We Are Given the Following Stress Distribution Inside a Continuous Domain

Von Mises Stress Distribution

When comparing the von Mises stress distribution around top and bottom circumferences, the magnitude of it becomes lower at the bottom circumference as compared to the top circumference.

From: The Laser Cutting Process , 2018

Multiphysics applications in automotive engineering

In Multiphysics Modeling, 2016

9.1.4 Results and discussions

The deformations and the von Mises stress distribution of solid parts of the HEM in one vibration cycle are shown in Figure 9.7a–d (f = 20 Hz), and the velocity distributions of the fluid domain are shown in Figure 9.8 a–d (f = 20 Hz). Although there is a limitation of demonstrating the behavior of the HEM by those figures, we know from the animations at different frequencies that, for the high frequency forced vibration, the uncoupler works well. On the other hand, the fluid flow in the inertial track provides more damping for the low frequency vibration. The time history curves of the reflection force on the base of HEM are plotted for all 14 frequency points (Table 9.4), combining with the corresponding displacement excitation curve. Using a Fourier transformation, we have obtained the dynamic stiffness curve, and the delay angle curve. The comparisons with the experimental results done by the R&D Center of China FAW Co., Ltd. are shown in Figures 9.9 and 9.10, respectively. Both figures show a good consistency between the coupled simulation results and the experimental results. The error between simulation results and the experimental results is less than 15%, for the dynamic stiffness and the delay angle. It demonstrates that the numerical capability proposed in this research, and implemented in INTESIM software, is applicable for the study of the dynamic characteristics of HEM in a fully coupled way.

Figure 9.7. Deformations and von Mises stress distributions of the solid parts (f = 20 Hz).

Figure 9.8. Velocity distributions of the fluid domain (f = 20 Hz).

Figure 9.9. Comparison of dynamic stiffness of the HEM of the computational results with experimental results.

Figure 9.10. Comparison of the delay angle of the HEM of the computational results with experimental results.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124077096000092

FEM for Two-Dimensional Solids

G.R. Liu , S.S. Quek , in The Finite Element Method (Second Edition), 2014

7.10.4 Results and discussion

Using the above ABAQUS input file that describes the problem, a static analysis is carried out. Figure 7.25 shows the Von Mises stress distribution obtained with 24 bilinear quadrilateral elements. It should be noted here that 24 elements (41 nodes) for such a problem may not be sufficient for accurate results. Analyses with a denser mesh (129 nodes and 185 nodes) using the same element type are also carried out. Their input files will be similar to that shown, but with more nodes and elements.

Figure 7.25. Analysis no. 1: Von Mises stress distribution using 24 bilinear quadrilateral elements.

Figure 7.26 and Figure 7.27 show the Von Mises stress distribution obtained using 96 (129 nodes) and 144 elements (185 nodes), respectively. Figure 7.28 also shows the results obtained when 24 eight-nodal elements (105 nodes in total) are used instead of four-nodal elements. The element type in ABAQUS for an eight-nodal, plane stress, quadratic element is "CPS8." Finally, linear, triangular elements (CPS3) are also used for comparison, and the stress distribution obtained is shown in Figure 7.29.

Figure 7.26. Analysis no. 2: Von Mises stress distribution using 96 bilinear quadrilateral elements.

Figure 7.27. Analysis no. 3: Von Mises stress distribution using 144 bilinear quadrilateral elements.

Figure 7.28. Analysis no. 4: Von Mises stress distribution using 24 eight-nodal, quadratic elements.

Figure 7.29. Analysis no. 5: Von Mises stress distribution using 192 three-nodal, triangular elements.

From the results obtained, it can be noted that analysis 1, which uses 24 bilinear elements, does not seem as accurate as the other three. Table 7.3 shows the maximum Von Mises stress for the five analyses. It can be seen that the maximum Von Mises stress using just 24 bilinear, quadrilateral elements (41 nodes) is just about 0.0139   GPa, which is a bit low when compared with the other analyses. The other analyses, especially from analyses 2 to 4 using quadrilateral elements, obtained results that are quite close to one another when we compare the maximum Von Mises stress. We can conclude that using just 24 bilinear, quadrilateral elements is definitely not sufficient in this case. The comparison also shows that using quadratic elements (eight-nodal) with a total of 105 nodes, yielded results that are close to analysis 3 with the bilinear elements and 185 nodes. In this case, the quadratic elements also have curved edges, instead of straight edges, and this would better define the curved geometry. Looking at the maximum Von Mises stress obtained using triangular elements in analysis 5, we can see that, despite having the same number of nodes as in analysis 2, the results obtained showed some deviation. This clearly shows that quadrilateral elements in general provide better accuracy than triangular elements. However, it is still convenient to use triangular elements to mesh complex geometry containing sharp corners.

