Chap 3 -Stress Mechanical Behavoir 4511

8/16/2019 Chap 3 -Stress Mechanical Behavoir 4511

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MECHANICAL BEHAVIOR

OF MATERIALS

strain & Stress


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Introduction

•

The two principal responses ofmaterials to external forces aredeformation and fracture.

The deformation may beelastic, plastic, or creep (time-dependent plastic deformation).Fracture may occur suddenly orafter repeated application of

loads (fatigue).


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Introduction

•

Fracture is the breaing of amaterial into two or morepieces. !f fracture occurs before

much plastic deformationoccurs, we say that thematerial is brittle . !n contrast, ifthere has been extensi"eplastic deformation preceding

fracture, the material is


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Introduction

uctile and Brittle material


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Introduction

Ductile Fracture Brittle Fracture


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Stress

!f the state of stress is the

same e"erywhere in a body

Stress, #, is defined as theintensity of force at a point$


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Stress


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Stress• % normal stress

(compressi"e or tensile)

is one in which the forceis normal to the area onwhich it acts.

# yy & F y '%y

• % shear stress , theforce is parallel to thearea on which it acts. # y

& F y '% is the shearstress caused by a shearforce in the y-directionacting on a plane normal

to .


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Stress

The nine components o stress actin! on anin"nitesimal element# The normal stresscomponents are σxx, σyy, and σzz. The shearstress components are σyz, σzx, σxy, σzy, σxz, and

σyx.


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Stress $ transormationo a%es


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Stress

rincipal stresses!t is always possible to nd a set of axes (*, +, ) alongwhich theshear stress components "anish. !n this case the normal

stresses, σ 1, σ 2, and σ 3, are called principal stresses


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Stress


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Stress


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Stress

• Mohr(s circle, named after

hristian tto /ohr, is a two-dimensionalgraphical representation of the stress tensor.

• The /ohr circle is used to determinegraphically the stress components acting on

a rotated coordinate system.

https://en.wikipedia.org/wiki/Christian_Otto_Mohrhttps://en.wikipedia.org/wiki/Circlehttps://en.wikipedia.org/wiki/Circlehttps://en.wikipedia.org/wiki/Christian_Otto_Mohr


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Stress

• )ra*in! Mohr(s circle$

• 0raw the artesian coordinate system (σn , τn)with a horiontal σ-axis and a "ertical τ-axis.

• 1lot two points %(σx , -τxy)and 2 (σy , +τxy) in the

space corresponding to the nown stresscomponents on both perpendicular planes.

• 0raw the diameter of the circle by 3oiningpoints % and 2 with a straight line %2.

• 0raw the /ohr ircle. The centre of thecircle is the midpoint of the diameter line%2 , which corresponds to the intersection of

this line with the σn axis.


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Stress+'lane stress,


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Stress+'lane stress,


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rientations


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Sample problem+'lanestress,

• body is loaded under stresses #x & 45 /1a,

#y & *5 /1a, 6 xy & -*4 /pa.

Find$

• the three principal stresses.

• the largest shear stress in the body.

• rientation of the principal stresses the

largest shear stress.• Sol"e the problem using setch of /ohr7s

circle diagram for this stress state.


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Sample problem+'lanestress,

• %t a point of interest on the free surfaceof an engineering component, thestresses with respect to a con"enient

coordinate system in the plane of thesurface are #x & 84, #y & +4, and 6xy &+5/1a. 0etermine the principal normaland shear stresses and their coordinate

system rotations. %lso determine themaximum normal stress and themaximum shear stress.


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Solution


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Sample problem

-od. is loaded under stresses:

Find:-the three principalstresses./the lar!est shear stress

in the -od.#-Orientation of the principal stresses thelargest shear stress.

-ol!e the pro"lem usings#etch of Mohr0s circle


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9xercises


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Strain

• %n innitesimal normal strain isdened by the change of length, :, ofa line$


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Strain


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Strain


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Strain

• There are se"eral reasons that true strainsare more con"enient than engineeringstrains.

*. True strains for e;ui"alent amounts ofdeformation in tension and compressionare e;ual except for sign.

+. True strains are additi"e. For a

deformation consisting of se"eral steps,the o"erall strain is the sum of the strainsin each step.


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Strain

• Due to a loading, the plate is deformed into the

dashed shape shown in Figure below. Determine• (a) the average normal strain along the side !,

• (b) the average shear strain in the plate at relative to

the x and y axes.


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Strain

• Part (a). "ine !, #oin#ident with the y axis,

be#omes line !$ after deformation, as shown

in Figure below.%he length of !$ is

he average normal strain for ! is therefore


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Strain


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Strain• Small strains $

Normal strain+1 ),

Strain


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Strain

• Shear strain+1 ),

The total engineering shear strain,


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Strain

Small strains can -e treated astensors

St i


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Strain

• Mohr0s strain circles

Chap 3 -Stress Mechanical Behavoir 4511

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