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Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II SOLUTION for QUIZ NO. 1 October 15, 2003 Problem 2 ( 20 points ) • For an imposed strain history consisting of a rapidly applied step-function jump in strain of magnitude , and applied at time t = 0, the stress relaxation function E r ( t ) is defined by σ ( t ) E r ( t ) = The stress relaxation function acknowledges that in a linear viscoelastic material stress/strain is a function of time. In linear viscoelastic behavior, the time dependance will die out and the stress will eventually reach a ”relaxed value”. • For an imposed stress history consisting of a rapidly applied step-function jump in stress of magnitude σ , and applied at time t = 0, the creep function J c ( t ) is defined by ( t ) J c ( t ) = σ The creep function is used when a constant stress is applied and the strain then changes with time (or ”creeps”.) In linear viscoelastic behavior, it will creep until the time dependance dies out and reaches a constant strain.) • The Correspondence Principle states that linear elastic models can be applied to lin- ear viscoelastic problems with equal geometries by replacing E with E r ( t ) or 1 /J c ( t ) depending on whether the displacement or load is held fixed in time. Example: For a simply supported beam with a constant point load in the center, the displacement at the center is modelled by − PL 3 δ = 48 EI If the beam has a is made of a linear-viscoelastic material and the load P is applied suddenly at time t = 0, the deﬂection will vary over time and E can be replaced by 1 J c ( t ) δ ( t ) = − P L 3 J c ( t ) 48 I 1 • The molecular structure of an amorphous polymer consists of long polymeric chains that are randomly oriented and intertwined. It is termed amorphous because there is no order to its structure. The solid is held together in two ways: through weak van der Waals and hydrogen bonding and also through the mechanical intertwining of the chains. The major feature of amorphous polymers is the glass transition temperature ( T g .) When the temperature of the polymer is greater than T g , there is enough thermal en- ergy in the chains of atoms to break free from most of the van der Waals bonds. Thus when an amorphous polymer is loaded, the chains move more freely than if the van der Waals bonds were present. Thus it takes less load to displace the material a certain amount when the material is above T g compared to below and a very large drop in E is noticed when T > T g . Also, E of amorphous polymers are less than the elastic modulus of crystalline and semi-crystalline polymers....
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- Spring '04