Chapter 5
Unified Mechanics of Thermo-mechanical
Analysis
5.1 Introduction
This chapter describes the implementation of unified mechanics theory (UMT) for
thermo-mechanical analysis. Implementation will use viscoplasticity, because
thermo-viscoplastic model is necessary for metals/alloys operating at 0.3T m or
higher where T m is the melting temperature in Kelvin.
5.2 Unified Mechanics Theory-Based Constitutive Model
5.2.1 Flow Theory
5.2.1.1 Newtonian Mechanics: Elastic Constitutive Relationship
(Hooke’s Law)
For a classical von Mises rate-independent plasticity, the elastic constitutive relationship is given by Hooke’s law in the rate form as follows:
_
σ ¼ C : _
ε
e
ð5:1aÞ
_
ε
e
¼ _
ε À _
ε
p
À _
ε
θ
À
Á
ð5:1bÞ
where _
ε, _
ε
p , and _
ε
θ are total strain rate, inelastic strain rate, and thermal strain rate
vectors, respectively, and C is the elastic constitutive tensor for the virgin material.
In Eqs. (5.1a) and (5.1b) “:” represents the inner product between the fourth-order
constitutive tensor C and the elastic strain rate vector, _
ε
e .
© Springer Nature Switzerland AG 2021
C. Basaran, Introduction to Unified Mechanics Theory with Applications,
https://doi.org/10.1007/978-3-030-57772-8_5
203