Mathematical modeling of electrorheological materials

Received May 11, 2000 / Published online January 23, 2001     

On constitutive equations for electrorheological materials

Constitutive equations for electrorheological (ER) fluids have been based on experimental results for steady shearing flows and constant electric fields. The fluids have been modeled as being rigid until a yield stress is reached. Additional stress is then proportional to the shear rate. Recent experimental results indicate that ER materials have a regime of solid-like response when deformed from a rest state. They behave in a viscoelastic-like manner under sinusoidal shearing and exhibit time-dependent response under sudden changes in shear rate or electric field. In this work, a constitutive theory for ER materials is presented which accounts for these recent experimental observations. The stress is given by a functional of the deformation gradient history and the electric field vector. Using the methods of continuum mechanics, a general three-dimensional constitutive equation is obtained. A sample constitutive equation is introduced which is then used to determine the response of an ER material for different shear histories. The calculated shear response is shown to be qualitatively similar to that observed experimentally.     

On the modeling of fiber dispersion in fiber-reinforced elastic materials

When an isotropic material is subject to a uniaxial tension, the principal strain transverse to the direction of applied load is always negative. However, in fiber reinforced materials the transverse principal strain can change its sign as the load increases, passing through the zero-points, known as perversions. We investigate how the number of perversions in a material reinforced by two symmetrically aligned families of distributed fibers depends both on the degree of fiber dispersion and the model used for fiber dispersion. Angular integration and three variants of the generalized structure tensor approach are considered and discussed. The study of perversions clearly demonstrates the qualitative difference between these approaches in the case of high dispersion of fibers. The results suggest that this difference is primarily due to the way compressive fibers are modeled.     

On the use of universal relations in the modeling of transversely isotropic materials

A material is of coaxial type   if the Cauchy stress tensor T$\mathit{T}$ and the strain tensor B$\mathit{B}$ are coaxial for all deformations. Clearly a hyperelastic material is of coaxial type if and only if it is isotropic. Here we present a weaker definition of materials of coaxial type. Anisotropic materials may be of a coaxial type in a weak sense if for a given specific  B$\mathit{B}$ we have that TB=BT$\mathit{TB}=\mathit{BT}$. We denote these materials B$\mathit{B}$-coaxial. We show that for transverse isotropic materials weak coaxial constitutive equations may be characterized using universal relations. We discuss the impact of B$\mathit{B}$-coaxial materials in the modeling of soft tissues. We conclude that B$\mathit{B}$-coaxial materials are a strong evidence that in real world materials two anisotropic invariants are always necessary to model in a meaningful and correct way single fiber reinforced materials.     

关于采用粗粒化提高颗粒材料多尺度模拟守恒特性的研究

连续体-颗粒耦合方法常用来描述连续-非连续颗粒行为或解决颗粒材料与其他可变形构件间相互作用问题。粗粒化coarse-graining (CG)是基于统计力学的均匀化方法,由离散的颗粒运动定义连续的宏观物理场。本文利用粗粒化(CG)推导有限元-离散元(FEM-DEM)表面和体积耦合的一般性表达式。对于表面耦合,CG可以将耦合力分布到颗粒-单元接触点以外的位置,如相邻的积分点;对于体积耦合,CG可以将颗粒尺度的运动均匀化到耦合单元上。由粗粒化推导出的耦合项仅包含一个参数,即粗粒化宽度,为均匀化后的宏观场定义了一个可调整的空间尺度。当粗粒化宽度为零时,表面和体积耦合表达式简化为常规局部耦合。本文通过弹性立方体冲击颗粒床和离散-连续介质间波传播两个数值算例,展示使用粗粒化方法提高耦合系统能量守恒的优势,并结合其他耦合参数(如体积耦合深度)讨论了粗粒化参数对数值稳定性和计算效率的影响。     

Nonlinear multiscale modeling of polymer materials

In this study, a hyperelastic multiscale modeling technique is used to predict elastic properties of polycarbonate and polyimide polymer systems using a set of widely accepted atomistic force fields. The model incorporates molecular simulations and a nonlinear, continuum mechanics-based, constitutive formulation that incorporates the behavior of the polymer materials as predicted from molecular simulations. The predicted properties of the polymers using multiple force fields are compared to experimentally measured values. Both static and dynamic molecular simulations are performed using molecular mechanics energy minimizations and molecular dynamics simulation techniques, respectively. The results of this study indicate that static molecular simulation is a useful tool to predict the bulk-level nonlinear mechanical behavior of polymers for finite deformations. It is found that the AMBER force field yields the most accurate predicted mechanical and physical properties of the modeled polymer systems compared to the other force fields used in this study.     

On the modeling of the Taylor cylinder impact test for orthotropic textured materials: experiments and simulations

Taylor impact tests using specimens cut from a rolled plate of tantalum were conducted. The tantalum was experimentally characterized in terms of flow stress and crystallographic texture. A piece-wise yield surface was interrogated from an ODF corresponding to this texture assuming two slip system modes, in conjunction with an elastic stiffness tensor computed from the same ODF and single crystal elastic properties. This constitutive information was used in EPIC-95 3D simulations of a Taylor impact test, and good agreement was realized between the calculational results and the experimental post-test geometries in terms of major and minor side profiles and impact-interface footprints.     

