Chap02

Materials

Chapter 2

Mechanical DesignAn Integrated Approach

大葉大學機械與自動化工程學系 2/52

2.1 Introduction

• Material– Crystalline A material is made up of a number of small units called

crystals or grains.– Noncrystalline

• Assumption in this text– Homogeneous– Isotropic

• Commonly used engineering materials

2.2 Material Property Definitions

• Mechanical properties of material– What are the mechanical properties?– How to get the mechanical properties?

• Material– Isotropic– Orthotropic– Anisotropic

• Material– Ductile material mild steel, nickel, brass, copper, magnesium, ----– Brittle material concrete, stone, cast iron, glass, ---

• The factor considered in the selection of a material in design– Functional

• Strength• Stiffness• Durability• Machinability• Workability• Malleability• Hardness

– Economic• Initial cost• Maintenance cost• Replacement cost

2.3 Static Strength• Tension test

– A specimen subjected to tensile loading, applied slowly or statically at room temperature.

– To provide basic information about overall response of specimens to the applied loads.

– To obtain a stress-strain diagram.

• Stress-strain diagram– The stress is found by dividing the force by the cross-

sectional area, and the strain is found by dividing the elongation by the gage length.

– The stress-strain diagram differ widely for different materials.

• Stress-Strain Diagrams for Ductile Materials– Conventional or engineering stress-strain diagram (OABCDE)– True stress-strain diagram (OABCF)

• Yield Strength– Elastic range (OA)– Proportional limit (Sp)– Yield point (yield strength) (Sy)– Perfectly plastic (BC)

• Strain Hardening (Cold working)– Range CD– An increase in stress is required for a continued increase in

strain

• Unloading at point g in region CD– Retain to the point h along line gh, and line gh // line OA– Permanent set (Oh)

• Reloading at point h– Stress-strain curve (hgDE)– New yield point (g)– The ductility is reduced after unloading and then reloading

process

• Ultimate Tensile Strength (Su)• Fracture Strength (Sf)

• Necking (DE)• Strain(OA) << Strain(BC)

• Definition of ductility of a material

Percent elongation= (100)

Percent reduction in area= (100)

Lf - L0

A0- Af

WhereA0 , L0 : Original cross-sectional area and gage length of the specimen

Af , Lf : Final cross-sectional area and gage length of the specimen

• Offset Yield Strength

– 0.2% offset methodDetermine the yield strength for certain materials which don’t show a distinctive yield point.

• Stress-Strain Diagram for Brittle Materials– No well-defined linear region.– No difference between the ultimate strength and the fracture

strength.– The strain at the rupture is much smaller.– Rupture occurs with no noticeable prior change in the rate of

elongation.– The fracture is associated with the tensile stresses.

• Stress-Strain Diagrams in Compression– For most ductile materials, the yield strength is about the

same in tension and compression (Even material).

– For brittle material, the stresses in compression are much greater than in tension (Uneven material).

2.4 Hooke’s Law and Modulus of Elasticity• Hooke’s Law ( the stress-strain relation)

where E : elastic modulus or Young’s modulus

– The law is valid only up to the proportional limit of the material.

– The E is the slope of the stress-strain curve in the linearly elastic range.

– The E represents the stiffness of material in tension or compression.

– Tangent modulus Et = d /d

Secant modulus Es = /

• Hooke’s Law for shear stress and shear strain

where G : shear modulus of elasticity

• Poisson’s ratio lateral strain = - axial strain– Valid only for a uniaxial state of stress

– The lateral strain is of sense opposite to that of the axial strain.

– For most materials, = 0.25 ~ 0.35.– For most materials, < 0. 5 .

2.5 Generalized Hooke’s Law

1 2 yxxE

• For a two-dimensional state of stress

1 2 xyyE

)]([1 zyxx E

)]([1 zxyy E

• For a three-dimensional state of stress

• The relation of E, G and

)]([1 yxzz E

– For an isotropic materials, they are only two independent elastic constants.

