Mecánica de Suelos Avanzada

• These are study notes on advanced soil mechanics, focusing on stresses, strains, and consolidation.
• The information is attributed to Prof. Nicolás Estrada M. from the Universidad de los Andes, Colombia.

Introducción

• Introduces the increasing complexity of projects, necessitating a deeper understanding of the mechanical behavior of earth materials.
• Discusses identifying scales within earth materials and the physical quantities used to describe relevant phenomena.

Escalas de Interés en un Suelo

• Presents various scales of observation or description in granular materials like soil: micro, meso, and macro.
• The micro scale corresponds to constituents (particles) and their interactions (contacts).
• The meso scale represents the group of particles, describing the geometry of interactions between particles.
• The macro scale includes the representative elementary volume, where sample properties are independent of system size.
• The macro scale is identified as most relevant for civil engineers.

Esfuerzos y Deformaciones

• Acknowledges the complexity of stresses, requiring a sophisticated physical quantity called a tensor for complete description.
• A tensor is a mathematical object describing physical properties and generalizing scalars and vectors. Scalars are tensors of order 0, vectors of order 1.
• The order of a tensor is the necessary number of directions to describe it.
• Second-order tensors are commonly used to describe stresses.

Tensor de Esfuerzos

• Illustrates an infinitesimal soil element in cubic form, showing normal and tangential stresses applied to its faces.
• The symbol \sigma is used for normal stresses, while \sigma or \tau can represent tangential stresses.
• Each stress has two subindices, where the first indicates the direction of the plane's normal, and the second indicates the stress direction.
• These nine stresses are organized into a matrix called the stress tensor.
• In most cases, this tensor is symmetrical, meaning: \sigma_xy = \sigma_yx, \sigma_yz = \sigma_zy, and \sigma_xz = \sigma_zx.

Esfuerzos Principales

• The values of stresses depend on the coordinate system used.
• A specific orientation exists where only normal stresses act on the element's faces; this is the state of principal stresses.
• From largest to smallest, these stresses are denoted as \sigma_1, \sigma_2, and \sigma_3, or as major, intermediate, and minor principal stresses.

Invariantes del Tensor de Esfuerzos

• The stress tensor has three invariants that remain constant regardless of the coordinate system's orientation.
• These invariants are used in equations to calculate the principal stresses.
• The equations are:
• I₁ = σxx + σyy + σzz
• I₂ = σxxyy + σyyzz + σzzxx - ⌧xy² - ⌧yz² - ⌧zx²
• I₃ = σxxyyzz + σxx⌧yz² + σyy⌧zx² - σzz⌧xy + 2⌧xy ⌧yz ⌧zx
• σ1 = (I1/3) + 2√(I1² - 3I2)/3 cos θ
• σ2 = (I1/3) + 2√(I1² - 3I2)/3 cos ((2π/3) - θ)
• σ3 = (I1/3) + 2√(I1² - 3I2)/3 cos ((2π/3) + θ)
• cos 3θ =(2I3 – 9I1I2 + 27I3)/(2(I1²-3I2)^(3/2))
• Invariants can be calculated from principal stresses:
• I₁ = σ₁ + σ₂ + σ₃
• I₂ = σ₁σ₂ + σ₂σ₃+ σ₃σ₁
• I₃ = σ₁σ₂σ₃

Contenido (Consolidación 2/2)

• Topics include primary consolidation (without time effects), secondary compression, and consolidation over time.

Consolidación Primaria (sin efectos de tiempo)

• Parameters used are: e₀, Cc, Cs ≈ Cr, σ'ᵥ₀
• Two types of soils are considered: normally consolidated and overconsolidated.

Suelos normalmente consolidados

• Presents a graph of void ratio (e) vs. log σ'ᵥ
• Includes the Normal Consolidation Line (NCL).
• The process is elasto-plastic and non-reversible.
• Formula for change in void ratio: Δε = Cc log(σ'ᵥf / σ'ᵥ₀)
• Formula for change in height: ΔH = H₀ * (Δε / (1 + e₀))

Suelos sobreconsolidados

• Shows a graph of void ratio (e) vs. log σ'ᵥ for overconsolidated soils.
• Features the Normal Consolidation Line (NCL).
• Includes the Unloading-Reloading Line (URL), indicating elastic and reversible processes.
• Formula for change in void ratio related to overconsolidation: Δε = Cr log(σ'ᵥc / σ'ᵥgeo)
• Formula for change in void ratio after overconsolidation: Cs log(σ'ᵥf / σ'ᵥvc)
• Formula for change in height: ΔH = H₀ * (Δε / (1 + e₀))