Plasticity
The continuum mechanics basics for the one-dimensional bar will be compiled at the beginning of this chapter. The yield condition, the flow rule, the hardening law and the elasto-plastic modulus will be introduced for uniaxial, monotonic loading condition
- PDF / 1,201,062 Bytes
- 39 Pages / 439.37 x 666.142 pts Page_size
- 62 Downloads / 244 Views
Plasticity
Abstract The continuum mechanics basics for the one-dimensional bar will be compiled at the beginning of this chapter. The yield condition, the flow rule, the hardening law and the elasto-plastic modulus will be introduced for uniaxial, monotonic loading conditions. Within the scope of the hardening law, the description is limited to isotropic hardening, which occurs for example for the uniaxial tensile test with monotonic loading. For the integration of the elasto-plastic constitutive equation, the incremental predictor-corrector method is generally introduced and derived for the fully implicit and semi-implicit backward-Euler algorithm. On crucial points the difference between one- and three-dimensional descriptions will be pointed out, to guarantee a simple transfer of the derived methods to general problems. Calculated examples and supplementary problems with short solutions serve as an introduction for the theoretical description.
11.1 Continuum Mechanics Basics The characteristic feature of plastic material behavior is that a remaining strain εpl occurs after complete unloading, see Fig. 11.1a. Solely the elastic strains εel return to zero at complete unloading. An additive composition of the strains by their elastic and plastic parts ε = εel + εpl
(11.1)
is permitted at restrictions to small strains. The elastic strains εel can hereby be determined via Hooke’s law, whereby ε in Eq. (3.2) has to be substituted by εel . Furthermore, no explicit correlation is given anymore for plastic material behavior in general between stress and strain, since the strain state is also dependent on the loading history. Due to this, rate equations are necessary and need to be integrated throughout the entire load history. Within the framework of the time-independent
A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_11, © Springer-Verlag Berlin Heidelberg 2013
273
274
11 Plasticity
(a)
(b)
Fig. 11.1 Uniaxial stress–strain diagrams for different isotropic hardening approaches: a arbitrary hardening; b linear hardening and ideal plasticity
plasticity investigated here, the rate equations can be simplified to incremental relations. From Eq. (11.1) the additive composition of the strain increments results in: dε = dεel + dεpl .
(11.2)
The constitutive description of plastic material behavior includes • a yield condition, • a flow rule and • a hardening law. In the following, solely the case of the monotonic loading1 is considered, so that solely the isotropic hardening is considered in the case of the material hardening. This important case, for example, occurs in the experimental mechanics at the uniaxial tensile test with monotonic loading. Furthermore, it is assumed that the yield stress is identical in the tensile and compressive regime: kt = kc = k.
11.1.1 Yield Condition The yield condition enables one to determine whether the relevant material suffers only elastic or also plastic strains at a certain stress state at a point of the relevant body. In the uniaxi
Data Loading...
