Flow stress

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Flow stress

Flow stress is the effective stress acting at uniaxial tension along sample axis at the stage of uniform deformation (up to necking).

During plastic deformation the flow stress usually increases. In QForm UK a strain hardening hypothesis is implemented, according to which the hardening quantity is determined with the effective plastic strain

The flow stress dependency on effective strain, strain rate and temperature can be described as follows:

is considered as description of constitutive behavior of material independently on the stress state mechanism, in other words this is a characteristic of material flow stress behavior. A single curve concept provides the possibility to use the compression and torsion tests for determination of material rheological properties.

Material flow stress is determinative of the deforming force obtained as a result of simulation.

Click to show/hide hidden textThe influence of process parameters on the flow stress

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Strain hardening

Strain rate hardening

Strain softening

 

Temperature increase at plastic deformation causes the flow stress reduction.

For the majority of materials at cold deformation the increase of flow stress with increase of effective strain and strain rate is inherent. This effect is called strain hardening and strain-rate hardening, respectively. The graph of dependence of flow stress on cumulative (effective) strain at constant temperature and constant strain rate

is called the hardening curve.

For some metals at elevated temperatures and small strain rate a phenomenon of softening occurs that is related to dynamic recrystallization.
Click to show/hide hidden textExperimental methods for determination of the flow stress

Flow stress of metal at simulation in QForm UK is chosen from the materials database. If chosen material is absent in database, its properties are determined experimentally and then stored in the database.

It is recommended to perform the experimental research on determination of material properties even, if the material is available in the database. The database in QForm UK is composed on the basis of literary sources analysis, however, the results of different authors' experiments often differ from each other even for the same material.

In order to determine the flow stress (hardening curves) different methods are used. The most common are tensile test, compression test, and torsion test.

It is recommended to perform processing of experimental results with the inverse method.

During tests it is required to maintain constant strain rate (not to be confused with testing machine grippers conveying speed) and sample temperature.

Tensile tests are carried out for standard specimens. In order to determine the strains the extensometers are used.

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At the stage of uniform deformation the area of specimen decreases. In this case the effective strain is determined from a formula:

Effective stress

Here l0 is the extensometer base (the distance between supporting elements), Dl - base increment during deformation, P - strain energy, F0 - specimen initial area.

Uniform deformation is completed with necking - local cross-section area reduction. Then the strain concentrates in the neck and it is impossible to define the stress directly from the test. For the majority of materials the effective strain in the moment of necking does not exceed the value of 0.2...0.5, that is significantly less than in the forming operations.

Extrapolation of hardening curve derived from tensile test results at uniform deformation stage for great values of effective strain can cause significant errors.

The main advantage of a compression test is the possibility of plotting the hardening curve for higher values of true strains (up to 0,8 and higher). Usually the cylindrical specimens are used for compression tests.

In ISO 6892 is suggested the method of determining hardening power during compression tests. It is assumed that the biggest contribution to error of hardening curve determination by means of compression test is made by contact friction. However, compression test simulation in a wide range of friction conditions change demonstrated that for samples with initial height-to-diameter ratio greater than 1.8 the graphs "strain energy - path" are rather similar up to value of true strain of 0.7. Friction build-up is to the a great extent compensated by the reduction of contact surface at the expense of barrel creation. The deviations of loading graphs obtained with simulation in this range do not exceed 5% for friction factor change from zero to its maximum value, that is equal to1. Thus, there arises the possibility of easy plotting the hardening curve on the basis of processing of results of compression test on cylindrical specimens up to true strain of 0.7...0.8. However, the reduction of friction impact remains pertinent, because it further reduces the plotting error.

Uniaxial tensile state is achieved by means of eliminating friction at contact surfaces. Polishing of the specimen and base plate flat ends and their lubrication allows sufficient reduction of friction at contact surfaces. However, in the process of compression the contact pressure between the specimen and plates increases, and the lubricant is pressed out, so the friction increases again. Sometimes teflon films with oil lubrication are used. This allows a great reduction of friction. However, this method can not be used for high-temperature tests because of teflon toxicity at elevated temperatures.

