Incremental Growth Analysis of a Cantilever Beam under Cyclic Thermal and Axial Loads
<p>Load diagram for the uniaxial stress model.</p> "> Figure 2
<p>(<b>a</b>) Cantilever beam and applied loads. (<b>b</b>) Constant mechanical load. (<b>c</b>) Cyclic thermal gradient.</p> "> Figure 3
<p>The schematic of stress–strain regimes.</p> "> Figure 4
<p>The steps in APDL.</p> "> Figure 5
<p>The stress–strain path corresponding to the stress regime R1 at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>3</mn> <mi>K</mi> </mrow> </semantics></math>.</p> "> Figure 6
<p>The stress–strain path corresponding to the stress regime R2 at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.366</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>6</mn> <mi>K</mi> </mrow> </semantics></math>.</p> "> Figure 7
<p>The stress–strain path corresponding to the stress regime P at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.167</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>6</mn> <mi>K</mi> </mrow> </semantics></math>.</p> "> Figure 8
<p>The stress–strain path corresponding to the stress regime S1 at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>2</mn> <mi>K</mi> </mrow> </semantics></math>.</p> "> Figure 9
<p>The stress–strain path corresponding to the stress regime S2 at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.25</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>3</mn> <mi>K</mi> </mrow> </semantics></math>.</p> "> Figure 10
<p>The stress–strain path corresponding to the stress regime R1 at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>3</mn> <mi>K</mi> </mrow> </semantics></math> with more cycles.</p> "> Figure 11
<p>The stress–strain path corresponding to the stress regime R1 at <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>0.9</mn> <mo>,</mo> <mo> </mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>d</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mstyle> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>∆</mo> <mi>T</mi> <mo>=</mo> <mn>14</mn> <mi>K</mi> </mrow> </semantics></math>.</p> "> Figure 12
<p>Variations in the ratchet strains with constant <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>t</mi> </mrow> </msub> </mrow> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>y</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> </semantics></math> for the plane stress model.</p> "> Figure 13
<p>Bree interaction diagram, along with results from the theory.</p> ">
Abstract
:1. Introduction
2. Methodology
2.1. Solution for Stresses and Strains
- Purely elastic regime (E);
- Shakedown regime one (S1);
- Shakedown regime two (S2);
- Plastic cyclic regime (P);
- Ratcheting regime one (R1);
- Ratcheting regime two (R2).
2.2. Material Behavior Assumptions
2.3. Problem Definition
2.4. The Study Procedure
3. Results and Discussion
- Its remarkable capability to examine the critical design parameters prior to the construction process reduces potential costs.
- Its virtual prototyping capability allows the design process to be conducted with the fewest physical prototypes. Consequently, companies are less likely to spend such huge amounts of money on unsuccessful experiments.
- This analysis considers various conditions and characteristics (stress, fatigue, creep, ratcheting, shakedown, etc.) for the prototypes.
- By exploiting FEA, we will have high productivity levels in manufacturing, and the industries will considerably prosper.
4. Conclusions
- For Regime R1, the plastic strain in the first applying thermal load equals the sum of the elastic and plastic strain of the applied load in the following cycles, as shown in Figure 5.
- As shown in Figure 5, the difference between the maximum and minimum stress related to a cyclic thermal load and the initial stress due to constant load in Regime R1 is equal.
- In contrast to the results of Regime R1, the absolute values of maximum and minimum stress are the same in Regime R2, as presented in Figure 6.
- Fully reversed plastic does not occur in Regime R2. When applying the thermal load, the plastic strain is larger than the plastic strain when removing the thermal load, whereas there is no plastic strain when removing the load in Regime R1.
- Low-cycle fatigue is found in Regime P, as shown in Figure 7.
- All of the regimes with more cycles could also be observed, as shown in Figure 10.
- Figure 11 shows that various amounts could be obtained from Bree’s diagram to reach the regimes (elastic, shakedown, fully plastic, and ratcheting).
