Constitutive description

We also want to point out the difference between simple rate-dependent phenomena and path-dependent effects. Simple rate dependence means that the internal micromechanical state (as possibly represented by some meso-scale variables) depends only on the current deformation and current rate of deformation the material has no memory of the past. In terms of dislocation dynamics and (7.1), a simple rate-dependent constitutive description would be one in which... 

The thermal diffusivity for aluminum is = 5.2 x 10 m s [50]. Use this value to determine the time necessary for substantial temperature change over the length scale of 10 following creation of shear bands in the shock front. Should the temperature evolution of the shear band be included in a constitutive description on time scales of compression and release ... 

P.S. Follansbee and U.F. Kocks, A Constitutive Description of the Deformation of Copper Based on the Use of the Mechanical Threshold Stress as an Internal State Variable, Acta Metall. 36, 81-93 (1988). 

P.S. Follansbee, U.F. Kocks A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Mater. 36(1) 81-93 (1988)... 

Modern developments in bifurcation and stability theory, which lead on to chaos or turbulence in dynamical systems, only serve to show why it is worth embedding as much complexity as possible within the constitutive description of material properties. There is a sound intellectual case for trying to subsume fine-scale motions of polymer melt molecules within an Oldroyd... 

The constitutional description of an alloy system consists of text and a table/diagram section which are separated by the bibliography referring to die original literature (see Fig. 1). The tables and diagrams carry the essential constitutional information and are commented on in the text if necessary. 

Constitutive Description of Polymer Melt Behavior K-BKZ and DE Descriptions. Although there are many nonlinear constitutive models that have been proposed, the focus here is on the K-BKZ model because it is relatively simple in structure, can be related conceptually to finite elasticity descriptions of elastic behavior, and because, in the mind of the current author and others (82), the model captures the major features of nonlinear viscoelastic behavior of polymeric fluids. In addition, the reptation model as proposed by Doi and Edwards provides a molecular basis for understanding the K-BKZ model. The following sections first describe the K-BKZ model, followed by a description of the DE model. 

Failure, however defined, should be a part of a complete constitutive description of a material as discussed in the previous sections. In other, words, the key to dealing effectively with the failure of time dependent or viscoelastic polymers lies in treating failure properties as a termination of a nonlinear viscoelastic process. Perhaps, for this reason, a number of investigators have suggested that modulus and strength laws should be related to each other for polymers (eg., Landel, (1964)). 

As pointed out in the introductory chapter, there are many multifunctional materials that can be applied to realize adaptive structural systems. They are distinguished by means of excitation mechanism, application range, and maturity of development. While research in material science will widen the choice in the future, currently only piezoelectric materials are suitable for the intended purpose. Therefore, the piezoelectric effect and its constitutive description is examined in detail in this chapter. 

In this chapter, the transition from voluminous to areal structures, as already prepared in Section 4.4, will be implemented with special regard to laminated composites and adaptive capabilities making use of the piezoelectric effect for actuation as well as sensing. Therefore, a comprehensive constitutive description is developed and appropriate kinematic relations are specified. Afterwards, possibilities of different complexity for the reduction to a less general description are considered in view of specialized application cases. 

As stated by Remark 6.4, the wall thickness of the considered beams is small in comparison to the cross-sectional dimensions, being reflected in r (s) and rs(s), and to the radius of curvature R s). These geometric orders of magnitude also enter the constitutive description of the laminated beam wall in Eq. (6.4b). The plate stiffness coefficients 613(5) and 633(5) and coupling stiffnesses 831(5) and 633(5) essentially depend upon the difference of cubed, respectively squared, laminae positions in the thickness direction, while the membrane stiffness kss s) is a function of the laminae thicknesses. To comply with Remark 6.4, it is necessary to revise the formulation of the warping... 

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