Time dependence of stress

Al-Saidi LF, Mortensen K, Almdal K (2003) Environmental stress cracking resistance behaviour of polycarbonate in different chemicals by determination of the time-dependence of stress at constant strains. Polym Degradat Stabil 82(3) 451—461... 

Consider a tensile specimen of an isotropic metal with elastic parameters E = 210 000 MPa and v = 0.3, and a yield strength (Tf = 210 MPa. The material hardens linearly and isotropically according to equation (3.50), with hardening parameter H = 10 000 MPa. The tensile specimen is elongated, starting with an unloaded state, at a constant strain rate of n = 0.001 s . We want to determine the time-dependence of stresses and strains. 

Sketch the time-dependence of stress and strain and explain the meaning of the parameter J ... 

Ganapathy R, Sood AK (2008) Nonlinear flow of wormlike micellar gels regular and chaotic time-dependence of stress, normal force and nematic ordering. J Non-Newt Ruid Mech 149(l-3) 78-86... 

The boundary value problem consisting of the partial differential equation (9.129), along with its initial and boundary conditions, has a well-known solution that includes the time-dependence of stress at the confined end of the line given by... 

The dependence of stress on strain, strain rate, and time for both shear and el-ongational deformation can also be described by various other constitutive equations. i4-ii9 Some of these involve specially defined measures of strain." " In some, the strain dependence and time dependence of stress are factored as in equations 66 and 67. Experimental data are fitted very closely but detailed molecular interpretations are not provided. 

The time dependence of stress under conditions of constant strain rate has been discussed for the case of linear viscoelasticity in Section FI of Chapter 3. For uniaxial extension at constant strain rate ei = (l/ ) d /dt), the time-dependent tensile stress ffriO is often expressed in terms of a time-dependent viscosity = An example of the stress growth in such elongational flow from... 

Fig. 5.3. Time dependence of stress Ozz and strain Czz in a dynamic-mechanical experiment (schematic)...

Many attempts have been made to obtain mathematical expressions which describe the time dependence of the strength of plastics. Since for many plastics a plot of stress, a, against the logarithm of time to failure, //, is approximately a straight line, one of the most common expressions used is of the form... 

The viscoelastic nature of the matrix in many fibre reinforced plastics causes their properties to be time and temperature dependent. Under a constant stress they exhibit creep which will be more pronounced as the temperature increases. However, since fibres exhibit negligible creep, the time dependence of the properties of fibre reinforced plastics is very much less than that for the unreinforced matrix. 

Non-Newtonian fluids vary significantly in their properties that control flow and pressure loss during flow from the properties of Newtonian fluids. The key factors influencing non-Newtonian fluids are their shear thinning or thickening characteristics and time dependency of viscosity on the stress in the fluid. 

K(l) is the function defining the time dependence of the creep. The constant ac is a critical stress characteristic of the material, and at stresses greater than (r< the creep compliance increases rapidly with stress.. ... 

So far we have employed in this discussion a critical shear stress as a criterion for fibre fracture. In Sect. 4 it will be shown that a critical shear strain or a maximum rotation of the chain axis is a more appropriate criterion when the time dependence of the strength is considered. 

The second approach used in first-principles tribological simulations focuses on the behavior of the sheared fluid. That is, the walls are not considered and the system is treated as bulk fluid, as discussed. These simulations are typically performed using ab initio molecular dynamics (AIMD) with DFT and plane-wave basis sets. A general tribological AIMD simulation would be run as follows. A system representing the fluid would be placed in a simulation cell repeated periodically in all three directions. Shear or load is applied to the system using schemes such as that of Parrinello and Rahman, which was discussed above. In this approach, one defines a (potentially time-dependent) reference stress tensor aref and alters the nuclear and cell dynamics, such that the internal stress tensor crsys is equal to aref. When crsys = aref, the internal and external forces on the cell vectors balance, and the system is subject to the desired shear or load. 

Peppas, N.A., Ponchel, G., and Duchene, D., Bioadhesive analysis of eontrolled-release systems II. Time-dependent bioadhesive stress in poly(acrylic acid)-eontaining systems, J. Control Rel, 5 143-150 (1987). 

The second additional requirement of CMP carriers is that they must allow the tool to polish a broad range of films with varying amounts of film stress. Film stress causes the wafer to deform, altering the pressure distribution across the wafer during CMP. These pressure variations cause characteristically fast or slow polishing across the wafer. This picture is further complicated by the time dependence of this stress during the course of the polish cycle. 

In polymers the time dependence of an modulus plays a more important role than in metals. If polymers are loaded with a constant stress they undergo a deformation e, which increases with time. This process is named creep. Conversely, if a test specimen is elongated to a certain amount and kept under tension, the initial stress s decreases with time. This decay is called stress relaxation. 

Some materials might produce a unique failure surface providing measurements could be conducted under first stretch conditions in a state of equilibrium. Tschoegl (110), at this writing, is attempting to produce experimental surfaces by subjecting swollen rubbers to various multiaxial stress states. The swollen condition permits failure measurements at much reduced stress levels, and the time dependence of the material is essentially eliminated. Studies of this type will be extremely useful in establishing the foundations for extended efforts into failure of composite materials. 

Since the shear-stress-shear-rate properties of pseudoplastic materials are defined as independent of time of shear (at constant temperature), the alignment or decrease in particle size occurring when the shear rate is increased must be instantaneous. However, perfect instantaneousness is not always likely if the foregoing causes of pseudoplastic behavior are correct, as they are believed to be. Pseudoplastic fluids are therefore sometimes considered to be those materials for which the time dependency of properties is very small and may be neglected in most applications. 

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

In Fig. 24(a) the purely elastic deformation and the plastic elastic flow processes are plotted and hatched in a different manner. Figure 24(b) shows the dependence of stress on time. It can also be seen, that with discharge at time t0 the purely elastic residual deformation disappears at once, whereas the plastic-elastic portion does so gradually (diffusion processes). 

Dependencies of stress a and elastic deformation a upon time t are measured in extension under conditions of x = const. 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

Important examples of stick-slip are earthquakes that have long been recognized as resulting from a stick-slip frictional instability. The use of a full constitutive law of rock friction that takes into account the time dependence of /is and the dependence of /j>k on speed and sliding distance can account for the rich variety of earthquake phenomena as seismogenesis and seismic coupling, pre- and post-seismic phenomena, and the insensitivity of earthquakes to stress transients [461],... 

The results of the calculations shown in Fig. 2.32 represent a complete quantitative solution of the problem, because they show the decrease in the induction period in non-isothermal curing when there is a temperature increase due to heat dissipation in the flow of the reactive mass. The case where = 0 is of particular interest. It is related to the experimental observation that shear stress is almost constant in the range t < t. In this situation the temperature dependence of the viscosity of the reactive mass can be neglected because of low values of the apparent activation energy of viscous flow E, and Eq. (2.73) leads to a linear time dependence of temperature ... 

Figure 9.7 The time-dependence of deformations under constant stress...

According to results reported in the literature [1-13] if the shear stress is canceled out after steady-state conditions are reached, the time dependence of the recoverable deformation [er(t)—cre(t)/r ] is obtained where e(t) is the shear strain, a is the stress and r is the viscosity (2.6). The higher temperature, the greater the unrecoverable contribution to the shear deformation i.e. the viscous deformation. Figure 2.3 shows the effect of temperature on the strain. 

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