Introductory Solid Mechanics

Material Properties
Stress-Strain Diagram
Material Properties

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):

\[\epsilon = \frac{\delta}{L_0} = \frac{L-L_0}{L_0}\]

Evaluation of average normal stress, or engineering stress (relative to underformed 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.

1. Elastic Behavior

For $\sigma \le \sigma_{pl}$ (proportional limit): linear Hooke's law $\sigma = E\varepsilon$ with Young's modulus $E$ deformation is linear and elastic. For $\sigma_{pl} < \sigma < \sigma_Y$ (below yield stress): deformation is nonlinear and elastic.

Typical values:

Material Young's Modulus [GPa]
Mild Steel 210
Copper 120
Bone 18
Plastic 2
Rubber 0.02

2. Yielding

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 offset method.

3. Strain Hardening

Atoms rearrange in plastic region, higher stress is sustained. Plastic strain remains after unloading as permanent set. Reloading is linear elastic up to the new, higher yield stress (at $A'$).

4. Failure

  • 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, the true maximum is the true fracture stress.

    Examples

    Concrete (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

    • Isotropic vs. anisotropic materials

  • Isotropic: material properties are independent of the direction
  • Anisotropic: material properties depend on the direction

    • Material Properties

    Poisson's Ratio

    Axial (normal) strain:

    \[\epsilon_{x} = \frac{\delta}{L}\]

    Poisson's ratio [$\nu$]:

    \[\nu = -\frac{lateral strain}{axial strain}\] where \[0 < \nu < 0.5\]

    Laterial strain:

    \[\epsilon_{z} = \epsilon_{y} = -nu\epsilon_{x}\]

    Negative Poisson's Ratio (auxetics)

    Possible because of non-trivial structure of the material.

    Hooke's Law for Shear Stress and Strain

    Only two of the three material constants are independent in isotropic materials.

    Hooke's law:

    \[\tau_{xy} = G\gamma_{xy}\]

    Shear Modulus:

    \[G = \frac{E}{2(1+\nu)}\]

    Strain Energy

    Deformation does work on the material: equal to internal strain energy (by energy conservation).

    \[\Delta U = \int F_{z}dz = \int \sigma_{z}(\Delta x \Delta y \Delta z)d\epsilon_{z}\]

    Energy Density:

    \[u = \frac{\Delta U}{\Delta V} = \int \sigma_{z}d\epsilon_{z}\]

    Can be generalized to any deformation: areas under stress strain curves

    \[u = \int \sigma d\epsilon\] or \[u = \int \tau d\gamma\]

    Fatigue - Repeated Loadings

    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