Phase relationships and elementary soil definitions
Context and course logistics
Today’s focus: elementary definitions and phase relationships in soils to prepare for the math and derivations later.
Upcoming lab logistics:
Two four-week lab sessions due to large class size; induction proof (MDO) required to attend the lab.
You’ll receive a lab manual/handbook; you can write notes on paper, iPad, or scan and submit.
A soil classification lab handout will be provided; read it before lab and annotate if you wish.
Geology quiz on Canvas next week; open-book/on notes; single 14-hour window from 9 a.m. Tuesday; 26 questions (25 scored) + an integrity question; one attempt; content up to slides 3.2; mostly geology with some sand/silt/clay questions.
If technical questions arise, use discussion board rather than asking for direct answers.
Core aim of the week: define elementary soil properties and phase relationships to enable later calculations and derivations.
Soils as a 2–3 phase material
Soils consist of solid grains plus voids that can contain either air or water, making a 2-3 phase material: solid phase, air phase, and water phase.
The key goal is to quantify relationships between mass, volume, and density/unit weight across these phases using phase diagrams.
Tomorrow (and the next few lectures) will cover example methods to define soil properties and perform calculations.
Practical framing: soils are in situ, variable, anisotropic, and heterogeneous; material properties vary across short distances and directions.
Because of variability, engineering design often uses ranges of properties rather than precise single values.
In later courses (Civil 300), you’ll explicitly estimate properties and report ranges rather than a single precise value.
Core references mentioned for reading concepts: Soil Mechanics, Section 1.6; Introduction to Geotechnical Engineering, Sections 2.2–2.3.
Phase diagrams: separating the soil into solids, air, and water
Phase diagram idea: separate a soil specimen into three pieces for analysis:
Solid phase: all soil grains (sand, silt, clay, gravel)
Air voids: empty spaces in the voids filled with air
Water voids: the same void spaces can be filled with water
Volumes in the phase diagram:
$V_s$ = volume of solids
$V_a$ = volume of air
$V_w$ = volume of water
$Vv$ (total void volume) = $Va + V_w$
$V$ (total soil volume) = $Vs + Vv = Vs + Va + V_w$
Mass in the phase diagram:
$M_s$ = mass of solids
$Ma$ = mass of air (negligible, typically $Ma o 0$ in standard analyses)
$M_w$ = mass of water
$M$ (total mass) = $Ms + Mw + Ma \,\approx\, Ms + M_w$
Mass of air is neglected (air has negligible weight for these purposes).
Mass, weight, and unit weight: basic definitions
Weight relation: $W = M g$ where $g \approx 9.81\ \mathrm{m\,s^{-2}}$ (metric).
Unit weight (weight density): $\gamma = \dfrac{W}{V}$; units typically in kN/m$^3$ or N/m$^3$.
In the lab and design, it’s common to switch between mass-based and weight-based formulations, keeping units consistent.
On earth, $1\ \text{kg}$ weighs $9.81\ \text{N}$; a $1\ \mathrm{m^3}$ column of water has weight $W = 1000\times 9.81 = 9.81\ \mathrm{kN}$ and unit weight $\gammaw = 9.81\ \mathrm{kN/m^3}$; density $\rhow = 1000\ \mathrm{kg/m^3}$.
For consistency, you may use either mass-based or weight-based formulations as long as units cancel properly.
Volume decomposition and total volume
Total volume: $V = Vs + Vv = Vs + Va + V_w$
Void volume: $Vv = Va + V_w$
The framework lets you relate volume to mass through densities/unit weights.
Phases in idealized soils: three common cases
Completely dry soil (no water in voids):
$Vv = Va$; $Vw = 0$; $M = Ms$; $W$ and $\gamma$ depend only on solids and total volume.
Partially saturated soil (air and water in voids):
$Vv = Va + Vw$; total volume $V = Vs + Va + Vw$; total mass $M = Ms + Mw$ (air has negligible mass).
Completely saturated soil (all voids filled with water):
$Vv = Vw$; $M = Ms + Mw$; $W = Ws + Ww$; $V = Vs + Vw$.
In practice, many analyses assume unsaturated soils to be more complex; unsaturated soil mechanics is an advanced topic pursued in research.
From mass and volume to unit weight and density
Density (mass-based): $\rho = \dfrac{M}{V}$
Unit weight (weight-based): $\gamma = \dfrac{W}{V}$
Density and unit weight relationship for solids, water, and air:
$\rhos = \dfrac{Ms}{Vs}$; $\gammas = \dfrac{Ws}{Vs}$
$\rhow = \dfrac{Mw}{Vw}$; $\gammaw = \dfrac{Ww}{Vw}$ (for water, $\gamma_w = 9.81\ \mathrm{kN/m^3}$ on Earth)
Air is assumed massless for practicality: $Ma \approx 0$, $\gammaa \approx 0$
Interconversion between mass and weight: $W = M g$; $\gamma = \dfrac{W}{V} = \dfrac{M g}{V}$; $\rho = \dfrac{M}{V}$; hence $\gamma = \rho g$ when using density.
