3. Elastic Constants.pptx

mechanics of solid
DR salahuddin

1. Dr. Salah Uddin, PhD (Geotech), (Univ of Nottingham, UK) ME (Structure), (NED UET Karachi, Pakistan) BE (Civil), (Balochistan UET Khuzdar, Pakistan) Associate Professor Email: [email protected] WhatsApp: [phone] Mechanics of Solids-I Relationship between Elastic Constants

2. Topics Contents Stress, Strain and Mechanical Properties of Materials Uniaxial state of stress and strain, Relationships between elastic Constants Response of materials under different sets of monotonic loading (including impact), Normal and shearing stress and strains, Distribution of direct stresses on uniform and non-uniform members, Thermal stresses and strains Serial No. of lectures: 01-08 (Total Classes: 08)

3. Goal of the lecture To understand and learn elastic constants and their relationship

4. Elasticity CLO-1 (PLO-1) Elasticity , ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed. A body with this ability is said to behave (or respond) elastically There is a limit to the magnitude of the  force  and the accompanying deformation within which elastic recovery is possible for any given material. This limit, called the  elastic limit , is the maximum  stress  or force per unit area within a solid material that can arise before the onset of permanent deformation.

5. Elastic Constants Elastic constants  are the constants describing mechanical response of a material when it is elastic. Elastic constants  measure the  proportionality  between strain and stress If this mechanical response is linear , one can define a set of constants that relate any applied stress to the corresponding strain

6. Young’s Modulus For a simple tension or compression test, the easiest  elastic constant  to define is  Young's modulus , E. Young's modulus is the elastic constant defined as the  proportionality constant  between stress and strain: The Young's modulus is the slope of the linear elastic response for a number of materials   Young’s Modulus is the ability of any material to resist the change along its length Modulus of elasticity is often called Young’s modulus, after another English scientist, Thomas Young (1773–1829).

7. Bulk Modulus When a body is subjected to mutually perpendicular direct stresses which are alike and equal, within its elastic limits, the ratio of direct stress to the corresponding volumetric strain is found to be constant. This ratio is called bulk modulus and is represented by letter “K”. Unit of Bulk modulus is MPa .  

8. Shear Modulus A type of stress called a shear stress,  , can also be defined which can produce a shear strain,  .     Just as for normal stresses, shear loading can result in a mechanical response that is elastic and nearly linear, but the shape change is called a shear strain.

9. The elastic constant that describes the  linear relation  between   and   is called the  shear modulus , G.      Shear Modulus G is the shear modulus of elasticity (also called the modulus of rigidity).

10. Poisson’s Ratio Another elastic constant that can be obtained in a tension or compression test wherein the strain along the loading direction and orthogonal to it is called  Poisson's ratio , ν. When a prismatic bar is loaded in tension, the axial elongation is accompanied by lateral contraction (that is, contraction normal to the direction of the applied load).

11. Poisson’s Ratio Lateral contraction is easily seen by stretching a rubber band, but in metals the changes in lateral dimensions (in the linearly elastic region) are usually too small to be visible. However, they can be detected with sensitive measuring devices. https://extrudesign.com/poisson-ratio-definition/

12. Poisson’s Ratio The lateral strain e at any point in a bar is proportional to the axial strain at that same point if the material is linearly elastic. The ratio of these strains is a property of the material known as Poisson’s ratio. This dimensionless ratio, usually denoted by the Greek letter (nu), can be expressed by the equation     The minus sign is inserted in the equation to compensate for the fact that the lateral and axial strains normally have opposite signs .

13. Relationship between Elastic Constants   This relationship, will be derived later in our discussion of Torsion, shows that E, G, and are not independent elastic properties of the material.   The relationship between Young’s modulus (E), rigidity modulus (G) and Poisson’s ratio ( ) is expressed as  

14. Relationship between Elastic Constants The relationship between Young’s modulus (E), bulk modulus (K) and Poisson’s ratio ( ) is expressed as :     Young’s modulus can be expressed in terms of bulk modulus (K) and rigidity modulus (G) as :  

15. Poisson’s ratio can be expressed in terms of bulk modulus (K) and rigidity modulus (G) as : Relationship between Elastic Constants  

16. Thanks