Table 7.3. Maximum Von Mises stress.

Analysis No.	Number/Type of Elements	Total Number of Nodes in Model	Maximum Von Mises Stress (GPa)
1	24 bilinear, quadrilateral	41	0.0139
2	96 bilinear, quadrilateral	129	0.0180
3	144 bilinear, quadrilateral	185	0.0197
4	24 quadratic, quadrilateral	105	0.0191
5	192 linear, triangular	129	0.0167

From the stress distribution, it can generally be seen that there is stress concentration at the corners of the rotor structure, as expected. Therefore, if structural failure is to occur, it would be at these areas of stress concentration.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080983561000072

Subsea Manifold Design

Yong Bai , Qiang Bai , in Subsea Engineering Handbook (Second Edition), 2019

20.4 Component Analysis

The manifold structural components, including straight tee and quarter bend, are designed by FE analysis with Abaqus. The assessment criteria are according to Part 5 of ASME VIII div. 2. Elastic stress analysis method is used to assess the structure strength against plastic collapse. To evaluate protection against plastic collapse, the calculated equivalent stresses are compared with below formulas.

$P_m\le S$

$P_L\le 1.5S$

$P_L+P_b\le 1.5S$

where P _m is general primary membrane stress, P _L is local primary membrane equivalent stress, P _b is primary bending equivalent stress, and S is allowable stress of material under design condition.

20.4.1 Straight Tee

Figure 20-11 shows the straight tee locating at the bypass line on the minor barrel of the manifold. The properties of the tee material are listed in Table 20-4.

Figure 20-11. Straight Tee

Table 20-4. Material Properties for Straight Tee

Material	Young's Modulus (MPa)	Poisson Ratio	Yield Stress (MPa)	Allowable Stress (MPa)
ASTM A694 F65	206,000	0.28	448	323

Figure 20-12 shows the meshed Abaqus FE model. A quarter of the tee is analyzed since the tee is symmetric with two planes. The corner of the tee is expected to be stress concentration area. The paths for linearization of stress at the stress concentration area are also marked in the figure. Figure 20-13 shows the analysis results of von Mises stress distribution in the tee under the given loads. After stress linearization, the stresses are compared to the allowable values, and the results are summarized in Table 20-5. The results show that all types of stresses satisfy the code requirements. Without separating secondary stress, the stresses still satisfy the requirements, and there is no requirement for limit-load analysis.

Figure 20-12. Path for Linearization of Stress

Figure 20-13. von Mises Stress Distribution of Tee

Table 20-5. Linearized Stress Compared with Design Criteria for Tee

Stress Classification		Mises Stress (MPa)	Design Criteria	Allowance Stress (MPa)	Code Requirement
Primary	General Membrane	275.4	S	323.0	OK
	Local Membrane	187.47	1.5S	484.5	OK
	Local Membrane + Bending	462.87	1.5S	484.5	OK
Primary + Secondary Stress		462.87	3S	969.0	OK

20.4.2 Quarter Bend

The quarter bend shown in Figure 20-14 is located at the branch pipe of manifold. The properties of the bend material are listed in Table 20-6.

Figure 20-14. Quarter Bend

Table 20-6. Material Properties of Steel for Quarter Bend

Material	Young's Modulus (MPa)	Poisson Ratio	Yield Stress (MPa)	Allowable Stress (MPa)
ASTM A694 F65	206,000	0.28	448	323

Figure 20-15 shows the meshed Abaqus FE model for a half bend since the structure is symmetry. The paths for stress linearization at the corner of bend are marked in the figure. Figure 20-16 shows the analysis results of von Mises stress distribution of the bend under the given loads. After linearization of stress, the stresses are compared to the allowable values. Analysis results are summarized in Table 20-7, which show that all types of stresses satisfy the code requirements. Without separating secondary stress, stresses still satisfy the requirements, and no need for further limit-load analysis.

