Deformation compatibility

The fundamental analysis of a laminate can be explained, in principle, by use of a simple two-layered cross-ply laminate (a layer with fibers at 0° to the x-direction on top of an equal-thickness layer with fibers at 90° to the x-direction). We will analyze this laminate approximately by considering what conditions the two unbonded layers in Figure 4-3 must satisfy in order for the two layers to be bonded to form a laminate. Imagine that the layers are separate but are subjected to a load in the x-direction. The force is divided between the two layers such that the x-direction deformation of each layer is identical. That is, the laminae in a laminate must deform alike along the interface between the layers or else fracture must existl Accordingly, deformation compatibility of layers is a requirement for a laminate. Because of the equal x-direction deformation of each layer, the top (0°) layer has the most x-direction ress because it is stiffer than the bottom (90°) layer in the x-direction./ Trie x-direction stresses in the top and bottom layers can be shown to have the relation... 

That is, although we required equal x-direction displacements of the two layers (the proportion of N in each layer was adjusted to create that equal-displacement condition), the lateral displacfiments arfi guitfijjjffer-ent. Those different displacements are a violation of the requireddeToT- mation compatibility of laminae in a laminatir Tolemedy this violation, the top layer must get wider by application of a lateral tensile stress aj, and the bottom layer must get narrower by application of a compressive stress Oy. The two deformations must result in equal-width laminae to satisfy deformation compatibility. Moreover, the lateral stresses in each layer must satisfy force equilibrium in the y-direction, i.e.,... 

DEFORMATION-COMPATIBLE LAMINATE WITH STRESSES FROM CLT... 

The deformation compatibility at the interface requires that the deformation gradient tensors of each phase be used in Equation (5.18). To take into account the molecular alignment in the amorphous phase, the deformation gradient of the amorphous phase is incorporated as follows ... 

Deformation compatibility of the foundations and bridge structure is an important consideration. Different types of foundation may behave differently therefore, same type of foundations should be used for one section of bridge structure. Diameter of the piles and inclined piles are two important factors to be considered in terms of deformation compatibility and are discussed in the following. 

It is worth to note that conventional beam theory cannot provide a reliable prediction for neither of the above structural systems. Uang et al. (2000) and Yu et al. (2000) proposed a simplified model that cmisiders the interaction of forces and deformation compatibility between the beam and the haunches. 

O Figure 29.3 shows the load equilibrium and deformation compatibility for the dynamic Volkersen s model of a lap-strap joint having a half-infinite length. The joint configuration has only one discontinuity at the edge of the joint, and its boundary condition is easier to treat than... 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

The wave functions for the two inner-core spherons can, of course, be described as the symmetric and antisymmetric combinations of l.t and Ip-functions. The Nilsson (19) treatment of neutron and proton orbitals in deformed nuclei is completely compatible with the foregoing discussion, which provides a structural interpretation of it. 

An interesting feature of polarized IR spectroscopy is that rapid measurements can be performed while preserving molecular information (in contrast with birefringence) and without the need for a synchrotron source (X-ray diffraction). Time-resolved IRLD studies are almost exclusively realized in transmission because of its compatibility with various types of tensile testing devices. In the simplest implementation, p- and s-polarized spectra are sequentially acquired while the sample is deformed and/or relaxing. The time resolution is generally limited to several seconds per spectrum by the acquisition time of two spectra and by the speed at which the polarizer can be rotated. Siesler et al. have used such a rheo-optical technique to study the dynamics of multiple polymers and copolymers [40]. 

Most probable positions of the chains are determined by the use of a characteristic vector r. This vector is representative of an average network chain of N links (the average links per chain). It deforms affinely whereas the actual network chains might not, and its value depends only upon network deformation. Crystallization leaves r essentially unaltered since the miniscule volume contraction brought about by crystallization can be ignored. But real network chains are severely displaced by crystallization. These displacements, however, must be compatible with the immutability of r. So in a sense, the characteristic vector r limits the configurational variations of the chains to those consistent with a fixed network shape and size at a given deformation. 

In this model, idealized skeletons are assumed, deformations of the skeleton by ligands being neglected. The ligands have a static or dynamic symmetry about their skeletal bond axes which is compatible with the skeletal symmetry. Here it should be noted that the conceptual dissection of molecules into skeleton and ligands has been a standard procedure developed by stereochemists quite some time ago. 

While conductivities of nanocarbons dispersed in polymers fall short of those of metals, a variety of applications can be unlocked by turning an insulating matrix into a conductor, which requires only small volume fractions that can therefore keep the system viscosity at a level compatible with composite processing techniques. Of particular interest are novel functionalities of these conductive matrices that exploit the presence of a conductive network in them, such as structural health monitoring (SHM) based on changes in electrical resistance of the nanocarbon network as it is mechanically deformed [30]. 

---