# Stress Strain Hooke’s Law Poisson’s ratio

The Strength of Material or SOM is a subject which deals with How solid bodies react when subjected to a various type of loadings. This article covers most basic topics of the strength of material. These topics act as the foundation stone of the complete civil engineering. Most of the subject like Structural Analysis, Design of Concrete Structures, Design of Steel Structures use these topics in one way or other. Hence the grip on these concepts is the must. This article covers Stress Strain Hooke’s Law Poisson’s ratio.

## Stress

It is the force of resistance offered by a body against the deformation. These deformations are caused by externals forces usually in terms of loads acting on the body. So, When a force is applied on the body it causes the deformation and as a result, stresses are induced.

Load is applied on body whereas Stresses are Induced.

Let’s consider a Rod with the uniform cross-sectional area “A” subjected to axial loads “F”. as shown in the figure below.

At every section, Resistance “R” offered is equal to the force “F”. If the resistance offered by the section against the deformation be assumed to be uniform across the section in called the “Intensity of Stress” or “Unit Stress”. Hence Intensity of Stress = p =F/A.

Every material is capable of inducing following types of Stresses based on the application of corresponding forces.

1. Tensile Stress
2. Compressive Stress
3. Shear Stress

The Increase in length induces the tensile stress while compression induces the compressive stress. Corresponding Strains are called  Tensile strain and Compressive strain. If the bottom face of a block is fixed and force is applied to the opposite face (i.e. top). It will bend the block as shown in the gif below.

Value of shear stress is (Shear Resistance)/(Shear Area)

## Strain

Now suppose that due to the application of Force. Change in length is “dL” making the total length “L+dL”.

The ratio of the change in length to the original length of the member is “Strain”.

Hence Strain is dL/L. So, The tensile strain is (Increase in length)/(Original Length). Compressive strain is (Decrease in length/Orignal Length). Shear strain is (Transverse displacement)/(Distance from the lower face).

A material is Elastic when it undergoes a deformation on the application of a force such that the deformation disappears on the removal of force.

## Hooke’s Law

If an object keeps on inducing stress on itself. It will behave elastic only till a certain limit of stress and this value of stress is the Elastic Limit. If the force is so large that the stress induced in beyond the elastic limit, The material will lose it elastic properties. In such case, the material will not come back to its initial shape again.

The Hook’s Law states that. applied

When the force is applied on a meterial such that the intensity of stress is within certain limit. The ratio of intensity of stress to the corresponding strain is constant and that constant is characteristic of material.

In  the case of axial loading, this constant ratio is called “Modulus of Elasticity” or “Young’s Modulus” and denoted by “E”. The product of cross-sectional area “A” of a body with “E” i.e. “AE” is axial rigidity. (Provided “E” corresponds to axial stress.)

In the case of Shear forces, this constant is called “Shear Modulus” or “Modulus of Rigidity”.

##### Young’s Modulus of Different Materials
 Material GPa Mpsi Material GPa Mpsi Rubber (small strain) 0.01–0.1 1.45–14.5×10−3 Tooth enamel (largely calcium phosphate) 83 12 Low density polyethylene 0.11–0.86 1.6-6.5×10−2 Stinging nettle fiber 87 12.6 Diatom frustules (largely silicic acid) 0.35–2.77 0.05–0.4 Bronze 96–120 13.9 – 17.4 PTFE (Teflon) 0.5 0.075 Brass 100–125 14.5 – 18.1 HDPE 0.8 0.116 Titanium (Ti) 110.3 16[4] Bacteriophage capsids 1–3 0.15–0.435 Titanium alloys 105–120 15 – 17.5 Polypropylene 1.5–2 0.22–0.29 Copper (Cu) 117 17 Polyethylene terephthalate (PET) 2–2.7 0.29–0.39 Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain) 181 26.3 Nylon 2–4 0.29–0.58 Silicon Single crystal, different directions 130–185 18.9 – 26.8 Polystyrene, solid 3–3.5 0.44–0.51 Wrought iron 190–210 27.6 – 30.5 Polystyrene, foam 0.0025-0.007 0.00036-0.00102 Steel (ASTM-A36) 200 29 Medium-density fiberboard (MDF) 4 0.58 polycrystalline Yttrium iron garnet (YIG) 193 28 wood (along grain) 11 1.6 single-crystal Yttrium iron garnet (YIG) 200 29 Human Cortical Bone 14 2.03 Cobalt-chrome (CoCr) 220-258 29 Glass-reinforced polyester matrix 17.2 2.49 Aromatic peptide nanospheres 230–275 33.4 – 40 Aromatic peptide nanotubes 19–27 2.76-3.92 Beryllium (Be) 287 41.6 High-strength concrete 30 4.35 Molybdenum (Mo) 329 – 330 47.7 – 47.9 Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric) 30–50 4.35 – 7.25 Tungsten (W) 400 – 410 58 – 59 Hemp fiber 35 5.08 Silicon carbide (SiC) 450 65 Magnesium metal (Mg) 45 6.53 Tungsten carbide (WC) 450 – 650 65 – 94 Glass (see chart) 50–90 7.25 – 13.1 Osmium (Os) 525 – 562 76.1 – 81.5 Flax fiber 58 8.41 Single-walled carbon nanotube 1,000 + 150 + Aluminum 69 10 Graphene 1050 152 Mother-of-pearl (nacre, largely calcium carbonate) 70 10.2 Diamond 1050 – 1210 152 – 175 Aramid 70.5–112.4 10.2 – 16.3 Carbyne 32100 4660

