BEYOND ELASTICITY. Sub-topics. Plasticity Toughness Hardness Design problems

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1 BEYOND ELASTICITY Sub-topics 1 Plasticity Toughness Hardness Design problems

2 PLASTICITY The stress is no longer proportional to strain 2

3 STRESS-STRAIN BEHAVIOR: PLASTICITY Tensile strength is the stress at the maximum Plastic deformation: ostress and strain are not proportional othe deformation is not reversible odeformation occurs by breaking and rearrangement of atomic bonds 3

4 YIELDING The position of this point may not be determined precisely. a convention has been established wherein a straight line is constructed parallel to the elastic portion of the stress strain curve at some specified strain offset, usually Yield strengths may range from 35 Mpa for a low-strength aluminum to over 1400 Mpa for high-strength steels. 4

5 TENSILE STRENGTH What do you think is more important property for structural applications: yield stress or tensile strength? TS corresponds to the maximum stress that can be sustained by a structure in tension 5

6 WHAT IS MECHANICAL BEHAVIOR? 6

7 DUCTILITY The ductility of a material is its ability to deform under load and can be measured by either a length change or an area change. 7

8 TYPICAL MECHANICAL PROPERTIES OF SOME METALS 8

9 ENGINEERING STRESS STRAIN: SUMMARY 9

10 TOUGHNESS Ductility, as percent elongation Ductility, as percent reduction in area Brittle materials are approximately considered to be those having a fracture strain of less than about 5%. 10

11 ELASTIC STRAIN RECOVERY 11

12 PLASTICITY AND TEMPERATURE The magnitudes of both yield and tensile strengths decrease with increasing temperature; just the reverse holds for ductility it usually increases with temperature. 12

13 WHAT IS WRONG WITH STRESS-STRAIN CURVE?? It should be realized that the ε and σ are based on areas and lengths that no longer exist at the time of measurement. To correct this situation true stress and true strain quantities are used. A is the instantaneous area at the time of measurement True strain T, defined by 13

14 TRUE STRESS - STRAIN Necking begins at M, which corresponds to M on the true curve. In the elastic region the relation between stress and strain is simply the linear equation coefficients The correct stress within the neck is slightly lower than the stress computed from the applied load and neck cross sectional area. The corrected true stress strain curve takes into account the complex stress state within the neck region. 14

15 TABULATION OF N AND K VALUES FOR SEVERAL ALLOYS True stress-true strain relationship in plastic region of deformation (to point of necking) The parameter n is often termed the strain hardening exponent and has a value less than unity K and n are constants, which values will vary from alloy to alloy, and will also depend on the condition of the material (i.e., whether it has been plastically deformed, heat treated, etc.). 15

16 PROBLEM Calculate the modulus of elasticity of the aluminium alloy for which the engineering stress strain curve is shown in Figure. Calculate the length of a bar of initial length 125 cm when a tensile stress of 207 Mpa is applied

17 HOW ARE AMORPHOUS SOLIDS (GLASS) DEFORMED? These materials deform by viscous flow, the same manner in which liquids deform; the rate of deformation is proportional to the applied stress. In response to an applied shear stress, atoms or ions slide past one another by the breaking and reforming of interatomic bonds. However, there is no prescribed manner or direction in which this occurs, as with dislocations. The characteristic property for viscous flow, viscosity, is a measure of a noncrystalline material s resistance to deformation. 17

18 DEFORMATION OF CERAMICS? 18

19 PROBLEM The flexural strength of a ceramic material is 310 MPa, and the flexural modulus is x 10 3 MPa. A sample, which is 1.25 cm wide, cm high, and 20 cm long, is supported between two rods 12.5 cm apart. Determine the force required to fracture the material and the deflection of the sample at fracture, assuming that no plastic deformation occurs. 19

20 WHAT ARE POLYMERS? Linear Branched The degree of crystallinity may range from completely amorphous to almost entirely (up to about 95%) crystalline; by way of contrast, metal specimens are almost always entirely crystalline, whereas many ceramics are either totally crystalline or totally noncrystalline. Cross-linked Three factors that influence the degree of crystallinity are: i) Chain length ii) iii) Chain branching Interchain bonding Network (3D) 20 The mechanical properties are also governed by the structure of the polymer chains.

21 HOW PLASTIC ARE POLYMERS? Lamellar POLYETHYLENE Schematic representation of the detailed structure of a spherulite. (From John, C. Coburn, Dielectric Relaxation Processes in Poly(ethylene terephthalate, 1984.) 21

22 DEFORMATION OF POLYMERS F Two adjacent chain-folded lamellae and interlamellar amorphous material before deformation. Separation of crystalline block segments during the third stage Elongation of amorphous tie chains during the first stage of deformation. Orientation of block segments and tie chains with the tensile axis in the final 22 deformation stage.

23 STRESS STRAIN CURVE FOR A SEMI-CRYSTALLINE POLYMER A neck occurs in amorphous thermoplastics as the polymer chains align. Instead of being weaker because of the smaller cross-sectional area, the neck is stronger because the chains are aligned stronger van der Waal s bonds and the covalent bonds of the backbone 23

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25 TYPICAL STRESS STRAIN BEHAVIOUR FOR POLYMERS The mechanical characteristics of polymers are highly sensitive to the rate of deformation (strain rate), the temperature, and the chemical nature of the environment (the presence of water, oxygen, organic solvents, etc.) Brittle Plastic Highly elastic ELASTOMERS 25

26 HARDNESS Material s resistance to localized plastic deformation Measured by forcing an indenter into the surface Methods to characterize hardness can be divided into three primary categories: 1) Scratch Tests 2) Rebound Tests 3) Indentation Tests 26

27 INDENTATION HARDNESS TESTS H = F/A Indentation tests actually produce a permanent impression in the surface of the material. The force and size of the impression can be related to a quantity (hardness), which can be objectively related to the resistance of the material to permanent penetration. Because the hardness is a function of the force and size of the impression, the pressure (and hence stress) used to create the impression can be related to both the yield and ultimate strengths of materials. Several different types of hardness tests have evolved over the years. These include macro hardness test such as Brinell, Vickers, and Rockwell and micro hardness tests such as Knoop and Tukon. 27

28 BRINELL HARDNESS TEST The Brinell hardness number is obtained by dividing the applied force, P, in kg, by the actual surface area of the indentation which is a segment of a sphere diameter of the ball in mm indentation depth from the surface in mm 28

29 VICKERS HARDNESS TESTS a four-sided diamond pyramid is implied as an indenter H V σ y / 3 29

30 CORRELATION BETWEEN HARDNESS AND TENSILE STRENGTH The deformations caused by indenter can be correlated to those produced at the yield and ultimate tensile strengths in a tensile test. An important difference is that the material cannot freely flow outward, 30 so that a complex triaxial state of stress exists under the indenter.

31 Copyright 2007, 2010 by Michael Ashby, Hugh Shercliff and David Cebon. Published by Elsevier Ltd. All rights reserved.

32 WHAT ARE THE LIMITS OF SAFE DEFORMATION? DESIGN/SAFETY FACTORS The material to be used for the particular application must be chosen so as to have a yield strength at least as high as σ d 32

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