Question

After the Maxwell and Voight models, the standard linear solid model is the next simplest model. Derive an equation which shows how the overall stress, stress rate, strain and strain rate are related by EI, E2, and n (i.e. one that takes into account the parallel and series properties) for the standard linear solid model provided below. 2

Answer 1

This model consists of spring E_1 which is placed in series with kelvin-vogit model represented by spring E_2 and a dashpot n . This allows the spring E_1 to be activated instantaneously during loading and unloading accompanied by a time dependent response of spring E_2 due to the load transfer through spring dashpot kelvin -vogit element The applied stress sigma produces strain in spring E_1 and stress relationship for spring is governed by sigma = E_1 elementsof_1 ----(1) The derivative of (1) gives strain rate Rightarrow E_1 = sigma/E_1 -(2) This applied stern(sigma) is also distributed in the kelvin vogit elements such that, sigma = sigma_1 + sigma_2 ---(3) The stresses and strains in kelvin - vogit elements has following relationship sigma_1 = E_2 E_25 ----(4) and sigma_2 = n E_2 ----(5) The time derivative of the total strain (E = E_1 + E_2) im the standard linear element is