A Mathematical Model for Amorphous Polymers Based on Concepts of Reptation Theory Lixiang Yang Department of Mechanical Engineering, Procter & Gamble and University of Cincinnati Simulation Center, Cincinnati 45206, Ohio Mechanical behaviors of amorphous polymers have been investigated in all aspects from macroscopic thermodynamics to molecular dynamics in past five decades. Most models either have too complex mathematics or can only explain mechanical behaviors of speci fic materials under certain defined conditions. In this article, a mathematical model is pro- posed to understand mechanical behaviors of amorphous polymers with aid of the concepts of reptation theory. This new model is capable to match most experimental results of different amorphous polymers for a wide range of time and temperature effect from rubber zone to glassy zone. Above glass transitional temperature, the model shows hyperelastic behavior. Below glass transitional temperature, elastic– viscoplastic properties can be obtained. In the proposed model, no yielding surface is assumed. Hyperelasticity and Mullin’ s effect are illustrated in a different way without assum- ing strain energy function in advance. Yielding stress is con- trolled by Young’ s moduli, defect density, and defect velocity of molecular chains. Anisotropic plasticity is simply controlled by anisotropic Young’ s moduli. Therefore, no additional aniso- tropic parameters are needed to define anisotropic yielding surface. Strain rate, temperature, and hydrostatic pressure effects on yielding stress are through their effect on Young’ s moduli. Linear elastic, hyperelastic, viscoelastic, and viscoplastic models are put into one single equation, which makes the mathematical structure very easy to understand and easy to use. This model is validated by comparing with five existed experimental data. Proposed model also shares some features similar to the old well-known large deformation models for amorphous polymers. POLYM. ENG. SCI., 00:000–000, 2019. © 2019 Society of Plastics Engineers 1. INTRODUCTION Mechanical behaviors of amorphous structures exposed to differ- ent temperature, impact speed, and hydrostatic pressure have been investigated for many decades. In macroscopic level, amorphous structure study is limited by the complex interactions of time, tem- perature, and hydrostatic pressure. In microscopic level, it becomes hard to analyze random motion or diffusion of molecular chains. It is not straightforward to explain elastic and inelastic large deformation of amorphous structures by using continuum mechanics framework. Many internal variables and assumptions are needed to get accepted experimental fitting [1]. Relating macroscopic mechanical behavior to molecular chain movement is far more challenging. Some investi- gators built viscoelastic constitutive models for amorphous polymers by combining continuum mechanics and modular chain dynamics [2] [3]. In their work, a rheological model is built composed of springs and dashpots. Nonlinear viscosity of dashpot is coming from the Brownian motion in a combination of reptation motion and con- tour length fluctuations, which is based on Doi and Edwards’s model [4]. For example, Bergstrom and Boyce (BB) [2] generalized the relationship between effective lengths of a chain with creep time. The effective creep rate or coefficient of the dashpot is finally given as a function of effective stress and the average chain stretch, which is shown in Eq. 24 in BB paper [2]. In this article, a new constitutive model will be built by the con- cepts from reptation theory. In our derivation, plastic strain rate equation is similar to the effective creep rate equation given by BB. Our model also has a similar mathematical structure as the Bernstein–Kearsley–Zapas (BKZ) equation [5]. However, we arrived at the similar mathematical structure with a different approach and will have plasticity included which is not shown in both BB model and BKZ model. In this model, not like BB model, we will not use eight chain hyperelastic model to model elastic part of materials. We will simply apply linear Hooke’s law for the elastic part of constitutive relationship. This constitutive model is built using traditional elastic and plastic framework. When deformation is small enough not to dramatically change original structure dimen- sions, linear stress–strain relationship will be applied. Meanwhile, all other definitions of stress and strain will converge to nominal stress and nominal strain for infinitesimal deformation. As strain increases, additional term from “defect density” and “defect veloc- ity” need be added to model nonlinear effects such as viscoelastic and viscoplastic properties. For large deformation theory, we can still use initial state as reference frame, for example, Lagrangian specification of the field where stress and strain relationship will become nonlinear. In rubber zone, “defect” generated in molecular chains will affect how fast molecular forces of polymers get stiffen- ing. In glassy zone, time-, temperature-, and hydrostatic-pressure- dependent Young’s moduli will affect yielding stress values. Doi and Edwards [4] built a constitutive equation which has the form of a BKZ equation [5]. BKZ equation states that true stress is a type of expression of functional polynomials. In the simplified version of BKZ model, the stress is given as the differ- ence between linear elastic model and convolutional integrals, shown in Eq. 2.11 by Bernstein et al. [5]. Although Doi and Edwards’ model looks like BKZ model, only a convolutional inte- gral is kept. In Doi and Edwards’ model, the kennel of con- volutional integral is obtained from one-dimensional Brownian motion. As Brownian motion is considered to be random move- ment, modeling it mathematically is related to random walk and statistical mechanics. We will not be interested in the details of how to model molecule chain motions by using random process. Instead, we are trying to find an alternative way to understand this molecule chain movements by using a deterministic mathematical function. From reptation theory given by de Gennes [6], reptations or movements of molecular chains correspond to the Correspondence to: L. Yang; e-mail: lxyoung12@gmail.com Grant sponsor: Procter & Gamble; Grant sponsor: University of Cincinnati. DOI 10.1002/pen.25237 Published online in Wiley Online Library (wileyonlinelibrary.com). © 2019 Society of Plastics Engineers POLYMER ENGINEERING AND SCIENCE—2019