Three-dimensional reduced stiffness

Two-dimensional off-axis reduced stiffness for twisted yarn was developed to three-dimensional reduced stiffness. As described earlier, if the migration structure in the twisted yarn is stationary along the yarn axis, in other words, the yarn axis is the same as the loading axis, the Young s modulus can be calculated through Eq. (10.14) as the orientation angle of each cylindrical element is invariable. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

The fundamental problem involved is always one of data quality, and consequently the back analysis approach must be applied with care and the results interpreted with caution. Back analysis is of use only if the soil conditions at failure are unaffected by the failure. For example, back-calculated parameters for a first-time slide in stiff overconsolidated clays could not be used to predict subsequent stability of the sliding mass, since the shear strength parameters will have been reduced to their residual values by the failure. In such cases, an assumption of c = 0 and the use of a residual friction angle ip is warranted (Bromhead, 1992). If the three-dimensional geometrical effects are important for the failed slope under consideration and a two-dimensional back analysis is performed, the back-calculated shear strength will be too high and thus unsafe. 

As shown in Figure 10.11, the theoretical and experimental results were compared at 30° off-axis angle. As can be seen, the Young s modulus of PLA composites is in good agreement with the theoretical results. Thus, it is concluded that the Young s modulus can be predicted through three-dimensional off-axis reduced stiffness for twisted yarn composites. 

This article has investigated the potential of cross-linked polyamides for tribological applications. Results have shown a significant influence of the structural properties decrease in crystallinity, cross-linking particularly within quasi-amorphous or less crystalline ranges and the formation of a three-dimensional network. This shows up in the macroscopic material behavior increased insolubility, increased elastic modulus, particularly in the glass transition region, increased stiffness, brittleness, reduced creep and an increase in surface polarity. 

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