Strain-concentration tensor

Estimates for the macroscopic drained stiffness tensor Chom(f) as a function of the morphological parameter can be derived from various micromechanical techniques. The micromechanical approach classically refers to the concept of strain concentration tensor, denoted here by A. By definition, in an evolution... 

Let us consider the case where the pore space is a network of saturated cracks. In order to implement the classical micromechanical estimates of the strain concentration tensor A introduced in (14), the cracks are modelled as flat oblate spheroids. For simplicity, a uniform crack radius a is considered. N denotes the crack density. For an isotropic distribution of crack orientations, the macroscopic behavior derived from (15) is isotropic as well (Deude et al., 2002) ... 

Once the inclusion assembly has been constmcted, the homogenised stiffness matrix of the composite is calculated as follows. First, calculate Eshelby tensors S,- for the inclusions [97,98] in local coordinates CS,. Transform the result in the global coordinate system GCS. Then calculate the strain concentration tensors for all the inclusions ... 

Using the above concepts and equations, the average composite stiffiiess can be obtained from the strain concentration tensor A and the filler and matrix properties ... 

Fisher, F. X, Bradshaw, R. D., and Brinson, L. C. Fiber waviness in nanotube-reinforced polymer composites— II modeling via numerical approximation of the dilute strain concentration tensor. Comp Sci and Tech., 63,1705-1722 (2003). 

Equation 6.20 is the required equation for the effective stiffness tensor Cyia-Since Cp, Cp and q/ are all known, one only needs to find the strain-eoneen-tration tensor Ayu- Different expressions of Ayi i represent different models. Many models have been reviewed by Tucker and Liang (1999). They recommend the Mori-Tanaka model as the best choice for injection molded composites. The model was proposed by Mori and Tanaka (1973) and has later been described by Benveniste (1987) and Christensen (1990) in a simpler direct way. The Mori-Tanaka strain-concentration tensor is given by... 

Curly brackets represent an average over all possible orientations of term ( ). In Eq. (1), the terms Cj and C2 are the elastic stiffness tensors of the matrix material and the particle, respectively, is the particle volume fraction and A2 is the strain concentration tensor defined as ... 

In order to prediet the stiffness, the strain concentration tensor A is needed. Mori and Tanaka (1973) eonsidered a composite model where the heterogeneities are diluted in the matrix. This model takes into account an interaction between the inclusion and the surroimdings (inelusions and polymer matrix) in their original model, the inclusions were considered to have the same shape and orientation. Benveniste (1987) made a reconsideration and reformulation of the Mori-Tanaka s theory in its application to the computation of the effective properties of composite. In this model the inclusions can be considered either aligned or randomly oriented. This formulation is more suitable for the morphology of clays dispersed in a polymer. The expression of the strain concentration tensor A is written as follows ... 

Another important concept is the strain-concentration and stress-concentration tensors A and B which are basically the ratios between the average filler strain (or stress) and the corresponding average of the composites. 

E Strain rate tensor for dispersed n Particle number concentration... 

When applied to partide-reinforced polymer composites, micromechanics models usually follow certain basic assumptions linear elasticity of fillers and polymer matrix the fillers are axisymmetric, identical in shape and size, and can be characterized by parameters such as aspect ratio well-bonded filler-polymer interface, so no interfacial slip is considered filler-matrix debonding and matrix microcraddng. Further details of some important preliminary concepts such as hnear elastidty, average stress and strain, composites average properties, and the strain concentration and stress concentration tensors can be found in preview literature [48-50]. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

The coupling of the mechanical field with the chemical field is realized as follows As there are bound charges present in the gel, a jump in the concentrations of the mobile ions at the interface between the gel and the solution is obtained. This difference in the concentrations leads to an osmotic pressure difference Ati between gel and solution. As a consequence of this pressure difference, the gel takes up solvent, which leads to a change of the swelling of the gel. This deformation is described by the prescribed strain e. This means that the mechanical stress is obtained by the product of the elasticity tensor C and the difference of the total (geometrical) strain e and the prescribed strain ... 

Doraiswamy and Metzner noted that use of the LCF approach is permissible at concentrations above that which would correspond to the transition from isotropic to aligned morphology, ( ) > 8/p. The theory provided fair description of the stress-strain dependence for systems containing 10 wt% GF and excellent agreement for those with 40 wt% GF. Also, the approach gave good predictions of the diagonal terms of the second-order orientation tensor. 

Since the early 1980 s, Princen s work was continued by several other authors, e.g., by Reinelt [1993]. The latter author considered theoretical aspects of shearing three-dimensional, highly concentrated foams and emulsions. Initially, the structure is an assembly of interlocked tetrakaid-ecahedra (which have six square surfaces and eight hexagonal ones). An explicit relation for stress tensor up to the elastic limit was derived. When the elastic limit is exceeded, the stress-strain dependence is discontinuous, made of a series of increasing parts of the dependence, displaced with a period of y = 2. ... 

For concentrated suspensions and incompressible liquids, the phenomenological relation connecting the stress tensor and the rate-of-strain tensor is... 

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