Principles of Stress-Strength Modelling of the Highly Thermally Loaded Materials—The Influence of an Effect of Strength Differential on the Material Effort

This paper presents an improvement in the Huber–Mises–Hencky (HMH) material effort hypothesis proposed by Burzyński. Unlike the HMH hypothesis, it differentiates the plastic effort between compression and tensile load states, and links shear with tensile limit. Furthermore, it considers the fact that construction materials do not have infinite resistance in the pure tensile hydrostatic load state, which was proved by the static load experiment performed on St12T heat-resistant steel. The asymmetry between tensile and compressive loads is captured by the elastic region asymmetry coefficient ϰ, which was established by experiment for St12T steel in the temperature range between 20 °C and 800 °C.


Introduction
Demonstrating thermal stresses and strains is generally a complex and challenging task. Typically, if the acceleration time is long enough transient thermal gradients are significantly reduced keeping thermal stresses well below the yield strength. On the other hand, in the case of steep and fluctuated loads containing rapid cooling-heating cycles, the thermal stresses can become much higher than yield strength inducing plastic deformation. This is particularly harmful for areas with a high stress concentration.
During the design process of a highly thermally loaded components such as combustion chambers, special attention should be paid to stress-strength modelling. It is even more important in cases such as the use of modern experimental materials, such as sintered metals or orthotropic reinforced ceramics, because of their limited current application. According to commonly accepted procedures in the gas turbine industry, it is good practice to apply additional non-standard thermal-strength models. The reasons for this approach are as follows: the first obvious purpose is the verification of standard models that have been widely used for years. The second, less obvious reason is to improve existing methods via the detection of phenomena which do not exist in classical materials. In such a case, classical tools need to be upgraded or replaced to follow the progress in materials engineering. The fluid-solid interaction approach is particularly important in improving the accuracy of load state prediction. The following paper proposes a novel model of the phenomena, with an emphasis on viewing strength variability as a function of material load (effort). This approach was used to model the stress and deformation in turbine components, and is successfully tailored to describe the physical phenomena present in the wet combustion liner designed in the project.
Heat-resistant materials such as chromium steel, are the subject of numerous testing campaigns aimed at improving the durability of gas turbine combustion chambers [1]. candidate to measure material effort [2,10]. Therefore, in our numerous papers on this subject, we differentiate two notions: "effort hypothesis", and "strength hypothesis". Note that the notion of "material effort" is fully consistent with the Maxwell energy approach. In the complete mathematical form, this energy-based approach was initiated by Beltrami (1885) in the form [9]: Φ ≤ K, where Φ is some elastic deformation energy (precisely, the volumetric density of energy) describing a state of material effort, and K is a critical value of this energy. Beltrami first found that critical energy K depends on the uniaxial yield k t or torsion k s . This approach also opened up the possibility of using many other forms of experimental data such as: Vigers hardness, Sharpy energy, toughness critical energy, cleavage energy, and so on (see Orłowski et al. [22,23]).
In this approach, it is important to know the value of critical energy: where 0.01 < K < 0.30 (as an example) for k t = 700 MPa, E = 2.1 GPa, and K = 0.23 MJ. From the perspective of required experimental parameters, the above criteria can be classified as: one-parameter (Beltrami, Huber, HMH), two-parameters (Schleicher), three-parameters (Burzyński, Zawadzki), and five-parameters (Pęcherski). Note that the energetic approach is dedicated to the question of multi-parameter criteria in cases where parameters represent physically different phenomena and dimensionalities. For instance, "limiting cleavage" is, from the very beginning, given in terms of [Joule] not in terms of [MPa]. Another example is "limiting hardness", which is given in Brinell or Vickers scale units. The other candidates in the multi-parameter effort hypothesis are: brittleness, adhesiveness, gumminess, chewiness, and resilience [13,15]. If robust scientific tools for measuring these limiting parameters can be obtained, then a general framework for finding the principles of energy-like effort modeling, even with 15 parameters, is possible. Therefore, our paper investigates this question from the very beginning.

