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Unformatted text preview: AE 321 - “Theoretical questions”
Last update: September 29, 2008
1) Vector calculus: TRUE/FALSE a) The inner product of two vectors is a vector b) The gradient of a scalar ﬁeld is a vector c) The divergence of a vector ﬁeld is a vector d) The curl of a vector ﬁeld is a vector 2) Indicial notation: how many expressions/equations are denoted by (a) Di , j B j (b) Cij + D ji (c) Bii (no sum) (d) Cij , j B j (e)
ij = Cijkl kl (f) C pq , rU pr = Bq
AE 321 Q1. 1 3) Let a, u and W be a scalar, vector and two-tensor field, respectively. Describe the following quantities in both vector/tensor and indicial notations: a) Gradient of a b) Divergence of u c) Curl of u d) Gradient of W e) W is symmetric f) W is skew-symmetric g) V is the inverse of W 4) With b) c) denoting the Kronecker delta, simplify/compute
om np ij a) Amnop
ij ij ij jp pi AE 321 Q1. 2 5) Compatibility equations a) What do the compatibility equations tell us? b) When do you need the compatibility equations? c) When don't you need the compatibility equations? 6) Let r, and z denote cylindrical coordinates. What is the physical significance of a) rr b) c) d) e)
r rz rr + + zz 7) Consider a circular string of radius a. Under some external load, the string expands while remaining circular to a radius b. What is the corresponding value of (the hoop strain) ? AE 321 Q1. 3 8) What is the physical significance of a) 11 b) 33 c) 12 d) e)
23 11 + 22 + 33 9) Define/give the physical significance of the following stress components a) 11 b) 33 c) 13 d) e) f)
rr r zz 10) What is Cauchy relation? What is it used for? AE 321 Q1. 4 11) Define a) Scalar field (0-tensor field) b) Vector field (1-tensor field) c) 2-Tensor field 12) Define and explain how, given an axis system and the corresponding expression of the stress field [ ij ], you would find the principal stresses and principal directions. What is the expression of the stress tensor in the principal directions? 13) What is Mohr's circle used for? How does it work? 14) Given a stress field [ ij ], define volumetric and deviatoric stresses 15) Describe the general form of the stress tensor for a state of a) Plane stress b) Uniaxial stress c) Biaxial stress d) Hydrostatic stress e) Pure shear stress 16) What is the difference between Lagrangian and Eulerian strains? AE 321 Q1. 5 17) What is the key assumption between Lagrangian and infinitesimal strain tensors? 18) Given a strain tensor [ ij ], define volumetric and deviatoric strains 19) Sketch the uniaxial stress-strain curve for a brittle and ductile material 20) Sketch the unixial stress-strain curve for a ductile material and define (in words and graphically) a) Yield stress b) Young's modulus c) Ultimate tensile stress d) Failure strain 21) What is the most general form of Hooke's law? 22) Define and provide units for a) Poisson's ratio b) Young's modulus c) Bulk modulus AE 321 Q1. 6 23) Describe Saint Venant's principle: What is it? Why is it useful? 24) Define/describe the creep response of a material 25) Define/describe stress relaxation 26) What is a viscoelastic material? Sketch it creep and relaxation response 27) What is the principle of superposition? 28) What is the warping function? 29) Define a) Principal stress criterion b) Tresca yield criterion c) Von Mises yield criterion d) Yield surface e) Fatigue f) S-N curve g) Miner's rule h) Anisotropic material i) Orthotropic material j) Isotropic material k) Linearly/nonlinearly elastic material
AE 321 Q1. 7 ...
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- Spring '08