Stress-strain curves simulations

The stress-strain curves simulate a homogeneous deformation process of the polymer. However, on the microscale above the linear part of the stress-strain curve (see Fig. 1.15, curves (b), (c), (d)), localized heterogeneous deformation mechanisms occur. Depending on the polymer chemical structure and entanglement molecular weight Mg and on the deformation conditions (temperature and strain rate), several types of heterogeneous deformation are observed micro plastic zones, crazes, deformation zones, and shear bands. Their main features are sketched in Fig. 1.18. 

The material properties used in the simulations pertain to a new X70/X80 steel with an acicular ferrite microstructure and a uniaxial stress-strain curve described by er, =tr0(l + / )", where ep is the plastic strain, tr0 = 595 MPa is the yield stress, e0=ff0l E the yield strain, and n = 0.059 the work hardening coefficient. The Poisson s ratio is 0.3 and Young s modulus 201.88 OPa. The system s temperature is 0 = 300 K. We assume the hydrogen lattice diffusion coefficient at this temperature to be D = 1.271x10 m2/s. The partial molar volume of hydrogen in solid solution is... 

Gas compression in closed-cell polymer foams was analysed, and the effect on the uniaxial compression stress-strain curve predicted. Results were compared with experimental data for a foams with a range of cell sizes, and the heat transfer conditions inferred from the best fit with the simulations. The lateral expansion of the foam must be considered in the simulation, so in subsidiary experiments Poisson s ratio was measured at high compressive strains. 13 refs. 

Figure 46a shows that a reasonable adaptation of the stress contribution of the clusters can be obtained, if the distribution function Eq. (55) with mean cluster size =25 and distribution width b= 0.8 is used. Obviously, the form of the distribution function is roughly the same as the one in Fig. 45b. The simulation curve of the first uniaxial stretching cycle now fits much better to the experimental data than in Fig. 45c. Furthermore, a fair simulation is also obtained for the equi-biaxial measurement data, implying that the plausibility criterion , discussed at the end of Sect. 5.2.2, is also fulfilled for the present model of filler reinforced rubbers. Note that for the simulation of the equi-biaxial stress-strain curve, Eq. (47) is used together with Eqs. (45) and (38). The strain amplification factor X(E) is evaluated by referring to Eqs. (53) and (54) with i= 2= and 3=(l+ ) 2-l.

Similar well fitting simulation curves for the experimental stress-strain data as those shown in Fig. 46b can also be obtained for higher filler concentrations and silica instead of carbon black. In most cases, the log-normal distribution Eq. (55) gives a better prediction for the first stretching cycle of the virgin samples than the distribution function Eq. (37). Nevertheless, adaptations of stress-strain curves of the pre-strained samples are excellent for both types of cluster size distributions, similar to Fig. 45c and Fig. 46b. The obtained material parameters of four variously filled S-SBR composites used for testing the model are summarized in Table 4, whereby both cluster... 

The success of the developed model in predicting uniaxial and equi-biaxi-al stress strain curves correctly emphasizes the role of filler networking in deriving a constitutive material law of reinforced rubbers that covers the deformation behavior up to large strains. Since different deformation modes can be described with a single set of material parameters, the model appears well suited for being implemented into a finite element (FE) code for simulations of three-dimensional, complex deformations of elastomer materials in the quasi-static Emit. 

Beside the consideration of the up-cycles in the stretching direction, the model can also describe the down-cycles in the backwards direction. This is depicted in Fig. 47a,b for the case of the S-SBR sample filled with 60 phr N 220. Figure 47a shows an adaptation of the stress-strain curves in the stretching direction with the log-normal cluster size distribution Eq. (55). The depicted down-cycles are simulations obtained by Eq. (49) with the fit parameters from the up-cycles. The difference between up- and down-cycles quantifies the dissipated energy per cycle due to the cyclic breakdown and re-aggregation of filler clusters. The obtained microscopic material parameters for the viscoelastic response of the samples in the quasi-static limit are summarized in Table 4. 

Small Debond Energy. For SDE, when cr< crs, the unloading modulus E depends on r0, but is independent of T, and Cl. However, the permanent strain e0 depends on T, and Cl, as well as r0. These differing dependencies of E and e0 on constituent properties have the following two implications. (1) To simulate the stress-strain curve, both e0 and E are required. Consequently, r0, T, and Cl must be known. (2) The use of unloading and reloading to evaluate the constituent properties has the convenience that the hysteresis is dependent only on tq. Consequently, precise determination of r0 is possible. Moreover, with t0 known from the hysteresis, both T,- and Cl can be evaluated from the permanent strain. The principal SDE results are as follows. 

The preceding constitutive laws may be used to simulate stress-strain curves for comparison with experiments. In order to conduct the simulations, the constituent properties, t, T, and ft are first assembled into the non-dimensional parameters9 , 2and 2r. For this purpose, it is necessary to have independent knowledge of d(a). When this does not exist, an estimation procedure is needed, based on Eqn. (45), through evaluation of ds, ermc and a . The first step is to use Eqn. (41) to evaluate the saturation crack spacing ds... 

Fig. l. 27 Simulated stress-strain curves for 1-D CMCs indicating the relative importance of constituent properties. 

When d a) has been established in this manner, stress-strain curves can be simulated for 1-D materials. Based on this approach, simulations have been used to conduct sensitivity studies of the effects of constituent properties on the inelastic strain. Examples (Fig. 1.27) indicate the spectrum of possibilities for CMCs. 

