Equations
Equations
You will regularly write down mathematical expressions for solving the problems we assign to you and for passing the exam. Please pay attention to precision when formulating equations and avoid possible ambiguity. In particular, it is important to identify which mathematical objects are hidden behind a symbol.
Vectors and tensors
The symbol \[x\] here denotes a scalar, unless you explicitly tell us that it is a vector, for example, \[x\in\mathbb{R}^3\]. In engineering and the natural sciences, however, it is more common to indicate what kind of object we are dealing with by the type of symbol. Vectors, for example, are represented by arrows, \[\vec{x}\]. Bold symbols, \[\mathbf{x}\], are an alternative. Components of a vector, \[x_\alpha\], have no arrow because these are numbers (scalars).
The lecture material uses arrows \[\vec{x}\] for vectors and underscores \[\underline{K}\] for matrices. If you deviate from this notation, you will need to explain your notation in the exercise sheets or your solution to an exam question. We use arrows and underscores because, unlike bold symbols, they can be easily implemented in the blackboard notation.
Inner and outer products
An inner product (scalar product) is expressed by a point \[\cdot\], \[\vec{a}\cdot\vec{b}\]. The expression \[\vec{a}\vec{b}\] is not a scalar product but the outer product. We recommend using a specific symbol for the outer product to avoid ambiguity, e.g. \[\vec{a}\otimes\vec{b}\]. Each point (in the scalar product or the double contraction \[\underline{A}:\underline{B}\]) represents a sum.
Typical Material Properties Reference
This table provides typical elastic properties of common engineering materials at room temperature (20°C). These values are used throughout the course for worked examples and exercises.
Metals
| Material | Young’s Modulus E (GPa) | Shear Modulus G (GPa) | Poisson’s Ratio ν | Yield Stress σ_y (MPa) | Density (kg/m³) |
|---|---|---|---|---|---|
| Steel (mild) | 200-210 | 80-85 | 0.27-0.30 | 250-350 | 7850 |
| Stainless Steel | 190-200 | 75-80 | 0.27-0.30 | 170-310 | 8000 |
| Aluminum | 69-72 | 26-27 | 0.33-0.34 | 40-90 | 2700 |
| Copper | 110-130 | 40-50 | 0.32-0.34 | 70-200 | 8960 |
| Titanium | 100-110 | 40-45 | 0.32-0.34 | 170-480 | 4510 |
| Cast Iron | 120-160 | 45-65 | 0.20-0.30 | 150-370 | 7100-7800 |
Polymers
| Material | Young’s Modulus E (GPa) | Shear Modulus G (GPa) | Poisson’s Ratio ν | Yield Stress (MPa) |
|---|---|---|---|---|
| PET | 2.7-4.1 | 1.0-1.5 | 0.35-0.40 | 48-72 |
| Polycarbonate | 2.3-2.4 | 0.8-0.9 | 0.37-0.38 | 62-70 |
| PMMA | 2.4-3.2 | 0.9-1.2 | 0.37-0.40 | 72-90 |
Ceramics and Composites
| Material | Young’s Modulus E (GPa) | Shear Modulus G (GPa) | Poisson’s Ratio ν | Compressive Strength (MPa) |
|---|---|---|---|---|
| Alumina (Al₂O₃) | 300-400 | 150-160 | 0.20-0.25 | 1000-2800 |
| Silicon Carbide | 350-450 | 190-210 | 0.14-0.19 | 2000-3900 |
| Glass | 50-90 | 20-35 | 0.20-0.27 | 30-100 |
| Concrete | 20-40 | 8-16 | 0.15-0.20 | 20-100 |
Conversion Formulas for Elastic Constants
Given any two of the four independent elastic constants (E, G, λ, ν), the others can be calculated:
\[\mu = G = \frac{E}{2(1+\nu)}\]
\[\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}\]
\[K = \frac{E}{3(1-2\nu)} \quad \text{(Bulk modulus)}\]
\[\nu = \frac{\lambda}{2(\lambda + \mu)}\]
Notes on Material Properties
- Young’s Modulus (E): Stiffness in tension/compression; higher values indicate stiffer materials
- Shear Modulus (G): Stiffness in shear; typically G ≈ 0.4E for most metals
- Poisson’s Ratio (ν): Ratio of transverse to axial strain; ranges from -1 to 0.5 (0.5 for incompressible materials)
- Yield Stress (σ_y): Stress at which permanent deformation begins; metals typically 100-1000 MPa, polymers 20-100 MPa
- Bulk Modulus (K): Resistance to volumetric compression; K = E/[3(1-2ν)]
Temperature effects: Most elastic moduli decrease with increasing temperature (~0.3-0.5% per °C for metals)