The Huber Material Effort
In 1903 Huber proposed an energetic measure of equivalent stress that is based on the energy of elastic deformation [9]: In terms of stresses: Note that, originally, his equation for energy density was expressed by three main strains [9], which is quite easy to decompose into distortional and volumetric parts: where H = E/3(1 − 2ν) and G = E/2(1 + ν) are the Helmholtz and Kirchhoff coefficients, respectively. It is important that energy can be split into volumetric and distortional (shear-like) components: Φ v + Φ f , where shear energy is: Three main stresses appear, q (distortion stress), τ n (octeadric stress), and σ HMH (Huber-Mises-Hencky stress) that are well known from the literature [17,24,25]. This is alongside another frequently used quantity, the second invariant of the deviatoric stress tensor J 2s .

An Extended Burzyński Material Effort
The necessity of extension of classical HMH life limiting hypothesis was proven by the heat resistant steels experiment. It indicated that the limit in uniaxial tension is different to the limit in uniaxial compression, so the one-parameter HMH hypothesis must be replaced with a two-parameter one. In general, according to the experimental data [1,4,28,29], there are several possible limiting parameters even in the single uniaxial probe: elastic limit, yield stress, maximum strength limit, and failure limit. In order to express the fact of generality within the energy approach, let us denote all of these limits collectively by k t . However, in the next section, focusing our attention only on plasticity, we replaced instances of k t with the letter σ t e . Burzyński indicates [6,7] that seven different load regimes should be tested: uniaxial tensile limit k t , uniaxial compression limit k c , torsional limit k s , bi-axial tensile k tt , bi-axial compression k cc , three-axial tensile k ttt , and three-axial compression k ccc . Note that, usually, four basic limits of the effort state k t are taken into consideration: elastic R e , plastic R pl , extremal R m , and rupture R z . The main motivation for extension of the one-parameter HMH hypothesis is the well-known Duguet-Mohr hypothesis for materials with k c > k t : In 1927, using the achievement of his supervisor Huber as a starting point, young Burzyński proposed a two-parameter extension of the Huber effort model [7]. Now, let us revalorize his line of reasoning by adding the thermal energy term Φ th , which derives from the thermal expansion of a solid material: where W is a measurement of effort (in terms of energy not stress), which is a quasi-linear composition of Φ f , Φ υ and the parameter η ν , which is a function of pressure p and two constants ω, δ: Equation (29) was created by inserting Equation (28) into Equation (27): Materials 2021, 14, 7449 6 of 14 which can be written in an extensive form as: Next, substituting: after multiplication by 12G(1 + ν) the following form is obtained: or, in brief: where: This means that a and b are: In terms of k t , κ, and ν our extended Burzyński hypothesis takes the following form: Another form can be obtained when the part: This is expressed as: Then, the extended (validated) Burzyński criterion has the form: If a and b → 0 , then the original Burzyński formulae is obtained [7]. If a = 0 and ν = 0.5, then the other Burzyński form arises [7]. Finally, when b = 0 (meaning κ = 1) a traditional HMH condition is formed.

Huber-Mises-Hencky and Burzyński Equivalent Stress
Numerical modelling in the design process needs to be applied for different structural elements to capture the state of material effort at every critical point in a structure. Usually, the design process assumes certain stress margins to ensure the robustness and safety of the final product. Strictly speaking, designers must always keep component effort below the strength limit. Typically for steels, when k t = k c , we use the well-established one-parameter Huber-Mises-Hencky (HMH) hypothesis to describe the effort of the given feature. In such a case, the equivalent stress could be written using the following relationships: where: J 2s is the second invariant of the stress deviator; s is the stress deviator; σ 1 , σ 2 , and σ 3 are the principal stresses of the stress tensor; and σ xx , σ yy , σ zz , σ xy , σ yz , and σ zx are the normal and shear components of the stress tensor. Burzyński understood that many materials are stronger in compressive load states rather than a tensile ones, so he proposed an improvement to the generalised Huber-Mises-Hencky hypothesis in the form below: or in a shorter form: where σ m is the mean normal stress and κ is the coefficient of asymmetry of the elastic area, which are determined by the following relationships: where, in turn: σ t e ≡ k t is the tensile yield limit, and σ c e ≡ k c is the compression yield limit. These parameters ought to be figured out during static compression and tension tests.
Note that some elements of this model have recently been developed in the literature, as in Banaś and Badur [8], where a numerical tool was prepared to extent the HMH surface modelling (of a cylinder) into the Burzyński surface (a paraboloid). The results of some numerical simulations alongside real experimental data have been examined in the available papers [15,28,[30][31][32][33].

