### Examine the stress at the boundaries in the plate vs time

**Assignment Detail:- ** Assignment: Impulse loaded plate
Load and geometry of the impulse loaded thin plate-
Consider a thin plate of length L = 100 mm, height H = 40 mm and thickness T = 2 mm- The plate has a straight stationary edge-crack of length L/2 positioned so that its tip coincides with the center of the plate, as illustrated in Fig- 1- The plate is initially stress free and at rest- Suddenly, at t = 0, the upper and lower horizontal boundaries becomes exposed to a remote tensile stress σ∞- The plate is made of an isotropic material with Young's modulus E = 32 GPa, a Poisson's ratio v = 0-2, and has a density ρ = 2450 kg/m3 and a viscous damping coefficient c =15 MPas- The theoretical longitudinal wave speed in the material is given by c0 = √E/ρ-- Develop a dynamic finite element model to analyze the mechanical behavior- For simplicity, a fully explicit time integration scheme should be used, and the mesh should at least have 4000 equally sized elements-
Problem formulation
a- Examine the stress at the boundaries in the plate vs time t- Explain the behavior-
b- Analyze how the stress waves propagates through the body- Examine the effective stress -von Mises- at the crack tip versus time- When does the stress wave reach the tip???? What happens at the crack tip at about 24 ps???? What is the maximum effective stress at the tip and when does it occur???? Is this physically reasonable???? Is there any practical implications of this????
c- Can one observe the theoretical longitudinal stress wave velocity in the simulation????
d- The material is changed form an isotropic linearly elastic material to an isotropic elastic-perfectly plastic material with a yield stress σs = 20 MPa-
One can assume that the yield surface is given by a J2 flow theory, f = 3/2 STS - σs = 0, s is the deviator stress- Everything else in the problem is the same- Perform the same analyses as above, i-e- a-c, for the elastic-perfectly plastic material- Explain the observed differences- Is the behavior physically trustworthy???? Comment why or why not????
Hint: An elastic-perfectly plastic material is conveniently implemented as a bi-linear strain hardening material with a plastic modulus H significantly lower than the Young's modulus E, i-e- H < E /100-

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