Anisotropic materials, deformations

The way in which a fully anisotropic material deforms when subjected to loading is much different than the deformation characteristics of isotropic materials. Consider, for example, a tensile specimen cut from a unidirectional (orthotropic) lamina. If the loading axis is parallel to one of the material axes (i.e., parallel or... 

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

Note 4 For an anisotropic material, // varies with the direction of the uniaxial deformation. 

The complete stress-strain relation requires the six as to be written in terms of the six y components. The result is a 6 x 6 matrix with 36 coefficients in place of the single constant, Twenty-one of these coefficients (the diagonal elements and half of the cross elements) are needed to express the deformation of a completely anisotropic material. Only three are necessary for a cubic crystal, and two for an amorphous isotropic body. Similar considerations prevail for viscous flow, in which the kinematic variable is y. 

As in the simple 3- or 4-point bending of a beam the vibrating reed device assumes the validity of the differential Eq. (2.2) which is due to Euler. Timoshenko25 included both rotary inertia and shear deformation deriving a more exact differential equation which reduces to the Euler equation as a special case. Use of the Timoshenko beam theory for anisotropic materials has been made by Ritchie et al.26 who derive a pair of equations for torsion-flexure coupling (which will always occur unless the axis of the beam coincides with the symmetry axis of the anisotropic material). 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

Table 1.13.1 Matrix entries for deformation of anisotropic materials...

The model used consisted of a half helmet with three element layers of composite material through the thickness. A half model was used to take advantage of the symmetry present in the problem. The elements used were eight-noded brick elements because these elements can represent the deformation sufficiently. Anisotropic material properties were defined for each element using a user subroutine and impact was designated at the front of the helmet. 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

These methods are able to analyze linear and nonlinear material behavior, isotropic or anisotropic, large deformations, etc. 

In addition to the strip tests, two-dimensional bursting tests should be carried out for the initial approval. The difficulties in carrying out bursting tests occur because the fabrics are anisotropic materials. This also means that the deformation shape only approximates to a calotte (clam) shell and the determination of the failure load has many elements of uncertainty. The problem has to be investigated further in future scientific work. 

In complex deformations, involving either unsteady strain or out of plane loading, fabrics are treated as two-dimensional continuum sheets or membranes, considered as anisotropic materials. Such materials are usually composed of viscoelastic building blocks, often revealing large displacement instead of small displacement and therefore rarely completely conform to the mles of elasticity. 

In many engineering applications short glass fiber reinforcement is added to the material. This has the advantage of increasing the strength and modulus of the material. As the material flows into the mold, the fluid deformation and interaction with other fibers alters the orientation of the fibers. The final orientation state will depend on the processing history of the material and may lead to highly anisotropic material properties. 

Axial compression of high modulus polymeric filaments results in the formation of so-called kinkbands. These are regions of sub-micron thickness where the compressive defonnation is concentrated. This mode of failure is characteristic for anisotropic materials. The processes involved in kinkband formation are not yet well understood. In this work the kinkband formation in single aramid filaments is measured as a function of the applied compressive strain. Above a critical compressive strain the kinkband density initially increases yery rapidly until eventually a maximum density is obtained. The experimental results are compared with an elastic stability model. The kinkbands form before elastic instability occurs and are therefore attributed to a plastic deformation process. A model is developed to describe the kinkband density as a function of the applied strain. 

Ten Thije, R., Akkerman, R., Huetink, J., 2007. Large deformation simulation of anisotropic material using an updated Lagrangian finite element method. Comput. Methods Appl. Mech. Eng. 196, 3141-3150. 

The highly anisotropic nature of PDA films allows us to show that in-plane properties of materials can be observed using IC-AFM. This is due to the tilt of the AFM cantilever which produces a small but significant in-plane component to the tip s motion. In the case of PDA monolayers, in-plane Action and shear deformation anisotropy leads to contrast in the IC-AFM phase image. The results can be explained using a sinq)le model that incorporates Hertzian contact mechanics with in-plane dissipation, and may be generalized to the study of other anisotropic materials. 

In most cases, the flow properties of polymers in solution or in a molten state are Newtonian, pseudoplastic, or a combination of both. In the case of liquid crystal polymer solutions, the flow behavior is more complex. The profound difference in the rheological behavior of ordinary and liquid crystalline polymers is due to the fact that, for the flrst ones, the molecular orientation is entirely determined by the flow process. The second ones are anisotropic materials already at equilibrium (Acierno and Brostow 1996). The spontaneous molecular orientation is already in existence before the flow and is switched on, varying in space, over distances of several microns or less (polydomain). If one ignores the latter, one can discuss the linear case (slow flow) as long as the rate of deformation due to flow (the magnitude of the symmetric part of the velocity gradient) is lower than the rate at which molecules rearrange their orientational spread by thermal motions. 

The adherends do not deform in shear, implying that the shear modulus of the adherends is much greater than that of the adhesive. This assumption becomes especially suspect with anisotropic materials such as wood- or fiber-reinforced composites. 

Let us consider the possible deformations (splay, twist and bend) of anisotropic materials. As illustrated in Figure C.l, without loosing generality we choose the coordinate system so tirat the z-axis be parallel to the imdistorted director field (iiIIz ). 

The inconsistency in the strain components in the a-plane QD samples reported in [57, 58] indicates a significant impact of the assumptions made in their estimation from the phonon frequencies. We note that in aU estimations of the strain components, the anisotropic phonon deformation potentials chj(to) and ce were assumed to be zero [57]. The results of the anisotropically strained GaN films presented in Sections 9.4.2 and 9.4.3 show clear splittings of the El and 2 indicating nonzero anisotropic deformation potentials for these phonon modes (Tables 9.4 and 9.5). We point out, however, that the values of the C j(xo) and c in Ref. 17 may be also affected by the assumptions made in the strain component determination by XRD (for instance the deviation from hexagonal symmetry is assumed to be small). The studies of the vibrational properties of nitride materials with nonpolar surface orientations are scarce and clearly, further experimental and theoretical investigations are needed to clarify these issues. 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

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