Modelling the curing process in magneto-sensitive polymers: Rate-dependence and shrinkage

This paper deals with a phenomenologically motivated magneto-viscoelastic coupled finite strain framework for simulating the curing process of polymers under the application of a coupled magneto-mechanical load. Magneto-sensitive polymers are prepared by mixing micron-sized ferromagnetic particles in uncured polymers. Application of a magnetic field during the curing process causes the particles to align and form chain-like structures lending an overall anisotropy to the material. The polymer curing is a viscoelastic complex process where a transformation from fluid to solid occurs in the course of time. During curing, volume shrinkage also occurs due to the packing of polymer chains by chemical reactions. Such reactions impart a continuous change of magneto-mechanical properties that can be modelled by an appropriate constitutive relation where the temporal evolution of material parameters is considered. To model the shrinkage during curing, a magnetic-induction-dependent approach is proposed which is based on a multiplicative decomposition of the deformation gradient into a mechanical and a magnetic-induction-dependent volume shrinkage part. The proposed model obeys the relevant laws of thermodynamics. Numerical examples, based on a generalised Mooney–Rivlin energy function, are presented to demonstrate the model capacity in the case of a magneto-viscoelastically coupled load.     

Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory

The paper outlines a constitutive model for finite thermo-visco-plastic behavior of amorphous glassy polymers and considers details of its numerical implementation. In contrast to existing kinematical approaches to finite plasticity of glassy polymers, the formulation applies a plastic metric theory based on an additive split of Lagrangian Hencky-type strains into elastic and plastic parts. The analogy between the proposed formulation in the logarithmic strain space and the geometrically linear theory of plasticity, makes this constitutive framework very transparent and attractive with regard to its numerical formulation. The characteristic strain hardening of the model is derived from a polymer network model. We consider the particularly simple eight chain model, but also comment on the recently developed microsphere model. The viscoplastic flow rule in the logarithmic strain space uses structures of the free volume flow theory, which provides a highly predictive modeling capacity at the onset of viscoplastic flow. The integration of this micromechanically motivated approach into a three-dimensional computational model is a key concern of this work. We outline details of the numerical implementation of this model, including elements such as geometric pre- and post-transformations to/from the logarithmic strain space, a thermomechanical operator split algorithm consisting of an isothermal mechanical predictor followed by a heat conduction corrector and finally, the consistent linearization of the local update algorithm for the dissipative variables as well as its relationship to the global tangent operator. The performance of the proposed formulation is demonstrated by means of a spectrum of numerical examples, which we compare with our experimental findings.     

Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers

Under certain conditions, such as sufficiently low temperatures, high loading rates and/or highly triaxial stress states, glassy polymers display an unfavorable characteristic—brittleness. A technique used for reducing the brittleness (increasing the fracture toughness) of these materials is rubber toughening. While there is significant qualitative understanding of the mechanical behavior of rubber-toughened polymers, quantitative modeling tools for the large-strain deformation of rubber-toughened glassy polymers are largely lacking.In this paper, we develop a suite of numerical tools to investigate the mechanical behavior of rubber-toughened glassy polymers, with emphasis on rubber-toughened polycarbonate. The rubber particles are modeled as voids in view of their deformation-induced cavitation early during deformation. A three-dimensional micromechanical model of the heterogeneous microstructure is developed to study the effects of initial rubber particle (void) volume fraction on the underlying elasto-viscoplastic deformation mechanisms in the material, and how these mechanisms influence the macroscopic response of the material. A continuum-level constitutive model is developed for the large-strain elasto-viscoplastic deformation of porous glassy polymers, and it is calibrated against micromechanical modeling results for porous polycarbonate. The constitutive model can be used to study various boundary value problems involving rubber-toughened (porous) glassy polymers. As an example, the case of an axisymmetric notched bar is simulated for the case of polycarbonate with varying levels of initial porosity. The quality of the constitutive model calibration is assessed using a multi-scale modeling approach.     