Experimental Procedure
To verify the Burzyński effort hypothesis and evaluate the load asymmetry coefficient (Equation (50)), experimental measurements were obtained for the tensile and compressive limits. That approach is an improvement of the classic Huber-Mises-Hencky hypothesis, which assumes an equal limit in the tensile and compressive load regimes. Figure 1 presents a set of ruptured samples made of St12T steel after the tensile limit test at 400 • C. The diameter and length are 5 mm and 52 mm, respectively (excluding threads). pothesis, which assumes an equal limit in the tensile and compressive load regimes. Figure 1 presents a set of ruptured samples made of St12T steel after the tensile limit test at 400 °C. The diameter and length are 5 mm and 52 mm, respectively (excluding threads). The experiment is based on the Heckert EUS-20 hydraulic universal testing machine shown in Figure 2. A test performed at an elevated temperature was undertaken with a 20 kN load and an accuracy of ±0.05 kN. The heating system contains the heating chamber that is powered by the transformer, and equipped with a double-digit temperature control system (PT-0102 NVO Termoprylad) and temperature gauges. The temperature gauge accuracy is equal to ±1 °C. The strain gauge (MTS 634 11F-24) has a measurement capability between −2.5 mm and 5 mm, and its measurement class is equal to 0.5 (ISO 9513). The conversion from an analogue to digital signal was achieved using the L-Card E440 converter coupled with the Power Graph 3.3.8 software for storing and analysing data. The experiment is based on the Heckert EUS-20 hydraulic universal testing machine shown in Figure 2. A test performed at an elevated temperature was undertaken with a 20 kN load and an accuracy of ±0.05 kN. The heating system contains the heating chamber that is powered by the transformer, and equipped with a double-digit temperature control system (PT-0102 NVO Termoprylad) and temperature gauges. The temperature gauge accuracy is equal to ±1 • C. The strain gauge (MTS 634 11F-24) has a measurement capability between −2.5 mm and 5 mm, and its measurement class is equal to 0.5 (ISO 9513). The conversion from an analogue to digital signal was achieved using the L-Card E440 converter coupled with the Power Graph 3.3.8 software for storing and analysing data. pothesis, which assumes an equal limit in the tensile and compressive load regimes. Figure 1 presents a set of ruptured samples made of St12T steel after the tensile limit test at 400 °C. The diameter and length are 5 mm and 52 mm, respectively (excluding threads). The experiment is based on the Heckert EUS-20 hydraulic universal testing machine shown in Figure 2. A test performed at an elevated temperature was undertaken with a 20 kN load and an accuracy of ±0.05 kN. The heating system contains the heating chamber that is powered by the transformer, and equipped with a double-digit temperature control system (PT-0102 NVO Termoprylad) and temperature gauges. The temperature gauge accuracy is equal to ±1 °C. The strain gauge (MTS 634 11F-24) has a measurement capability between −2.5 mm and 5 mm, and its measurement class is equal to 0.5 (ISO 9513). The conversion from an analogue to digital signal was achieved using the L-Card E440 converter coupled with the Power Graph 3.3.8 software for storing and analysing data.

