Micromechanical constitutive

Although the difference in final strength f, integrated through both the actual shock wave and the computational shock wave, will be mitigated by dynamic recovery (saturation) processes, this is still a substantial effect, and one that should not be left to chance. These are very important practical considerations in dealing with path-dependent, micromechanical constitutive models of all kinds. 

An important aspect of micromechanical evolution under conditions of shock-wave compression is the influence of shock-wave amplitude and pulse duration on residual strength. These effects are usually determined by shock-recovery experiments, a subject treated elsewhere in this book. Nevertheless, there are aspects of this subject that fit naturally into concepts associated with micromechanical constitutive behavior as discussed in this chapter. A brief discussion of shock-amplitude and pulse-duration hardening is presented here. 

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

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... 

Kliippel M, Schramm J (1999) An advanced micromechanical model of hyperelasticity and stress softening of reinforced rubbers. In Dorfmann A, Muhr A (eds) Constitutive models for rubber. A.A. Balkema, Rotterdam... 

Unfortunately, the initiation and evolution of crazes do not concern only the majority of thermoplastic glassy polymers, which exhibit brittle behavior. Crazes usually also constitute the dominant micromechanism for failure when many polymers generally considered tough are subjected... 

Mavrovouniotis GM, Breimer H, Edwards DA, Ting L (1993) A Micromechanical Investigation of Interfacial Transport Processes. II. Interfacial Constitutive Equations. Phil Trans R Soc Lond A 345 209-228... 

We present now the extension of the constitutive equation (7) to partially saturated porous media. The material is assumed to be saturated by a liquid phase (noted by index w) and a gas mixture (noted by index g ). The gas mixture is a perfect mixture of dry air (noted by index da) and vapour (noted by index va). Based on most experimental data of unsaturated rocks and soils (Fredlund and Rahardjo 1993), and on the theoretical background of micromechanical analysis (Chateau and Dormieux 1998), the poroelastic behaviour of unsaturated material should be non-linear and depends on the water saturation degree. We consider here the particular case of spherical pores which are dried or wetted under a capillary pressure equal to the superficial tension on the air-solid interface. By adapting the macroscopic non-linear poroelastic model proposed by Coussy al. (1998) to unsaturated damaged porous media, the incremental constitutive equations in isothermal conditions are expressed as follows ... 

In Section 5.2, we treated the SMPF as a microphase segregated structure and, based on their chemical composition and shape memory mechanisms, these phases are classified as soft phase and hard domain, and both may be semicrystalline. From the point of view of micromechanics modeling, however, this is not convenient because the RVE, which is made of a semicrystalline soft phase and a semicrystalline hard domain, will be difficult to analyze. Therefore, in the following, we will treat the SMPF as a two-phase composite with a crystalline phase and an amorphous phase that is, the RVE will be a two-phase element. Clearly, both the soft phase and hard domain contribute to the amorphous phase and crystalline phase. Selection of the RVE in such way makes it easier to utilize existing constitutive relations for both amorphous polymer and crystalline polymer. 

In the literature, the constitutive equation for both the amorphous polymer and crystalline polymer has been well established. Therefore, we can direcdy use these relations to model the amorphous phase and crystalline phase of the SMPFs. We then need to consider the cychc texture change of both subphases because the mechanical behaviors of the individual microconstituents may vary when they are packed in a multiphase material system and a certain deviation in their mechanical responses may exist between the individual and their assembled configurations. Since this is a shape memory material, we also need to model the shape recovery behavior. After that, we can use the above micromechanics relation to assemble the macroscopic constitutive relation. In order to determine the parameters used in the constitutive model, we need to consider the kinematic relations under large deformation. Finally, we will discuss the numerical scheme to solve the coupled equations. 

Once the constitutive equations and texmre changes of the two subphases are established, we can assemble them together in the RVE by the micromechanics approach. Figure 5.23 shows a schematic of the RVE. The detailed discussion can be found in Section 5.3.1 and Equation (5.2). 

Kinematics analysis is a very important step to relate the micromechanics result with the macroscopic constitutive behavior. The reason is that the material parameters involved in the subphases and RVE need to be determined through macroscopic testing. For example, a... 

The elastoplastic multiscale analysis requires several computational modules, including (1) a microscale computation module, which consists of a set of numerical solutions for the local constitutive equation of each subphase, (2) a micromechanical computation module, which provides numerical tools to link the mechanical properties of each of the local subphases to the macroscopic responses, and (3) a macroscale computation module, in which the continuum mechanics governing equations are enforced to simulate the overall mechanical response of the material and to identify the local loading conditions over the R VE. Each of these computational modules is discussed in the following. A flowchart of the multiscale analysis is shown in Figure 5.24. 

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. 

To address these limitations, a new constitutive model was developed for conventional and highly crosslinked UHMWPEs (Bergstrom, Rimnac, and Kurtz 2003). This model, which is inspired by the physical micromechanisms governing the deformation resistance of polymeric materials, is an extension of specialized constitutive theories for glassy polymers that have been developed during the last 10 years, is discussed later. 

The properties for the basic layers of CSM and WR constituting the different parts of the panel were computed using micromechanics as detailed in the EUROCOMP Design Code. More specifically, elastic constants of a monolayer of CSM were derived using the theory of composites with randomly orientated continuous fibres, and those of WR were computed using the concept of ply efficiency equal to 0.5 for a bi-directional balanced cloth ply. Table 1 shows the respective proportions and properties of the glass fibres and resin used in the fabrication of the panels and the subsequent elastic constants for a monolayer of CSM and WR. 

Predicting fiber orientation. Isotropic constitutive models are not valid for injection-molded fiber-reinforced composites. Unless the embedded fibers are randomly oriented, they introduce anisotropy in the thermomechanical properties of the material. The fiber orientation distribution is induced by kinematics of the flow during filling and, to a lesser extent, packing. An extensive literature deals with flow-induced fiber orientation while much other work has been devoted to micromechanical models which estimate anisotropic elastic and thermal properties of the fiber-matrix system from the properties of the constituent fiber and matrix materials based on given microstructures. Comprehensive reviews of both research areas have been given in two recent books edited, respectively, by Advani and by Papathanasiou and Guell where many references can be foimd. 

Fig. 20 Thermomechanical model for covalently crosslinked SMPs. (a) Schematic diagram of the micromechanics foundation of the 3-D SMP constitutive model (1). Existence of two extreme phases in the polymer is assumed. The diagram represents a polymer in the glass tiansition state with a predominant active phase (b) In the 1-D model, the frozen fraction (pf = Lf (T) /L(T) is defined as a physical internal state variable that is related to the extent of the glass transition, (c) Frozen fraction, (j>f (T), as a function of temperature, derived from curve fitting of the modified recovery strain curve divided by the predeformation strain, (d) Prediction of the free strain recovery responses during heating for polymers predeformed at different levels. Fig. (a) and (b) reprinted with permission from ref. [92], Copyright 2005, Materials Research Society, Warrendale, PA. Fig. (c) and (d) reprinted from [71], Copyright 2006, with permission from Elsevier.

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