Figure 20-15. Path for Linearization of Stress of Quarter Bend

Figure 20-16. von Mises Stress Distribution of Quarter Bend

Table 20-7. Linearized Stresses Compared with Design Criteria for Bend

Stress Classification		Stress (MPa)	Design Criteria	Allowance Stress (MPa)	Code Requirement
Primary	General Membrane	236.78	S	323.0	OK
	Local Membrane	260.99	1.5S	484.5	OK
	Local Membrane + Bending	440.09	1.5S	484.5	OK
Primary + Secondary Stress		440.09	3S	969.0	OK

20.4.3 Hub Support Structure

The lifting analysis and in-plane analyses of the manifold frame system show that the design criteria of API RP 2A are satisfied under different load conditions. Local FE analyses of the hub support structure are required to verify that the local strength of the structure is in the allowable range. Abaqus software is used to analyze the local strength of the hub support structures under the loads from jumper and piping. Figure 20-17 shows the model boundary and external loading conditions.

Figure 20-17. Boundary Conditions and Applied Load of Hub Support Structure

Figure 20-18 shows the von Mises stress distribution under loads from jumper and piping. The stress is presented in the unit of MPa. The analysis results show that the maximum local strain is less than 1% of the allowable values, and the corresponding von Mises stresses are below 60% allowable value.

Figure 20-18. Contour Plot of von Mises Stress in Hub Support Structure

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128126226000208

Laser cutting quality assessment and numerical methods for modeling of cutting

Bekir Sami Yilbas , in The Laser Cutting Process, 2018

4.4.3 Numerical solution

A finite element model is used to solve governing equations of heat transfer and thermal stress. COMSOL Finite Element Code [7] is used in the simulations. A nonuniform rectangular grid is used with 100   ×   150   ×   100 cells. The grids are dense near the two heat sources in order to accurately resolve for temperature distribution. The solver was run until the converged results were obtained. In this case, the residual error for the energy equation was less than the limit set in the simulations. It should be noted that the error related to the predictions is minimized through setting the residual error in the energy equation in COMSOL [7].

In order to demonstrate temperature and stress fields in the heated region, a case study is introduced; in which case, a laser delivering three beams with varying intensity is used and the intensity at the first, the second, and third spots are given in Table 4.2 . The steel sheet is used in the simulations, and Table 4.3 gives the thermal properties of the steel used in the simulations.

Table 4.2. Laser parameters used in the simulations

First spot coordinate	Second spot coordinate	Third spot coordinate	Feed rate (m/s)	Intensity (W/m2)	Gaussian parameter "a" (m)
x 1  =   0 y 1  =   0 z 1  =   0	x 2  =   0.5   mm y 2  =   0.25   mm z 2  =   0	x 3  =   0.5   mm y 3  =   −   0.25   mm z 3  =   0	0.40	0.1   ×   1012–0.2   ×   1012	0.0002

Table 4.3. Thermal properties of the workpiece used in the simulations [8]

Specific heat capacity (J/kg   K)	=   109.2   +   2.57   × T  −   0.00653   × T 2
Thermal conductivity (W/mK)	=   6.742   +   0.0286   × T
Density (kg/m3)	=   7945.3   −   0.2   × T
Absorption coefficient (1/m) ×   107	6.17
Thermal expansion coefficient (1/K)	=   1.356   ×   10−   5  +   6.95   ×   10−   9  × T
Elastic modulus (Pa)	=   2.24x  ×   1011  −   8.93   ×   107  × T
Poisson's ratio	=   0.265   +   8.22   ×   105  × T

Fig. 4.7A shows the temperature distribution along the x-axis, and the y-axis location is at y  =   0.25   mm, where at the center of the second and third spots, (Fig. 4.6), different laser intensities appear at the irradiated spots. The temperature increases to reach its maximum at the second spot center. The presence of the first spot modifies a slight temperature rise around the second spot. In this case, the rise of temperature between the first and second spots is influenced by the first spot. Consequently, the temperature gradient varies in this region. As the intensity at the first spot reduces, this influence diminishes gradually. The location of the peak temperature along the x-axis is modified by the power intensity at the irradiated surface. Therefore, reducing the power intensity at the second spot results in the shift of the temperature peak along the x-axis. In addition, decay of temperature from its peak value is sharper along the x-axis. This is because of the high cooling rates behind the second spot as compared to that of the first spot.