• Dilatation

Where V : the change in volume V0 : original volume

– e represents the unit change in volume– The shear strains cause no change in volume.– For incompressible materials, e=0.

zyxVVe

• Bulk modulus of elasticity

– K represents the modulus of volumetric expansion

• The stress-strain relationships may be expressed as

eG xx 2

eG yy 2

xyxy G

eG zz 2

yzyz G

xzxz G

)(21zyxzyx E

)21)(1(

Ex. 2.1• A steel panel of a device is approximated by a plate of

thickness t, width b, length a, subjected to stress x and y Calculate:– The value of x for which length a remains unchanged– The final thickness t’ and b’ – The normal strain for the diagonal AC

Given: a=400 mm, b=300 mm, t=6 mm, E=200 GPa, =0.3, y= 220 MPa

Assumptions: The plate is in plane state of stress

Solution:1. length a remains unchanged : x= 0 2. plane state of stress : z= 0

)(1 xyy E

)( yxz E

(a) x = 0.3(220106)=66 MPa

0)(1 yxx E

1001)]66(3.0220[)10(200

429)22066()10(200

3.03 z

t’ = t(1+z ) = 6(0.9996)=5.998 mm

b’ = b(1+y ) = 300(1.001)=300.3 mm

(c) AC = (3002+4002 )1/2 = 500 mm

A’ C’ = (300.32+4002 )1/2 = 500.18 mm

AC = (500.18-500)/500=360

)(1 xyy E

)( yxz E

2.6 Thermal Stress-Strain Relations

TE yxx )(1

TEEyxx

• For a two-dimensional state of stress

xyxy G

TE xyy )(1

TEExyy

T : the temperature change

: the coefficient of expansion

• Thermal stress-strain relation– Because free thermal expansion causes no distortion in an

isotropic material, the shear strain is uneffected. (均勻熱膨脹不影響剪應變 )– In statically determinate structures, a uniform temperature

change will not cause any stresses, as thermal deformations are permitted to occur freely.

(靜定結構，均勻溫昇不產生應力 )– A temperature change in a structure supported in a statically

indeterminate manner induces stresses in the members. (靜不定結構，均勻溫昇產生應力 )

2.7 Temperature and Stress-Strain Properties• Short-Time Effects of Elevated and Low Temperatures

– The ultimate strength, yield strength and modulus of elasticity are lowered with increasing temperature.

– At low temperature, there is an increase in ultimate strength, yield strength, modulus of elasticity, and hardness and a decrease in ductility for metal.

AISI 304 stainless steel

• Long-Time Effects of Elevated Temperatures : Creep– Creep : most metals under a constant load at elevated

temperatures over a long period develop additional strain.

– Creep is time dependent because deformation increases with time until a rupture occurs.

– Creep deformation must be maintained small.

– For some applications and within certain temperatures, stress, and time limits, creep effects need not be considered.

• Creep curve for structural steel – 1st region: the material is becoming stronger because of

strain hardening, and creep rate decreases continuously.

– 2nd region: strain rate remains constant because of the balancing effects of strain hardening and annealing .

– 3rd region: the annealing effect predominates, and the deformation occurs at an accelerated creep rate until a rupture results.

2.8 Moduli of Resilience and Toughness• Modulus of Resilience(彈力 )

– Resilience: the capacity of a material to absorb energy within the elastic range.

– Modulus of resilience (Strain energy density): the energy absorbed per unit volume of material. The area under the straight-line portion of the stress-strain diagram

U yr 2

where Ur: Modulus of resilience; Sy: yield strength E: Modulus of elasticity

• Modulus of Toughness(強健 )– Toughness: the capacity of a material to absorb energy

without fracture.– Modulus of toughness: the energy absorbed per unit volume

of material up to the point of fracture. The area under the stress-strain diagram

f dU t

where Ut: Modulus of toughness; f : strain at fracture

– Toughness of a material is related to its ductility as well as to its ultimate strength.

