Composite isostrain

On the other hand, ultimate properties are largely used to evaluate the adhesion and the chemical treatment efficiency but, again, the relevant theories are in an elementary stage they often consider the inclusion of one filler only and assume isostrain behaviour of the two phases and a a, e linear behaviour until the ultimate properties are reached in the case of brittle composite materials. In the case of thermoplastics, the yield point appears to substitute adequately, with the same approximation, the break point. 

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. 

In general, SMPF is perceived as a two-phase composite material with a crystalline phase mixed with an amorphous phase. A multiscale viscoplasticity theory is developed. The amorphous phase is modeled using the Boyce model, while the crystalline phase is modeled using the Hutchinson model. Under an isostrain assumption, the micromechanics approach is used to assemble the microscale RVE. The kinematic relation is used to link the micro-mechanics constitutive relation to the macroscopic constitutive law. The proposed theory takes into account the stress induced crystallization process and the initial morphological texture, while the polymeric texture is updated based on the apphed stresses. The related computational issue is discussed. The predictabihty of the model is vahdated by comparison wifli test results. It is expected that more accurate measurement of the stress and strain in the SMPF with large deformation may further enhance the predictability of the developed model. It is also desired to reduce the number of material parameters in the model. In other words, a deeper understanding and physics based theoretical modeling are needed. 

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. 

Stress Shielding. Beyond the traditional biocompatibility issues, hard tissue biomaterials must also be designed to minimize a phenomenon known as stress shielding. Due to the response of bone remodeling to the loading environment, as described by Wolffs law, it is important to maintain the stress levels in bone as close to the preimplant state as possible. When an implant is in parallel with bone, such as in a bone plate or a hip stem, the engineered material takes a portion of the load— which then reduces the load, and as a result, the stress, in the remaining bone. When the implant and bone are sufficiently well bonded, it can be assumed that the materials deform to the same extent and therefore experience the same strain. In this isostrain condition, the stress in one of the components of a two-phase composite can be calculated from the equation ... 

When using the creep model for duplex microstructure, as developed by French et al. [98], it is possible to fabricate a composite with a defined creep resistance that will control the grain size, the width of the layers, and the compression axis. In the case of isostrain, where the strain and strain rate are the same for each phase, and assuming the creep equation in a compact form as, e = Ajo the stress for this configuration, o, is given as ... 

Prediction of the modulus of a short-fiber composite needs to take into account end effects since isostrain conditions are not satisfied at the fiber ends. Stress builds up along each fiber from zero at its end to a maximum at its center. As shown in Figure 1.4, at the interface, the matrix is severely sheared at the fiber... 

Loading parallel to fibers. First, if we applied an overall load, F, in N on the composite along the direction of the fibers, this load is carried either by the fibers and the matrix. Moreover, assuming a good bond between matrix and fibers, both stretch similarly and if the loading is isostrain, i.e., all strains are equals, we have ... 

It is important to note that isostress and isostrain loading conditions represent theoretical limits for the design of a composite material reinforced by continuous fibers. In practice, most of the time, mechanical performances fall between these limits. On the other hand, in the isostrain loading situation, a lower volume fraction of fibers is required to obtain a similar stiffness of the composite. 

Effective material coefficients for the hydrophone application of 1-3 composite (Smith 1983) could be estimated in isostress/isostrain approximation as a function of the ceramic volume content v... 

Effective material properties could be easily calculated using the isostress/isostrain approximation (see parallel connection of phases in Eqs.(7.21), (7.22), and (7.23)). More complicated estimates include also non-homogeneous stress/strain distribution over each phase (Cao et al. 1993). Effective piezoelectric coefficient depends on the aspect ratio (i.e. on the ratio of ID-period length vs. composite thickness). It is much smaller than ceramic s J33 for low ceramics content in PZT/epoxy composite (Cao et al. 1993). On the contraiy, the effective uniaxial figure of merit is higher for aspect ratios lower than 1 and... 

The maximum stiffness is obtained when a uniaxial stress is applied parallel with the layers, as indicated in Figure 8.1. ft is assumed that the strain is the same in all the composite layers, a form of loading known as the isostrain (or homogeneous strain) condition. 

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. 

Let us now consider the elastic behavior of a continuous and oriented fibrous composite that is loaded in the direction of fiber ahgnment. First, it is assumed that the fiber-matrix interfacial bond is very good, such that deformation of both matrix and fibers is the same (an isostrain situation). Under these conditions, the total load sustained by the composite is equal to the sum of the loads carried by the matrix phase and the fiber phase Ff, or... 

The mechanical properties (modulus, tensile strength and strain at break) as well as the electrical conductivity of the various monofilaments are shown in Figure V.1. The data clearly indicate that the tenacity and modulus systematically increase with increasing PPTA content. In fact, a simple, nearly linear, relation was observed between the mechanical properties and the fiber composition. This behavior is in accord with that generally found for composites loaded under isostrain conditions [71]. The enhancement of the mechanical properties with increased PPTA (not unexpectedly) came at the expense of the electrical conductivity of the polyblend fibers. 

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