Constitutive Micro-modelling

Although microstructure and mechanical properties of polymers have been the subject of many studies, it is only during the last two decades or so that deformation mechanisms have been properly described. The earlier models aid in the interpretation of structural observations under different loading conditions; however, they cannot predict the microstructural damage caused by deformation. The proposed model is expressed in the framework of viscoplasticity coupled with degradation at large deformations. This involves the concepts of damage mechanics considering the original microstructure, the particular irreversible rearrangements, and the deformation mechanisms. As mentioned above, molecules in polyethylene materials can adopt two basic arrangements. Nevertheless, to simplify the complex microstructure, the model regards the material exclusively as crystalline; the amorphous phase is ignored. Thus, a three-dimensional aggregate of randomly oriented and perfectly bonded crystals is used to describe the behaviour. In an aggregate, crystals are much larger than molecules, but smaller than material points. Initially, the material is assumed to be micro – and macro-scopically homogeneous.

The theoretical formulation of the micromechanical model has been presented in detail in Alvarado – Contreras et al. (2005). Consequently, we limit ourselves to a brief description of the most important features of the model, as summarized in Table 1. The model solves the stress-strain problems in two stages, one for the microscopic problem (local problem) and the other for coupling the micro – and macro-scopic problems (global problem).

Figure 2. Slip systems in polyethylene crystals.

At the local problem, deformation mechanisms in single crystals are considered. These mechanisms are based on the theory of shear slip on crystallographic planes (Asaro & Rice, 1977). The material is initially isotropic and homogeneous, and the elastic deformations are ignored (realistic for large deformation analyses). The slip plane normal n“ and slip direction s“ define the slip systems (Figure 2). The second-order tensors R “ and A“ represent symmetric and skew-symmetric orientation tensors associated with the slip systems, as shown in Eq. 1. The deformation rate D and plastic spin Wp are consequences of the shear rates у “ on all slip systems, as indicated in Eq. 2 (Asaro, 1979). The constitutive equation connecting the microscopic deviatoric stress S and deformation rate D is expressed in Eq. 3. Here, the fourth-order compliance tensor M, based on a power-law model for the shear rates, is characterized by the inverse of the rate sensitivity n, the reference shear rate у0 , and the current critical shear strengths g“ (Eq. 4). Furthermore, Q“ is a simple scalar damage variable associated with the atomic debonds of the slip planes, and it evolves according to the evolution law given in Eq. 5, where Q 0 and m are the reference damage rate and rate exponent. Noting that damage and hardening processes may be considered independent of each other (Leimatre & Chaboche, 1994), the rate of hardening of a slip system evolves according to Eq. 6, where h0 and c characterize the slip system hardening and saturation strength, respectively. To estimate the crystal orientation evolution, the local spin W is calculated as the sum of the rigid-body spin W* and the plastic spin Wp , where the plastic component is affected by an empirical release parameter Y which is a function of the material parameter £ and slip system damage Q“ , as specified in Eq. 7.

In general, microstructure and its changes are non-deterministic. The same is true for the fluctuations in stresses and strains. Thus, for the global problem, the macroscopic mechanical behaviour is calculated as a volume average of the corresponding crystal responses. First, let D and W be, respectively, the macroscopic deformation rate and spin applied to the crystal aggregate at some material point. The partition of the deformation rate among the crystals is calculated through Eq. 8, where P is a local fourth-order projection tensor that considers the inextensibility of the crystal molecules, and P -1 represents the inverse of its volume average (Parks & Ahzi, 1990). Conversely, crystal spins, having only three independent components, simply equal the global spin, as in Eq. 9. Based on the above steps, the global response is estimated by averaging the stresses in all crystals. Then, if the homogenized reduced stress S satisfies the boundary conditions to adequate tolerance, the solution is accepted. Otherwise, corrections based on the successive substitution scheme are performed, and the procedure is repeated using S as a new trial stress. Once an estimate of S has been found, it can be related to the global deformation rate D throughout the relationship given in Eq. 10, where M is the global average compliance tensor.

Table 1. Summary of the micro-mechanical model.

W = W* +TWp , where T = tanh(-1 CEQ“)

Local problem

Global problem


D = P : P -1 : D

Deformation rate partition


w = W

Spin partition


D = M: S

Global constitutive equation

3. Numerical Results (Micro-mechanical Approach)

This section shows the capability of the model to represent the mechanical behaviour of high-density polyethylene based only on its crystalline microstructure. Calculations are carried out for an initially isotropic aggregate of 100 crystals in uniaxial tension. In the simulations, the deformation rate is prescribed, and the crystal stresses and orientation changes are determined. From them, the macroscopic stress state is calculated. Considering axisymmetric boundary conditions, a constant macroscopic strain rate of 10-3/s and null macroscopic spin are applied. For each crystal, all eight slip systems are active and the initial resolved shear strengths are known, as listed in Table 2 (Parks & Ahzi, 1990). The material parameters are fitted from two tensile tests obtained from the literature (Hillmansen et al., 2000; G’Sell et al., 2002).

Figure 3 shows the normalized equivalent macroscopic stress (ct/t0) versus the equivalent strain. The stress values are normalized using a t0 = 7.2MPa. This figure shows most of the features of the experimental responses are reproduced up to certain deformation values. The strengthening observed in the experimental tests is accounted for by adjusting the saturation strength c while the other parameters (see Table 3) remain unchanged. Two values for the saturation strength c in Eq. 6 were used, c = 0.75 and 5. As shown, the higher the c-value, the higher the stress rate (represented by the curve slope for a given equivalent deformation). Comparing, the calculated responses are stiffer than those observed in real polyethylene. This earlier strengthening is obvious at large deformations. As shown in Figure 3, an upturn in the stress-strain curve occurs beyond a strain of 1.3 for c = 0.75. This
illustrates the importance of both the crystal lattice spin and the amorphous phase for large deformations; this last feature is ignored in the current formulation.

Figure 4 shows the evolution of the damage average for the eight slip systems for the case with c = 0.75. The solid and broken lines represent the damage evolution for the transverse and chain slip systems, respectively. The profiles depicted in the figure indicate that the largest damage values occur on the (100)[001], (110)[1 1 0], and (1 1 0)[001] slip systems. All slip systems show three damage stages; decreasing, constant, and increasing. Thus, the model represents both the variations of damage gradients for the different slip systems as material is drawn as well as the sharp gradients near failure.

Table 2. Resolved shear strengths.


g“ [MPa]

(100) [001]


(010) [001]


(110) (001)


(100) [010]


(010) [100]




Table 3. Material parameters for the simulations.


Y0 [/s]


Q 0

h0 [MPa]








Figure 3. Normalized equivalent stress (a/x0) as a function of the equivalent strain for the two materials analyzed; о =