Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05,

Weak form equation–based finite-element modeling of viscoelastic asphalt mixtures

The objective of this study is to demonstrate using weak form partial differential equation (PDE) method for a finite-element (FE) modeling of a new constitutive relation without the need of user subroutine programming. The viscoelastic asphalt mixtures were modeled by the weak form PDE-based FE method as the examples in the paper. A solid-like generalized Maxwell model was used to represent the deforming mechanism of a viscoelastic material, the constitutive relations of which were derived and implemented in the weak form PDE module of Comsol Multiphysics, a commercial FE program. The weak form PDE modeling of viscoelasticity was verified by comparing Comsol and Abaqus simulations, which employed the same loading configurations and material property inputs in virtual laboratory test simulations. Both produced identical results in terms of axial and radial strain responses. The weak form PDE modeling of viscoelasticity was further validated by comparing the weak form PDE predictions with real laboratory test results of six types of asphalt mixtures with two air void contents and three aging periods. The viscoelastic material properties such as the coefficients of a Prony series model for the relaxation modulus were obtained by converting from the master curves of dynamic modulus and phase angle. Strain responses of compressive creep tests at three temperatures and cyclic load tests were predicted using the weak form PDE modeling and found to be comparable with the measurements of the real laboratory tests. It was demonstrated that the weak form PDE-based FE modeling can serve as an efficient method to implement new constitutive models and can free engineers from user subroutine programming.

Physical and chemical kinetics of structural build-up of cement suspensions

Abstract : The structural build-up of fresh cement-based materials has a great impact on their structural performance after casting. Accordingly, the mixture design should be tailored to adapt the kinetics of build-up given the application on hand. The rate of structural build-up of cement-based suspensions at rest is a complex phenomenon affected by both physical and chemical structuration processes. The structuration kinetics are strongly dependent on the mixture’s composition, testing parameters, as well as the shear history. Accurate measurements of build-up rely on the efficiency of the applied pre-shear regime to achieve an initial well-dispersed state as well as the applied stress during the liquid-solid transition. Studying the physical and chemical mechanisms of build-up of cement suspensions at rest can enhance the fundamental understanding of this phenomenon. This can, therefore, allow a better control of the rheological and time-dependent properties of cement-based materials. The research focused on the use of dynamic rheology in investigating the kinetics of structural build-up of fresh cement pastes. The research program was conducted in three different phases. The first phase was devoted to evaluating the dispersing efficiency of various disruptive shear techniques. The investigated shearing profiles included rotational, oscillatory, and combination of both. The initial and final states of suspension’s structure, before and after disruption, were determined by applying a small-amplitude oscillatory shear (SAOS). The difference between the viscoelastic values before and after disruption was used to express the degree of dispersion. An efficient technique to disperse concentrated cement suspensions was developed. The second phase aimed to establish a rheometric approach to dissociate and monitor the individual physical and chemical mechanisms of build-up of cement paste. In this regard, the non-destructive dynamic rheometry was used to investigate the evolutions of both storage modulus and phase angle of inert calcium carbonate and cement suspensions. Two independent build-up indices were proposed. The structural build-up of various cement suspensions made with different cement contents, silica fume replacement percentages, and high-range water reducer dosages was evaluated using the proposed indices. These indices were then compared to the well-known thixotropic index (Athix.). Furthermore, the proposed indices were correlated to the decay in lateral pressure determined for various cement pastes cast in a pressure column. The proposed pre-shearing protocol and build-up indices (phases 1 and 2) were then used to investigate the effect of mixture’s parameters on the kinetics of structural build-up in phase 3. The investigated mixture’s parameters included cement content and fineness, alkali sulfate content, and temperature of cement suspension. Zeta potential, calorimetric, spectrometric measurements were performed to explore the corresponding microstructural changes in cement suspensions, such as inter-particle cohesion, rate of Brownian flocculation, and nucleation rate. A model linking the build-up indices and the microstructural characteristics was developed to predict the build-up behaviour of cement-based suspensions The obtained results showed that oscillatory shear may have a greater effect on dispersing concentrated cement suspension than the rotational shear. Furthermore, the increase in induced shear strain was found to enhance the breakdown of suspension’s structure until a critical point, after which thickening effects dominate. An effective dispersing method is then proposed. This consists of applying a rotational shear around the transitional value between the linear and non-linear variations of the apparent viscosity with shear rate, followed by an oscillatory shear at the crossover shear strain and high angular frequency of 100 rad/s. Investigating the evolutions of viscoelastic properties of inert calcite-based and cement suspensions and allowed establishing two independent build-up indices. The first one (the percolation time) can represent the rest time needed to form the elastic network. On the other hand, the second one (rigidification rate) can describe the increase in stress-bearing capacity of formed network due to cement hydration. In addition, results showed that combining the percolation time and the rigidification rate can provide deeper insight into the structuration process of cement suspensions. Furthermore, these indices were found to be well-correlated to the decay in the lateral pressure of cement suspensions. The variations of proposed build-up indices with mixture’s parameters showed that the percolation time is most likely controlled by the frequency of Brownian collisions, distance between dispersed particles, and intensity of cohesion between cement particles. On the other hand, a higher rigidification rate can be secured by increasing the number of contact points per unit volume of paste, nucleation rate of cement hydrates, and intensity of inter-particle cohesion.

