Composite materials, stress

The preceding expressions, Equations (3.84) through (3.89), are more easily understood when they are plotted as in Figure 3-48. There, the composite material strength (i.e., the maximum composite material stress) is plotted as a function of the fiber-volume fraction. When V, is less than the composite material strength is controlled by the... 

The maximum composite material stress (i.e., the strength) is then... 

The FRC specimens were tested using the same testing procedure as described above. Optimal composite material stress-strain behaviour was obtained with the specimens referred to... 

Alexander, C., Bedoya, J., Repair of dents subjected to cychc pressure service using composite materials. Stress Engineering Services, Inc., Proceedings of the 8th International Pipehne Conference IPC2010, Calgary, September 27—October 1, 2010. 

More recently, Raman spectroscopy has been used to investigate the vibrational spectroscopy of polymer Hquid crystals (46) (see Liquid crystalline materials), the kinetics of polymerization (47) (see Kinetic measurements), synthetic polymers and mbbers (48), and stress and strain in fibers and composites (49) (see Composite materials). The relationship between Raman spectra and the stmcture of conjugated and conducting polymers has been reviewed (50,51). In addition, a general review of ft-Raman studies of polymers has been pubUshed (52). 

The design of smart materials and adaptive stmctures has required the development of constitutive equations that describe the temperature, stress, strain, and percentage of martensite volume transformation of a shape-memory alloy. These equations can be integrated with similar constitutive equations for composite materials to make possible the quantitative design of stmctures having embedded sensors and actuators for vibration control. The constitutive equations for one-dimensional systems as well as a three-dimensional representation have been developed (7). 

K. Kabe, M. Koishi, and T. Akasaka, "Stress Analysis for Twisted Cord and mbbet of FRR," presented at 6th Japan—U.S. Conference on Composite Materials, Odando, Fla., June 1992. 

A composite material used for rock-drilling bits consists of an assemblage of tungsten carbide cubes (each 2 fcm in size) stuck together with a thin layer of cobalt. The material is required to withstand compressive stresses of 4000 MNm in service. Use the above equation to estimate an upper limit for the thickness of the cobalt layer. You may assume that the compressive yield stress of tungsten carbide is well above 4000 MN m , and that the cobalt yields in shear at k = 175 MN m . What assumptions made in the analysis are likely to make your estimate inaccurate ... 

The inherent anisotropy (most often only orthotropy) of composite materials leads to mechanical behavior characteristics that are quite different from those of conventional isotropic materials. The behavior of isotropic, orthotropic, and anisotropic materials under loadings of normal stress and shear stress is shown in Figure 1-4 and discussed in the following paragraphs. 

Several experiments will now be described from which the foregoing basic stiffness and strength information can be obtained. For many, but not all, composite materials, the stress-strain behavior is linear from zero load to the ultimate or fracture load. Such linear behavior is typical for glass-epoxy composite materials and is quite reasonable for boron-epoxy and graphite-epoxy composite materials except for the shear behavior that is very nonlinear to fracture. 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

For a unidirectionally reinforced composite material subject to uniaxial load at angle 0 to the fibers (the example problem in Section 2.9.1 on the maximum stress criterion), the allowable stresses can be found from the allowable strains X, Y , etc., in the following manner. 

As with the maximum stress failure criterion, the maximum strain failure criterion can be plotted against available experimental results for uniaxial loading of an off-axis composite material. The discrepancies between experimental results and the prediction in Figure 2-38 are similar to, but even more pronounced than, those for the maximum stress failure criterion in Figure 2-37. Thus, the appropriate failure criterion for this E-glass-epoxy composite material still has not been found. 

Finally, for the off-axis composite material example of this section, substitution of the stress-transformation equations. 

The Tsai-Hill failure criterion appears to be much more applicable to failure prediction for this E-glass-epoxy composite material than either the maximum stress criterion or the maximum strain failure criterion. Other less obvious advantages of the Tsai-Hill failure criterion are ... 

For E-glass-epoxy, the Tsai-Hill failure criterion seems the most accurate of the criteria discussed. However, the applicability of a particular failure criterion depends on whether the material being studied is ductile or brittle. Other composite materials might be better treated with the maximum stress or the maximum strain criteria or even some other criterion. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

N. J. Pagano and P. C. Chou, The Importance of Signs of Shear Stress and Shear Strain in Composites, Journal of Composite Materials, January 1969, pp. 166-173. 

Robert M. Jones ar Harold S. Morgan, Analysis of Nonlinear Stress-Strain Behavior of Fiber-Reinforced Composite Materials, AIAA Journal, December 1977, pp. 1669-1676. 

The average stress acts on cross-sectional area A of the representative volume element, oj acts on the cross-sectional area of the fibers Af, and acts on the cross-sectional area of the matrix A. Thus, the resultant force on the representative volume element of composite material is... 

The apparent Young s modulus, E2, of the composite material in the direction transverse to the fibers is considered next. In the mechanics of materials approach, the same transverse stress, 02, is assumed to be applied to both the fiber and the matrix as in Figure 3-9. That is, equilibrium of adjacent elements in the composite material (fibers and matrix) must occur (certainly plausible). However, we cannot make any plausible approximation or assumption about the strains in the fiber and in the matrix in the 2-direction. 

The nonlinear shear stress-shear strain behavior typical of fiber-reinforced composite materials is ignored, i.e., the behavior is regarded as linear. 

In a uniaxial tension test to determine the elastic modulus of the composite material, E, the stress and strain states will be assumed to be macroscopically uniform in consonance with the basic presumption that the composite material is macroscopically Isotropic and homogene-ous. However, on a microscopic scSeTBotFTfhe sfre and strain states will be nonuniform. In the uniaxial tension test,... 

We know full well that such j fprrnstress state carmot exist throughout the composite material, yefw seek the implication of such an approximation. The strain energy for the stresses in Equation (3.41) is... 

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