chap 4 elasti
TRANSCRIPT
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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$ected only by thecorres!ondin shear stresses, so that
G is the shear modulus
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Volumetric train andH!drostatic tress
• "olume chane due to normal strains
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The bul% modulus
The average normal stress is called the hydrostaticstress and is given by
the volumetric strain is proportional to the h!drostatic stress" Theconstant of proportionalit! relating these is called the bulkmodulus
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ELASTICITY
• The four elastic constants, E, &, ',and (, can be e)!ressed in terms oftwo others*
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• A sam!le of material subected to acom!ressive stress /0 is confined so that it
cannot deform in the ydirection, as shown in1iure below. Assume that there is no frictionaainst the die, so that deformation can freelyoccur in the )direction. Assume further thatthe material is isotro!ic and e)hibits linearelastic behavior
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• 2etermine the followin in terms of /0 and theelastic constants of the material*
• #a$ 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 =