Fig. 4.7. Temperature and the von Mises stress variation along the z-axis [4].

Fig. 4.7B shows the von Mises stress distribution along the x-axis at the y-axis location; y  =   0.25   mm for different beam intensities at the irradiated spots. The von Mises stress distribution has two peaks. The first peak corresponds to the high temperature gradient due to first and second spots, and the second peak is because of the temperature gradient due to the second peak alone. Because the y-axis location is at y  =   0.25   mm, the temperature gradient is high around the second spot, which triggers the formation of high levels of von Mises stresses in this region. The location of peak von Mises stress is not same as the location of the peak temperature. As the laser intensity reduces at the second peak, the von Mises stress reduces due to attainment of the low-temperature gradients. Moreover, location of the peak stress due to the first spot moves away from the first spot center as the intensity reduces at the first spot.

Fig. 4.8A shows the temperature distribution along the z-axis at the location x  =   0, and y  =   0 for different power intensities at the first and second spots. Temperature decay is sharp in the surface region and temperature decay becomes gradual as the depth below the surface increases toward the solid bulk. The sharp decay of temperature in the surface region is attributed to convection cooling at the surface. Moreover, the high temperature gradient at the surface enhances the conduction heat transfer from the surface region toward the solid bulk, which contributes to sharp decay of temperature in the surface region. As the laser intensity at the first spot reduces, the value of the maximum temperature reduces, giving rise to relatively small temperature gradients than that at high intensities. In addition, a low temperature gradient suppresses conduction heat transfer from the surface region toward the solid bulk. Consequently, temperature decay becomes more gradual along the x-axis than that of high intensities.

Fig. 4.8. Temperature and the von Mises stress variation along the z-axis [4].

Fig. 4.8B shows the von Mises stress distribution inside the substrate along the z-axis at x and y-axes locations at x  =   0 and y  =   0 for different laser intensities. The von Mises stress reduces sharply from its maximum value as the depth below the surface increases. In this case, the von Mises stress distribution follows almost the same as temperature distribution in the surface region. However, as the depth below the surface increases, decay of the von Mises stress becomes gradual. This is attributed to the thermal strain developed in this region. Consequently, in this region, the von Mises stress does not follow temperature distribution. The influence of laser intensity on the von Mises stress is significant; in which case, increasing intensity enhances the temperature gradient while increasing the von Mises stress levels, particularly in the surface region.

Fig. 4.9A shows the temperature distribution along the z-axis at x, and y-axis locations at x  =   0 and y  =   0.25   m. The y-axis location corresponds to the center of the second irradiated spot. The maximum temperature of z  =   0 remains high, which is attributed to the location of the second spot, as it is on the scanning direction of the first spot. Therefore, the region of the second spot is heated by the first spot during the laser scanning at the workpiece surface. Increasing intensity at the second spot enhances the temperature increase in the surface region; however, increasing intensity at the first spot modifies the temperature distribution inside the substrate material slightly.

Fig. 4.9. Temperature and the von Mises stress variation along the z-axis [4].

Fig. 4.9B shows the von Mises stress distribution along the x-axis at x and y-axis location at x  =   0 and y  =   0.25   mm as similar to those shown in Fig. 8A. The von Mises stress reduces sharply in the surface region, which is more pronounced at high laser intensities at the second spot. The von Mises stress remains almost the same value at the same depth below the surface for all intensities used in the simulations. This is attributed to thermal strain developed in this region. It should be noted that tensile stress is formed at the surface and stress becomes compressive in the region below the surface. Because the von Mises stress incorporates normal stress compounds and shear stresses, the von Mises stress becomes almost constant in this region.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128129821000047

Sport biomechanics: Experimental and computer simulation of knee joint during jumping and walking

Radivoje Radaković , Nenad Filipovic , in Computational Modeling in Bioengineering and Bioinformatics, 2020

3.2 Noninvasive determination of knee cartilage deformation

Deformation distribution in the three-dimensional model for cartilage during one jump has been presented in Fig. 12.13.

