Mechanical properties of solid bodies Presentation

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Mechanical properties of solid bodies

Today’s Theme:

Learning Objectives

To define Young’s modulus under elastic deformation

Success Criteria

Students have achieved the objective if they are able to
Know the correct definitions for the three terms (stress, strain, and Young's modulus);
explain the relationship between the values (stress, strain, and Young's modulus);
solve problems using the Young's modulus formula;
Analyze the strain of wires of different cross-sections and lengths.

Vocabulary

Young modulus, stiff, stiffness, strong, weak, tough, brittle, ductile, ultimate tensile stress, intrinsic, extrinsic
Юнг модулі, қатты, қаттылық, күшті, әлсіз, қатты, морт, пластикалық, шектік созушы жүктеме, қатысты, сыртқы
Модуль Юнга, жесткая, жесткость, сильная, слабая, жесткая, хрупкая, пластичная, предельная растягивающая нагрузка, свойственный, внешний

Strain

The ratio of the extension of an object to its original length
Symbol: ε
𝜀𝜀= 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑒𝑒𝑥𝑥𝑡𝑡𝑒𝑒𝑛𝑛𝑠𝑠𝑖𝑖𝑜𝑜𝑛𝑛 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑜𝑟𝑟𝑖𝑖𝑔𝑔𝑖𝑖𝑛𝑛𝑎𝑎𝑙𝑙 𝑙𝑙𝑒𝑒𝑛𝑛𝑔𝑔𝑡𝑡ℎ 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ
Since strain is the ratio of two lengths, it does not have a unit.

Stress

The strain in an object is produced by stress or tensile stress.
Symbol: σ
𝜎𝜎= 𝑓𝑜𝑟𝑐𝑒 𝑎𝑟𝑒𝑎 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑜𝑟𝑐𝑒 𝑓𝑓𝑜𝑜𝑟𝑟𝑐𝑐𝑒𝑒 𝑓𝑜𝑟𝑐𝑒 𝑎𝑟𝑒𝑎 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑜𝑟𝑐𝑒 𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎 𝑛𝑛𝑜𝑜𝑟𝑟𝑚𝑚𝑎𝑎𝑙𝑙 𝑡𝑡𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑓𝑓𝑜𝑜𝑟𝑟𝑐𝑐𝑒𝑒 𝑓𝑜𝑟𝑐𝑒 𝑎𝑟𝑒𝑎 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑜𝑟𝑐𝑒 = 𝑓𝑜𝑟𝑐𝑒 𝑐𝑟𝑜𝑠𝑠−𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑓𝑓𝑜𝑜𝑟𝑟𝑐𝑐𝑒𝑒 𝑓𝑜𝑟𝑐𝑒 𝑐𝑟𝑜𝑠𝑠−𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑐𝑐𝑟𝑟𝑜𝑜𝑠𝑠𝑠𝑠−𝑠𝑠𝑒𝑒𝑐𝑐𝑡𝑡𝑖𝑖𝑜𝑜𝑛𝑛𝑎𝑎𝑙𝑙 𝑎𝑎𝑟𝑟𝑒𝑒𝑎𝑎 𝑓𝑜𝑟𝑐𝑒 𝑐𝑟𝑜𝑠𝑠−𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎
Unit: N m-2 or Pa

Stress v. strain graphs

It is often useful to create a graph of stress v. strain.
Because stress is related to force and strain is related to extension, a stress v. strain graph will have the same basic shape as a force v. extension graph.
There is a straight line region between the origin and the proportionality limit P. In this region, we can say that 𝑠𝑠𝑡𝑡𝑟𝑟𝑒𝑒𝑠𝑠𝑠𝑠∝𝑠𝑠𝑡𝑡𝑟𝑟𝑎𝑎𝑖𝑖𝑛𝑛

The Young modulus

𝑠𝑠𝑡𝑡𝑟𝑟𝑒𝑒𝑠𝑠𝑠𝑠∝𝑠𝑠𝑡𝑡𝑟𝑟𝑎𝑎𝑖𝑖𝑛𝑛
in equation form, 𝑠𝑠𝑡𝑡𝑟𝑟𝑒𝑒𝑠𝑠𝑠𝑠=𝐸𝐸×𝑠𝑠𝑡𝑡𝑟𝑟𝑎𝑎𝑖𝑖𝑛𝑛
This constant E is known as the Young modulus of the material
𝐸𝐸= 𝑠𝑡𝑟𝑒𝑠𝑠 𝑠𝑡𝑟𝑎𝑖𝑛 𝑠𝑠𝑡𝑡𝑟𝑟𝑒𝑒𝑠𝑠𝑠𝑠 𝑠𝑡𝑟𝑒𝑠𝑠 𝑠𝑡𝑟𝑎𝑖𝑛 𝑠𝑠𝑡𝑡𝑟𝑟𝑎𝑎𝑖𝑖𝑛𝑛 𝑠𝑡𝑟𝑒𝑠𝑠 𝑠𝑡𝑟𝑎𝑖𝑛 = 𝐹 𝐴 𝑥 𝐿 0 𝐹 𝐴 𝐹𝐹 𝐹 𝐴 𝐴𝐴 𝐹 𝐴 𝐹 𝐴 𝑥 𝐿 0 𝑥 𝐿 0 𝑥𝑥 𝑥 𝐿 0 𝐿 0 𝐿𝐿 𝐿 0 0 𝐿 0 𝑥 𝐿 0 𝐹 𝐴 𝑥 𝐿 0 = 𝐹 𝐿 0 𝐴𝑥 𝐹𝐹 𝐿 0 𝐿𝐿 𝐿 0 0 𝐿 0 𝐹 𝐿 0 𝐴𝑥 𝐴𝐴𝑥𝑥 𝐹 𝐿 0 𝐴𝑥
Where
F is the tensile force exerted on the solid
A is the cross sectional area of the solid
x is the extension of the solid
L0 is the original length of the solid
Units: N m-2 or Pa
The measure of a stiffness of a material
Note: Because the Young modulus is a proportionality constant, it can only be calculated from the linear part of a stress v. strain graph.

Intrinsic v. Extrinsic properties

Because the units of Young’s modulus is Pa, it is independent of the shape of a material (as cross-sectional area increases, so does the force needed to extend it a given amount)
Young’s modulus is independent of the size or shape of the material
An intrinsic property
By comparison, the spring constant k will change based on the size and shape of the material
Steel springs will have different spring constants as the thickness of the wire, size of coils, etc. changes.
An extrinsic property

Experimental work – Estimating the Young modulus

Follow the instructions on the paper

Ultimate Tensile Stress (UTS)

Definition: the maximum stress that an object can experience without breaking
The maximum stress on a stress – strain graph

Example:

Descriptors of materials

Definition: has a large ultimate tensile stress (difficult to break)
Example: iron

Definition: has a small ultimate tensile stress (breaks easily)
Example: clay

Descriptors of materials

Definition: Undergoes very little plastic deformation before breaking
Example: Glass

Definition: Undergoes a large plastic deformation before breaking
Example: Copper

Descriptors of materials

Definition: Absorbs a lot of energy before breaking (i.e. the area under the graph is large)
Example: steel

Summary reflection

Reflection: Students will answer the following questions with a thumbs up/middle/down
1) I know the definitions for stress, strain, and Young’s modulus
2) I can explain the relationship between stress, strain, and Young’s modulus.