The document discusses the stress-strain relationships and essential elastic material properties such as Young's modulus, Poisson's ratio, and Hooke's law. It provides examples and calculations for the deformation of materials under various loads, emphasizing the importance of considering deformations in design to prevent structural failures. Additionally, it highlights the relationships between stress, strain, and volumetric changes in materials.

1. Section 3 Stress-Strain Relationships The relationship between stress and deformation will be covered in this section, and some of the important elastic material properties such as Young’s modulus and the modulus of rigidity will be defined. © Loughborough University 2010. This work is licensed under a Creative Commons Attribution 2.0 Licence .

2. Contents Introduction Young’s Modulus Deformation of a Bar Young’s Modulus for Bars in Compression Poisson’s Ratio Hooke’s Law Shear Stress-Strain Relationship Volumetric (Dilatational) Strain Bulk Modulus Credits & Notices

3. Introduction Stress values do not always provide the limiting factor in design Deformations that accompany stress need to be considered Large deformations can lead to structural failure Deformations occur for a variety of reasons External loads Change in temperature Irradiation effects There are many modes of deformation Bending, twisting, compression, shear… Stress and deformation (strain) are related, and this relationship needs to be considered when solving certain structural problems

4. Young’s Modulus In the elastic range, a stress-strain curve is linear – i.e. the slope of the stress-strain curve is constant This constant is defined as the modulus of elasticity or Young’s modulus

5. Deformation of a Bar Using the definitions of stress and strain in the expression for Young’s modulus: The deformation of the bar with known geometry can be determined for the prescribed external load

6. Example 3.1 A 200mm long aluminium alloy strip has a cross section 25mm wide by 2mm thick, and a Young’s modulus of 70 GPa. If a tensile load of 2kN is applied to the strip, how much will it elongate? Answer:  =0.11 mm

7. Young’s Modulus for Bars in Compression All of the discussion so far also apply to the case of an axial compression of a bar For ductile materials such as aluminium alloy and mild steel, E t  E c For cast iron, ceramics, and thermosetting polymers, E t << E c For fibre-reinforced composites, E t > E c

8. Example 3.2 A mild steel shaft of 1.5m length, 100mm diameter, and with a Young’s modulus of 215 GPa is designed to have a maximum elongation of 1mm under load. What is the maximum permissible tensile load on the shaft? Answer: P=1126 kN

9. Poisson’s Ratio When we dealt with the axial elongation of a slender bar in tension, its lateral contraction was NOT considered In the linear elastic range, the ratio of the axial elongation to the lateral contraction is constant and is called Poisson’s ratio: For homogeneous and isotropic materials, Poisson’s ratio is the same in all directions Poisson’s ratio is a dimensionless quantity L L ′  x /2 P x P x x y

10. Example 3.3 A thin titanium rod 100mm long and 2mm in diameter has a Young’s modulus of 110GPa and Poisson’s ration of 0.314. Determine the lateral contraction of the rod under a tensile load of 4.4kN. Answer:  y =-0.4%

11. Hooke’s Law Hooke’s Law – linear relationship between components of stress and strain Consider a 3-dimensional block subjected to a uni-axial stress  x The three orthogonal strains produced in the block are  x , -  x , -  x For a tri-axial state of stress: x y z

12. Shear Stress-Strain Relationship Similar to Hooke’s law in tension / compression, shear stress and strain in the x-y plane is related by the shear modulus of elasticity or modulus of rigidity Similarly for the other two shear planes, we have For homogeneous and isotropic materials, all three shear modulii are the same For pure shear (i.e. normal stresses are not present on the relevant planes) the shear modulus is related to the Young’s modulus by:

13. Example 3.4 A rectangular piece of foam sandwich panel 30mm wide is formed by bonding two carbon-epoxy skins either side of the foam which has a modulus of rigidity of 60 MPa. The other dimensions of the panel are given in the figure below. The two skins are assumed to be macroscopically isotropic and homogenous and are almost rigid when compared to the foam. The lower skin is rigidly fixed, and a horizontal force P is applied to the upper skin as shown. If the upper skin moves through 0.5mm under the action of the load, determine (a) the average shear strain in the foam and, (b) the force exerted on the upper skin Answer: (a)  xy =0.72º (b) P=45 kN 150mm 40mm P P y x 0.5mm

14. Volumetric (Dilatational) Strain Consider a unit block of dimensions 1  1  1 If the block is subject to strains  x ,  y and  z in the associated three orthogonal directions, its dimensions increase to: Thus, the new volume is: Neglecting higher orders of strain And the volumetric strain is therefore:

15. Bulk Modulus Consider a block subjected to equal stresses on all six faces (i.e. a block submersed in a fluid at great depth) The hydrostatic stress / pressure is negative since it tends to compress the block. From Hooke’s Law: The volumetric strain is therefore: The Bulk modulus is defined as the ratio of the hydrostatic stress to the volumetric strain , therefore

16. This resource was created by Loughborough University and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme. © 2010 Loughborough University. Except where otherwise noted this work is licensed under a Creative Commons Attribution 2.0 Licence . The name of Loughborough University, and the Loughborough University logo are the name and registered marks of Loughborough University. To the fullest extent permitted by law Loughborough University reserves all its rights in its name and marks, which may not be used except with its written permission. The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence. All reproductions must comply with the terms of that licence. The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher. Credits