Lecture Notes 2.1

Lecture Notes 2.1 - Introduction to deformable bodies Thus...

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1 Introduction to deformable bodies • Thus far we have covered statics, and we have assumed that structural elements behave as rigid bodies • What does this mean? – Mathematically, the location of any point in the body is know if we know: center of mass, angular position (r, theta) about c of m – In real life, forces have no consequence on a body other than to move (or not move) it • Is this realistic?
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2 Compare 1 Ton 1 Ton FBD’s of these two cases are the same!!! Is the result the same (in terms of static, yes, in terms of actual observed behavior, no).
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3 • Statics tells us that these 2 cases are the same in terms of force equilibrium, but obviously they are not. • If you examine the basketball, a point’s angular position relative to its c of m has changed • The characteristic feature of a deformable body is that parts of it can move relative to other parts • As we will learn throughout this section, it is the characteristics of deformation that will dictate the ability of biological materials and their replacements to function mechanically.
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4 Strain L 0 L Consider a string originally of length L 0 , stretched a distance Δ L to a new length L: We have stretched it L/L0 times itself. We define this as the STRETCH RATIO, and designate it as λ . Note that stretch ratio is UNITLESS. This is important because it eliminates the absolute length from consideration, and makes this a geometry-independent calculation of deformation. We relate lambda to an approximation of the strain , which is sort of like deformation, but unitless (like deflection). Δ L L L L Δ + = 0
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5 Strain It turns our we can define strain in many ways…for instance: λ ε 1 1 1 0 0 0 = = = = l l l l l l Do we “normalize” to the original length, or the current length? Who decides what is current? Engineering strain True strain
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6 Why is strain important? Very soon we will talk about stress, which is related to the degree of internal forces in a deformable body. We can relate stress to strain through material properties. What kind of strains are there? Tension Compression Shear Compression and tension can be approximated by the equations from the previous page…shear can be approximated by the angle made by the deformation θ 2 θ 1 γ = tan 2 −θ 1 )
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7 These were pretty simple cases….what about something more complex? Suppose you have something that goes from P to Q…how can you figure out (and quantify) the strains when it potentially experiences translation and rotation?
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10 Example From the initial to deformed state (P->Q, a->x), we can write the ‘x’ positions as functions of the ‘a’ positions (or vice versa). In other words: x1=f(a1,a2,a3) a1=f(x1,x2,x3) x2=f(a1,a2,a3) or a2=f(x1,x2,x3) x3=f(a1,a2,a3) a3=f(x1,x2,x3) Note: 1 = ‘x’ 2 = ‘y’ 3 =‘z’
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11 • Let’s start easy….the z-position of any point in Q is the same as it was in P, so x3=a3 • It is also the same in y, so x2=a2 • What about in x?
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12 • Look at r,s,t, and u (and primed) In P r=0 s=1 t=1 u=0 In Q r’=0 s’=1 t’=1+1tan 30 u’=tan 30 So we can write: x1=a1+(1)(tan30) Or more generally x1=a1+a2*tan30
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This note was uploaded on 01/30/2012 for the course 125 208 taught by Professor Shreiber during the Spring '08 term at Rutgers.

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Lecture Notes 2.1 - Introduction to deformable bodies Thus...

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