The nonlinear hereditary-type stress–strain relations for metals with temperature effects

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Abstract

The objective of this paper is to present the approach to the development of hereditary-type theory with due account of temperature influencing the time dependent mechanical behavior of metals and alloys. The primary step included the designing of the model with the hereditary-type governing equation structured to account for the thermal effect. It is shown that the whole temperature influence can be modeled by introducing the special power function with one additional parameter, which is put under the integral. Isothermal and quasi-isothermal processes with abrupt change of temperature are considered. The proposed hereditary-type model with temperature effect is verified experimentally.

Introduction

The first publication on hereditary-type theory application for the description of metals behavior appeared in the International Journal of Solids and Structures in 1978. Following the major principles of that historic publication we will show that in some cases the analytical results, which this theory helps to receive, are quite simple for interpretation and can not be obtained by any other known theories.

It was shown [1], [2], [3] that with some modifications the nonlinear-hereditary theory of viscoelasticity might be applied to the description of the behaviour of metals. Moreover, the theory allows to analyze and understand some controversial experimental phenomena, which used to be viewed and interpreted in different ways. This work is devoted to construction of the hereditary type constitutive equation for metals, which accounts for the thermal effects influencing the mechanical behavior.

The hereditary theory of viscoelasticity is based on the hypothesis of the upper instantaneous straining diagram existence from which “slipping down” takes place with time. This circumstance which allows to eliminate the static diagram from the constitutive equation and on the other hand account for the elastic mechanical history gave reason to apply the hereditary theory to the analysis of metals.

The innovative specific feature for the adaptation of the theory for metals consists in replacement the values of total strain in the governing equation by plastic componentsφ(p)=(1+K)σ,Kσ=0tK(t-τ)dτ.Equation σ=φ(p) defines the instantaneous strain diagram, p=ε-σ/E, ε is the total strain. The domain of the function φ(p) consists of nonnegative values of total strain, while φ(0) is the dynamic yield stress at an infinitely high strain rate.

At the initial moment, corresponding to the start of loading in (1), t=0; consequently, the integral term takes into account the entire history of loading in the elastic range. However, the substantial experimental data indicates the apparent evidence of internal processes in elastic materials leading to the development of subsequent plastic strain. Eq. (1) appears to be the simplest nonlinear hereditary type relationship between the values of p and σ.

Section snippets

Constitutive equation with temperature effects

The analysis of thermal factor influencing the metals mechanical behavior had been the subject of long going discussion and had been covered in a variety of publications. Majority of research efforts in this field consist generally in constructing stress–strain curves at various temperatures and in attempts to describe temperature effect by artificial introduction of parameters dependent on temperature into the governing equations. Some empirical or semi-empirical relationships allowing to

Quasi-isothermal processes of loading

The quasi-isothermal processes can be defined as such that the whole time interval can be divided into subintervals for which the process is isothermal (i.e. the temperature is constant) and the change from one constant level to the other is abrupt (i.e. the rate of change being infinite).

It is known that in ordinary hereditary-type viscoelasticity is characterized by the same initial moduli for all stress–strain curves with various loading programs, which are equal to instantaneous curve

Some experimental examples

In case of abrupt change of temperature the condition (7) or (9) is disturbed and material will exhibit the elastic behavior, as was mentioned above. The whole history of straining would not effect the further elastic behaviour. The only point of particular importance is the point with the coordinates (σ,p), starting from which the new linear segment of the stress–strain diagram starts off. For the further analysis of materials behaviour it is necessary to built the new curve of instantaneous

Conclusion

It is well known that mechanics of hereditary media is the most perspective method for the description of viscoelastic behaviour for various materials, especially for polymers and polymeric composites. The presented research shows that this method can be applied to the metals too if they posses the viscoelastic properties, for example for low carbon steels, titan, copper and some others at room temperature (and various rate of loading) and almost for every metal and alloy at high temperatures

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