Fig. 12.13. Deformation distribution [%] in the three-dimensional FE model for cartilage during one jump.

More details of this parameter are given in. Performing this system of equation using finite element method and applying the boundary condition for tibia and femur, we obtained mean value of stress distribution during contact period defined in the Fig. 12.13. For the special interest is average value of maximal von Mises stress given by relation:

$\overline{\sigma }=\frac{1}{T_C}\underset{0}{\overset{T_c}{\int }}{\sigma }_{\mathit{max}}\left(t\right)\mathit{dt}$

where T _C is contact period and σ _max (t) is maximal value of von Mises stress on the knee cartilage in some time moment t (Fig. 12.14).

Fig. 12.14. Maximal value of von Mises stress distribution in [Pa].

This procedure is performed on the six subjects.

Effective von Mises stress distribution for patient-specific femoral cartilage, meniscus, and tibial cartilage during one gate cycle has been presented in Fig. 12.15.

Fig. 12.15. Effective von Mises stress distribution for patient-specific femoral cartilage, meniscus, and tibial cartilage during one gate cycle.

The volunteer performs walking along 2.5-meter distance path away with his own ordinary velocity and attached infrared marker and accelerometers sensor on the left leg. Results for marker coordinates and corresponding accelerations, measured by the three-axial accelerometer, are presented in Fig. 12.16.

Fig. 12.16. Position of reflective marker during walk and corresponding three-axial acceleration.

According to Garling et al. (2007), measured values for marker position are influenced by noise due to the wobbling of the participant's skin. The value of this uncertainty is in range of ±   2   mm.

The force plate is positioned in the first half of the walking path. During the experiment, the participants were asked to walk along so the force plate records the value of the ground reaction force (Fig. 12.17).

Fig. 12.17. Ground reaction forces during standing on the force plate.

The force value is zero in the beginning of the walk, and when the participant stands on the force plate, starting with the heel, the force gradually increases and reaches the maximum and then drops to zero again when the foot is detached by the force plate.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128195833000126

ON THE MECHANICAL INTEGRITY OF AEROENGINE COMPRESSOR DISC ASSEMBLIES

S.A. Meguid , in Current Advances in Mechanical Design and Production VII, 2000

2.3.1 Straight dovetail slot

To create the three dimensional model for a straight dovetail slot, use was made of the two dimensional model. The three dimensional geometry was created by extruding the two dimensional sector in the direction normal to its plane. In view of symmetry of geometry and loading, only one half of the disc (t10 - mm) was modelled.

The von Mises stress distribution along the blade/disc interface at two different thickness locations is shown in Fig. 4 together with the stress distribution obtained from the two dimensional analysis. The stress distribution for the three dimensional model reveals that the stress level in the middle of the disc is much higher than that at the disc surface. As can be seen more clearly from Fig. 5, the magnitude of the peak stress at the lower contact line increases by as much as 40% from the disc surface to the disc central plane. This trend of the stress field was verified using three dimensional photoelastic stress freezing technique [12]. The two dimensional analysis underestimates the stress level along the contact region. The maximum value of the von Mises stress obtained from the two dimensional investigation at the lower contact point is 7% smaller in magnitude than the maximum value predicted by the three dimensional analysis (Fig. 5).

Fig. 4. Von Mises stress along the interface at different thickness locations

Fig. 5. Von Mises stress across the thickness at the lower contact line

The three dimensional model with the straight dovetail slot was also analyzed for different values of the coefficient of friction. The results reveal that the blade and the disc experience sliding contact for all values of the coefficient of friction investigated (0 to 1.5). As can be seen in Fig. 6, an increase in the coefficient of friction decreases the disc boundary peak stresses. This effect of the coefficient of friction on the stress field at the blade/disc interface was identified earlier by the two dimensional model.

