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1)Frictional slip of a cantilever beam the cantilever beam in Example 5

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1)Frictional slip of a cantilever beam the cantilever beam in Example 5.4 is now under additional axial load P=100 N at the tip, after the distributed load q is applied. Using the variation of the penalized potential energy, determine the stick or slip condition and calculate the tip displacement. Use friction penalty parameter ω_t=10⁶, axial rigidity E_A=105 N, and friction coefficient μ=0.5. Assume the axial displacement in the form of u(x)=a₀ + a₁x.

2)(Plastic Deformation of a Bar) Consider a bar under a uniaxial tension load. At load step tn, the axial stress σ₁₁=300 MPa, and the material is purely elastic before tn. At load step t_n+1, a strain increment Δε₁₁ = εΔt = 0:1 is given, determine stress and plastic variables. The material is combined linear isotropic/kinematic hardening, and the material parameters are given in Table 4.2.

3)(Isotropic/Kinematic Hardenings) A uniaxial bar is under proportional loading with axial stress σ. When the effective plastic strain is e_p=0.1, calculate the value of axial stress. Consider three different hardening models:  (a) isotropic, (b) kinematic, and (c) combined hardening with β=0.5. Assume that the initial yield stress is 400 MPa and the plastic modulus is H=200 MPa.

4)(Uniaxial Tensile Test) An axial stress σ is applied to a uniaxial bar. The yield stress of the material is σ_Y. Calculate the tensile stress σ when the material yields using the yield function formula in Eq. (4.70).

5)(Stress calculation in the perturbed Lagrangian formulation) When the dilatational strain energy density function is defined as Eq. (3.123), write the expression of stress as in Eq. (3.119) for the perturbed Lagrangian formulation. Also, show that the perturbed Lagrangian stress becomes identical to that of the penalty method when the pressure variable is eliminated in the element level.

6) A) (Uniaxial bar: updated Lagrangian formulation) Solve the uniaxial bar in Example 3.8 using the updated Lagrangian formulation. Assume that the change in cross-sectional area is ignored in one-dimensional bar. B) (Invariants) Show that the three invariants of the left Cauchy–Green deformation tensor G are equal to those of C when the three eigenvalues of the deformation gradient are λ₁, λ₂, and λ₃.