Framework using functional forms of hardening internal state variables in modeling elasto-plastic-damage behavior

This work gives the thermodynamically consistent theoretical formulations and the numerical implementation of a plasticity model fully coupled with damage. The formulation of the elasto-plastic-damage behavior of materials is introduced here within a framework that uses functional forms of hardening internal state variables in both damage and plasticity. The damage is introduced through a damage mechanics framework and utilizes an anisotropic damage measure to quantify the reduction of the material stiffness. In deriving the constitutive model, a local yield surface is used to determine the occurrence of plasticity and a local damage surface is used to determine the occurrence of damage. Isotropic hardening and kinematic hardening are incorporated as state variables to describe the change of the yield surface. Additionally, a damage isotropic hardening is incorporated as a state variable to describe the change of the damage surface. The hardening conjugate forces (stress-like terms) are general nonlinear functions of their corresponding hardening state variables (strain-like terms) and can be defined based on the desired material behavior. Various exponential and power law functional forms are studied in this formulation. The paper discusses the general concept of using such functional forms. however, it does not address the relevant appropriateness of certain forms to solve different problems. The proposed work introduces a strong coupling between damage and plasticity by utilizing damage and plasticity flow rules that are dependent on both the plastic and damage potentials. However, in addition to that the coupling is further enhanced through the use of the functional forms of the hardening variables introduced in this formulation.The use of this formulation in solving boundary value problems will be presented in future work. The fully implicit backward Euler scheme is developed for this model to be solved in a Newton–Raphson solution procedure.     

Non-isothermal polymer melt flow in sudden expansions

This paper presents numerical results for laminar, incompressible and non-isothermal polymer melt flow in sudden expansions. The mathematical model includes the mass, momentum and energy conservation laws within the framework of a generalized Newtonian formulation. Two constitutive relations are adopted to describe the non-Newtonian behavior of the flow, namely Cross and Modified Arrhenius Power-Law models. The governing equations are discretized using the finite difference method based on central, second-order accurate formulas for both convective and diffusive terms. The pressure–velocity coupling is treated by solving a Poisson equation for pressure. The results are presented for two commercial polymers and demonstrate that important flow parameters, such as pressure drop and viscosity distribution, are strongly affected by heat transfer features.     

Constitutive modeling of biodegradable polymers: Hydrolytic degradation and time-dependent behavior

A large range of biodegradable polymers has been used to produce implantable medical devices. Apart from biological compatibility, these devices shall be also functional compatible and perform adequate mechanical temporary support during the healing process. However, the mechanical behavior of biodegradable materials during its degradation, which is an important aspect of the design of these biodegradable devices, is still an unexplored subject. Based on the literature, the mechanical behavior of biodegradable polymers is strain rate dependent and exhibits hysteresis upon cyclic loading. On the other hand, ductility, toughness and strength of the material decay during hydrolytic degradation. In this work, it is considered a three-dimensional time-dependent model adapted from the one developed by Bergström and Boyce to simulate the performance of biodegradable structures undergoing large deformations incorporating the hydrolysis degradation. Since this model assumes that the mechanical behavior is divided into a time independent network and a non-linear time-dependent network, it enables to simulate the monotonic tests of a biodegradable structure loaded under different strain rates. The hysteresis effects during unloading–reloading cycles at different strain levels can be predicted by the model. A parametric study of the material model parameters evolution during the hydrolytic degradation was conducted to identify which parameters are more sensible to this degradation process. The investigated model could predict very well the experimental results of a blend of polylactic acid and polycaprolactone (PLA–PCL) in the full range of strains until rupture during hydrolytic degradation. From these results and analyses, a method is proposed to simulate the three-dimensional mechanical behavior during hydrolytic degradation.     

