2.1. Strain-Hardening Cross-Effect
Strain-hardening cross-effect is the effect of pre-stressing on the yield limit in the direction perpendicular to that of pre-stressing. In three-dimensional principal stress space, the yield condition for isotropic material is usually shown as a cylinder inclined at some angle (equal for all axes) to all three axes of the coordination system. The cross-section of that cylinder corresponding to the plane stress can be illustrated as an ellipsis shown in
Figure 1. If when applying uniaxial tension along the
σ11 axis we yield material reaching the new yield limit
σY11, then due to strain hardening we can expect some shift (kinematic hardening) and change of size (isotropic hardening or softening) of the ellipsis representing the yield condition. The center of the ellipsis is shifted by
α11 which can be interpreted as the component of the back-stress tensor. Enlarged and shifted along the
σ11 axis, the subsequent yield surface is shown in
Figure 1. As we can see in this figure, in the direction
σ33 perpendicular to the specimen gage part surface we can expect some change in the yield stress due to strain-hardening cross-effect. Subsequent yield stress in direction
σ33 is denoted as
σY33 and can be determined using an appropriate indentation test.
To investigate strain-hardening cross-effect, one has to perform testing in a complex stress state. Most of such experiments were performed on tubular specimens subject to axial load, torque, and internal pressure. The evolution of the yield surface due to strain hardening of material subject to complex stress state was investigated by many researchers, especially in the second half of the 20th century. Recently, such tests are rarely performed due to the high cost and high work effort necessary to produce credible results. Some of the papers regarding the verification of strain-hardening hypotheses also report results regarding strain-hardening cross-effect.
In the paper [
10], the effect of tensile and torsional pre-strain on the subsequent yield surface was reported. Experiments were performed on thin-walled copper tubes subjected to simultaneous tension and torsion. The authors reported pronounced, positive cross-effect for the material pre-strained in either tension and torsion (tensile yield stress was increased by torsional pre-strain, and yield stress in shear was increased by tensile pre-strain). A similar effect was reported in [
11] for En 25 steel tubular specimens subjected to pre-stress in either tension or torsion. In this case, the cross-effect of the tensile pre-stress is much greater than that of a shear pre-stress.
The results reported in the above-mentioned and other papers indicate that we can always expect some strain-hardening cross-effect due to initial pre-stressing. This kind of effect is expected to be useful for the quantification of deformation-induced damage in the phase preceding the nucleation of material discontinuities.
2.2. Damage Indicators and Damage Parameter
It was assumed, as proven in a formerly published paper [
7], that modified Johnson’s proposition [
12] of parameter quantifying damage (damage parameter) D can be used as the reference for the validation of other damage indicators. The definition of this parameter is as follows:
where
stands for the current value of the accumulated plastic strain intensity and
is the final value of the accumulated plastic strain intensity (corresponding to the failure of the material). Such a reference damage parameter is calculated as the ratio of the current plastic strain intensity at a given spot of the material to the value of plastic strain intensity corresponding to the failure of the material. In the case of an arbitrary loading path in the stress space, the damage indicator is calculated as integral to the plastic strain intensity along this path (index notation used):
The definition of this parameter, first given by Odquist in his paper [
13], can be used for a proportional as well as a non-proportional path in the complex stress state.
If the material is subject to uniaxial load, which is the case in question, three non-zero strain tensor components can be identified as principal strains. In the case of a proportional strain path (monotonic deformation or ratcheting in the case of cyclic loading) and in terms of principal strain, plastic strain intensity
can be expressed by the following formula [
14]:
For deformation obtained as the result of an applied, uniaxial loading program, this parameter can be easily calculated on the base of the measurements of the initial and deformed geometry of the specimens’ gage part cross-sections marked in
Figure 2.
The patented [
15] method of calibration (and validation) of damage indicators was described in another paper [
7]. The idea behind this procedure was to produce, in a single test, a plastic strain intensity field varying from zero (undamaged state) to that corresponding to failure in the gage part of the specimen, as shown in
Figure 2. This gage part has a variable width
b and constant thickness
a. The deformation was produced by monotonic and cyclic loading to rupture. The plastic strain intensity at selected and marked cross-sections of the gage part was calculated on the base of measurements of the cross-section geometry before and after the test, assuming the incompressibility of the material. The failure strain
corresponds to the plastic strain intensity at the fracture surface. Other details of the method used for the introduction of deformation-induced damage into the investigated material can be found in the patent description [
15] and also in the aforementioned paper [
7].
