Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 506 Objective Rate Of Stress 0 2k3 0 0 0 0 0 0 0 A1 A2 0 0 0 A2 A1 2k3 0 0 A A2J A2 A 0 2k4 0 0 0 0 0 0 0 0 0 0 iii 0 4Ắ 5 0 0 0 0 0 0 0 A1 A2 0 0 0 Aằ AJ 4Ắ 5 0 0 A A1 0 4k6 0 0 0 0 0 0 0 0 0 0 iv Therefore trAj 2k2 trAi O trA2 2k2 trA2 4fc4 trA2 8fc6 trAiA2 0 trA A2 2k4 trAjA2 0 trA A2 4k6 Example In a simple shearing flow compute the stress components for the Rivlin -Ericksen liquid. Solution. From Eqs. and the results of the previous example we have note A3 A4 0 711 -p W2 r22 2 fc2 2 k4 2 Pl k2 k2 fc2 Tìĩ -p T12 2k2ns k2 4k 6 k2 i Wk2 s k T23 T13 0 where Pi k2 indicates that Hl is a function of k2 etc. The normal stress differences Tq - r22 and T22 - T33 are even functions of k rate of shear whereas the shear stress function s k is an odd function of k. Objective Rate of Stress The stress tensor T is objective therefore in a change of frame T Q i TQr i Non-Newtonian Fluids 507 Taking material derivative of the above equation we obtain note D Dt D Dt tqT q qT Ợr ff 8 18 2 Dt dt Dt dt I The above equation shows that the material derivative of stress tensor T is not objective. That the stress rate D T Dt is not objective is physically quite clear. Consider the case of a time-independent uni-axial state of stress with respect to the first observer. With respect to this observer the stress rate Ơĩ Dt is identically zero. Consider a second observer who rotates with respect to the first observer. To the second observer the given stress state is rotating respect to him and therefore to him the stress rate DT Dt is not zero. In the following we shall present several stress rates at time t which are objective A Jaumann derivative of stress Let us consider the tensor Xr R r r T r R r 8-18-3 We note that since R r Rf t I therefore the tensor J and the tensor T are the same at time t. That is Xi T i i However while DT Dt is not an objective stress rate we will show that pỹ ii Dr L J T t is an objective stress rate. To show this we note that in .