一种化学-力学耦合连续介质理论与数值格式

对化学驱动的连续介质化学-力学耦合系统进行研究,从热力学定律和化学势角度出发,推导了等温过程的化学-力学耦合本构关系和控制方程,利用变分方法建立了化学-力学耦合系统的能量泛函,得到化学-力学耦合控制方程的等效积分形式和相应的有限元列式. 结合算例,对连续介质的化学-力学耦合行为进行了数值计算,数值结果反映了化学与力学系统的相互耦合作用,即浓度变化能引起介质的变形,同样力学作用也能引起浓度重分布. 从全新的角度建立了描述连续介质的化学-力学耦合行为的基本理论和数值方法,能够较好地反映一类连续介质的化学-力学耦合行为.      

Aspects on the finite-element implementation of the Gurson model including parameter identification

In this contribution, various aspects on the finite-element implementation of the Gurson model are considered. In particular, a linear representation for the plastic potential is used, which shows superior convergence property in the local iteration procedure compared to the original quadratic representation. The formulation of the model is performed in the spatial configuration based on the multiplicative decomposition of the deformation gradient, and for integration an exponential map scheme is used. A further important aspect is the sensitivity analysis consistent with the underlying integration scheme necessary for minimizing a least-squares functional for parameter identification by use of a gradient-based optimization algorithm. In a numerical example the local convergence behavior for the two versions of the Gurson model, linear and quadratic are compared. Furthermore material parameters are determined by least-squares minimization based on experimental data obtained for an axisymmetric tensile bar for a ferritic steel.     

Shearing and mixing effects on synthesis and properties of organoclay/polyester nanocomposites

Mixing of solid nanoparticles in viscous fluids is a key stage in synthesis of nanocomposites and can affect their final properties. A multi-step preparatory mixing is developed to synthesize the nanocomposites of layered silicate in thermosetting polymers. This study aims to investigate the influences of mixing conditions and steps taken to process the thermosetting nanocomposites on the viscoelastic properties of suspensions. We also examine subsequent influences of mixing on the microstructure and dispersion state of cured hybrids of organically modified clays in a polyester resin. The nanocomposites were prepared in a sequential mixing process developed for the model nanocomposites of organoclays and thermoset resin. Depending on the mixing conditions, the final nanocomposites showed mixed intercalated and moderately to highly delaminated structure. TEM images show that the nanoclay galleries are dispersed in the polymer phase after curing reactions. The startup viscosities and linear viscoelastic properties of the nanocomposites are significantly influenced by the extent and the time duration of mixing. These observations indicate that extensive mechanical mixing combined with a stationary step followed by moderate shear mixing can improve the polymer and nanoparticle interactions at the interface. In the last part of this work, we develop a simple but efficient mathematical formulation on the flow of oblate spheroids in viscous media and compare selected model predictions with the measured startup shear viscosities of suspensions.     

Liquid seeping into porous medium

 A mathematical model is presented to describe the hydrodynamics behavior of liquid seeping into porous medium. The model takes into account the inertia and the evaporation effects. Analytical solution is obtained for the hydrodynamics behavior of the seeping process when the inertia effects are excluded and numerical solutions are obtained when these effects are included. It is found that both evaporation and inertia have significant effects on the seeping process. These effects are investigated qualitatively and quantitatively. Received on 10 June 1999     

SECTIONAL FINITE ELEMENT ANALYSIS OF COUPLED DEFORMATION BETWEEN ELASTOPLASTIC SHEET METAL AND VISCO-ELASTOPLASTIC BODY

The present paper is devoted to developing a new numerical simulation method for the analysis of viscous pressure forming (VPF), which is a sheet flexible-die forming (FDF) process. The pressure-carrying medium used in VPF is one kind of semisolid, flowable and viscous material and its deformation behavior can be described by the visco-elastoplastic constitutive model. A sectional finite element model for the coupled deformation analysis between the visco- elastoplastic pressure-carrying medium and the elastoplastic sheet metal is proposed. The resolution of the Updated Lagrangian (UL) formulation is based on a static explicit approach. The frictional contact between sheet metal and visco-elastoplastic pressure-carrying medium is treated by the penalty function method. Coupled deformation between sheet metal and visco-elastoplastic pressure-carrying medium with large slip is analyzed to validate the developed algorithm. Finally, the viscous pressure bulging (VPB) process of DC06 sheet metal is simulated. Good agreement between numerical simulation results and experimental measurements shows the validity of the developed algorithm.     