Characterization of Non-linear Polymer Properties to Predict Process Induced Warpage and Residual Stress of Electronic Packages

Nonlinear thermo-mechanical properties of advanced polymers are crucial to accurate prediction of the process induced warpage and residual stress of electronics packages. The Fiber Bragg grating (FBG) sensor based method is advanced and implemented to determine temperature and time dependent nonlinear properties. The FBG sensor is embedded in the center of the cylindrical specimen, which deforms together with the specimen. The strains of the specimen at different loading conditions are monitored by the FBG sensor. Two main sources of the warpage are considered: curing induced warpage and coefficient of thermal expansion (CTE) mismatch induced warpage. The effective chemical shrinkage and the equilibrium modulus are needed for the curing induced warpage prediction. Considering various polymeric materials used in microelectronic packages, unique curing setups and procedures are developed for elastomers (extremely low modulus, medium viscosity, room temperature curing), underfill materials (medium modulus, low viscosity, high temperature curing), and epoxy molding compound (EMC: high modulus, high viscosity, high temperature pressure curing), most notably, (1) zero-constraint mold for elastomers; (2) a two-stage curing procedure for underfill materials and (3) an air-cylinder based novel setup for EMC. For the CTE mismatch induced warpage, the temperature dependent CTE and the comprehensive viscoelastic properties are measured. The cured cylindrical specimen with a FBG sensor embedded in the center is further used for viscoelastic property measurements. A uni-axial compressive loading is applied to the specimen to measure the time dependent Young’s modulus. The test is repeated from room temperature to the reflow temperature to capture the time-temperature dependent Young’s modulus. A separate high pressure system is developed for the bulk modulus measurement. The time temperature dependent bulk modulus is measured at the same temperatures as the Young’s modulus. The master curve of the Young’s modulus and bulk modulus of the EMC is created and a single set of the shift factors is determined from the time temperature superposition. The supplementary experiments are conducted to verify the validity of the assumptions associated with the linear viscoelasticity. The measured time-temperature dependent properties are further verified by a shadow moiré and Twyman/Green test.

Molecular Origins of Higher Harmonics in Large-Amplitude Oscillatory Shear Flow: Shear Stress Response

Recent work has focused on deepening our understanding of the molecular origins of the higher harmonics that arise in the shear stress response of polymeric liquids in large-amplitude oscillatory shear flow. For instance, these higher harmonics have been explained by just considering the orientation distribution of rigid dumbbells suspended in a Newtonian solvent. These dumbbells, when in dilute suspension, form the simplest relevant molecular model of polymer viscoelasticity, and this model specifically neglects interactions between the polymer molecules [R.B. Bird et al., J Chem Phys, 140, 074904 (2014)]. In this paper, we explore these interactions by examining the Curtiss-Bird model, a kinetic molecular theory designed specifically to account for the restricted motions that arise when polymer chains are concentrated, thus interacting and specifically, entangled. We begin our comparison using a heretofore ignored explicit analytical solution [Fan and Bird, JNNFM, 15, 341 (1984)]. For concentrated systems, the chain motion transverse to the chain axis is more restricted than along the axis. This anisotropy is described by the link tension coefficient, ε, for which several special cases arise: ε = 0 corresponds to reptation, ε > 1/8 to rod-climbing, 1/2 ≥ ε ≥ 3/4 to reasonable predictions for shear-thinning in steady simple shear flow, and ε = 1 to the dilute solution without hydrodynamic interaction. In this paper, we examine the shapes of the shear stress versus shear rate loops for the special cases ε = (0,1/8, 3/8,1) , and we compare these with those of rigid dumbbell and reptation model predictions.