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Material Properties (Images need updating)

Stress-Strain Diagram

A stress-strain diagram is the relationship of normal stress as a function of normal strain. One way to collect these measurement is a uniaxial tension test in which a specimen at a very slow, constant rate (quasi-static). A load \( P \) and distance \( L \) are measured at frequent intervals. Evaluation of average normal strain, or engineering strain (relative to undeformed length):
$$ \varepsilon = \frac{\delta}{L_0} = \frac{L-L_0}{L_0}\ $$
Evaluation of average normal stress, or engineering stress (relative to undeformed cross section):
$$ \sigma = \frac{P}{A_0}\ $$
Note that two stress-strain diagrams for a particular material will be similar, but not identical, e.g. because of imperfections, different composition, rate or loading, or temperature.
Fig: Stress Strain Diagram
Stress-strain diagram showing key properties

Elastic Behavior

Loading in this region results elastic behavior, meaning the material returns to its original shape when unloaded. This region of the diagram is mostly a straight line, limited by the proportional limit). This region ends at \( \sigma_Y \) (yielding). For \( \sigma \le \sigma_{pl} \), the diagram is linear, and the behavior is elastic. For \( \sigma_{pl} < \sigma < \sigma_Y \), the diagram is nonlinear, but the behavior is still elastic. Young's Modulus Hooke's law is used for small deformations in the elastic region. The Young's modulus is the slope \( E \).
$$ \sigma = E\varepsilon\ $$
Fig: Modulus Examples
Table screenshot from ref pages. Approximate elastic modulus values for different material families.
Shear Modulus
Fig: Shear Diagram

Taken from TAM251 Lecture Notes - L3S14

Hooke's law:
$$ \tau_{xy} = G\gamma_{xy}\ $$
Shear Modulus:
$$ G = \frac{E}{2(1+\nu)}\ $$
Only two of the three material constants (ie; \( G \), \( E \), \( \nu \)) are independent in isotropic materials.

Plastic Behavior

Stresses above the plastic limit (\( \sigma_Y < \sigma \)) cause the material to permanently deform. Yield Strength Perfect plastic or ideal plastic: well-defined \( \sigma_Y \), stress plateau up to failure. Some materials (e.g. mild steel) have two yield points (stress plateau at \( \sigma_{YL} \)). Most ductile metals do not have a stress plateau; yield strength \( \sigma_{YS} \) is then defined by the 0.002 (0.2\%) offset method. Strain Hardening
Fig: Strain Hardening

Taken from TAM251 Lecture Notes - L3S8

Atoms rearrange in plastic region of ductile materials when a higher stress is sustained. Plastic strain remains after unloading as permanent set, resulting in permanent deformation. Reloading is linear elastic up to the new, higher yield stress (at A') and a reduced ductility. Ultimate Strength The ultimate strength (\( \sigma_u \)) is the maximum stress the material can withstand. Necking
Fig: Necking

Taken from TAM251 Lecture Notes - L3S5

After ultimate stress (\( \sigma_u < \sigma \)), the middle of the material elongates before failure. Failure Also called fracture or rupture stress (\( \sigma_f \)) is the stress at the point of failure for the material. Brittle and Ductile materials fail differently.
Fig: Ductile vs Brittle

Taken from TAM251 Lecture Notes - L9S19

  • Brittle materials: small plastic region between yield and failure (fracture), no necking, primary fail by normal stress.
  • Ductile materials: large region of plastic deformation before failure (fracture) at higher strain, necking; often fails under 45° cone angles by shear stress.
Note the difference between engineering and true stress/strain diagrams: ultimate stress is a consequence of necking, and the true maximum is the true fracture stress. Example: Concrete is a brittle material.
  • Maximum compressive strength is substantially larger than the maximum tensile strength.
  • For this reason, concrete is almost always reinforced with steel bars or rods whenever it is designed to support tensile loads.

Other Derived Properties

Directional Materials

  • Isotropic: material properties are independent of the direction
  • Anisotropic: material properties depend on the direction (ie; composites, wood, and tissues)

Poisson's Ratio

Fig: Poisson's Ratio

Taken from TAM251 Lecture Notes - L3S15

Axial (normal) strain:
$$ \varepsilon_{x} = \frac{\delta}{L}\ $$
Poisson's ratio \( \nu \):
$$ \nu = -\frac{\text{lateral strain}}{\text{axial strain}} $$
Typical Range:
$$ 0 < \nu < 0.5\ $$
Lateral strain:
$$ \varepsilon_{z} = \varepsilon_{y} = -\nu\varepsilon_{x}\ $$
Negative Poisson's Ratio (auxetics) Possible because of non-trivial structure of the material.

Strain Energy /!\ BSM: we do not cover this topic in class/hw/exams. ME 330 is covered in an introductory way in ME 330; not sure where else it might come up in ME curriculum.

**Reference pages have a broken link image here**

Deformation does work on the material: equal to internal strain energy (by energy conservation).
$$ \Delta U = \int F_{z}dz = \int \sigma_{z}(\Delta x \Delta y \Delta z)d\varepsilon_{z}\ $$
Energy Density:
$$ u = \frac{\Delta U}{\Delta V} = \int \sigma_{z}d\varepsilon_{z}\ $$
Can be generalized to any deformation: areas under stress-strain curves
$$ u = \int \sigma d\varepsilon \quad \text{or} \quad u = \int \tau d\gamma\ $$

Fatigue - Repeated Loading /!\ BSM: we do not cover this topic in class/hw/exams. Fatigue is covered in ME 330 briefly, and ME 371 more in depth.

Fig: SN Curve

Taken from TAM251 Lecture Notes - L3S6

If stress does not exceed the elastic limit, the specimen returns to its original configuration. However, this is not the case if the loading is repeated thousands or millions of times. In such cases, rupture will happen at stress lower than the fracture stress - this phenomenon is known as fatigue.