Multiplicative decomposition

The nitramines, which have found practical application as explosives are multi-functional nitramines, usually cyclic (Fig. 7). Nitramines have been studied by many groups and multiple decomposition pathways... 

Multiple decomposition pathways observed Possible reaction channels of the enol ion in 1029... 

TGA experiments on polymeric systems often show complex TGA mass/temperature curves in which multiple decomposition products correspond with the weight change observed (see, for example, Figure 2.10). TGA has thus proven to be an excellent quantitative technique but less suitable for specification. This drawback can be eliminated if the components which are causing the mass losses detected, are also analysed simultaneously, the so-called evolved gas analysis (EGA). Several TGA-EGA systems are described in literature, analysing the evolved gases with different techniques i.e. thermal conductivity, cold-trapping followed by GC, mass spectrometry (MS) and infrared (FTIR). MS and FTIR have proven to be the most powerful techniques [3, 10]. 

Fig. 5 A 2-D DWT image wavelet coefficients for multiple decomposition levels (shown for...

Figure 4.4 shows a schematic of mapping from an undeformed body to a deformed body, as well as the decomposition of the deformation gradient to the plastic deformation gradient, which leads to an intermediate relaxed body with pure plastic deformation and is obtained by conceptually unloading the deformed body elastically to zero stress, and the elastic deformation gradient, which leads the intermediate body to the deformed body. If the total deformation gradient is F, the plastic component is F, and the elastic component is F, similar to polar decomposition, the multiplicative decomposition leads to... 

Figure 4.4 Multiplicative decomposition of an undeformed body to an intermediate body by elastically unloading tbe deformed body to zero stress...

Figure 4.6 Schematic of multiplicative decomposition of the deformation gradient with n intermediate configurations...

Figure 4.18 (a) Multiplicative decomposition of the deformation gradient of each component. Source [86] Reproduced with permission from ASME and (b) 1-D rheological analogy... 

Lubarda, V.A. (2004) Constitutive themies based on the multiplicative decomposition of deformation gradient thermoelasticity, elastoplasticity, and biomechanics. ASME Applied Mechanics Review, 57, 95-108. 

As discussed in Chapter 4, a strain description based on displacement gradients becomes nonlinear, which may result in computational difficulties for elastoplastic analysis. Therefore, the deformation gradient and multiplicative decomposition technique are usually used ... 

Stress Definition in Solid Mechanics Multiplicative Decomposition of Deformation Gradient Constitutive Modeling of Cold-Compression Programmed Thermosetting SMP... 

Figure 5 Multiple Decomposition Types for Individual Variables.

In particular, we are interested in the case of finite strain plasticity characterized by the multiplicative decomposition F = F F of the deformation gradient in an elastic and plastic part see Armero (2006) and references therein. The stresses are then given in terms of an elastic potential W" C") for C = F F as... 

Lubarda VA (2004) Constitutive theories based on the multiplicative decomposition of deformation gradient Thermo-elasticity elastoplasticity and biomechanics. Appl Mech Rev 57 95-108 Lubliner J (1990) Plasticity theory. Collier-MacmiUan, New York McQuarrie DA, Simon JD (1997) Physical chemistry. University Science Books, Sausahto Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-HaU, Englewood Cliffs... 

Meggyes A (2001) Multiple decomposition in finite deformation theory. Acta Mech... 

It should be noted that TGA will not always be accurate because various components in polymeric formulations are not observed as independent weight loss in TG curves (e.g. sulfur, accelerators, antioxidants and antidegradants in elastomers) and may undergo weight loss over a large temperature range. Low-MW volatile products (e.g. oils, waxes, plasticisers and resins) tend to overlap with polymer decomposition for most choices of method parameters. In the presence of multiple decomposition... 

The discussion so far has demonstrated that redox shuttles can have multiple decomposition pathways, making the design of a stable redox shuttle... 

Another study addressed the uncertainty of extracting the kinetic parameters solely from thermograms [a.9, a.47]. Thus, modification of the Ozawa method [a.49] was introduced to tackle complex TG curves, since the traditional approach of the integral method has major limitations in extracting reliable kinetic parameters. The method implemented in extracting the multiple decomposition kinetics was based on subtraction of the mass of a specific event from the total mass loss, and then addressed the following event on the TG curve. For PMMA, the results demonstrated that decomposition could be described in terms of both depolymerisation and vaporisation, while for polytetrahydrofuran (PTHF) it can be described in terms of vaporisation only. 

If the plastic deformation is non-uniform, then the permanent change of the shape of the volume elements are generally incompatible. This implies the existence of some additional elastic deformations E, associated witli self stresses which maintain the body continuity. Consequently, the elastic deformation characterizes the deformation of the crystalline lattice and thus aquires a non-permanent, reversible character. Hence, in the model we obtain a multiplicative decomposition into E and P and we assume that a relaxation is possible. 

The basic framework for Ihe Hybrid Model (HM) has evolved from its initial formulation [10] into a more complex constitutive model, the augmented Hybrid Model [11], which has been shown to predict, with great accuracy, the behavior of UHMWPE specimens subjected to several different loading scenarios. The model was developed based on the micromechanical behavior of sarticrystalline polymers at the molecular level. At the core of the augmented HM is the multiplicative decomposition of the applied deformation gradient (F) into two separate components—elastic... 

In 2009, Xu et al. completed a comprehensive study of the C2H5O potential energy surface, reporting on the decomposition and isomerization reactions of all three C2H5O radicals and reviewing the previous hterature. Multiple decomposition paths were found for all species, with a main channel accounting for >75% of reaction for each species at moderate to h h pressures and temperatures below about 800 K the secondary processes were predicted to increase in importance at higher temperatures and lower... 

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