Thermomechanics of Cosserat medium: modeling adiabatic shear bands in metals

Abstract

During most metal manufacturing processes, the medium deforms by generating large quantities of plastic strain at relatively high strain rates, inevitably inducing rises in temperature. Metals characterized by low thermal conductivity properties might locally retain high temperatures, consequently undergoing thermal softening. The classical balance laws governing the continuum equilibrium show severe mesh sensitivity if they were numerically discretized through finite element methods. Furthermore, the plastic deformation tends to localize in narrow areas whose characteristic length is comparable to grain size, thereby requiring the adoption of theories able to predict size-effects. In this manuscript we demonstrate that the Cosserat medium is able to overcome these issues related to manufacturing processes simulation. We first provide a thermodynamically-consistent description of the Cosserat medium, and then we propose a method to calibrate the two additional characteristic lengths introduced by the Cosserat medium description by enriching the model with the TANH stress flow rule under adiabatic conditions.

Derivation of \(\dot{T}\)

The derivation of the temperature rate can be derived starting from the definition of the entropy from Eq. (27) and assuming the Helmholtz free energy function as in Eq. (21):

(60)

then, by then taking the time variation of the entropy:

(61)

where:

(62)

(63)

(64)

and:

(65)

and by plugging them it into the heat Equation Eq. (33), we obtain:

(66)

from which the temperature rate can be evaluated as:

(67)

Table 3 Order of magnitudes of the terms in Eq. (67)

The one reported in Eq. (67) is an expression of the temperature rate which contains many terms whose magnitude is well below the magnitude of the larger terms as for example, therefore assumptions were made on some of the quantities populating Eq. (67) in order to make it usable, and here follows the list of hypotheses:

• 
• 

• $\dfrac{\partial C_{\varepsilon }}{\partial T}$ = 0;

• 

• T = 500 K ;

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• 
• 
• 
• 
• 
• 
• 
• 
• 

Furthermore, by assuming the material elastic and plastic models to be the ones described in Tables 1 and 2 , the terms in Eq. (67) have the orders of magnitudes listed in Table 3.

From the comparison of the order of magnitudes of the terms in Eq. (67) that are presented in Table 3, the temperature flow rule can assume the following form:

(68)