Limit Properties of the St12T Steel
The most popular heat-resistant material used in power plants is St12T steel [4,8,[14][15][16]28]. As mentioned in the previous section, static tensile and compressive tests were made to figure out the steel limit properties that are required for further calculations. To account for experimental variability three points were tested at a single temperature ( Table 1). The only exception is the 800 • C case, where a single test was completed. Table 1 contains averaged values for each test condition: Every model predicting the load and effort of power plant components ought to contain certain material properties that are dependent on temperature: Young's modulus E = E(T); yield limit σ e = σ e (T) or offset yield limit σ 0.2 = σ 0.2 (T) (see Figure 3); and tensile strength R m = R m (T). From the perspective of the Burzyński hypothesis, the impact of temperature on the elastic region coefficient of asymmetry (introduced by Burzyński) should also be taken into account according to the equation below:  Elastic region asymmetry coefficients presented in Figure 4 were calculated according to Equation (50) and the yield strength limits presented in Table 1. In the following sections, the method based on the measured elastic region asymmetry coefficient (Equation (50)) will be called the vB (validated Burzyński) method. Elastic region asymmetry coefficients presented in Figure 4 were calculated according to Equation (50) and the yield strength limits presented in Table 1. Elastic region asymmetry coefficients presented in Figure 4 were calculated according to Equation (50) and the yield strength limits presented in Table 1. The polynomial interpolation curve presented on Figure 4 is described by Equation (51) and is valid for the temperature range of 20 °C -800 °C. This is a convenient form of the variable input parameter that is accepted by numerical analysis software. In the following sections, the method based on the measured elastic region asymmetry coefficient (Equation (50)) will be called the vB (validated Burzyński) method. The polynomial interpolation curve presented on Figure 4 is described by Equation (51) and is valid for the temperature range of 20 • C-800 • C. This is a convenient form of the variable input parameter that is accepted by numerical analysis software.
In the following sections, the method based on the measured elastic region asymmetry coefficient (Equation (50)) will be called the vB (validated Burzyński) method.

Thermal Shifts of Burzyński Plastic Regions
The Burzyński hypothesis formulates plasticity as a function of three principal stresses (σ 1 , σ 2 , σ 3 ) and is described by the following formula: where: ν = σ c e σ t e 2(σ s e ) 2 − 1 is a coefficient of plasticity, and σ s e denotes torsional yield limit. In the general case, the coefficient ν should also be modified to account for temperature dependency. However, ν is assumed to be 0.5 to govern the plasticity of St12T steel. The reason for this is the lack of measurements of torsional yield limit within the required temperature range. The same simplification of ν was adopted by Burzyński for brittle and plastic materials.
The plastic limits governed by Equation (52) are plotted in Figure 5. Each surface accounts for a temperature (Table 1), a cycle asymmetry coefficient, and a tensile yield (elastic) offset [17]. The intersection curves between paraboloids and the reference plane that contains the deviatoric stress axis σ D ( Figure 5) and the hydrostatic axis σ m (σ 1 = σ 2 = σ 3 ) are presented in Figure 6. Figures 5 and 6 are based on the average normal stress σ m at the hydrostatic axis, and the deviatoric stress σ D on the deviator stress axis. A 3D type of chart is required to mark the dependency between principal stresses and the deviatoric/hydrostatic axes.
The paraboloid shape of the tensile strength surfaces on Figure 5, and their crosssection on Figure 6 reveal the dependency between temperature and the plastic evolution of St12T steel (especially for regions of tension σ 1 + σ 2 + σ 3 > 0). The asymmetry coefficient κ clearly controls the shape and position of critical surfaces. It is very clear around 400 • C, where peak σ m stress is higher than at 200 • C ( Figure 5). That inversion was observed for the first time in the literature. Additionally, Figure 3 proves that a plastic region also occurs at low temperatures (below 200 • C).