Practical note: choose a consistent set of units (SI) and cancel units correctly.
Solid density, bulk density, and dry density
Bulk density (overall soil density):
$\rho{bulk} = \dfrac{M}{V}$; in weight terms, $\gamma{bulk} = \dfrac{W}{V}$.
Dry density (mass of solids per total volume, ignoring water):
$\rho{dry} = \dfrac{Ms}{V}$; $\gamma{dry} = \dfrac{Ws}{V}$.
Note: water mass is excluded from the numerator; total volume still includes voids.
Solid density (density of the solid grains only, ignoring voids):
$\rhos = \dfrac{Ms}{Vs}$; $\gammas = \dfrac{Ws}{Vs}$.
Unit weights summary:
$\gamma = \dfrac{W}{V}$ (overall unit weight)
$\gamma{dry} = \dfrac{Ws}{V}$ (dry unit weight, solids per total volume)
$\gammas = \dfrac{Ws}{V_s}$ (unit weight of solids)
Why these matter: connect mass/weight to density/unit weight, relate to water content and mineralogy, and influence stress distribution in soils.
Note on variability: soils vary by mineralogy, moisture, density, and void distribution; expect ranges rather than exact numbers in design.
Saturation concepts and unit weights under different moisture states
Saturated unit weight: when all voids are filled with water
$\gamma{sat} = \dfrac{Ws + Ww}{Vs + V_w}$
Equivalent density form: $\gamma{sat} = \dfrac{\rhos Vs g + \rhow Vw g}{Vs + V_w}$
Dry unit weight vs saturated unit weight:
Completely dry: $\gamma = \gamma_{dry}$ (no water in voids to contribute weight)
Saturated: $\gamma = \gamma_{sat}$ (water adds weight in voids)
Solid unit weight vs bulk unit weight:
Solid unit weight: $\gammas = Ws / V_s$
Bulk unit weight: $\gamma_{bulk} = W / V$ (includes solids, water, and possibly air if not dry)
Specific gravity of solids (G_s): relates solid density to water density
$Gs = \dfrac{\gammas}{\gammaw} = \dfrac{\rhos}{\rho_w}$
Typical values: $G_s \approx 2.65$ for many mineral soils (roughly 2.65 ± a bit)
Relationship to particle density: $\rhos = Gs \rhow$ (with $\rhow = 1000\ \mathrm{kg/m^3}$)
Notes on unit weights of water and soils:
$\gammaw = 9.81\ \mathrm{kN/m^3}$ in metric units (from $\rhow = 1000\ \mathrm{kg/m^3}$ and $g = 9.81\ \mathrm{m/s^2}$)
In practice, you may adopt a commonly used approximate value (e.g., 10 kN/m^3) for rough estimates, but this course emphasizes consistency with $\gamma_w = 9.81$.
Interrelating mass, volume, and density: the phase diagram toolkit
The phase diagram lets you relate the three pieces (volumes) to the three masses and to unit weights:
Mass balance: $M = Ms + Mw + Ma$ (with $Ma$ negligible)
Volume balance: $V = Vs + Vv = Vs + Va + V_w$
Useful derived relationships:
Porosity $n$ and void ratio $e$ relate voids to solids:
Porosity: $n = \dfrac{V_v}{V}$
Void ratio: $e = \dfrac{Vv}{Vs}$
Relationship between porosity and void ratio (derived by assuming $V_s = 1$ for simplicity):
$n = \dfrac{e}{1+e}$
Inverse: $e = \dfrac{n}{1-n}$
These relationships let you switch between phase fractions and mass/density quantities for calculations.
Void ratio and porosity: definitions and connections
Void ratio (e):
$e = \dfrac{Vv}{Vs}$
Porosity (n):
$n = \dfrac{V_v}{V}$ (fraction of total volume that is voids)
Relationship between n and e (via the phase diagram):
$n = \dfrac{e}{1+e}$, hence $e = \dfrac{n}{1-n}$
Practical interpretation:
Porosity depends on total volume; void ratio depends on solid volume.
Both describe how much empty space exists in a soil sample, which influences density, unit weight, and stiffness.
Degree of saturation and air content
Degree of saturation (Sr): fraction of voids filled with water
$Sr = \dfrac{Vw}{V_v}$ (dimensionless, 0–1 or 0–100% when expressed as percent)
Air content (Ca): fraction of total volume that is air
$Ca = \dfrac{Va}{V}$ (dimensionless, 0–1 or 0–100% when expressed as percent)
Related extremes:
Completely dry, unsaturated case: $Sr = 0$, $Ca = 1$ (all voids contain air)
Partially saturated: $0 < S_r < 1$, some air and some water in voids
100% saturated: $Sr = 1$, $Ca = 0$ (all voids filled with water, no air)
Definitions in terms of volumes:
$Sr = \dfrac{Vw}{V_v}$
$Ca = \dfrac{Va}{V}$ (note: includes solids in the total volume)
From these, you can connect moisture and saturation to total volume and voids.
