Mechanical properties 1 Elastic behaviour of materials


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1 MME131: Lecture 13 Mechanical properties 1 Elastic behaviour of materials A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Deformation of material under the action of a mechanical force Concept of stress and strain The stressstrain behaviour of materials Elastic behaviour of materials References: 1. Callister. Materials Science and Engineering: An Introduction 2. Askeland. The Science and Engineering of Materials Lec 13, Page 1/15
2 Mechanical properties of materials From an application standpoint, one of the most important topics within Materials and Metallurgical Engineering is the study of how materials respond to external loading or deformation. Most components, even if used primarily for other property (electronic properties, for example) have to fulfill certain mechanical functions as well. Important mechanical properties are: strength, hardness, stiffness and ductility. Laboratory testing to measure mechanical properties attempts to replicate the service conditions. Consistency is accomplished by using standardised test, so people are measuring same thing in the same way American Society for Testing Materials (ASTM) maintains and updates standards for mechanical properties. Several other standards organizations exist, e.g. SAE, ANSI, BS, ISO, JS, DIN... What happens to material when it is loaded with a mechanical force? pulling or stretching squeezing or squashing sliding twisting Material deforms, either elastically or plastically, depending on the magnitude of the force applied. xsectional area reduced due to tensile deformation Lec 13, Page 2/15
3 Elastic deformation Initial state Small load applied Load removed bond stretch return to initial state d F Elastic means reversible!! This happens when strains are small (~0.5%) (except for the case of polymers) F Linear elastic Nonlinear elastic d Plastic deformation At lower temperatures, T < T m /3 1. Initial state 2. Large load load applied 3. Unload Load removed bond stretch and planes sheared planes still stretched F d e+p d p F d e linear elastic d p d e d Plastic means permanent!! Lec 13, Page 3/15
4 The concept of stress and strain The mechanical behaviour of material under applied force may be ascertained by a simple stress strain diagram or, load deformation diagram Stress  Force or load per unit area of crosssection over which the force or load is acting Strain  Change in dimension (elongation) per unit length Stress and strain are considered positive for tensile loads, negative for compressive loads One of the most commonly performed mechanical stressstrain test is known as the tensile test. The machine Two categories of machines are available: Screwdriven: allows selection and control of the strain rate (de/dt) Hydraulically driven: allows selection and control of the loading rate (ds/dt) The sample 505 bar Nickname for the ASTM standard specimen most commonly used in tensile testing; a cylindrical specimen, 0.505" dia. along 2" gauge length (i.e., the length of the straight section between threaded ends). This diameter gives a convenient 0.20 in 2 crosssectional area. Tensile testing Lec 13, Page 4/15
5 The material s response to the applied tensile or compressive load is a change in length. During tension test, instantaneous applied load/force (F) and elongation or deformation (d) data are recorded, and the output of test is given as F d chart We can monitor very precisely the applied load using a load cell and the change in length (d) with an extensometer. F  d characteristics are dependent on the size of specimen For example, for a doubled crosssectional area, to generate the same elongation, the load must be doubled. To minimise these geometric factors, load and elongation parameters are normalised to the respective parameters of engineering stress and engineering strain. Engineering stress Tensile stress, s Shear stress, t s = F t A o original area before loading t = F s A o Stress units: Pa (N/m 2 ) or psi (lb/in 2 ) Lec 13, Page 5/15
6 Engineering strain Tensile strain : e = d L o d/2 w o L o Lateral strain : e L = d L w o d L /2 d/2 d L /2 q/2 Shear strain : g = tan q p/2  q Strain is always dimensionless p/2 q/2 Lec 13, Page 6/15
7 stress, s The stress strain behaviour s uts s y s p s p proportional limit max. stress in linear region s y yield strength, or proof stress stress that initiate permanent deformation or results in a specific amount ( 0.2 or 0.35%) of permanent strain Slope = E e f s uts ultimate tensile strength max. engineering stress on curve e f elongation or strain to failure total strain at break 0.2 % strain, e E modulus of elasticity slope of curve in linear region the stressstrain curve for an aluminum alloy Lec 13, Page 7/15