Micromechanics modeling the solute diffusivity of unsaturated granular materials

This work is devoted to modeling the evolution of the homogenized solute diffusion coefficient within unsaturated granular materials by means of micromechanics approach. On the basis of its distinct role in solute diffusion, the liquid water within unsaturated granular materials is distinguished into four types, namely intergranular layer (interconnected capillary water), isolated capillary water, wetting layer and water film. Application on two sands shows the capability of the model to accurately reproduce the experimental results. When saturation degree is higher than the residual saturation degree Sr^r, the evolution of homogenized solute diffusion coefficient with respect to the saturation degree depends significantly on the connectivity of the capillary water. Below Sr^r, depending on the connectivity of the wetting layer, the homogenized solute diffusion coefficient within unsaturated sands decreases by 2–6 orders of magnitude with respect to that in bulk liquid water. The upper bound of the solute diffusion coefficient contributed by the water films is 4–6 orders of magnitude lower than that in bulk liquid water.     

Homogenized modeling for vascularized poroelastic materials

A new mathematical model for the macroscopic behavior of a material composed of a poroelastic solid embedding a Newtonian fluid network phase (also referred to as vascularized poroelastic material), with fluid transport between them, is derived via asymptotic homogenization. The typical distance between the vessels/channels (microscale) is much smaller than the average size of a whole domain (macroscale). The homogeneous and isotropic Biot’s equation (in the quasi-static case and in absence of volume forces) for the poroelastic phase and the Stokes’ problem for the fluid network are coupled through a fluid-structure interaction problem which accounts for fluid transport between the two phases; the latter is driven by the pressure difference between the two compartments. The averaging process results in a new system of partial differential equations that formally reads as a double poroelastic, globally mass conserving, model, together with a new constitutive relationship for the whole material which encodes the role of both pore and fluid network pressures. The mathematical model describes the mutual interplay among fluid filling the pores, flow in the network, transport between compartments, and linear elastic deformation of the (potentially compressible) elastic matrix comprising the poroelastic phase. Assuming periodicity at the microscale level, the model is computationally feasible, as it holds on the macroscale only (where the microstructure is smoothed out), and encodes geometrical information on the microvessels in its coefficients, which are to be computed solving classical periodic cell problems. Recently developed double porosity models are recovered when deformations of the elastic matrix are neglected. The new model is relevant to a wide range of applications, such as fluid in porous, fractured rocks, blood transport in vascularized, deformable tumors, and interactions across different hierarchical levels of porosity in the bone.     

Constitutive modeling of particle reinforced rubber-like materials

The present study is focused on the constitutive modeling for the mechanical behavior of rubber reinforced with filler particles. A filler-dependent energy density function is proposed with all the continuum mechanics-based necessities of an effective hyperelastic material model. The proposed invariant-based energy function comprises a single set of material parameters for a material subjected to several modes of loading conditions. The model solution agrees well with existing experimental results. Later, the effect of varying concentrations of filler particles in the rubber matrix is also studied.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

Modulus-switching viscoelasticity of electrorheological networks

To form an electrorheological network (ERN), semiconducting nanoparticles were embedded in a polymer that was cross-linked to restrict particle motion. The microstructure ranged from random to aligned, depending on the degree of field-induced particle alignment during chemical network formation. We investigated in detail the softness effects of the matrix, having a relatively low storage modulus, on the dynamic rheological behavior of the ERN and analyzed its anisotropy. The anisotropy of the microstructure was probed rheologically with the modes of small-amplitude oscillatory shear (loading perpendicular to the field direction) and compression (loading in the field direction). The storage shear modulus was found to be a function of the applied electric field, particle volume fraction, and the pre-alignment electric field strength during the cross-linking reaction of the matrix, which governs the thickness of particle columns and intercolumn distance. Nonlinear behavior at small strain (below 0.1%) was conspicuous; this nonlinear viscoelasticity was accompanied by only a limited deformation of ordered connectivity. Throughout this study, we fabricated the ERN with the highly controllable modulus-switching effect acting in a shear-mode operation. Managing this anisotropy of an ERN by the electrical and chemical process is important in the design of smart materials that will provide improved stability and mechanical strength compared with fluid-type electrorheological materials and faster response time compared with that of conventional charged polymer gel.     

Computational modeling of surface phenomena in soft-wet materials

We consider the problem of extracting tribological information from experimental observations of contact with soft-wet materials. Particular attention is placed on simulating the response of two rotating cylinders of soft specimens placed in frictional contact, with a variable coefficient of friction dependent on the relative sliding velocity. The bulk behavior is modeled by means of a finite deformation viscoelasticity formulation, with constitutive parameters taken to be representative of hydrogels. We focus on the modeling of the surface behavior and employ a mortar-finite element contact formulation. Through a series of numerical studies, we demonstrate the strong sensitivity of the results to the choice of interfacial constitutive parameters. The difficulties of extracting such parameters using only experimental data and approximate analytical expressions are also examined.