Moisture content (water content) and practical measurement
Water content (often denoted w or moisture content):
Definition by mass ratio: $w = \dfrac{Ww}{Ws}$
In practice (laboratory): weigh moist sample, dry it (oven-dry), weigh solids, compute $w$ as $Ww / Ws$. Then express as a percentage: $w\% = w \times 100$.
Alternative mass-based form used in some textbooks: $w = \dfrac{Mw}{Ms}$ (if you use masses directly rather than weights).
Physical meaning: how much water is present relative to the solid particle mass; organic soils can have very high moisture contents, sometimes exceeding 100%.
Relationship to phase diagram and other properties:
Moisture content affects the water portion of $Vw$ and thus impacts $Vv$, $n$, $e$, and $S_r$.
Representative soil parameters and typical ranges
Soils vary widely; typical illustrative values (representative ranges) include:
Loose sand: $e \approx 0.8$ (void ratio when loose)
Moisture content at full saturation: $w_{sat} \approx 30\%$ (approximate for some sands)
Dry unit weight: $\gamma_{dry} \approx 14.5\ \mathrm{kN/m^3}$ (loose sand)
Dense, well-graded sand: $\gamma_{dry} \approx 18\ \mathrm{kN/m^3}$
Material-type implications:
Angular sands (e.g., silty sand) vs transported sands may have different ranges.
Stiff clays (residual clays from weathered volcanic material) tend to have narrower ranges.
Wind-driven silt (loose, fine-grained) has characteristic properties.
Soft organic clays: very high moisture content and high void ratio; very low shear strength.
Glacial till: dense, well-graded mixtures of gravel, sand, and silt with high unit weights.
These representative values are used for intuition and teaching; actual engineering design relies on measured properties with appropriate uncertainty ranges.
Practical implications and design philosophy
Real-world soils are variable and anisotropic; a single value is rarely adequate for design.
Engineering practice often reports ranges of plausible values (e.g., settlement estimates between 10–20 cm) rather than precise numbers.
Precision management: keep decimal precision until the final step; avoid over-precise numbers in reports due to inherent material uncertainty.
In labs, you’ll move between mass-based and weight-based measurements; ensure unit consistency throughout calculations.
The course emphasizes understanding phase relationships and basic definitions first, then applying them to calculations in subsequent lectures.
Quick synthesis: key formulas to remember (LaTeX)
Mass balance (general): M = Ms + Mw + M_a
Volume balance (general): V = Vs + Vv = Vs + Va + V_w
Void volume: Vv = Va + V_w
Weight and unit weight: W = Mg,\qquad \gamma = \frac{W}{V}
Saturated unit weight: \gamma{sat} = \frac{Ws + Ww}{Vs + V_w}
Dry unit weight: \gamma{dry} = \frac{Ws}{V}
Unit weight of solids: \gammas = \frac{Ws}{V_s}
Solid density and water density:
\rhos = \frac{Ms}{Vs}, \quad \rhow = \frac{Mw}{Vw}
\gammaw = \rhow g = 9.81\ \mathrm{kN/m^3} (on Earth)
Specific gravity of solids: Gs = \frac{\gammas}{\gammaw} = \frac{\rhos}{\rho_w}
Porosity and void ratio:
n = \frac{Vv}{V}, \quad e = \frac{Vv}{V_s}
Relationship between n and e: n = \frac{e}{1+e} \quad\text{or}\quad e = \frac{n}{1-n}
Degree of saturation and air content:
Sr = \frac{Vw}{Vv}, \quad Ca = \frac{V_a}{V}
Moisture (water) content: w = \frac{Ww}{Ws}\quad\text{(dimensionless)},\; w\% = 100\times w
Next steps on the course plan
Next week: start applying these definitions to concrete calculations and derivations.
Instructor will post example calculation sheets on Canvas for practice in the upcoming week.
You’ll practice converting between mass/volume and weight/volume using the phase diagram framework.
Quick takeaway
Soil is a 2–3 phase material (solids, air, water) with volumes and masses that relate through a set of interlocking definitions:
$V = Vs + Vv$, $Vv = Va + V_w$
$M = Ms + Mw + Ma$ (with $Ma\approx 0$)
Unit weights and densities connect mass and volume and enable stress analyses in soils.
Key dimensionless quantities to master early: $n$, $e$, $Sr$, $Ca$, $G_s$, and their practical interrelations.