The accumulated plastic strain intensity can be used as a damage indicator for any mechanism of damage and arbitrary loading path in the stress space as suggested in this paper [
3]. In the case of LCF and HCF tests, accumulated plastic strain consists of ratcheting and inelastic response in the load cycle (double hysteresis loop width) [
3]. The sum of inelastic response is in this case (R = 0.05—only tensile load) negligible and can be disregarded. We can assume that all accumulation of plastic strain is the result of ratcheting. In the case of static tension, the damage is induced by monotonic deformation. The most important question in the case of the presented investigations is: how the uniaxial stress state resulting from the applied load program influences the yield limit in the perpendicular (normal to specimen surface) direction. We can assume that no residual stress acts in this direction and changes in yield stress and flow stress are the sole results of changes in crystal defect density, their agglomeration, and locking at obstacles (e.g., grain or phase boundaries). This allows us to eliminate, for example, the influence of the Bauschinger effect on the results of indentation tests.
The parameters selected to be used in the presented investigations as damage indicators—Brinell hardness HB and work of indentation limited by constant maximum load W—were calculated on the base of indentation test results. The correlation between the mentioned damage indicators and the reference damage parameter (1) was investigated and will be reported in the following paragraphs.
Hardness, by definition, is related to yield limit. The definition of hardness takes the following form [
4]:
where
F is the indentation force and
S is the curved surface area of the indentation [
16]. According to Lemaitre [
4], the coefficient of proportionality has to be introduced into the relation between the hardness and the yield limit
, so finally it takes the following form:
where
c is the coefficient of proportionality.
Hardness is influenced both by strain hardening and damage in the form of discontinuities. This is also valid if we assume that the indentation is performed in the direction perpendicular to the initial, tensile loading. In the general case, both effects of dislocation structure change, and the creation and growth of micro-discontinuities contribute to damage progress; however, in the early phase of the process (before nucleation of micro-discontinuities) we can disregard the influence of discontinuities and focus on the change of the dislocation structure.
Loading can be performed along a non-proportional path in the stress space and micro-discontinuities of arbitrary orientation and form (planar cracks, voids) can be formed. In spite of this, the damage parameter
D in this case shall be coupled with the isotropic component of the strain hardening law [
4] and the yield condition shall be written as suggested in [
17] in the following form:
where
is the stress tensor,
is the back-stress tensor,
is the accumulated plastic strain intensity (2), and
k is the yield stress in shear increasing with progressing plastic deformation (isotropic hardening). In the initial phase of damage, before the nucleation of micro-cracks, we can assume that yield stress is increasing due to the generation and mutual locking of dislocations. After the balance between the effect of strain hardening and the formation of discontinuities on the yield condition is achieved, yield stress shall start decreasing.
Since no back stress is expected in the direction perpendicular to the material surface (
= 0, see
Figure 1) we can assume that the yield stress in this direction, for the initial phase of damage, is dominated by the change of dislocation density and its structure (isotropic hardening).
where
k is the yield stress in shear (the initial value of this parameter is shown in
Figure 1). In the later phase, the mechanism of damage is dominated by the creation and growth of physical micro-discontinuities limiting the ratcheting resulting from the dislocation movement. Such micro-discontinuities grow with applied load cycles and finally achieve a size enabling the triggering of uncontrolled crack extension leading to the final rupture of the specimen. If we can determine, on the basis of appropriate experiments, the form of both functions
and
we can find the critical (maximum) value of the hardness differentiating function (7):
It has to be mentioned at that point, that use of the linear damage function (1) doesn’t give reasonable results. The maximum value of hardness determined experimentally corresponds to a much higher value of
than the value determined with the use of Equation (8). This leads to the conclusion that damage is not a linear function of the accumulated equivalent plastic strain. Bearing in mind that fact, the definition of the damage parameter proposed by Johnson doesn’t seem to be sufficient to describe all the phenomena responsible for the progress of material damage. A more appropriate measure of the damage shall be defined, for example, on the base of measurements of the residual ability to achieve permanent deformation. Using definition (1) the residual failure strain intensity can be given by a simple relation:
Assuming, for example, the Hancock–Mackenzie ductile failure criterion [
18], we can postulate that the damage criterion for ductile materials can be modified to the following form:
where
a is the constant characterizing the material in question and
η is the triaxiality factor characterizing the stress state (ratio of first and second invariant of stress tensor). A simple measuring procedure for the determination of
as well as a description of the necessary equipment can be found in the paper [
19].
Summarizing, the usefulness of hardness measurements for damage assessment shall be applicable only to the initial phase of the process (before crack nucleation) and must be verified in an experimental way.