## Stress Strain relationship for Mild Steel

The above figure shows the Stress-Strain diagram for a mild steel specimen subjected to the tensile test. The plot is a straight line till point “A”. So, “A” is the limit of proportionality. In this range young’s modulus, Hooke’s Law etc are applicable. From “A” to “B”  curve is not linear but steel is still elastic. “B” is the Elastic limit. So, If Stresses more than “B” are induced. Steel will not come back to its initial length. i.e Plastic deformation are taking place (Plastic deformations are those which are not restored back).In “B” to “C” the strain increases with the almost constant rate. At “C”, there is considerable extension corresponding to decreasing in load. Stress at “C” is Upper Yield Point. “D” shows the condition in which mild steel again offers a greater extension. “D” is lower yield point. Stress at “E” is

“D” shows the condition in which mild steel again offers a greater extension. “D” is lower yield point. Stress at “E” is the ultimate tensile stress of the material. Increasing extension, the load required decreases and the specimen breaks. This condition is shown at “F”.”F” is Stress at failure.

For a compound section, one can also calculate the corresponding equivalent area. This is often used in Designing of Concrete Structures. Which uses “M” Modular Ratio. “M” is “Es/Ec“.  You can use the spreadsheet given below to find equivalent areas in case of concrete and steel composite section.

## Temperature Stresses

As we know that the variation of temperature can cause a material to shrink or expand. In such case, If a material can freely expand or contract then no stresses will generate. But if we restrict the expansion or contraction of material. The stresses corresponding to the amount of change restricted shall be induced in the material.

Let’s say If we heat a material to cause a temperature difference “dT” with “α” coefficient of linear thermal expansion and “L”. The change in length will be α*(dT)*L. If the material expands or contract freely, its length will simply change and there will be no stress in it. but if we restrict the material from changing the length, there will be stresses in material corresponding to change in length “dL” which is α*(dT)*L.

##### Coefficient of linear thermal expansion (per  °C)
 Aluminium 23×10-6 Wrought Iron 12×10-6 Brass and Bronze 18×10-6 Copper 17×10-6 Steel 13×10-6 Concrete 10×10-6 Cast Iron 11×10-6 Glass 8.5×10-6

## Poisson’s ratio

Whenever an object is under strain from one direction (Let’s say “Lateral”). The variation of that direction induces the variation in other directions as well. (say “longitudinal”)

Ratio of Lateral Strain to Longitudinal Strain is Poisson’s Ratio.

Poisson’s Ratio usually lies between 0.25-0.35. Concrete have Poisson’s Ratio of only 0.1-0.2. Cork have Zero Poisson’s Ratio. The Value of Poisson’s Ratio is constant for the Elastic range of any material. (Provided the material is uniform i.e Homogenous). However, variation can be seen beyond elastic range.

With the help of Poisson’s ratio, one can calculate strain in the different direction if strain along one is known. It can also calculate the Volumetric strain of various materials. For ex. the volumetric strain of a sphere is 3 times the strain of the diameter.

##### Poisson’s Ratio of some Common Materials
 Rubber 0.4999 Stainless steel 0.30–0.31 Gold 0.42–0.44 Steel 0.27–0.30 Saturated Clay 0.40–0.49 Cast iron 0.21–0.26 Magnesium 0.252-0.289 Sand 0.20–0.45 Titanium 0.265-0.34 Concrete 0.1-0.2 Copper 0.33 Glass 0.18–0.3 Aluminium-alloy 0.32 Foam 0.10–0.50 Clay 0.30–0.45 Cork 0

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