Strain elastic, structural

The non-linear response of plastic materials is more challenging in many respects than pseudoplastic materials. While some yield phenomena, such as that seen in clay dispersions of montmorillonite, can be catastrophic in nature and recover very rapidly, others such as polymer particle blends can yield slowly. Not all clay structures catastrophically thin. Clay platelets forming an elastic structure can be deformed by a finite strain such that they align with the deforming field. When the strain... 

Since the stiffness of the bonds transfers to the stiffness of the whole filler network, the small strain elastic modulus of highly filled composites is expected to reflect the specific properties of the filler-filler bonds. In particular, the small strain modulus increases with decreasing gap size during heat treatment as observed in Fig. 32a. Furthermore, it exhibits the same temperature dependence as that of the bonds, i.e., the characteristic Arrhenius behavior typical for glassy polymers. Note however that the stiffness of the filler network is also strongly affected by its global structure on mesoscopic length scales. This will be considered in more detail in the next section. 

Accordingly, we expect a power law behavior G,0 (O/Op)3 5 of the small strain elastic modulus for 0>0. Thereby, the exponent (3+df [j)/(3—df)w3.5 reflects the characteristic structure of the fractal heterogeneity of the filler network, i.e., the CCA-clusters. The strong dependency of G 0 on the solid fraction Op of primary aggregates reflects the effect of structure on the storage modulus. 

There have been a number of studies that demonstrate that crystallized AMF and butter exhibit linear (ideal) viscoelastic behavior at low levels of stress or strain (4), where the strain is directly proportional to the applied stress. For most materials, this region occurs when the critical strain (strain where structure breaks down) is less than 1.0%, but for fat networks, the strains typically exceed 0.1% (4, 66). Ideally, within the LVR, mUkfat crystal networks will behave like a Hookean solid where the stress is directly proportional to the strain (i.e., a oc y), as shown in Figure 15 (66, 68). Within the elastic region, stress will increase linearly with strain up to a critical strain. Beyond that critical strain (strain at the limit of linearity), deformation of the network will occur at a point known as the yield point. The elastic limit quickly follows, beyond which permanent deformation and sample fracture occurs. Beyond these points, the structural integrity of the network is compromised and the sample breaks down. 

Representation of the Nonlinear "Dynamic Viscoelasticity in Terms of the Effects of Nonlinear Elasticity Strain Rjote-Dependent Viscosityy and Reversible Strain- lnduced Structural Change... 

The above analysis shows that the nonlinear dynamic viscoelastic behavior of polymers can be resolved into three components the nonlinear elasticity resulting from the variation of modulus with the phase angle or strain during the cycle nonlinear internal friction resulting from strain and strain-rate dependence and eflFects associated with the reversible, strain-induced structural changes. 

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

Hadley, D. W. (1975) Small strain elastic properties, in Structure and Properties of Oriented Polymers, edited by Ward, I. M., New York Halsted Press, J. WUey Sons, pp. 290-325. 

Structure has a low-strain elastic response such that its behavior is predictable under all design loadings. 

Freund, L. B. (1995a), Evolution of waviness on the surface of a strained elastic solid due to stress-driven diffusion. International Journal of Solids and Structures 32, 911-923. 

For linear elastic structures under static loads, the conventional finite-element solutions for the nodal displacements, u, strains, e, and stresses, cr, are expressed by... 

Gere, J. M. 2004. Mechanics of Materials, 6th ed. London Brooks/Cole. Describes the fundamentals of mechanics of materials. Principal topics are analysis and design of structural members subjected to tension, compression, torsion, and bending as well as stress, strain, elastic behavior, inelastic behavior, and strain energy. Transformations of stress and strain, combined loadings, stress concentrations, deflections of beams, and stability of columns are also covered. Includes many problem sets with answers in the back. 

Generally viscoelastic problems can be solved using relations between internal stresses and external loads subject to the geometry of the structure in a similar manner as for elastic materials in the subject areas mentioned above. For both elastic and viscoelastic materials, the state of the material or equations of state must be included. Here elastic and viscoelastic materials are different in that the former does not include memory (or time dependent) effects while the latter does include memory effects. Because of this difference, stress, strain and displacement distributions in polymeric structures are also usually time dependent and may be very different from these quantities in elastic structures under the same conditions. 

The stress and strain in an elastic structure may vary with time providing external loads vary with time. Therefore, it is possible to transform time dependent stresses and strains for elastic structures to give. 

Lee, J.D., "Finite Element Procedures for Large Strain Elastic-Plastic Theories", Computers Structures, 28, pp.395-406, 1988. 

The distinction is quantitatively significant for an infinite homogeneous Poisson solid, A,V = (5/9) A V. Furthermore, the stiffness seen by the source, and thus the true strain, depends on the elasticity structure outside the source volume. 

These techniques have very important applications to some of the micro-structural effects discussed previously in this chapter. For example, time-resolved measurements of the actual lattice strain at the impact surface will give direct information on rate of departure from ideal elastic impact conditions. Recall that the stress tensor depends on the elastic (lattice) strains (7.4). Measurements of the type described above give stress relaxation directly, without all of the interpretational assumptions required of elastic-precursor-decay studies. 

Although the initial elastic and the primary creep strain cannot be neglected, they occur quickly, and they can be treated in much the way that elastic deflection is allowed for in a structure. But thereafter, the material enters steady-state, or secondary creep, and the strain increases steadily with time. In designing against creep, it is usually this steady accumulation of strain with time that concerns us most. 

The information presented in this work builds upon developments from several more established fields of science. This situation can cause confusion as to the use of established sign conventions for stress, pressure, strain and compression. In this book, those treatments involving higher-order, elastic, piezoelectric and dielectric behaviors use the established sign conventions of tension chosen to be positive. In other areas, compression is taken as positive, in accordance with high pressure practice. Although offensive to a well structured sense of theory, the various sign conventions used in different sections of the book are not expected to cause confusion in any particular situation. 

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