Damage characterization in non-isothermal stretching of acrylics. Part I: Theory
Highlights
► Damage evolution and quantification in polymer processing operations. ► Thermodynamically consistent elastic–viscoplastic constitutive model. ► Thermo-mechanical response of amorphous polymer below and above glass transition. ► Damage evolution in terms of entropy based damage mechanics.
Introduction
Amorphous polymers have become widely popular in many domestic and industrial applications. Manufacturing of products for such applications include stretching of polymeric materials to large deformation at high temperatures during which material response is highly sensitive to rate and temperature. Numerical models are faster and cheaper in evaluation of different materials and process parameters compared to experimental studies while experimental work is also essential to identify material parameters involved in constitutive models. Polymer processing simulations are complex thermo-mechanical problems which require appropriate description of operation conditions and accurate constitutive models describing temperature and rate sensitivity of material.
Several researchers have studied yield behavior of PMMA under different testing conditions and test temperatures (Arruda et al., 1995, Arruda and Boyce, 1993, Hasan et al., 1993, Dooling et al., 1998, Dooling et al., 2002, Richeton et al., 2006, Richeton et al., 2007, Bauwens-Crowet, 1973, Palm et al., 2006, Ames et al., 2009, Srivastava et al., 2010). It was observed that mechanical response characteristics such as initial elastic modulus, yield stress and strain hardening increase with increasing strain rate or decreasing temperature, but there is a dramatic change between temperatures below and above θg (Arruda et al., 1995, Richeton et al., 2006). However, these transitions were observed to take place over some temperature range around θg which was also different for each individual aspect of response. For PMMA, effect of transition from solid to rubbery state on elastic modulus was observed to take place between approximately θg ± 10 °C (Richeton et al., 2005a). In literature, these transitions in material properties between rubbery state and solid glassy state have been unrealistically modeled as a significant change in a small temperature region (Ames et al., 2009, Srivastava et al., 2010). A proper definition of material property should be continuous and smooth over glass transition region (Richeton et al., 2007). Decrease in modulus with increasing temperature has been successfully modeled by Mahieux and Reifsnider, 2001, Mahieux and Reifsnider, 2002 in terms of Weibull statistics of failure of secondary bonds during different relaxation processes for a wide class of polymers over entire range of temperature that extends far below and substantially above θg. Later, Richeton et al. (2005a) incorporated effect of strain rate on modulus through some strain rate sensitivity factors which depends on type of amorphous polymer. Other experimental studies on amorphous polymers show that yield stress vanishes at temperatures larger than θg (Richeton et al., 2005b). Accordingly, it is widely assumed that internal stress resisting to plastic flow also drops to zero at θg and stays at zero for temperatures above θg (Richeton et al., 2006). As glass transition does not take place at a single temperature but over a large temperature region (Mahieux and Reifsnider, 2002), change in yield characteristic will have a similar behavior.
Many molecular theories have been proposed for modeling of amorphous polymers such as state transition theory of Eyring, 1936, John et al., 2009, conformational change theory of Robertson (1966), inter-molecular shear resistance model of Argon (1973), Ree–Eyring Theory (Bauwens-Crowet, 1973, Ree and Eyring, 1955, Bauwens-Crowet et al., 1969, Bauwens-Crowet et al., 1972, Bauwens, 1972), rubber-elastic network model (Arruda et al., 1995, Palm et al., 2006, Boyce et al., 2000, Bergström and Boyce, 1998). All these theories are fundamentally based on the same elementary mechanism of thermally activated molecular movement and include effects of both strain rate and temperature, yet yield stress predictions are only valid for certain range of temperature and strain rate. It was later discovered that mechanical response of amorphous polymers in a wider range of strain rates and temperatures can be predicted through Eyring Cooperative model with time/temperature principle (Richeton et al., 2005b) based on the work of Fotheringham et al., 1976, Fotheringham and Cherry, 1978. Yet, piece-wise definition of plastic flow rule with respect to glass transition in this approach (Richeton et al., 2007) create a non-smooth change in viscoplastic response when implemented for the case of a temperature change around glass transition. Alternatively, Anand et.al described change in viscoplastic response with temperature through temperature dependence of activation energy term (Srivastava et al., 2010). Though the model has a continuous definition for activation energy in temperature domain, viscoplastic response abruptly changes over a very small temperature region due to substantial difference between activation energies in “glassy” and “rubbery” region.
