Micromechanical constitutive models

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. 

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... 

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. 

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. 

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.

The basic framework for Ihe Hybrid Model (HM) has evolved from its initial formulation [10] into a more complex constitutive model, the augmented Hybrid Model [11], which has been shown to predict, with great accuracy, the behavior of UHMWPE specimens subjected to several different loading scenarios. The model was developed based on the micromechanical behavior of sarticrystalline polymers at the molecular level. At the core of the augmented HM is the multiplicative decomposition of the applied deformation gradient (F) into two separate components—elastic... 

Modeling of damage processes such as fatigue, fracture, and wear have not been explored in this chapter but could be important to consider depending on the particular research objectives. Constitutive models for UHMWPE will continue to evolve as our understanding of the micromechanics and damage behavior of this versatile material are more fuUy understood and as the availability of computational resources continues to increase. 

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 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 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. 

In this chapter, we reviewed some of the recent developments in modeling and predicting the mechanical properties of polyurethane elastomers as a function of their formulation. Based on the knowledge of the formulation and processing history, one can roughly predict the degree of the microphase separation between the hard and soft phases. That, in turn, enables the construction of constitutive micromechanical models for the calculation of elastic, viscoelastic, and nonlinear tensile and compressive properties. 

When applied to particle reinforced polymer composites, micromechanics models usually follow such basic assumptions as (i) linear elasticity of fillers and polymer matrix (ii) the fillers are axisymmetric, identical in shape and size, and can be characterized by parameters such as aspect ratio (iii) well-bonded filler-polymer interface and the ignorance of interfacial slip, filler-polymer debonding or matrix cracking. The first concept is the linear elasticity, that is, the linear relationship between the total stress and infinitesimal strain tensors for the filler and matrix as expressed by the following constitutive equations ... 

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