Stiffnesses reduced

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

Tsai and Pagano [2-7] ingeniously recast the stiffness transformation equations to enable ready understanding of the consequences of rotating a lamina in a laminate. By use of various trigonometric identities between sin and cos to powers and sin and cos of multiples of the angle, the transformed reduced stiffnesses. Equation (2.85), can be written as... 

The reduced stiffnesses, Qy, are defined in terms of the engineering constants in Equation (2.66). In any other coordinate system in the plane of the lamina, the stresses are... 

The laminate is now degraded to the point where the outer layers carry stress only in the x-direction and the inner layers can carry stress only in the y-direction. In both cases, the stress is parallel to the fibers. Thus, the laminate is completely decoupled, both thermally and mechanically. The only nonzero reduced stiffnesses are... 

The example considered to illustrate the strength-analysis procedure is a three-layered laminate with a [4-15°/-15°/+15°] stacking sequence [4-10]. The laminae are the same E-glass-epoxy as in the cross-ply laminate example with thickness. 005 in (.1270 mm), so that the total laminate thickness is. 015 in (.381 mm). In laminate coordinates, the transformed reduced stiffnesses are... 

In their pioneering paper on laminated plates, Reissner and Stavsky investigated an approximate approach (in addition to their exact approach) to calculate deflections and stresses for antisymmetric angie-ply laminated plates [5-27]. Much later, Ashton extended their approach to structural response of more general unsymmetrically laminated plates and called it the reduced stiffness matrix method [5-28]. The attraction of what is now called the Reduced Bending Stiffness (RBS) method is that an unsymmetrically laminated plate can be treated as an orthotropic plate using only a modified D matrix in the solution, i.e.,... 

The topic of invariant transformed reduced stiffnesses of orthotropic and anisotropic laminae was introduced in Section 2.7. There, the rearrangement of stiffness transformation equations by Tsai and Pagano [7-16 and 7-17] was shown to be quite advantageous. In particular, certain invariant components of the lamina stiffnesses become apparent and are heipful in determining how the iamina stiffnesses change with transformation to non-principal material directions that are essential for a laminate. 

Also, relate this invariant to the reduced stiffness invariant Qgg - Q,2-... 

To reduce stiffness at the beginning, an appropriate initial value of the steam density is calculated in the FORTRAN subroutine START, which uses the halfinterval method for the non-linear algebraic equation. Note that the execution may be very slow because of equation stiffness. Increasing the value of CINT during the initial heating period may terminate ISIM execution. 

NSAIDs act primarily by inhibiting prostaglandin synthesis, which is only a small portion of the inflammatory cascade. They possess both analgesic and antiinflammatory properties and reduce stiffness but do not slow disease progression or prevent bony erosions or joint deformity. They should seldom be used as monotherapy for RA instead, they should be viewed as adjuncts to DMARD treatment. Common NSAID dosage regimens are shown in Table 4-4. 

Finally, the use of low-molecular-weight species to improve flow properties called plasticizers normally reduces stiffness, hardness, and brittleness. Plasticization is usually restricted to amorphous polymers or polymers with a low degree of crystallinity because of the limited compatibility of plasticizers with highly crystalline polymers. Other additives, such as antioxidants, do not affect the mechanical properties significantly by themselves, but can substantially improve property retention over long periods of time, particularly where the polymer is subject to environmental degradation. 

Equation (5.7) related the stress tensor to the strains throngh the modulus matrix, [Q, also known as the reduced stiffness matrix. Oftentimes, it is more convenient to relate the strains to the stresses through the inverse of the rednced stiffness matrix, [2] called the compliance matrix, [5]. Recall from Section 5.3.1.3 that we introduced a quantity called the compliance, which was proportional to the inverse of the modulus. The matrix is the more generalized version of that compliance. The following relationship then holds between the strain and stress tensors ... 

Here, the array [g]/ is the transformed reduced stiffness matrix, and each gy term can be related to the gy terms of the reduced stiffness matrix, [g], through the angle, 6. 

Enhancement of mechanical properties is of interest only if it is not accompanied by a loss of other important properties of the blend. Of particular concern for such polymer blends is stiffness, because most means of increasing impact strength also reduce stiffness (14-19). But this is not the case for the iPS-fc-iPP-iPS-iPP blends studied here as seen in Table II. It is clear that the enhancement in toughness just described is not accompanied by a loss of stiffness, but it is essentially unaffected by the compatibilizer. And the stiffness of iPS-fc-iPP-iPS-iPP is higher than that of iPP and HIPS. The impact-modulus behavior seems to be due to the tough (or rigid) characteristics, morphologies of phases, and semicrystalline isotactic structure of each block in the iPS-b-iPP diblock copolymer. 

Cheng et al. [1990] refer to several possible polymeric blends, consisting of fluorinated and non-fluorinated, elastomeric and non-elastomeric, polymers and copolymers, which may also contain crosslinking agents and other additives, for use in cable jackets and heat-recoverable e.g., heat-shrinkable) articles. They point out that PVDF, and copolymers of ethylene and tetrafluo-roethylene (ETFE) have many desirable properties but their high stiffness can be undesirable for certain applications. However, their blends with some thermoplastic elastomers show markedly reduced stiffness, while substantially retaining their other desirable properties. Cheng et al. 

Both the Goto et al. [35] and the Lorenzini et al. [34] models are well suited to fit the experimental long-chain branching index /lcb (number of branches per 1000 chain carbon atoms), the measured Afw, and also the full MWD. In LDPE modeling, the scheme of Goto et al., eqs (4.6-9) and (4.6-10), is frequently favored because of a reduced stiffness of the differential equations associated with this approach. If the type and the concentration of branched macroradicals need to be specified, as is desirable in copolymerization modeling, the scheme of Lorenzini et al. [34] appears to be more suitable. 

The saturated acid (phthalic anhydride) helps to reduce the cross-link density and, hence, the brittleness of the cured polyester resin. Resin composition can be varied so that product properties can be tailored to meet specific end-use requirements. For example, a resin with enhanced reactivity and improved stiffness at high temperatures is obtained by increasing the proportion of unsaturated acid. On the other hand, a less reactive resin with reduced stiffness is obtained with a higher proportion of the saturated acid. 

The reported study demonstrates that it is possible to increase Izod impact strength without reducing stiffness. This can be done by the use of high levels of acrylic impact modifier in combination with ultrafine talc. The modifier imparts impact strength, while the talc maintains stififiiess that would have been lost if the impact modifier was used alone. 

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