– The modulus of toughness is approximated by

this equation is not suitable for brittle material.

– For brittle material, the modulus of toughness is approximated by

where Su: Ultimate strength; u: strain at the ultimate strength

– Fracture toughness: the ability of a material to resist at the tip of a crack.

uut SU 32

Ex. 2.2• A prismatic round steel rod must acquire an elastic strain

energy of Uapp = 200 in.lb. Determine the required yield strength Sy for a factor of safety of n=2.5 with respect to permanent deformationGiven: E=30106 psi, d=7/8 in., L= 4 ft.

Solution: 1. The volume of the member

2. Strain energy U=nUapp=2.5(200)=500 in.lb 3. Strain energy density Ur=U/V=500/28.9=17.3 lb/in3

U yr 2

)1030(2

3.17 6

32 .9.28)124()8/7(4

Sy = 32.2 ksi

2.9 Dynamic and Thermal Effects: Brittle-Ductile Transition

• Transition temperature (Tt): the temperature at a material’s behavior changes from ductile to brittle.

– Failure occurs by fracture at T < Tt

– At temperature close to Tt, the material generally exhibits some yielding prior to a partially brittle fracture.

• The transition temperature is defined as the temperature at which there is a sudden decrease in impact toughness.

• The principle factors governing whether failure occurs by fracture or yielding

– Temperature: If the temperature increase ( T > Tt), the specimen yields. If the temperature decreases ( T < Tt), the specimen fractures without yielding.

– Load rate: Increasing the rate at which the load is applied increases a metal’s ability to resist yielding.

– Triaxiality: The effect on the transition of a three-dimensional stress condition around the notch is similar to that of loading rate.

– Fatigue– Creep– Severe quenching in heat treatment

• Impact load (Shock load): the time of application on the structure is less than one-third of the lowest natural period of the structure.

2.10 Hardness

• Hardness: the ability of a material to resist indentation and scratching.

– Hardness testing is one of the principle methods for ascertaining the suitability of a material for its intended purpose. It is also a valuable inspection tool for maintaining uniformity of quality in heat-treated parts.

– Indentation hardness tests: Brinell, Rockwell, or Vickers.• Brinell Hardness

– The test uses a spherical ball in contact with a flat specimen of the material and subjected to a selected compressive load. The diameter of the indentation is measured with an optical micrometer.

– Brinell hardness number (Bhn) HB: The applied load divided by the area of the surface of indentation. The harder is the material, the higher the Brinell number.

• Rockwell Hardness– The test uses a indenter (steel ball or diamond cone) pressed

into the material. The thickness should be at least 10 times the indentation diameter.

– Rockwell hardness number RB; RC: The relationships of the total test force to the depth of indentation.

– RB scales used for soft metals. RC scales used for hard metals.

• Vickers Hardness

– The test is similar to the Brinell test. It uses a four-sided inverted diamond pyramid with an apex angle of 1360.

– Vickers hardness number HV: The ratio of the impressed load to the square indented area.

• Shore Scleroscope– The test uses a small diamond-tipped pointer or hammer that

is allowed to fall from a fixed height onto the specimen.– Hardness is measured by the height of rebound.

• Relationships among Hardness and Ultimate strength of steel– For most steels Su = 500HB psi

– For stress-relieved steels Sy = 1.05Su - 30000 psi

Sy = 525HB -30000 psi

2.11 Processes to Improve Hardness and the Strength of Metals

• The way to increase the hardness and strength of metals– Suitably varying the composition or alloying

– Mechanical treatment

– Heat treatment

• Mechanical Treatment– Hot-working processes

– Cold-working processes• Recrystallization temperature: the temperature what the

metal can be shaped and formed– Hot-working processes: above its recrystallization temperature

– Cold-working processes: below its recrystallization temperature

• Cold working (Strain Harding)– An increase in hardness and yield strength, with a loss in

toughness and ductility.