The constituent properties from Table 1.3 can, in turn, be used to simulate the stress-strain curves (Fig. 1.31). The agreement with measurements affirms the simulation capability whenever the constituent properties have been obtained from completely independent tests (Table 1.1). This has been done for the SiC/CAS material, but not yet for SiC/SiC. While the limited comparison between simulation and experiment is encouraging, an unresolved problem concerns the predictability of the saturation stress, crs. In most cases, ab initio determination cannot be expected, because the flaw parameters for the matrix (processing sensitive. Reliance must therefore be placed on experimental measurements, which are rationalized, post facto. Further research is needed to establish whether formalisms can be generated from the theoretical results which provide useful bounds on as. A related issue concerns the necessity for matrix crack density information. Again, additional insight is needed to establish meaningful bounds. Meanwhile, experimental methods that provide crack density information in an... 

Fig. 1.31 Simulated stress-strain curve for unidirectional SiC/SiC and comparison with experiment.

Fig. 1.35 A simulated stress-strain curve for a 2-D SiC/SiC in which the plastic strains are dominated by the 0° plies. The experimental results for a SiC/SiC composite are shown for comparison.

Using this simplified approach, simulations of stress-strain curves have been conducted.89,97 These curves have been compared with experimental measurements for several 2-D CMCs. One result is summarized in Fig. 1.35. It is apparent that the simulations lead to somewhat larger flow strengths than the experiments, especially at small inelastic strains. To address this discrepancy, further modeling is in progress, which attempts to couple the behavior of the tunnel cracks with the matrix cracks in the 0° plies. 

A detailed atomistic approach was used to investigate the molecular segment kinematics of a glassy, atactic polypropylene system dilated by 30%. ° The microstructural stress—dilation response consists of smooth, reversible portions bounded by sudden, irreversible stress jumps. But compared to the micro-structural stress—strain curve of the shear simulation, the overall trend more closely resembles macroscopic stress—strain curves. The peak negative pressure was in the neighborhood of 12% dilatation, with a corresponding secondary maximum in the von Mises shear stress. The peak negative pressure was re-... 

Molecular dynamics simulations of stress relaxation on idealized models [208] have shown, in agreement with the experimental trends, similarities between the simulated stress relaxation (stress versus time) curves of metals and polymers despite some essential differences between their stress-strain curves. The simulations also show the existence of domains in which motions are highly correlated, suggesting that cooperative motions are important in stress relaxation. 

The numerical simulation method of Termonia [67-72] was reviewed in Section 20.C.1 since it can be used in calculating the elastic moduli of composites. As described in that discussion, this method actually allows the calculation of complete stress-strain curves for fiber-reinforced composites. It must be emphasized that the ability of this method to simulate the mechanical properties of composites under large deformation by using a reasonable physical model is of far greater importance and uniqueness than its ability to model the elastic behavior. 

It was mentioned above that the simulation method of Termonia [67-72] can be used to calculate the stress-strain curves of many fiber-reinforced or particulate-filled composites up to fracture, including the effects of fiber-matrix adhesion. Such systems are morphologically far more complex than adhesive joints. Many matrix-filler interfaces are dispersed throughout a composite specimen, while an adhesive joint has only the two interfaces (between each of the bottom and top metal plates and the glue layer). If one considers also the fact that there will often he a distribution of filler-matrix interface strengths in a composite, it can be seen that the failure mechanism can become quite complex. It may even involve a complex superposition of adhesive failure at some filler-matrix interfaces and cohesive failure in the bulk of the matrix. 

Next we assess the effect of the initial levels of (p- on the shapes of stress-strain curves by considering the two extreme quench histories at rates and 4 in the simulation of Si by recalling from Section 7.6.2 that the slowest quench rate, q, gives = 0.22 and that the fastest, q, results in (p = 0.6. To make the comparison more dramatic, we also include a structure for a hypothetically slow quench, q, with = 0.05. 

Fig. 8.5 The computer-simulated equivalent stress-strain curve of amorphous polypropylene in a cubic simulation cell, strained by static energy minimization at 235 K, showing a number of unit plastic events as the system stress drops (O) and as the system pressure drops (O). Arrows show directions of forward and reverse straining (from Mott et al. (1993) courtesy of Taylor and Francis).

Stress-strain curves for amorphous SiC samples (a -SiC) are indicated in Fig. 3.21 for various values of the chemical disorder, x- Experimental simulation and MD calculations indicate a strong correlation between chemical and topological disorders. To maintain topological perfection for x > 0.54 seems impossible and so a stable amorphous structure is achieved. 

Fig. 13. Measured stress-strain curves for various temperatures including the implemented curves used in the ANSYS simulation (exemplarily, lower right).

The calculated temperature distribution is subsequently used to simulate the thermal deformation and, consequently, the resulting stresses and strains within the welded material. To this end the (thermo-) mechanical materials data, viz. a (cf. Figure 12, lower right), E (cf. Figure 15, right), the stress-strain curves (cf. Figure 13), and Poisson s ratio v = 0.3 are applied. Furthermore (linear) 8-node hexahedral elements are chosen. Note, that during... 

Great importance is attached to the determination of required materials data, since simulations can only be as reliable as the underlying input data. Here all caloric materials data were taken from literature or databases. Only mechanical data, in particular temperature depending stress-strain curves, are rarely available. Therefore one-dimensional tensile tests were performed at different temperatures by using the Gleeble3500 equipment. 

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