Rule isostrain

Li and Chou [73,74] have reported a multiscale modeling of the compressive behavior of CNT/polymer composites. The nanotube is modeled at the atomistic scale, and the matrix deformation is analyzed by the continuum FEM. The nanotube arrd polymer matrix are assttmed to be bonded by vdW interactions at the interface. The stress distributiorrs at the nanotube/polymer interface under isostrain and isostress loading conditiorrs have been examined. They have used beam elements for SWCNT using molectrlar structural mechanics, truss rod for vdW links and cubic elements for matrix. The rule of mixtrrre was used as for comparision in this research. The buckling forces of nanotube/polymer composites for different nanotube lengths and diameters are computed. The results indicate that continuous nanotubes can most effectively enhance the composite buckling resistance. 

The rule of mixtures equations have several drawbacks. The isostrain assumption in the Voight model implies strain compatibility between the phases, which is very unlikely because of different Poisson s contractions of the phases. The isostress assumption in the Reuss model is also unrealistic since the libers cannot be treated as a sheet. Despite this, these equations are often adequate to predict experimental results in unidirectional composites. A basic limitation of the rule of mixtures occurs when the matrix material yields, and the stress becomes constant in the matrix while continuing to increase in the fiber. 

Failure starts when the stress in matrix or fibre reaches the yield or tensile strength. Similar to the elastic case, we can apply an isostrain rule of mixtures (equation (9.2)) to calculate the stress in the composite ... 

In component design, the maximum stress in the material is usually the quantity of interest. If the fraction of fibres is so large that the matrix cannot bear a given load after the fibres have fractured (this is the case in figure 9.3), the failure stress is given by the isostrain rule of mixtures, equation (9.7), where Cf is the failure stress of the fibres and failure strain of the fibres. If the volume fraction of the fibres is small, the matrix can still bear the load even after the fibres have fractured. In this case, the failure stress of the composite is (1 — /f)<7m, with am being the failure stress of the matrix. The failure stress is thus reduced, compared to the pure matrix material. ... 

The failure strain of the fibre may also be larger than that of the matrix in some cases, for example in carbon-fibre reinforced duromers or in ceramic matrix composites. After the strain has exceeded the failure strain of the matrix, the complete load has to be borne by the fibres. Similar to the previous case, the maximum stress in the composite depends on the volume fraction of the fibres. If it is sufEciently large, the fibres do not break but can take a load of /tat. If the volume fraction is too small, the maximum stress is again determined by the isostrain rule of mixtures, equation (9.7), but now taking am as failure stress of the matrix and at as the stress in the fibre at the failure strain in the matrix. In ceramic matrix composites, the matrix frequently does not fail completely, but forms many small cracks bridged by the fibres. The stress-strain curve for this case will be discussed in section 9.3.3. 

The stress a is averaged over fibre and matrix and will not occur at any point in the component. AtjA and Am/A are the area fractions of fibre and matrix and thus equal the volume fractions ft and fm because the fibres extend throughout the component. If we insert ft and /m = 1 — /t into the equation, we find the isostrain rule of mixtures, equation (9.2) ... 

When >oo, the Halpin-Tsai model equations reach the upper bound, which is normally called Voigt rule of mixtures (ROM) [54] where fiber and matrix have the same uniform strain (i.e., isostrain approach) ... 

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