The Finite Element Method for the Analysis of Non-Linear and ...

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Example B. Jaumann stress rate formulation This formulation uses the following constitutive relationship: t˜τ ij = t C ijrs t D rs (7) where the velocity strain tensor t D is computed using the velocity gradient L (see Lecture 4) : [ ] 0 2 L = ẊX −1 = 0 0 L can be decomposed to s symmetric part D = D T (the velocity strain tensor) and a skew symmetric part W = −W T (the spin tensor): L = D + W [ 0 1 D = 1 0 which in this case yields ] [ ] 0 1 , W = −1 0 Now we use the same constitutive matrix C and Eqn (7) to obtain the Jaumann stress at time t: ⎡ ⎤ ⎡ ⎤ ˜τ 11 0 ⎢ ⎥ ⎢ ⎥ ⎣ ˜τ 22 ⎦ = ⎣ 0 ⎦ ˜τ 12 3846 Institute of Structural Engineering Method of Finite Elements II 16
Example The Jaumann stress is connected to the Cauchy stress through: ˜τ ij = ˙τ ij + τ ip W pj + τ jp W pi which from the above yields the follwoing formula for the Cauchy components τ ij ⎡ ⎢ ⎣ ˙τ 11 ˙τ 22 ˙τ 12 ⎤ ⎡ ⎥ ⎢ ⎦ = ⎣ 2τ 12 ⎤ −2τ 12 ⎥ ⎦ The above system of ordinary differential equations can be solved to finally get: ⎡ ⎢ ⎣ τ 11 τ 22 τ 12 ⎤ ⎡ ⎥ ⎢ ⎦ = ⎣ 1900(1 − cos2t) −1900(1 − cos2t) 1900sin2t ⎤ ⎥ ⎦ The results from methods A and B are rather close for small values of the deformation measure t but grow quite different as t gets larger than 0.1, indicating that the same C can no longer be used. Institute of Structural Engineering Method of Finite Elements II 17
Page 1 and 2: The Finite Element Method for the A
Page 3 and 4: Constitutive Relations It is necess
Page 5 and 6: Solution Flowchart General Solution
Page 7 and 8: Elastic Material For an elastic mat
Page 9 and 10: Elastic Material Important Note “
Page 11 and 12: Example Consider the four node elem
Page 13 and 14: Example Therefore, we ultimately ha
Page 15: Example Then, the Cauchy stress at
Page 19 and 20: Hyperelastic Material Hyperelastic
Page 21 and 22: Inelasticity Elastoplasticity, Cree
Page 23 and 24: Elastoplasticity The strain and str
Page 25 and 26: Elastoplasticity Isotropic & Kinema
Page 27 and 28: Thermoelastoplasticity and Creep Th
Page 29 and 30: Relaxation A relaxation test is def
Page 31 and 32: Viscoplasticity The elastic respons
Page 33 and 34: NL FE Special Considerations - The
Page 35 and 36: Contact Conditions Usual term Cont
Page 37 and 38: Definitions used in Contact Analysi
Page 39 and 40: Contact Analysis-Interface Conditio
Page 41 and 42: Contact Analysis Complexities In th

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