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## Thermodynamics model for localized failure and modified balance equation.

When the localized failure happens, the free energy is decomposed into a regular part in the fracture process zone and the irregular part of free energy at the localized failure point:where ∗ denotes the regular part and ∗ represents the singular part of the potential, denotes the temperature in any position and denotes the temperature at the localizedfailure point . In (2- 23) above, the irregular part of energy is limited to the localized failure point by using , the Dirac delta function: = ∞; = 0; (2-24) The regular part of the free energy pertains to the fracture process zone, and it keeps the same form as written in (2-1). The localized free energy is assumed to be equal to: (� , ) = 1 2 ()� 2.

### „Adiabatic‟ operator splitting solution procedure

Due to the positive experience of Kassiotis et al. (see [50]), we choose the operator split method based upon adiabatic split to solve this problem. In the most general case with active localized failure, the coupled thermomechanical problem is described by a set of mechanical balance equations defined in (2-39) and (2-40), accompanied by the energy balance equations in (2-42) and (2-43). Solving all of these equations simultaneously is certainly not the most efficient option. In order to increase the solution efficiency, we can choose between two possible operator split implementations: isothermal and adiabatic (see [17]). We note in passing that the isothermal operator split is not capable of providing the stability of the computation (see [50]). Therefore, we focus only upon the adiabatic operator split method. In this method, the problem is divided into two phases, with each one contribution to change of temperature.

#### Simple tension imposed temperature example with fixed mesh

In this section we consider several numerical examples in order to illustrate the satisfying performance of the proposed model. We consider a steel bar 5 mm long. The bar is built-in at left end and subjected to an imposed displacement at right end. The imposed displacement increases 1.6 ×10-4 mm in each step. Simultaneously, right end of the bar is heated and its temperature is raised from 00C to 10000C, with 100C increase in each step. The temperature at left end is kept equal to 0oC. The loading increases until localized failure of the bar. The problem geometric data and loading program are described in Figure 2-7and Figure 2-8, respectively.

**Material properties independent on temperature**

In this case, the material properties of the bar are assumed to be constant with respect to any change in temperature. The chosen values for material parameters are given in Table 2-1. The computed results for stress-strain curves in two sub-domains are presented in Figure 2-9, while the force-displacement curve of the bar is given in Figure 2-10. In Table 2-2 and Figure 2-11, we show the resulting time evolution of temperature and its distribution along the bar. For this case with material properties independent on temperature, we can conclude that there is no difference in the strain values between two sub-domains. The „jump‟ in temperature gradient ( ), which appears at localized failure point, also remains very small. The computed dissipation due to plasticity in fracture process zone is 36.63Nmm, while the dissipation due to localized failure is 29.44Nmm. In summary, the total mechanical dissipation in the bar is equal to 66.07Nmm.

**Heating effect of mechanical loading**

In this example, we would like to illustrate the heating effect produced by mechanical dissipation in a bar when localized failure occurs. Consider a steel bar of 10mm long, fixed at left end and subjected to an increasing displacement (0.045mm/s) at right end until collapse. The initial temperature is constant along the bar and equal to 00C. Material properties of the bar are given inTable 2-1. Due to a problem in manufacturing, the ultimate stress at the middle point reduces to 299MPa instead of 300MPa in other part (see Figure 2-24).

**Table of contents :**

Remerciements

Lời cảm ơn đến gia đình

Abstract

Résumé

Table of Figures

List of Tables

List of Publications

Journals

Conferences and Workshops

**1 Introduction **

1.1 Problem statement and its importance

1.2 Literature review

1.2.1 Previous works on stress-resultant model

1.2.2 Previous works on multi-dimensional thermodynamics model

1.3 Aims, scope and method

1.4 Outline

**2 Thermo-plastic coupling behavior of steel: one-dimensional simulation **

2.1 Introduction

2.2 Theoretical formulation of localized thermo-mechanical coupling problem

2.2.1 Continuum thermo-plastic model and its balance equation

2.2.2 Thermodynamics model for localized failure and modified balance equation.

2.3 Embedded-Discontinuity Finite Element Method (ED-FEM) implementation

2.3.1 Domain definition

2.3.2 „Adiabatic‟ operator splitting solution procedure

2.3.3 Embedded discontinuity finite element implementation for the mechanical part

2.3.4 Embedded discontinuity finite element implementation for the thermal part

2.4 Numerical simulations

2.4.1 Simple tension imposed temperature example with fixed mesh

2.4.2 Mesh refinement, convergence and mesh objectivity

2.4.3 Heating effect of mechanical loading

2.5 Conclusions

**3 Behavior of concrete under fully thermo-mechanical coupling conditions **

3.1 Introduction

3.2 General framework

3.2.1 General continuum thermodynamic model

3.2.2 Localized failure in damage model

3.2.3 Discontinuity in the heat flow

3.2.4 System of local balance equation

3.3 Finite element approximation of the problem

3.3.1 Finite element approximation for displacement field

3.3.2 Finite element interpolation function for temperature

3.3.3 Finite element equation for the problem

3.4 Operator split solution procedure

3.4.1 Mechanical process

3.4.2 Thermal process

3.5 Numerical Examples

3.5.1 Tension Test and Mesh independency

3.5.2 Simple bending test

3.5.3 Concrete beam subjected to thermo-mechanical loads

3.6 Conclusion

**4 Thermomechanics failure of reinforced concrete frames **

4.1 Introduction

4.2 Stress-resultant model of a reinforced concrete beam element subjected to mechanical and thermal loads

4.2.1 Stress and strain condition at a position in reinforced concrete beam element under mechanical and temperature loading.

4.2.2 Response of a reinforced concrete element under external loading and fire loading.

4.2.3 Effect of temperature loading, axial force and shear load on mechanical moment-curvature response of reinforced concrete beam element.

4.2.4 Compute the mechanical shear load – shear strain response of a reinforced concrete element subjected to pure shear loading under elevated temperature

4.3 Finite element analysis of reinforced concrete frame

4.3.1 Timoshenko beam with strong discontinuities

4.3.2 Stress-resultant constitutive model for reinforced concrete element

4.3.3 Finite element formulation

4.4 Numerical example

4.4.1 Simple four-point bending test

4.4.2 Reinforced concrete frame subjected to fire

4.5 Conclusion

**5 Conclusions and Perpectives **

5.1 Main contributions

5.2 Perpectives

**6 Bibliography .**