Question

5.2 A material can be characterized with a standard linear model (see figure below). The material is loaded with a step in the force at time t-0. At time t-t1 the material is suddenly unloaded stepwise. C1 (a) Derive the strain response for this loading history. (b) Make a drawing of the strain response as a function of time (c) Calculate de/dt for0.

Answer 1

The material is loaded with a step in the force t time t = 0, at time t = t_1 the material is suddenly uploaded stepwise. Device the strain response for this loading history consider the diagram calculate the stress for parallel components sigma_tot = sigma_1 + sigma_2 substitute the values sigma_tot = sigma = sigma_m + sigma_s_1 calculate the stress for series components sigma_m = sigma_m + sigma_s_2 calculate the strain for series components sigma_m - sigma_w + sigma_s_2 By the above relationship and the series relationship, the system can be modelled is d epsilon(t)/dt = E_2/n ((n/E_2 middot dm(t)/dt sigma(t) - E_1 E(t)))/E_1 + E_2 dE(t)/dt = (E_1 + E_2)^-1 [E_2/n ((n/E_2 middot d sigma(t)/dt) + sigma(t) - E_1 E(t))] \r\n By the laplace transformation exaction sigma + n/E_2 sigma = E_1 E + n(E_1 + E_2/E_2)E the initial deformation for the spring is E = sigma/E_1 + E_2 The creep response of the model is E(t) = (1/E_1 - E/E_1(E_1 + E_2) e^-t/2c) sigma_0