8 Tensile stressstrain curves for different materials. Note that these are qualitative. Properties obtained from the tensile test Elastic limit Tensile strength, Necking Hooke s law Poisson s ratio Modulus of resilience Tensile toughness Ductility Lec 13, Page 8/15
9 In the elastic region Initially, stress and strain are directly proportional to each other atoms can be thought of as masses connected to each other through a network of springs In tensile test, if the deformation is elastic, the stressstrain relationship follows the Hooke s law: s = E e E is known as the Young s modulus, the modulus of elasticity, or simply the modulus E has the same unit as those of stress, MPa or psi, although GPa (10 9 Pa) is commonly used. E is a measure of : bond strength (on the atomic level) intrinsic stiffness of material Very stiff materials : Ceramics, steels, W. Medium stiff materials : Cu, Al,. Low stiff materials : Plastics,. E is decreased with increasing T Young s modulus of elasticity For single phase (or, nearly single phase) materials, E is insensitive to : degree of plastic deformation microstructure (i.e., grain size, inclusion) Hooke s law applied for only a small value of e (typically < ~ %) ceramic materials follow Hooke s law up to fracture In the elastic region, E does not vary with the applied stress, i.e., E E(s) Lec 13, Page 9/15
10 In atomic scale, the microscopic elastic strain can be related to the small changes in the interatomic spacing and the stretching of interatomic bonds. Magnitude of modulus of elasticity is a measure of the resistance of separation of adjacent atoms. This is proportional to the slope of interatomic force separation curve at the equilibrium spacing, r 0, i.e., E df dr r 0 Lec 13, Page 10/15
11 Comparison of Young s modulus of elasticity Ceramics > Metals >> Polymers Example Design of a Suspension Rod An aluminum rod is to withstand an applied force of 45,000 pounds. To assure a sufficient safety, the maximum allowable stress on the rod is limited to 25,000 psi. The rod must be at least 150 in. long but must deform elastically no more than 0.25 in. when the force is applied. Design an appropriate rod. SOLUTION Using the definition of engineering stress, the required crosssectional area of the rod F A 0 = = (45000 lbs) / (25000 psi) = 1.8 in s 2 p d A 0 = 2 = 1.8 in 2 or d = 1.51 in 4 Lec 13, Page 11/15
12 Stress, s However, the minimum length or rod is specified as 150 in. To produce a longer rod, we might make the crosssectional area of the rod larger. The minimum strain allowed for the 150 in rod is Dl 0.25 in e = = = in/in 150 in l 0 Now, using the Hook s law s = E e = (10x10 6 psi) ( in/in) = psi Then, the area required to withstand this stress F lbs A 0 = = = 2.70 in 2 s psi Thus, in order to satisfy both the maximum stress and the minimum elongation requirements, crosssectional area of the rod must be at least 2.7 in 2, or a minimum diameter of 1.85 in. Nonlinear elastic stressstrain behaviour The stressstrain curve does not follow linear relation in the elastic limit Ds De Tangent modulus (at any stress s 2 ) Common materials showing such behaviour: grey cast iron concrete polymers In such cases, instead of Young s modulus, either a Tangent modulus or a Secant modulus is used. s 2 s 1 Ds De Secant modulus (between origin and any stress s 1 ) Strain, e Lec 13, Page 12/15
13 Poisson s ratio When pulled in tension (along zdirection), a sample gets longer (along zdirection) and thinner (contraction along x and ydirection). If compressed, sample gets fatter. Poisson s ratio defines how much strain occurs in the lateral x and ydirections when strained in the zdirection. e n =  x e =  z e y e z Theoretical value for isotropic material: 0.25 Maximum value: 0.50, Typical value: Metals: ~ 0.33, Ceramics: ~ 0.25, Polymers: ~ 0.40 Many materials are elastically anisotropic. y z d 0 d i x Problem A tensile stress is to be applied along the long axis of a cylindrical brass rod that has a diameter of 10 mm. Determine the magnitude of the load required to produce a 2.5x103 mm change in diameter if the deformation is entirely elastic. Data for brass: n = 0.34, E = 97 Gpa. z e x = Dd d 02.5x103 mm = =  2.5x10 10 mm 4 x d 0 = 10 mm Dd = 2.5x103 mm e z =  e x n s = e z E 2.5x104 =  = 7.35x = (7.35x104 ) (97x10 3 MPa) = 71.3 MPa Lec 13, Page 13/15
14 Other elastic properties Anelasticity The anelastic deformation is time dependent. Upon release of load, strain is not totally recovered. the time dependent microscopic and atomistic process occur during this stage For most metals, anelastic behaviour is negligible. For some polymers, its magnitude is significant, and this behaviour is then called viscoelastic behaviour. Lec 13, Page 14/15
15 Next Class MME131: Lecture 14 Mechanical properties 2 Plastic behaviour of materials Lec 13, Page 15/15
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