Characterization of precipitative self-healing materials by mechanokinetic modeling approach

A non-deterministic multiple scale approach based on numerical solution of the Monte-Carlo master equation on atomic lattices solved together with a standard finite-element formulation of solid mechanics is discussed. The approach is illustrated in application to long-term evolutionary processes of volume diffusion, precipitation and creep cavity self-healing in nanocrystalline austenite (Fe fcc) samples. A two-way mechanokinetic coupling is achieved through implementation of strain-dependent diffusion rates and dynamic update of the finite element model based on atomic structure evolution. Effect of macroscopic static loading and cavity geometry on the total healing time is investigated. The approach is widely applicable to the modeling and characterization of advanced functional materials with evolutionary internal structure, and emerging behavior in material systems.     

Adiabatic decohesion in a thermoplastic craze thickening at constant or increasing rate

When a crack in a thermally non-diffusive material is impact loaded—or propagates at high speed—a cohesive process which resists slow crack extension may itself cause decohesion by adiabatic heating. By assuming that decohesion ultimately occurs by low-energy disentanglement within a melt layer of critical thickness, the fracture resistance of craze-forming crystalline polymers can be estimated quantitatively. Previous estimates used a simple, thermomechanically linear representation of craze fibril drawing. This paper presents a more physically realistic, numerical formulation, and demonstrates it for constant craze thickening rate (as imposed by an ideal full-notch tension test) and for linearly increasing thickening rate (as at the tip of an impact-loaded or rapidly propagating crack). For a linear material, the numerical formulation gives results which asymptotically approach those from analytical solutions, as craze density approaches zero. In more realistic model polymers, the enthalpy of fusion increasingly delays decohesion as impact speed increases, although the temperature distribution of an endotherm appears to have little effect. Increasing molecular weight, heuristically associated with decreasing craze density and increasing structural dimension, increases the predicted impact fracture resistance. In every case, fracture resistance passes through a minimum as impact speed increases. The conclusions encourage the use of impact fracture tests, and discourage the use of the full-notch tension test, to assess the dynamic fracture resistance of a craze-forming polymer.     

A model for the nonisothermal behavior of viscoelastic media

Summary A new constitutive model is derived for the viscoelastic behavior of polymers under non-isothermal loading. The model extends the concept of adaptive links (entanglements) between polymeric molecules to thermoviscoelastic media. By using experimental data for Nylon-6 and polyisobutylene in the vicinity of the glass-transition temperature, we find parameters of the model and study their dependence on temperature. The model is employed for the numerical analysis of the material response to time-periodic loads under isothermal conditions and to time-varying loads under heating. The results of numerical simulation demonstrate fair agreement with experimental data. Accepted for publication 23 May 1996     

Thermoviscoplastic modelling of asymmetric effects for polymers at large strains

Glassy polymers such as polycarbonate exhibit different behaviours in different loading scenarios, such as tension and compression. To this end a flow rule is postulated within a thermodynamic consistent framework in a mixed variant formulation and decomposed into a sum of weighted stress mode related quantities. The different stress modes are chosen such that they are accessible to individual examination in the laboratory, where tension and compression are typical examples. The characterisation of the stress modes is obtained in the octahedral plane of the deviatoric stress space in terms of the Lode angle, such that stress mode dependent scalar weighting functions can be constructed. Furthermore the numerical implementation of the constitutive equations into a finite element program is briefly described. In a numerical example, the model is used to simulate the laser transmission welding process.     

