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Chapter 3: Plasticity. Tests for Mechanical Strength of Materials. Common tests used to determine the monotonic strength of metals. (a) Uniaxial tensile test. (b) Upsetting test. (c) Three-point bending test. (d) Plane-strain tensile test. (e) Plane-strain

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slide2

Tests for Mechanical Strength of Materials

Common tests used to determine the monotonic strength of metals. (a) Uniaxial tensile test.

(b) Upsetting test. (c) Three-point bending test. (d) Plane-strain tensile test. (e) Plane-strain

compression (Ford) test. (f)‏ Torsion test. (g) Biaxial test.

slide3

Mechanical Testing: Servohydraulic Machine

A servohydraulic

universal testing machine linked to

a computer. (Courtesy of MTS

Systems Corp.)‏

slide4

Stress-Strain Curves at Different Heat Treatments

Stress–strain curves for

AISI 1040 steel subjected to

different heat treatments; curves

obtained from tensile tests.

slide5

Uniaxial Stress-Strain Curve

Idealized shapes of

uniaxial stress–strain curve. (a)‏

Perfectly plastic. (b) Ideal

elastoplastic. (c) Ideal elastoplastic

with linear work-hardening. (d)‏

Parabolic work-hardening (σ =

σo + Kεn).

slide6

Plasticity

Ludwik-Hollomon equation

Voce equation

Johnson-Cook equation

slide7

True Stress and True Strain Curve

Schematic

representation of the change in

Poisson’s ratio as the deformation

regime changes from elastic to

plastic.

slide8

Stress-Strain Curve

True- and

engineering-stress–strain curves

for AISI 4140 hot-rolled steel. R.

A. is reduction in area.

slide9

Engineering Stress and Engineering Strain

Engineering- (or nominal-) stress–strain curves (a) without and (b) with a yield

point.

slide10

Tensile tests

Tensile specimen being tested; arrows show onset of necking.

slide11

Work hardening vs. Strain

Log dσ/dε versus log ε

for stainless steel AISI 302.

(Adapted with permission from A.

S. de S. e Silva and S. N. Monteiro,

Metalurgia-ABM, 33 (1977) 417.)‏

slide12

Necking

Correction factor for

necking as a function of strain in

neck, ln(A0/A), minus strain at

necking, εu. (Adapted with

permission from W. J. McGregor

Tegart, Elements of Mechanical

Metallurgy (New York: MacMillan,

1964), p. 22.)‏

Stress–strain curves for Fe–0.003% C alloy wire, deformed to increasing

strains by drawing; each curve is started at the strain corresponding to the prior

wire-drawing reduction. (Courtesy of H. J. Rack)‏

slide13

Strain Rate Effects

(a) Effect of strain rate

on the stress–strain curves for

AISI 1040 steel. (b) Strain-rate

changes during tensile test. Four

strain rates are shown: 10−1,

10−2, 10−3, and 10−4 s−1.

slide14

Plastic Deformation in Compressive Testing

(a) Compression

specimen between parallel platens.

(b) Length inhomogeneity in

specimen.

slide15

Stress-Strain Curve for Compression

(a) Stress–strain

(engineering and true) curves for

70–30 brass in compression. (b)‏

Change of shape of specimen and

barreling.

slide16

Finite Element Method

(a) Distortion of Finite Element Method (FEM) grid after 50% reduction in

height h of specimen under sticking-friction conditions. (Reprinted with permission from H. Kudo and S. Matsubara, Metal Forming Plasticity (Berlin: Springer, 1979),p. 395.) (b) Variation in pressure on surface of cylindrical specimen being

compressed.

slide17

Bauschunger Effect

Ratio of compressive

flow stress (0.2% plastic strain) and

tensile flow stress at different

levels of plastic strain for different

steels. (After B. Scholtes, O.

V¨ohringer, and E. Macherauch,

Proc. ICMA6, Vol. 1 (New York:

Pergamon, 1982), p. 255.)‏

The Bauschinger effect.

slide18

Plastic Deformation of Polymers

Schematic of the

different types of stress–strain

curves in a polymer.

Effect of strain rate

and temperature on stress–strain curves.

slide19

Glassy Polymers

Schematic of necking

and drawing in a semicrystalline polymer.

slide20

Neck Propagation in Polyethylene

(a) Neck propagation

in a sheet of linear polyethylene.