– Cold-working operations include cold rolling, drawing, turning, grinding, and polishing.

– Shot peening: the surface is bombarded with high-velocity iron or steel shot discharged from a rotating wheel or pneumatic nozzle.

• Hot working– It reduces the strain hardening of a material but avoids the

ductility and toughness loss.

– Hot-rolled metal tend to have greater ductility, low strength, and a poorer surface finish than cold-worked metals.

– Hold-working processes include rolling, forging, hot extrusion, and hot pressing.

• Heat Treatment– Quenching: The rapid cooling of a metal from an elevated

temperature by injecting or spraying the metal with a suitable cooling medium, such as oil or water, to increase harness.

– Tempering or Drawing: A process of stress relieving and softening by heating then quenching.

– Annealing: A process involving heating and slowly cooling, applied usually to induce softening and ductility.

– Normalizing: A process that includes annealing, except that the material is heated to a slightly higher temperature than annealing.

– Case hardening or Carburizing: A process where the surface layer is made substantially harder than the metal’s interior core.

– Through hardening: With a sufficiently high carbon content, the material is quenched and drawn at suitable temperatures to obtain the desired physical properties.

2.12 General Properties of Metals

• Iron and Steel– Iron, in its pure form, has almost no commercial use.

– The addition of other elements to iron essentially changes its characteristics, resulting in a variety of a cast and wrought irons and steel.

• Cast Irons– An iron alloy containing over 2% carbon.

– Brittle, and cannot be cold worked.

– Inexpensive, easily cast, and readily machined.

– Superior vibration characteristics and resistance to wear.

• Steels– An alloy of carbon containing less than 2% of carbon.

– Plain Carbon steels:• Contain only carbon (less than 1%)• Desired characteristics can be gained by heat treatment• The least expensive steel• Low-carbon, Medium and High-carbon steel

Type Carbon content Area of useLow-carbon 0.03 % ~ 0.25 % Plate, sheet

Medium 0.3 % ~ 0.55 % Machine partsHigh-carbon steel 0.6 % ~ 1.4 % Spring, tools

– Alloy steels:

• Hardness and strength can be controlled by the carbon content

• Alloying elements include manganese, molybdenum, chromium, vanadium, and nickel.

– Stainless Steels:

• Elements contain at least 12% chromium.

• Corrosion resistance and heat-resisting applications

• Three types – Austenitic (18% chromium, 8% nickel); Ferritic (17% chromium) Martensitic (12% chromium)

– Steel Numbering Systems:

• AISI / SAE system

AISI XX XX

Principal alloying element

Carbon content

Example:

AISI C 10 20

Process of manufacture

Plain carbon steel

0.20 % carbon

• ASTM system

• Aluminum Alloys– Good electrical and thermal conductivity; light reflectivity;

High strength-to-weight ratio

– The temper is a main factor governing its strength, hardness, and ductility.

• Copper Alloys– high electrical and thermal conductivity; resistance to

corrosion; low strength-to-weight ratio

– Brass: Copper-zinc alloyBronze: Copper and tin

2.13 General Properties of Nonmetals

• Plastics– Plastics are synthetic materials known as polymers

– Two principle classes:(1) Thermoplastic- material repeatedly soften when heated and harden when cooled.(2) Thermosets- structural change during processing to become permanently insoluble and infusible.

– Easily molded into complicated shapes; Allow large elastic deflection; making assembly fast and inexpensive.

• Ceramics– Compounds of nonmetalic as well as metallic elements,

mostly oxides, nitrades, and carbodes.

– High hardness and brittle; High compressive but low tensile strengths; high temperature and chemical resistance; high dielectric strength.

• Composite– Composite is made up of two or more unique element. A

high-strength reinforcement material embedded in a surrounding material.

– Relatively large strength-to-weight ratio.

• 2.3• 2.6• 2.10

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