- Theoretical results were compared to results obtained by FEM, and it turned out that there is a good agreement between them, as shown in Figure 13.
- Based on the striking benefits of finite element analysis, exploiting this solution is recommended.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Reference | Materials Studied | Experimental or Modeled | Main Tools Used | Key Results |
---|---|---|---|---|
Bingjun Gao [5] | Low carbon steel | Modeled | C-TDF approach, Jiang–Sehitoglu model | Isolated the shakedown area accurately. |
Shariati et al. (2012) [6] | Polyacetal | Experimental | Uniaxial cyclic loading | Ratcheting strain sensitive to stress amplitude and mean stress. |
Zhu et al. (2017) [7] | Metals | Modeled | Fatigue life prediction models | Proposed a highly accurate fatigue life prediction model. |
Shariati and Kolasangiani (2017) [13] | SS304L cylindrical shells | Experimental | Servo-hydraulic Instron 8802 machine | Plastic deformation increases with cylindrical shell angle. |
Bree (1967) [30] | Thin tubes | Modeled | Analytical methods | Introduced regimes related to strain behavior under cyclic loads. |
Servatan et al. (2023) [26] | Al 4043, SS316L, Ti–6Al–4V alloys | Modeled | Finite element analysis (Chaboche’s model) | Close agreement between predicted and simulated ratcheting curves. |
Yu et al. (2023) [27] | Ti6Al4V alloy (for medical implants) | Experimental and Modeled | Electron beam melting, fatigue life modeling | Mapping ratio significantly affects fatigue properties. |
Young’s Module | 30 × 106 MPa |
---|---|
Thermal conductivity | 0.01 W/mK |
Coefficient of thermal expansion | |
Poisson’s ratio | 0.3 |
Yield stress | 30,000 MPa |
(Theory) | ||||
---|---|---|---|---|
4 | 0.3 | 0.04048 × 10−2 | 0.24936 | 0.25 |
0.4 | 0.12044 × 10−2 | |||
3.5 | 0.29 | 2.71 × 10−5 | 0.28618 | 0.28571 |
0.39 | 0.07374 × 10−2 | |||
3 | 0.5 | 0.10019 × 10−2 | 0.33298 | 0.3333 |
0.6 | 0.16017 × 10−2 | |||
2.5 | 0.48 | 0.04052 × 10−2 | 0.39841 | 0.4 |
0.58 | 0.09018 × 10−2 | |||
2 | 0.6 | 0.042242 × 10−2 | 0.51151 | 0.5 |
0.7 | 0.09036 × 10−2 | |||
1.5 | 0.68 | 0.02308 × 10−2 | 0.63155 | 0.625 |
0.78 | 0.07042 × 10−2 | |||
1 | 0.8 | 0.02119 × 10−2 | 0.75937 | 0.75 |
0.9 | 0.7335 × 10−2 | |||
0.5 | 0.9 | 0.10594 × 10−3 | 0.87969 | 0.875 |
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Shahrjerdi, A.; Heydari, H.; Bayat, M.; Shahzamanian, M. Incremental Growth Analysis of a Cantilever Beam under Cyclic Thermal and Axial Loads. Materials 2024, 17, 4550. https://doi.org/10.3390/ma17184550
Shahrjerdi A, Heydari H, Bayat M, Shahzamanian M. Incremental Growth Analysis of a Cantilever Beam under Cyclic Thermal and Axial Loads. Materials. 2024; 17(18):4550. https://doi.org/10.3390/ma17184550
Chicago/Turabian StyleShahrjerdi, Ali, Hamidreza Heydari, Mehdi Bayat, and Mohammadmehdi Shahzamanian. 2024. "Incremental Growth Analysis of a Cantilever Beam under Cyclic Thermal and Axial Loads" Materials 17, no. 18: 4550. https://doi.org/10.3390/ma17184550