Among the methods of holding lubrication at contact surface used is the used turning of grooves of different length on the specimen flat edges. The grooves hold a small volume of grease creating hydrostatic pad with almost zero friction.

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Geometry of specimen for compression test: A – cylindrical specimen; B – Hocket specimen; c – Herberts specimen;
D - Lintermans-Fender specimen; E - Zolotarevsky's specimen.

In the conditions of uniaxial (frictionless) compression the reduction of height and increase of cross-section area are uniform throughout the specimen. The effective stresses and strains are determined from the formulas

Here h0, h - initial and current specimen height, P - strain energy, V0 - specimen volume.

Torsion tests allow obtaining of hardening curves in the area of considerable deformations (up to 3). In these tests it is just enough to create the conditions for obtaining a constant strain rate and temperature. Solid specimens are usually used for the plotting of hardening curves.

However, during the testing of solid specimens a considerable inhomogeneity of strain distribution, strain rates and stresses in the specimen cross-section occurs. At considerable strains in the specimens axial stresses occur as well. These factors complicate the direct simulation of hardening curves by means of torsion tests without use of any additional design functions in which the different assumptions on stressed and strained state are accepted.
Click to show/hide hidden textThe inverse method of flow stress determination

Flow stress depends on cumulative deformation, strain rate and temperature. In order to obtain exact results the tests on flow stress determination are carried out at constant values of parameters. However, it is unreachable in practice. However, it is unreachable in practice.

The finite result is influenced by the deviations of testing conditions from simple stress states (uniaxial tension, uniaxial compression, uniaxial shear), which are related to influence of friction and inhomogeneity of strain and strain rate, heat generation in the course of plastic deformation. High accuracy of stress-strain state evaluation by means of finite element method permits using of simulation to specify calculation results.

Traditionally the FE method is used for the prediction of metal flow and determination of strain energy and stress energy using the material rheological properties determined from testing as input data.

Conversely, in the inverse method the FEM simulation is used for determination of material rheological properties by means of iterative approximation to experimental data on loads and specimen geometry.

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Typical inverse method technique [H. Cho et al (2005)]

For determination of rheological model parameters by inverse method it is necessary to pre-set the mathematical relation describing the model and determine the variable parameters. Then by varying, these parameters will achieve coincidence of test and simulation results.

For example, if we use Hensel-Spittel formula as a material model

then these parameters are the coefficients A1,m1,m2,m3,m4.

As far as it is impossible to eliminate the friction impact on experiments completely, then the friction factor m and coefficient b in Levanov's friction law should be considered as additional variable parameters

Then it is necessary to carry out the experiments by means of compressing the cylindrical specimen and determine the load graph ( the dependence of strain energy on strain path P=f(S)) and the specimen geometry ( the diameters of characteristic cross-sections).

After selecting the initial approximation of variable parameters (e.g. from technical literature) the experiment simulation with FE method is carried out, for example, by means of QForm UK. Due to results of mathematical simulation the loading graph and the specimen geometry is determined.

Error of calculation of strain energy can be determined, for example [see. Hyunjoong Cho (2007)], as a root-mean-square deviation of strain energy according to the experiment results Pexp and calculation of Pcom for several points of the loading graph.

The error for prediction of geometric shape can be determined by comparison of maximum diameters (barrel diameter) in the experiment and in calculations

Cumulative error is determined as a sum of

If the simulation error exceeds the preset accuracy, a new set of variable parameters is generated and the simulations are repeated.

The generation of new variable parameters is carried out either by means of gradient methods [Cho, 2007], or by means of downhill simplex method [Yun (2011)].

Simulation cycle is completed if the simulation error does not exceed the preset value and the variable parameters increments at the next step are negligible.

Cho (2007) recommends selecting the parameters in sequence - first, selecting the parameters of the hardening curve due to strain energy graph, and then selecting the parameters of friction law due to workpiece shape.