# ELASTICITY

The MODULUS OF ELASTICITY (E) is the STIFFNESS of a material. This property is very consistent - almost every steel has E=200 GPa, whether hardened or not.

**Lecture Notes**: Elasticity.pdf Elasticity.one

Image | Video Lesson Description and Link | Duration | Date | Download |

Elasticity | 24:55 min | 20140731 |

**Elasticity**

The following animation shows a lattice of atoms (such as in a metal). There are only 2 ways to distort the atoms - axial (tension and compression) and shear (sideways).

This animation shows only the elastic portion of the stress/strain curve, where no atomic slip occurs.

Positive axial stress is tensile, positive shear stress is "clockwise" (left side up, right side down)

This next animation shows plastic deformation, where permanent deformation occurs through slip. If the was no slip, the force required to pull the atoms apart would be enormous. However, small imperfections and irregularities initiate miniscule slip planes that propagate in shear. Since atoms are so small, there is a relatively large number of these slip planes, which means a significant amount of plasticity can occur if slip is not constrained.

DEFINE | Formula | Units | Diagram |

Axial Stress (Tension or Compression) | Stress = Force / Area | MPa | |

Axial Strain (Tension or Compression) | Strain = extension / original Length | - | |

Shear Stress |
Stress = Force / Area | MPa | |

Modulus of Elasticity (Young's Mod) | E = Stress / Strain | GPa | Slope of Stress:Strain diagram |

Modulus of Rigidity (Shear Mod.) =~ 0.4E | G = S. Stress / S. Strain | GPa | Slope of S.Stress:S.Strain diagram |

Shear Strain | Strain = movement / original Depth | - | |

Shear in Detail: Shear Strain is usually small enough to ignore the changes in L with angle. Angle is in radians. Area is the zone that would slide apart assuming it broke in shear. |

## Strain

A material in tension or compression changes in length, and the change in
length compared to the original length is referred to as the ‘strain’,
symbol^{1} ε (Greek letter epsilon). Since strain is a ratio of two
lengths it has no units and is frequently expressed as a percentage: a strain of
0.005 corresponds to a ½% change of the original length.

^{1} Ivanoff uses e for strain:

e = x / l_{o} Where;

- e = strain (no units)
- x = extension (mm)
- l
_{o}= original length (mm)

### Hooke’s Law

As you know from a spring, if you gradually stretch it, the force needed
increases, but the material springs back to its original shape when the force is
released. Materials which react in the same way as a spring are said to be
‘elastic’. Typically if we measure the extension of different forces and plot
the graph of this, we will find that the extension is proportional to the force
applied. Materials that obey **Hooke’s Law** exhibit a linear
relationship between the strain and the applied stress (Figure 1).

Stress-strain graph for an elastic solid

Many metals follow Hooke’s Law until a certain level of stress has been
applied, after which the material will distort more severely. The point at which
straight line behaviour ceases is called the **limit of
proportionality**: beyond this the material will not spring back to its
original shape, and is said to exhibit some **plastic** behaviour
(Figure 2). The stress at which the material starts to exhibit permanent
deformation is called the **elastic limit** or **yield
point**.

Stress-strain graph for mild steel

For mild steel, if the stress is increased beyond the yield point the
sample will eventually break. The term (**ultimate**)
**tensile strength** is used for the maximum value of tensile
stress that a material can withstand without breaking, and is calculated at the
maximum tensile force divided by the original cross-sectional area.

Note that there may be substantial differences between the stress at the yield point and on breaking – for example, one source quotes the ‘ultimate tensile strength’ for AISI304 stainless steel as 505 MPa, and the ‘yield tensile strength’ as 215 MPa. For most engineering purposes, metals are regarded as having failed once they have yielded, and are normally loaded at well below the yield point.

With some materials, including mild steel, the stress/strain graph shows a
noticeable dip beyond the elastic limit, where the strain (the effect of the
load) increases without any need to increase the load. The material is said to
have ‘yielded’, and the point at which this occurs is the yield point. Materials
such as aluminium alloys on the other hand don’t show a noticeable **yield
point**, and it is usual to specify a ‘proof’ test. As shown in Figure 3,
the 0.2% **proof strength** is obtained by drawing a line parallel
to the straight line part of the graph, but starting at a strain of 0.2%.

Stress-strain graph for an aluminium alloy

## Modulus of ELASTICITY

This mechanical property measures stiffness - how much stress it takes to stretch it by a certain amount. It goes by several names:

- Modulus of Elasticity
- Modulus of Stiffness
- Young's Modulus

Modulus of Elasticity (MPa) = Stress (MPa) / Strain (No Units)

**E = f / e**

The best thing about E is that it stays very constant. Steel is about 200000 MPa, whether mild steel, carbon steel, heat treated etc. This makes it very easy to predict how much steel will stretch.

The slope of
the stress/strain graph varies with stress, so we take only the slope
of the initial straight-line portion (which is in the elastic range). The stress/strain ratio is referred to as
the **modulus of elasticity** or **Young’s Modulus**.
The units are those of stress, since strain has no units. Engineering materials
frequently have a modulus of the order of 10^{9}Pa, which is usually
expressed as GPa. Some approximate figures for typical electronic materials are
given in Table 1.

Material | Tensile strength MPa | Modulus of elasticity GPa |
---|---|---|

304 stainless steel | 500 | 200 |

copper | 270 | 120 |

96% alumina | 200 | 340 |

aluminium | 90 | 70 |

Sn63 solder | 35 | 30 |

epoxy resin | 40 | 3 |

silicone rubber | 10 | 0.003 |

First: Some revision of Simple Stress

Summary of the three central formulae used for strengths of materials.

Strength (how much stress a material can take) varies wildly depending on alloying elements, heat treatment, work hardening etc.

Stiffness is extremely constant, so it is predictable.

Stress vs Strain diagram. Bolts of increasing strength show decreasing ductility. As strength increases the yield point gets closer to the ultimate tensile strength.

## Example: (Ivanoff)

We usually know the material, which means we have E (modulus of Elasticity).

### Thermal Expansion.

The coefficient of thermal expansion is a property of a material (constant regardless of the size of the specimen)

Thermal stress is created when the specimen is prevented from expanding. To solve this, calculate the expansion, then compress the specimen back to the initial size. This will determine stress.

#### Questions:

Assignment: Do
all questions

Questions 25:13 to 25:21 (Read Chapter 25.5: Tensile
Elasticity)

Questions
26:6 to
26:11 (Read Chapter 26.3: Compressive Elasticity)

Questions 26:12 to 26:21 (Read Chapter 26.4-6: Poisson's Ratio, Thermal
Expansion)