Notes Mass-spring problems � No lecture Thursday (apologies) � [anisotropy] � [stretching, Poisson � s ratio] � So we will instead look for a generalization of “percent deformation” to multiple dimensions: elasticity theory cs533d-term1-2005 1 cs533d-term1-2005 2 Studying Deformation Going back to 1D � Let � s look at a deformable object � Worked out that dX/dp-1 was the key • World space: points x in the object as we see it quantity for measuring stretching and • Object space (or rest pose): points p in some compression reference configuration of the object � Nice thing about differentiating: constants • (Technically we might not have a rest pose, but (translating whole object) don � t matter usually we do, and it is the simplest parameterization) � So we identify each point x of the continuum with � Call A= � X/ � p the deformation gradient the label p, where x=X(p) � The function X(p) encodes the deformation cs533d-term1-2005 3 cs533d-term1-2005 4
Strain Why the half?? � A isn � t so handy, though it somehow encodes � [Look at 1D, small deformation] exactly how stretched/compressed we are � A=1+ � • Also encodes how rotated we are: who cares? � A T A-I = A 2 -1 = 2 � + � 2 � 2 � � We want to process A somehow to remove the � Therefore G � � , which is what we expect rotation part � Note that for large deformation, Green strain � [difference in lengths] grows quadratically � A T A-I is exactly zero when A is a rigid body - maybe not what you expect! rotation � Whole cottage industry: defining strain differently � Define Green strain ( ) 2 A T A � I G = 1 cs533d-term1-2005 5 cs533d-term1-2005 6 Cauchy strain tensor Analyzing Strain � Strain is a 3x3 “tensor” � Get back to linear, not quadratic (fancy name for a matrix) � Look at “small displacement” • Not only is the shape only slightly deformed, but it only slightly � Always symmetric rotates � What does it mean? (e.g. if one end is fixed in place) � Diagonalize: rotate into a basis of eigenvectors � Then displacement x-p has gradient D=A-I ( ) G = 1 2 D T D + D + D T • Entries (eigenvalues) tells us the scaling on the � Then different axes � And for small displacement, first term negligible • Sum of eigenvalues (always equal to the trace=sum ( ) � = 1 2 D + D T � Cauchy strain of diagonal, even if not diagonal): approximate volume change � Symmetric part of displacement gradient • Rotation is skew-symmetric part � Or directly analyze: off-diagonals show skew (also known as shear) cs533d-term1-2005 7 cs533d-term1-2005 8
Force Conservation of Momentum � In other words F=ma � In 1D, we got the force of a spring by simply multiplying the strain by some � Decompose body into “control volumes” material constant (Young � s modulus) � Split F into • f body (e.g. gravity, magnetic forces, …) � In multiple dimensions, strain is a tensor, force per unit volume but force is a vector… • and traction t (on boundary between two chunks of � And in the continuum limit, force goes to continuum: contact force) dimensions are force per unit area (like pressure) zero anyhow---so we have to be a little more careful � � � + = � ˙ ˙ f body dx tds X dx � W � � W � W cs533d-term1-2005 9 cs533d-term1-2005 10 Cauchy Stress Cauchy’s Fundamental Postulate � Traction t is a function of position x and normal n � From conservation of angular momentum can • Ignores rest of boundary (e.g. information like curvature, etc.) derive that Cauchy stress tensor � is symmetric: � Theorem � = � T • If t is smooth (be careful at boundaries of object, e.g. cracks) � Thus there are only 6 degrees of freedom (3D) then t is linear in n: t= � (x)n • In 2D, only 3 degrees of freedom � � is the Cauchy stress tensor (a matrix) � What is � ? � It also is force per unit area • That � s the job of constitutive modeling � Diagonal: normal stress components • Depends on the material � Off-diagonal: shear stress components (e.g. water vs. steel vs. silly putty) cs533d-term1-2005 11 cs533d-term1-2005 12
Divergence Theorem Constitutive Modeling � Try to get rid of integrals � This can get very complicated for � First make them all volume integrals with complicated materials divergence theorem: � Let � s start with simple elastic materials � � � nds = � � � dx � We � ll even leave damping out � � W � W � Then stress � only depends on strain, � Next let control volume shrink to zero: however we measure it (say G or � ) f body + � � � = � ˙ ˙ X • Note that integrals and normals were in world space, so is the divergence (it � s w.r.t. x not p) cs533d-term1-2005 13 cs533d-term1-2005 14 Linear elasticity Young’s modulus � Obvious first thing to do: if you pull on material, � Very nice thing about Cauchy strain: it � s resists like a spring: linear in deformation � =E � • No quadratic dependence � E is the Young � s modulus • Easy and fast to deal with � Let � s check that in 1D (where we know what � Natural thing is to make a linear should happen with springs) relationship with Cauchy stress � � � � = � ˙ ˙ x � Then the full equation is linear! � � � � � x E � X � � p � 1 � = � ˙ � � � ˙ x � � � � cs533d-term1-2005 15 cs533d-term1-2005 16
Example Young’s Modulus Poisson Ratio � Some example values for common materials: � Real materials are essentially incompressible (VERY approximate) (for large deformation - neglecting foams and • Aluminum: � =0.34 E=70 GPa other weird composites…) • Concrete: � =0.2 E=23 GPa � For small deformation, materials are usually � =0.2 • Diamond: E=950 GPa somewhat incompressible � =0.25 • Glass: E=50 GPa � Imagine stretching block in one direction • Nylon: � =0.4 E=3 GPa • Measure the contraction in the perpendicular • Rubber: � =0.49… E=1.7 MPa • Steel: � =0.3 directions E=200 GPa • Ratio is � , Poisson � s ratio � = � � 22 � [draw experiment; ] � 11 cs533d-term1-2005 17 cs533d-term1-2005 18 What is Poisson’s ratio? � Has to be between -1 and 0.5 � 0.5 is exactly incompressible • [derive] � Negative is weird, but possible [origami] � Rubber: close to 0.5 � Steel: more like 0.33 � Metals: usually 0.25-0.35 � What should cork be? cs533d-term1-2005 19
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