UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering,Department of Civil Engineering Mechanics and MaterialsFall 2010 Professor: S. GovindjeeHW 10: Due December 11. In viscoelastic materials the Poisson function is defined to be ν(t) = −ε22(t)/ε11(t)in a uniaxial constant load experiment where a cylinder of material is exposed to astep stress in the 1-direction and ν(t) is the ratio of the axial strain to the transversestrain. As sume a material that has elastic bulk behavior and Maxwell fluid deviatoricbehavior:˜σ = Ke (1)d`eijdt=12µd`σijdt+`σijη, (2)where`ε and`σ are the strain and stress deviators, respectively, ˜σ =13tr[σ] is thepressure, and e = tr[ε] is the volumetric strain. Find the Poisson function. Evaluateyour answer for the two limits ν(0) and ν(∞). Note: the given stress state is simplyσ(t) → H(t)σo0 00 0 00 0 0and you need to determine the strain response.2. (20pts) In the accompanying file you will find stress data for a stress relaxation exper-iment. The stress response is generated by an imposed strain historyε(t) =10−1t t < 10−1s10−2t ≥ 10−1s .This strain history involves a rise time of 100 ms to reach the hold strain of 10 mstrain.The first column of the data file gives a time value (in seconds) and the second columna corresponding stress value (in MPa). Note that the time spacing of the data pointsis not uniform.Find an expression for the relaxation modulus E(t). Make sure you use a suitablenumber of terms to adequately model the data. Note that the data file includes therise time response.3. (20pts) Consider a thermorheologically simple viscoelastic material with reference tran-sition temperature To= 200 K. Further assume a WLF model for the shift factorlog(a) =−C1(T − To)C2+ T − To,1with C1= 20 and C2= 100 K. Consider now that the material is subjected (fromquiescent conditions) to a strain history of ε(t) = 0.1 sin(t/τo) and a thermal historyof T (t) = To+ βt + γ cos(t/2τo), where β = 0.01 K/s and γ = 5 K. Assume that thematerial is to be mo de led as a standard linear solid with E∞= 10 MPa, Eo= 100 MPa,and τo= 1.5 s and find the stress response over the time interval I = [0, 100] s. Makethe following plots (a) time versus strain, (b) time versus temperature, (c) time versusstress, (d) time versus fictitious time, and (e) stress versus strain. To help you debugyour solution below is the answer to the question for the case where β = γ = 00 20 40 60 80 100−0.1−0.0500.050.1Time (s)Strain [−]0 20 40 60 80 100199199.5200200.5201Time (s)Temperature (K)0 20 40 60 80 100−10−50510Time (s)Stress (MPa)0 20 40 60 80 100050100150Time (s)Fictitious Time (s)−0.1 −0.05 0 0.05 0.1−10−50510Strain [−]Stress