From constitutive modeling point of view, large deformation behavior of amorphous polymers has been widely described in terms of dual decomposition of material response into two parallel working mechanisms of inter-molecular structure and molecular network structure (Richeton et al., 2006, Richeton et al., 2007, Palm et al., 2006, Ames et al., 2009, Boyce et al., 2000). More recently, Anand et.al. proposed a trial-mechanism including a single mechanism for intermolecular structure and two sub-mechanisms for molecular network structure which are selectively activated at different temperature ranges (Srivastava et al., 2010). Both dual- and trial-mechanism models were proven to be successful in describing large deformation behavior of amorphous polymers at different isothermal test conditions. It was observed that both mechanisms are not always active and contributions may vary significantly depending on temperature and rate. At high temperatures, material response is dominated by molecular network resistance (Sweeney and Ward, 1996), while inter-molecular resistance controls deformation at low temperatures (Srivastava et al., 2010). Since test temperature and strain rate were also kept constant in experimental studies for verification of trial- or dual-mechanism constitutive models, actual performance for cases where specimen temperature changes from above θg to below θg remains unpredictable. In Anand’s recent study (Srivastava et al., 2010), non-isothermal conditions was implemented for blow forming and micron-scale hot embossing, yet performance and accuracy of model was presented merely in terms of conformation of deformed shapes in experiments and simulations, while no other results were presented explicitly, regarding stress or force or state of material. Material models for amorphous polymers which have remarkable change in behavior around glass transition need to be validated and verified in non-isothermal tests. It is, therefore, essential to improve definitions of all temperature dependent aspects of constitutive model for true prediction of response of amorphous thermoplastic polymers under conditions of changing temperature causing rubbery to solid transition in the course of loading.
Damage quantification in any process can be determined based on experimental measures such as degradation in elastic modulus, variation in micro-hardness, density, hysteresis energy, etc. (Lemaitre, 1996). Since sample temperature continuously changes during polymer processing operations, direct comparison of material properties between virgin (undeformed) state and damaged (deformed) state becomes infeasible without isolating effect of temperature which is practically impossible. Instead of parameters depending on process variables (stress, temperature …), state variables (entropy, free energy) can provide a strong basis for damage quantification. Ye et al., 2003a, Ye et al., 2003b developed a new method for damage quantification based on purely thermodynamics of physical problem for numerous materials such as solder alloys and particle filled composites. For any irreversible process, entropy production is the increase in disorder which could be clearly considered as an indication of material degradation or damage in the material. Damage evolution based on irreversible entropy production has been proven to predict material degradation accurately for different processes such as thermomigration, electromigration or thermo-mechanical loading (Basaran and Nie, 2007, Li and Basaran, 2009, Gomez and Basaran, 2005, Basaran and Lin, 2007, Basaran et al., 2004). Entropy based damage model completely relies on entropy production formulation. Damage accumulation evolves with irreversible energy expenditures including only plastic work in intermolecular network and molecular network mechanisms. A similar relation between plasticity and damage was also developed for steel and concrete (Abu Al-Rub and Voyiadjis, 2003, Cicekli et al., 2007, Voyiadjis et al., 2002).
In this study, an improved version of dual-mechanism constitutive model is proposed to describe thermo-mechanical response of amorphous polymer below and above glass transition based on Anand’s work (Ames et al., 2009, Srivastava et al., 2010, Anand et al., 2009) which was taken as a physical fundament in our model. Entropy based damage approach was considered for damage evolution in polymer processing operations.
Section snippets
Constitutive relation, flow rule and entropy production
For large deformations of polymeric materials, it is necessary to study kinematics of constitutive model based on finite deformation tensors. Consider a body with a volume of Vo in undeformed configuration (Ωo) at time to which deforms into a volume V in current configuration (Ω) at time t. The motion of a point in the body can be uniquely defined with a continuous one-to-one mapping (χ) of position vectors X and x in original configuration and current configuration, respectively (Eq. 1). The
Concluding remarks
Applicability of material models developed for large deformation behavior of amorphous polymers over a wide range of temperatures and strain rates was verified so far under completely controlled cases (constant strain rate, constant temperature). However, temperature drop from temperatures above θg to temperatures below θg causes a transition from rubbery state to solid state and a significant change in material response in actual polymer processing operations. Accordingly, large deformation
Acknowledgment
This project has been sponsored by DuPont Surface Corporation. Help received from Dr. Clyde Hutchins and Dr. Keith W. Pollak is appreciated.
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