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The interfacial contribution, related to the strain concentration tensor within each material phase and inside the average strain field, is described by a modified Mori–Tanaka scheme.

Likewise, all brackets modeled consisted from a single material phase, and no different materials were used for the tie-wings and base of the bracket, as is sometimes done for metallic brackets [37].

Moreover, the interfacial contribution to the strain concentration tensor within each material phase and inside the average strain filed is described by a modified Mori-Tanaka scheme.

The results showed that the coating material phase and purity as well as interlayer material and the mismatch of the thermal and mechanical properties between coating and substrate materials influences the level of the thermal stresses.

Compared with standard level set methods using n scalar functions to represent 2n phases, each constant value in the present method denotes one material phase and 2n phases can be represented by 2n pre-defined constants.

Compressive loads equivalent to 0.5% of compression applied to the solid material phase and interstitial fluid flows with inlet velocities of 1, 10 and 100 μm/s applied to the interconnected pores were simulated, changing also the inlet side and the viscosity of the medium.

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These discoveries have certainly yielded vast opportunities in designing material phases and related properties in the topological domain.

In-depth discussions are also given on the effects of the base material phases and the assignment of the volume fractions on each scale.

Accordingly, the number of design variables depends only on one active phase in each of those sub-problems regardless of the number of material phases and is significantly decreased in comparison with the original problem.

The multiscale model is based on the mix design, i.e., volume fractions of different material phases and the intrinsic viscoelastic material behavior of the latter, making goal-oriented optimization of bituminous mixtures feasible.

The effective viscoelastic material behavior of a polymeric particulate composite is obtained from the micromechanics-based modeling with the properties of the individual material phases and corresponding volume fractions.

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