An interface damage model accounting for in-plane effects

The present work deals with the formulation of a kinematic enriched model for cohesive interface. In fact, the interface kinematics is defined by the relative displacement occurring between the two surfaces of the interface and, even, by the strain arising in the plane of the interface. A damage model which accounts for the mode I and mode II and for the axial deformation of the interface is proposed starting from the Drucker–Prager failure criterion. A numerical procedure is developed implementing the proposed interface model into a new finite element. The nonlinear evolutive problem is solved adopting a predictor–corrector technique within the backward time integration scheme. Simple numerical simulations are presented in order to assess the features of the model. Moreover, numerical applications are carried out in order to demonstrate the ability of the proposed model in reproducing the mechanical behavior of the cohesive elements strengthened with external FRP reinforcements. Comparisons between available experimental data and numerical results obtained using the proposed model show the effectiveness of the presented formulation.     

A thermo-mechanically-coupled large-deformation theory for amorphous polymers in a temperature range which spans their glass transition

Amorphous thermoplastic polymers are important engineering materials; however, their non-linear, strongly temperature- and rate-dependent elastic-viscoplastic behavior is still not very well understood, and is modeled by existing constitutive theories with varying degrees of success. There is no generally agreed upon theory to model the large-deformation, thermo-mechanically-coupled, elastic-viscoplastic response of these materials in a temperature range which spans their glass transition temperature. Such a theory is crucial for the development of a numerical capability for the simulation and design of important polymer processing operations, and also for predicting the relationship between processing methods and the subsequent mechanical properties of polymeric products. In this paper we extend our recently published theory [Anand, L., Ames, N. M., Srivastava, V., Chester, S. A., 2009. A thermo-mechanically-coupled theory for large deformations of amorphous polymers. Part I: formulation. International Journal Plasticity 25, 1474–1494; Ames, N. M., Srivastava, V., Chester, S. A., Anand, L., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: applications. International Journal of Plasticity 25, 1495–1539] to fill this need.     

含有分层损伤的复合材料加筋层合板的屈曲性态研究

基于Ｍｉｎｄｉｎ假定推导了考虑剪切的复合材料加筋层板的有限元列式,并在此基础上计算出筋间基板含嵌入分层以及筋与基板连接处含穿透分层的加筋层合板在受压缩载荷情况下的屈曲模式和临界力。本文所给出的有限元方法及结论对从事复合材料结构设计的工程人员具有参考价值。     

Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space

The paper outlines a new constitutive model and experimental results of rate-dependent finite elastic–plastic behavior of amorphous glassy polymers. In contrast to existing kinematical approaches to finite viscoplasticity of glassy polymers, the formulation proposed is constructed in the logarithmic strain space and related to a six-dimensional plastic metric. Therefore, it a priori avoids difficulties concerning with the uniqueness of a plastic rotation. The constitutive framework consists of three major steps: (i) A geometric pre-processing defines a total and a plastic logarithmic strain measures determined from the current and plastic metrics, respectively. (ii) The constitutive model describes the stresses and the consistent moduli work-conjugate to the logarithmic strain measures in an analogous structure to the geometrically linear theory. (iii) A geometric post-processing maps the stresses and the algorithmic tangent moduli computed in the logarithmic strain space to their nominal, material or spatial counterparts in the finite deformation space. The analogy between the formulation of finite plasticity in the logarithmic strain space and the geometrically linear theory of plasticity makes this framework very attractive, in particular regarding the algorithmic implementation. The flow rule for viscoplastic strains in the logarithmic strain space is adopted from the celebrated double-kink theory. The post-yield kinematic hardening is modeled by different network models. Here, we compare the response of the eight chain model with the newly proposed non-affine micro-sphere model. Apart from the constitutive model, experimental results obtained from both the homogeneous compression and inhomogeneous tension tests on polycarbonate are presented. Besides the load–displacement data acquired from inhomogeneous experiments, quantitative three-dimensional optical measurements of the surface strain fields are carried out. With regard to these experimental data, the excellent predictive quality of the theory proposed is demonstrated by means of representative numerical simulations.