Fig. 6. Effect of coefficient of friction on the von Mises stress across thickness at the lower contact line.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080437118500179

Tensile failure of composite scarf repair

E.V. Iarve , ... E.R. Ripberger , in Structural Integrity and Durability of Advanced Composites, 2015

26.5 Conclusions

The mechanics of composite repair were examined under tensile loading with and without overlay plies for nontraditional patch ply orientations. Three-dimensional nonlinear analysis was performed for the prediction of repair failure occurring as a result of adhesive failure. Only cohesive failure inside the adhesive was taken into account, with tensile strength properties measured on bulk adhesive and KGR-1 shear response. The failure of the adherend was predicted by a statistical criterion applied to the tensile fiber failure mode only. Good baseline comparisons for open-hole scarfed panels and panels repaired by standard ply-by-ply replacement patch composition were achieved.

Von Mises stress distributions in the adhesive were examined as a function of thickness and orientation of the overply for standard repair patch composition. Similar to the flush repair case, the adhesive stress peaked at the junction of the 0° plies, which coincided with the loading direction. It was shown that the overply only reduced the stress concentration at the 0° ply joint nearest to itself. Increasing the overply thickness had only a small effect on the stress concentration at the juncture of the 0° plies on the opposite side of the plate.

Multidimensional optimization was performed to calculate the repair patch ply orientations that reduce von Mises stress in the adhesive. These optimal stacking sequences achieved significant reduction of stress levels and resulted in predictions of up to 85% and 90% strength restoration, respectively, for flush and single-ply thickness overply repair. Experimental results agree with the predicted trends and show higher strength retention in the flush optimized repair than in the standard repair with the overply, despite the latter having an extra ply to carry the load.

Practical realization of optimized composite repair requires further study of the sensitivity of strength retention to the orientation of the patch plies. Field conditions often result in repair ply orientation accuracy error within ±5°.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780081001370000262

The biomechanics of lower human extremities

Aleksandra Z. Vulović , Nenad Filipovic , in Computational Modeling in Bioengineering and Bioinformatics, 2020

7.4 Step 4: Results and discussion

FE analysis was performed using ANSYS 14.5.7 software. The goal of the study was to analyze the influence of the ruptured ACL on the stress distribution in the knee joint. The results presented here are von Mises stress (Vulović et al., 2016) and maximum principal stress. Fig. 6.12 shows von Mises stress distribution for the knee joint model based on the previously explained material properties and boundary conditions.

Fig. 6.12. Von Mises stress distribution.

The medial and lateral menisci have the highest stress value, calculated to be around 10   MPa. The obtained results are significantly higher than the values when only the healthy knee joint is analyzed. This can be partially explained not only by the used boundary conditions but also with the fact that the analyzed situation included ruptured ACL. Bone (femur, tibia, and fibula) stress values are in the range from 45   Pa to 7   MPa. Near the lower tibia surface is one area with stress values above 7   MPa, which can be seen in the figure. This is the result of the used boundary conditions (node fixing). Ligament stress values are lower than the bone stress values, and they are in the range from 45   Pa to 6.5   MPa.

Maximum principal stress distribution is shown in Fig. 6.13.

Fig. 6.13. Maximum principal stress distribution.

The lowest maximum principal stress value was near the lower tibia surface, and it was calculated to be −   5   MPa, which was the result of the boundary conditions. The highest value was calculated to be on the place of the connection between the PCL and the femur, and it was around 10   MPa. This is the area of the high interest, and it will require further analysis.

This static analysis indicated that in the case of the ACL rupture, a person could be able to stand on his/her legs, without worrying that it will have any effect on the other parts of the joint. However, the obtained results require additional studies, where both dynamic analyses will be not only performed (e.g., gait cycle) but also improved material properties taken into consideration.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128195833000060

Multiphysics Modelling of Structural Components and Materials

Murat Peksen , in Multiphysics Modelling, 2018

4.5 Problems

4.1

A plate with a hole in the mid region is subjected to a tensile load of 50   kN (Fig. 4.P1).

a.

Determine the maximum stress, assuming E=200   GPa, v=0.28.

b.

Calculate the stress concentration factor and the maximum stress in example in 4.1 using an analytical expression. Compare the two results by determining the difference in percentage.

c.

The sample in 4.1 shows symmetry in both geometry and loading. By applying the appropriate boundary conditions, solve the same problem using a quadrant of the same model.

Figure 4.P1. Problem schematic.

4.2

A square steel plate of 200   mm has a hole of 40   mm at the centre. The plate has been subjected to a mechanical tensile load of 100   kN. Utilising a linear elastic approach, determine the following:

a.