(b) Neck formation and

propagation in a specimen, shown in schematic fashion.

slide22

Plastic Deformation of Glasses

Compression

stress–strain curves for

Pd77.5CU6Si16.5. (Adapted with

permission from C. A. Pampillo and H. S. Chen, Mater. Sci. Eng., 13 (1974) 181.)‏

slide23

Shear Steps

Shear steps

terminating inside material after

annealing at 250◦C/h, produced by (a) bending and decreased by (b)‏

unbending. Metglas

Ni82.4Cr7Fe3Si4.5B3.1 strip. (Courtesy of X. Cao and J. C. M. Li.)‏

slide24

Dislocations

(a) Gilman model of

dislocations in crystalline and

glassy silica, represented by

two-dimensional arrays of polyhedra. (Adapted from J. J. Gilman, J. Appl. Phys. 44 (1973)‏

675) (b) Argon model of displacement fields of atoms (indicated by magnitude and

direction of lines) when

assemblage of atoms is subjected to shear strain of 5 × 10−2, in

molecular dynamics computation. (Adapted from D. Deng, A. S.

Argon, and S. Yip, Phil. Trans. Roy. Soc. Lond. A329 (1989) 613.)‏

slide25

Viscosity of Glass

Viscosity of

soda–lime–silica glass and of

metallic glasses (Au–Si–Ge,

Pd–Cu–Si, Pd–Si, C0P) as a

function of normalized

temperature. (Adapted from J. F.

Shakelford, Introduction to Materials

Science for Engineers, 4th ed.

(Englewood Cliffs, NJ: Prentice

Hall, 1991), p. 331, and F. Spaepen

and D. Turnbull in Metallic Glasses,

ASM.) 1P=0.1 Pa · s.

Viscosity of three

glasses as a function of

temperature. 1 P=0.1 Pa · s.

slide26

Rankine, Tresca, and von Mises

Maximum-stress Criterion

Maximum-Shear-Stress Criterion

Maximum-Distortion-Energy Criterion

slide27

Comparison of the Rankine, von Mises, and Tresca

(a) Comparison of the

Rankine, von Mises, and Tresca

criteria. (b) Comparison of failure

criteria with test. (Reprinted with

permission from E. P. Popov,

Mechanics of Materials, 2nd ed.

(Englewood Cliffs, NJ:

Prentice-Hall, 1976), and G.

Murphy, Advanced. Mechanics of

Materials (New York: McGraw-Hill,

1964), p. 83.)‏

slide28

Displacement of the Yield Locus

Displacement of the

yield locus as the flow stress of the

material due to plastic

deformation. (a) Isotropic

hardening. (b) Kinematic

hardening.

slide30

Mohr-Coulomb failure criterion

Griffith Failure Criterion

McClintock-Walsh Crtierion

slide31

Failure Criteria for Brittle Material

(a) Simple model for solid with cracks. (b) Elliptical flaw in elastic

solid subjected to compression loading. (c) Biaxial fracture

criterion for brittle materials initiated from flaws without (Griffith)‏

and with (McClintock and Walsh) crack friction.

slide32

von Mises Ellipse

Translation of von

Mises ellipse for a polymer due to

the presence of hydrostatic stress.

(a) No hydrostatic stress, (b) with

hydrostatic stress.

slide33

Shear Yielding and Crazing for Amorphous Polymer

Envelopes defining

shear yielding and crazing for an

amorphous polymer under biaxial

stress. (After S. S. Sternstein and L.

Ongchin, Am. Chem. Soc., Div. of

Polymer Chem., Polymer Preprints, 10

(1969), 1117.)‏

slide34

Failure Envelope

Failure envelope for unidirectional E-glass/epoxy composite under biaxial

loading at different levels of shear stress. (After I. M. Daniel and O. Ishai, Engineering Mechancis of Composite Materials (New York: Oxford University Press, 1994), p. 121.)‏

slide35

Plane-Stress Yield Loci for Sheets with Planar Isotropy

Plane-stress yield loci

for sheets with planar isotropy or

textures that are rotationally

symmetric about the thickness

direction, x3. (Values of R indicate

the degree of anisotropy =

σ2/σ1.)‏

slide37

Hardness Tests

Comparison of the impression sizes produced by various hardness tests on

material of 750 HV. BHN = Brinell hardness number, HRC = Rockwell hardness

number on C scale, HRN = Rockwell hardness number on N scale, VPN = Vickers

hardness number. (Adapted with permission from E. R. Petty, in Techniques of Metals

Research, Vol. 5, Pt. 2, R. F. Bunshah, ed. (New York: Wiley-Interscience, 1971), p. 174.)‏

slide38

Impression

Impression caused by

spherical indenter on metal plate.

slide39

Rockwell Hardness Tester

Procedure in using

Rockwell hardness tester.

