Mechanical Properties of Matter PDF

# Mechanical Properties of Matter ## Dr. Ibrahim Elmashad Lecturer of Physics at Physics Department Faculty of Science, Benha University. ## Outlines - Introduction. - Stress. - Strain. - Elasticity and plasticity. - Examples in living Cell. - Work done in deforming a body. - Poisson's ratio....

# Mechanical Properties of Matter ## Dr. Ibrahim Elmashad Lecturer of Physics at Physics Department Faculty of Science, Benha University. ## Outlines - Introduction. - Stress. - Strain. - Elasticity and plasticity. - Examples in living Cell. - Work done in deforming a body. - Poisson's ratio. ## Introduction - When an external force acts on a body, there is change in its length, shape and volume. The body is said to be strained. - When this external force is removed, the body regains its original shape and size. - Such bodies are called **elastic bodies**. - **Examples:** Rubber - Therefore, elasticity is defined as the property by which a body regains its original position when the forces are removed. - The bodies which do not regain their original shape and size are called **plastic bodies**. - **Examples:** Steel, glass, quartz etc. - Nobody is either completely elastic or completely plastic. - The property of elasticity is different in different substances. - Steel is less elastic than rubber. - Liquids and gases are highly elastic. ## Stress - When a force _F_ is applied on a body, there will be relative displacement of the particles. - Due to the property of elasticity, the particles tend to regain their original position. - **Stress** is defined as the acting force per unit area. $Stress = F/A$. ## Strain The term **strain** refers to the relative change in dimensions or shape of a body that is subjected to stress. There are three types of strain: - Longitudinal strain. - Volume strain. - Shearing strain ### Longitudinal strain This Figure shows a bar of length _l_ that elongates to a length _l + Δl_ when equal and opposite forces _F_ are exerted at its ends. The **Longitudinal strain** is defined as the ratio of change in length to original length. $Longitudinal Strain = \frac{Δl}{l_0}$ ### Volume strain - This figure shows a block of volume _V_ that decreases to a volume _V - ΔV_ when a hydrostatic pressure acts on the faces of a block. - The **volume strain** defined as the ratio of change in volume to original volume. $Volume Strain = -\frac{ΔV}{V}$ ### Shearing strain **Shearing strain** is defined as the angle of shear measured in radians. The angle _θ_ measured in radians is called the **shearing strain = θ**. ## Elasticity and plasticity - When any stress is plotted against the appropriate strain, the resulting stress strain diagram is found to have a shape as given in the following figure: - The diagram is represented by a graph in which stress is plotted on y-axis and strain is plotted on x-axis. - The diagram shows that stress is proportional to strain up to the yield point, which is also called the **elastic limit**. - The area under the curve represents the work done for deforming the body. - Beyond the elastic limit, the body is deformed plastically, and stress is no longer proportional to strain. - The body is not completely elastic. - For most materials, stress continues to increase after the yield point until it reaches its ultimate yield strength, which is the maximum stress that the material can withstand before it starts to break. ## Elasticity of living cell materials - In many types of cells, the protoplasm has been pulled apart with micro needles until strands of it extended to great lengths. - On release of these strands of protoplasm, they snapped back to their original lengths, indicating that they are elastic. - Nuclei of cells have been stretched between micro needles and the indication is that nuclear material is elastic as long as it remains alive. - Experiments have been performed on individual chromosomes, it was found that, the chromosomes