Von Mises stress distribution of the plate.

b.

The normal stress components in x-, y- and z-directions.

c.

Calculate the equivalent strain.

d.

Calculate the total deformation of the component (Fig. 4.P2).

Figure 4.P2. Problem schematic.

4.3

Compare your results in Problem 4.2 for each section using an aluminium alloy material with the properties of E=72   GPa, v=0.33.

4.4

A rod specimen made of low alloy steel is desired to have a life of 15,000   h at 1100°F and is subjected to a load of 150   MPa.

a.

Calculate the required temperature for the same specimen for an equivalent of 48   h experiment.

b.

Consider the sample made of high alloy steel with a Larson–Miller parameter $CL$ of 30.

4.5

Consider a plate component according the given dimensions. Assume that the smaller hole is centrally located between the larger hole and the loaded face. The plate shall have elastic properties of E=90   GPa, v=0.30. The applied load is 150   kN (Fig. 4.P3).

a.

Determine the stresses around the hole edges and the total deformation.

b.

By adding one additional hole, observe if the maximum stress achieved in the component could be reduced. Calculate the difference in percentage if the same material properties are used. Consider equal distances between the holes (Fig. 4.P4).

Figure 4.P4. Problem schematic.

Figure 4.P3. Problem schematic.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128118245000043

Some applications of laser cutting

Bekir Sami Yilbas , in The Laser Cutting Process, 2018

5.5.1 Laser straight cutting of alumina tiles

The predictions of temperature and stress fields, together with the experimental results in relation to straight laser cutting alumina tiles, are presented in line with the previous study [3]. The data used in the simulations are given in Table 5.4A–C.

Tables 5.4. (A)–(C) Properties of alumina used in the simulations [22] and simulation conditions. L is the thickness, I_o is the peak power intensity, α is the thermal expansion coefficient; T_solidus and T_liquidus are the liquidus and solidus temperatures of alumina, respectively, and L_m is the latent heat of alumina tiles:

(A) Simulation conditions
L (m)	Io (W/m2)	α (1/T)	Tsolidus (K)	Tliquidus (K)	Lm (kJ/kg)
0.0015	8.3   ×   10 12	7.5   ×   10−   6	2260	2323	900

(B) Thermal and mechanical properties of alumina
T (K)	Cp (J/kg   K)	k (W/m   K)	E (GPa)	Poisson's ratio
300	780	37.06	380	0.270
400	950	28.19	375	0.272
500	1000	21.81	371	0.274
600	1100	17.23	366	0.276
700	1110	13.93	362	0.277
800	1140	11.56	357	0.279
900	1165	9.86	353	0.281
1000	1190	8.63	348	0.283
1100	1210	7.75	343	0.285
1200	1235	7.12	339	0.287
1300	1255	6.66	334	0.289

(C) Flow stress of alumina at different temperatures
Temp. (°K)	Flow stress (GPa)
300	4.70
473	4.20
673	3.60
873	3.00
1073	1.80
1273	1.40
1473	1.00

Fig. 5.21 shows the temperature distribution along the cut edge corresponding to the top and bottom of the cut edges for different cooling times while Fig. 5.22 shows the temperature contours in the cutting section. The cooling cycle begins when the laser power is turned off at the end of cutting process; therefore, the cooling cycle begins at t  =   0.05   s soon after the cutting cycle ends. The temperature increases to reach maximum at the irradiated spot. Temperature decay is low along the initially-cut section (0   m   ≤ x  ≤   0.005   m) because of the convective, radiative, and conductive heat loss from this region. The temperature remains almost the same in the region next to the peak temperature at the onset of cooling cycle. This indicates the phase change, which takes place almost at a constant temperature due to the difference between the solidus and liquidus temperature is small (Table 5.4A). The temperature exceeds the melting temperature of the substrate material before reaching its peak in the irradiated region. This shows the superheating of the liquid phase taking place in the irradiated region. As the cooling progresses, the peak temperature reduces and the temperature decay along the x-axis becomes gradual except in the region of the superheated liquid where the decay remains high. As the cooling progresses further, temperature decays almost uniformly along the x-axis, reducing the initial temperature of the substrate material when the cooling cycle ends. In the case of temperature distribution at the bottom surface of the cut edge, temperature behavior is similar to that corresponding to the top surface. However, the peak temperature reduces slightly at the surface of the cut edge because of: (i) the laser energy reaching the bottom region of the cutting section is lower than that at the surface, and (ii) the assisting gas cools the bottom edge prior to exiting the kerf. Nevertheless, the reduction in the peak temperature is not significantly high.