(Reprinted with permission from

H. E. Davis, G. E. Troxel, and C. T.

Wiscocil, The Testing and Inspection

of Engineering Materials, (New

York: McGraw-Hill, 1941), p. 149.)‏

slide41

Vickers Hardness Test

Relationships Between Yield Stress and Hardness

slide42

Hardness Distance Profile

(a) Hardness–distance

profiles near a grain boundary in

zinc with 100-atom ppm of Al and

zinc with 100-atom ppm of Au

(1-gf load). (b) Solute

concentration dependence of

percent excess boundary

hardening in zinc containing Al, Au,

or Cu (3-gf load). (Adapted with

permission from K. T. Aust, R. E.

Hanemann, P. Niessen, and J. H.

Westbrook, Acta Met., 16 (1968)‏

291.)‏

slide43

Knoop Indenter

Some of the details of

the Knoop indenter, together with

its impression.

slide44

Nanoindenter apparatus

A schematic of a

nanoindenter apparatus.

slide45

Topographic Feature of the Berkovich Indentation

An impression made

by means of Berkovich indenter in

a copper sample. (From Deng,

Koopman, Chawla, and Chawla,

Acta Mater., 52 (2004) 4291.) (a)‏

An atomic force micrograph,

which shows very nicely the

topographic features of the

indentation on the sample surface.

The scale is the same along the

three axes. (b) Berkovich

indentation as seen in an SEM.

slide46

Load vs. Indentation Displacement

A schematic

representation of load vs. indenter

displacement.

slide47

Simple Formability Tests for Sheets

Simple formability

tests for sheets. (a) Simple bending

test. (b) Free-bending test. (c)‏

Olsen cup test. (d) Swift cup test.

(e) Fukui conical cup test.

slide48

Plastic Anisotropy

“Ears” formed in

deep-drawn cups due to in-plane

anisotropy. (Courtesy of Alcoa,

Inc.)‏

slide49

Fibering

Effect of “fibering” on formability. The bending operation is often an integral

part of sheet-metal forming, particularly in making flanges so that the part can be

attached to another part. During bending, the fibers of the sheet on the outer side of

the bend are under tension, and the inner-side ones are under compression. Impurities

introduced in the metal as it was made become elongated into “stringers” when the

metal is rolled into sheet form. During bending, the stringers can cause the sheet to fail

by cracking if they are oriented perpendicular to the direction of bending (top). If they

are oriented in the direction of the bend (bottom), the ductility of the metal remains

normal. (Adapted with permission from S. S. Hecker and A. K. Ghosh, Sci. Am., Nov.

(1976), p. 100.)‏

slide50

Punch-Stretch Test

Sheet specimen

subjected to punch–stretch test

until necking; necking can be seen

by the clear line. (Courtesy of S. S.

Hecker)‏

slide51

Punch-Stretch Test

Schematic of sheet

deformed by punch stretching. (a)‏

Representation of strain

distribution: ε1, meridional strains;

ε2, circumferential strains; h, cup

height. (b) Geomety of deformed

sheet.

slide52

Forming-Limit Curve

Construction of a

forming-limit curve (or

Keeler–Goodwin diagram).

(Courtesy of S. S. Hecker.)‏

slide53

Different Strain Patterns in Stamped Part

Different strain

patterns in stamped part. (Adapted

from W. Brazier, Closed Loop, 15,

No. 1 (1986) 3.)‏

slide55

Strength of Biological Materials

Stress–strain response

fore a number of biological

materials.

slide57

Stress-Strain Response of Elastin

Stress–strain response

for elastin; it is the ligamentum

nuchae of cattle (Adapted from Y.

C. Fung and S. S. Sobin, J. Biomech.

Eng., 1103 (1981) 121. Also in Y.

C. Fung, Biomechanics: Mechanica

properties of Living Tissues

(NewYork: Springer, 1993) p. 244.)‏

slide58

Stress-Strain Response of Cortical Bone

Tensile and

compressive stress–strain curves

for cortical bone in longitudinal

and transverse directions.

(Adapted from G. L. Lucas, F. W.

Cooke, and E. A. Friis, A Primer on

Biomechanics (New York: Springer,

1999).)‏

slide59

Strain Rate Response of Cortical Bone

Strain-rate

dependence of tensile response of

cortical bone. (Adapted from J. H.

McElhaney, J. Appl. Physiology,

21(1966) 1231.)‏