.

• Heavier or denser sediments apply a greater load, increasing vertical stress dramatically.
• The depth of sediment accumulation directly affects the overall pressure applied to the original surface.
• This load needs calculation to understand stress effects at depth, as in our exercise, with a sediment density of \(2500 \, \mathrm{kg/m^3}\) over \(5000 \, \mathrm{m}\) depth.

Sediment load considerations are crucial for evaluating geo-stability, resource extraction potential, or engineering projects needing precise stress evaluations.

Horizontal stress calculation

Calculating horizontal stresses in geomechanics is essential to understanding how the subsurface environment reacts to vertical load, especially when the surface is laterally constrained. The horizontal stress components, \( \sigma_x \) and \( \sigma_y \), show the lateral pressure induced by vertical stress amplified by Poisson's ratio. The formula:\[\sigma_x = \sigma_y = u \cdot \frac{\sigma_z}{1 - u}\]demonstrates this calculation approach. Here:

• \( u \) is Poisson's ratio, representing material properties related to deformation.
• \( \sigma_z \) is the calculated vertical stress.

In the example provided, with \( u = 0.25 \), both horizontal stresses equaled \(40.875 \, \mathrm{MPa}\).Correct horizontal stress calculation helps in evaluating engineering integrity and safety under sediment load scenarios.