Chap 4 Elasti

8/16/2019 Chap 4 Elasti

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ELASTICITY


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ELASTICITY

• Elastic deformation isreversible. When a bodydeforms elasticallyunder a load, it will

revert to its oriinalsha!e as soon as the

load is removed


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Hooke’s Law for one Dimension

An isotropic material is one that has the same properties inall directions


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oo e s aw or reeDimensions


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Hooke’s Law for ThreeDimensions

Adding the columns of the abovetable to obtain the total strain ineach direction gives the following

equations:


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Elastic Constants for "arious #aterials at Ambient Tem!erature


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Shear modulus

Shear strains are a$ected only by thecorres!ondin shear stresses, so that

G is the shear modulus


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Volumetric train andH!drostatic tress

• "olume chane due to normal strains


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The bul% modulus

The average normal stress is called the hydrostaticstress and is given by

the volumetric strain is proportional to the h!drostatic stress" Theconstant of proportionalit! relating these is called the bulkmodulus


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ELASTICITY

• The four elastic constants, E, &, ',and (, can be e)!ressed in terms oftwo others*


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+roblem-

• A sam!le of material subected to acom!ressive stress /0 is confined so that it

cannot deform in the ydirection, as shown in1iure below. Assume that there is no frictionaainst the die, so that deformation can freelyoccur in the )direction. Assume further thatthe material is isotro!ic and e)hibits linearelastic behavior


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• 2etermine the followin in terms of /0 and theelastic constants of the material*

• #a$ The stress that develops in the !% direction"

• #b$ The strain in the &%direction"

• #c$ The strain in the '%direction"

• #d$ Assume that the compressive stress inthe &%direction has a magnitude of ()*+aand

•

The bloc% is made of a co!!er alloy, and thencalculate /y , 30 , 3). E 4 -56 '+a 4 -56,666#+a,

• and 7 4 6.585.


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Strains are measured on the surface of abrass alloy !art as follows* 3 ) 4 6.6668, 3 y 4 6.66-6, and : )y 4 6.666;. Estimate the

in!lane stresses / ) , / y , and


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A !late of metal is subected to stresses

,' - -;B *+a and ,! - -@9 *+a"The strains that occur as a result of these

stresses are measured to be*

.' - -66C-6 DB and .! - -9@6C-6 DB "

o yieldin occurs in the !late, the behavior iselastic. Estimate the elastic modulus E and+oissonFs ratio 7 for the metal. What ty!e of

metal is itG Answer*

E4 74


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Answer:σz=-56 MPa εx=-53.5 1-6 ε! = -5"" 1-6ε z =

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Chap 4 Elasti