could be extended to five times their original length while they were in the nucleus, when removed from it, they could be stretched to twenty--five times their original length. - The red blood cell has also been stretched and has been observed to snap back into its original position. In this case, the elasticity is primarily that of the red cell membrane. ## Types of elasticity constant - **Young's modulus of elasticity (Y):** It is defined as the ratio of normal stress to longitudinal strain. $Y = \frac{Normal Stress}{Longitudinal Strain} = \frac{F/A}{Δl/l_0}$ - **Shear modulus (S):** It is defined as the ratio of tangential stress to shearing strain. $S = \frac{Tangential Stress}{Shearing strain} = \frac{F/A}{θ}$ - **Bulk modulus of elasticity (B):** It is defined as the ratio of normal stress to volume strain. $B = \frac{Normal Stress}{Volume Strain} = -\frac{F/A}{ΔV/V}$ Where P is the change in pressure, the minus sign is included in the definition of B because an increase of pressure always causes a decrease in volume. | Material | Young Modulus Y 10¹¹ N/m² | Shear Modulus η 10¹¹ N/m² | Bulk Modulus B 10¹¹ N/m² | |:---|:---:|:---:|:---| | Aluminium | 0.70 | 0.30 | 0.70 | | Brass | 0.91 | 0.36 | 0.61 | | Copper | 1.1 | 0.42 | 1.4 | | Iron | 1.9 | 0.70 | 1.0 | | Steel | 2.0 | 0.84 | 1.6 | | Tungsten | 3.6 | 1.5 | 2.0 | ## Hook's law Consider a spring as shown in Figure, the stress required to produce a given strain depends on the nature of the material under stress. **Hook's law** states that within the elastic limit, stress is proportional to strain. Stress α Strain Stress = constant x Strain Where the constant is called the modulus of elasticity. ## The force constant Hook's law states that "The elongation of a body is directly proportional to the stretching force”. In the particular case of longitudinal stress and strain, we find that: $F_1 = \frac{YA}{l}Δl$ Then the quantity $\frac{YA}{l}$ is represented by a single letter _k_, and the elongation _Δl_ is renamed _x_, we have: $F_1 = kx$ ## Work done in deforming a body When a body is deformed by the application of external forces, the body gets strained. The work done is stored in the body in the form of energy and is called the **energy of strain**. Work done per unit volume w' = W/V $W' = \frac{1}{2}x stress x strain $ ## Example: calculate the work done in stretching a uniform metallic wire of area of cross-section 10⁶ m² and length 1.5 m through 4 x 10^-3 m. Given Y = 2 x 10¹¹ N/m². Solution l = 1.5 m, A = 10^-6 m² , dl=4 x 10^-3 m, Y = 2 x 10¹¹ N/m² $strain = \frac{dl}{l} = \frac{4 x 10^{-3}}{1.5}$ $ stress = Y. strain = \frac {2 x 10^{11} x 4 x 10^{-3}}{1.5} N/m^2$ volume of the wire = Al = 10^-6 x 1.5 m³ $work done per unit volume = \frac{1}{2}. stress. strain $ $total work done W = \frac{1}{2}. stress. strain. volume$ $W = (\frac{2 x 10^{11} x 4 x 10^{-3}}{1.5} ). (\frac{4×10^{-3}}{1.5}). (1.5. 10^{-6}). = 1.066 J $ ## Poisson's ratio σ Whenever a body is subjected to a force in a particular direction, there is change in dimensions of the body in the other two perpendicular directions (i.e. secondary strain). **Poisson's ratio** is defined as the ratio of secondary strain per unit stress to the longitudinal strain per unit stress. $Longitudinal Strain = \frac{d(l)}{l}$ $ Secondary Strain = \frac{dr}{r}$ ## The Poisson’s ratio σ has no units as it is a ratio of two numbers. For most of the materials, the value of σ is 1/2. Prove that (σ = 1/2) - The initial volume of the wire is $V = πr^2l$ - If the volume of the wire remains unchanged (dV = 0) after the force has been applied, then dV = 0 - 0 = π(r²dl + 2r drl) - rdl = -2ldr $ σ = \frac{-dr/r}{dl/l} = \frac{1}{2} $ This is the maximum possible value of Poisson's ratio. ## Assignment A metallic rod of **diameter 6 mm**. Find the force which must be exerted on a rod to expand it by 20% from its original length. [Taking Young's modulus for the rod material is Y = 9x10¹¹ N.m^-2.] ## Thank you.