Thermal Shifts of Burzyński Plastic Regions
The Burzyński hypothesis formulates plasticity as a function of three principal stresses ( 1 , 2 , 3 ) and is described by the following formula: where: ̃= 2( ) 2 − 1 is a coefficient of plasticity, and denotes torsional yield limit. In the general case, the coefficient ̃ should also be modified to account for temperature dependency. However, ̃ is assumed to be 0.5 to govern the plasticity of St12T steel. The reason for this is the lack of measurements of torsional yield limit within the required temperature range. The same simplification of ̃ was adopted by Burzyński for brittle and plastic materials. The plastic limits governed by Equation (52) are plotted in Figure 5. Each surface accounts for a temperature (Table 1), a cycle asymmetry coefficient, and a tensile yield (elastic) offset [17]. The intersection curves between paraboloids and the reference plane that contains the deviatoric stress axis ( Figure 5) and the hydrostatic axis ( 1 = 2 = 3 ) are presented in Figure 6.    6 are based on the average normal stress at the hydrostatic axis, and the deviatoric stress on the deviator stress axis. A 3D type of chart is required to mark the dependency between principal stresses and the deviatoric/hydrostatic axes.
The paraboloid shape of the tensile strength surfaces on Figure 5, and their crosssection on Figure 6 reveal the dependency between temperature and the plastic evolution of St12T steel (especially for regions of tension 1 + 2 + 3 > 0). The asymmetry coefficient clearly controls the shape and position of critical surfaces. It is very clear around 400 °C, where peak stress is higher than at 200 °C ( Figure 5). That inversion was observed for the first time in the literature. Additionally, Figure 3 proves that a plastic region also occurs at low temperatures (below 200 °C).

Conclusions
The Burzyński thermal effort hypothesis was created to capture the complex cycleplastic behaviour of thermally loaded materials. In the presented paper, experimental confirmation was obtained for a heat-resistant St12T steel. The quasi-static load was applied at several thermal conditions to capture the heat-resistant steel parameters for the 20-800 °C range. As a result, it was found that yield limit values do not drop proportionally with an increase in temperature. Since both compressive and tensile stresses act during every thermal loading of a structure, important changes between the Huber-Mises-Hencky and Burzyński methodologies can be captured.
The evolution of the plastic deformation in St12T steel ( Figure 6) clearly captures the impact of temperature on yield strength. That is particularly strong in the region of tension ( 1 + 2 + 3 > 0). Temperature equal to 800 °C can be assumed as critical for tension, as the tip of that paraboloid ( Figure 5) is close to zero at hydrostatic axis.
In summary, the authors have proved the advantage of the Burzyński three-parameter material effort model over the Huber-Mises-Hencky single-parameter approach. As a result, its use is recommended in industrial applications because it properly captures the interaction between temperature and yield surface position. One particularly valuable outcome is the strict mathematical description of the dependency between the temperature, shape, and position of the yield surface, which can be used in numerical simulations.

Conclusions
The Burzyński thermal effort hypothesis was created to capture the complex cycleplastic behaviour of thermally loaded materials. In the presented paper, experimental confirmation was obtained for a heat-resistant St12T steel. The quasi-static load was applied at several thermal conditions to capture the heat-resistant steel parameters for the 20-800 • C range. As a result, it was found that yield limit values do not drop proportionally with an increase in temperature. Since both compressive and tensile stresses act during every thermal loading of a structure, important changes between the Huber-Mises-Hencky and Burzyński methodologies can be captured.
The evolution of the plastic deformation in St12T steel ( Figure 6) clearly captures the impact of temperature on yield strength. That is particularly strong in the region of tension (σ 1 + σ 2 + σ 3 > 0). Temperature equal to 800 • C can be assumed as critical for tension, as the tip of that paraboloid ( Figure 5) is close to zero at hydrostatic axis.
In summary, the authors have proved the advantage of the Burzyński three-parameter material effort model over the Huber-Mises-Hencky single-parameter approach. As a result, its use is recommended in industrial applications because it properly captures the interaction between temperature and yield surface position. One particularly valuable outcome is the strict mathematical description of the dependency between the temperature, shape, and position of the yield surface, which can be used in numerical simulations.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to project restrictions.