Fig. 5.21. Temperature variation at the top and bottom edges of the kerf along the x-axis during the cooling cycle. The cooling cycle begins immediately after the cutting ends [3].

Fig. 5.22. Temperature contours inside the cutting section at the cooling cycle initiation [3].

Fig. 5.23 shows the von Mises stress distribution along the x-axis for different cooling times while Fig. 5.24 shows the von Mises stress contours in the cut section. The von Mises stress attains low values in the region where temperature is high (Fig. 5.21), particularly in the irradiated region during the early cooling periods. This is associated with the low-elastic modulus of the substrate material at high temperatures (Table 5.4). Consequently, thermal softening of the irradiated region results in attainment of low von Mises stress in the high temperature region. The von Mises stress attains high values in the region where temperature decay is sharp; that is, x  >   0.005   m. This is because of high thermal strain developed in this region despite the fact that elastic modulus is relatively lower than that at room temperature. As the cooling period progresses, the von Mises stress increases along the x-axis, which is more pronounced in the initially-cut edges along the kerf. As the cooling period progresses further, temperature reduces to initial temperature of the substrate material and the cooling cycle ends. In this case, the stress field becomes the residual stress. When comparing the von Mises stress distribution to the top and bottom surfaces of the cut edges, both stress curves behave the same; however, the maximum von Mises stress magnitude reduces slightly at the bottom surface.

Fig. 5.23. The von Mises stress distribution at the top and bottom edges of the kerf along the x-axis during the cooling cycle. The cooling cycle begins immediately after the cutting ends [3].

Fig. 5.24. The von Mises stress contours inside the cutting section at the cooling cycle initiation [3].

Fig. 5.25 shows temperature variation along the y-axis for different cooling periods. It should be noted that the y-axis location y  =   0 represents the cut edge surface (Fig. 5.1). Temperature decay in the surface vicinity 0   ≤ y  ≤   5   ×   10^{−   6}  m is gradual and it becomes sharp for 5   ×   10^{−   6}  ≤ y  ≤   200   ×   10^{−   6}  m. As the distance increases further away from the cut edge surface, temperature decay becomes gradual during the early cooling period (t  =   0.05   s). The gradual temperature decay in the surface vicinity is associated with the internal energy gain and the occurrence of the superheated liquid in this region. As the distance increases in the region next to the surface vicinity, conduction losses from this region to the solid bulk lowers the temperature sharply, resulting in high-temperature gradients in this region. As the cooling period increases, temperature decay becomes gradual, which is more pronounced in the surface vicinity. In this case, the superheated liquid cools at a relatively slower rate as compared to that of the solid phase in the neighborhood of the surface vicinity. Consequently, the slow cooling rate in the surface vicinity causes self-annealing of the solidified material in this region. As the cooling period progresses, temperature reduces to initial temperature along the y-axis in the cutting region.

Fig. 5.25. Temperature variation along the y-axis during the cooling cycle. The cooling cycle begins immediately after the cutting ends [3].

Fig. 5.26 shows the von Mises stress distribution along the y-axis for different cooling periods. The von Mises stress remains high in the early cooling periods in the region where temperature decay is sharp, while the temperature is relatively lower than the melting temperature of the substrate material. The attainment of a high von Mises stress is attributed to the temperature dependent on elastic modulus, which is high at low temperatures. Consequently, balance between the high elastic modulus and a low temperature gradient results in a high-stress field in the region next to the surface vicinity, and a low-stress field in the surface vicinity. In addition, the annealing effect due to slow cooling rate in the cut edge surface vicinity contributes to the attainment of von Mises stresses in the surface vicinity. As the cooling progresses further, the temperature reduces to the initial temperature and the stress field becomes the residual stress. The maximum von Mises stress at the end of the cooling cycle occurs in the region next to the surface vicinity and its value is on the order of 1   GPa.

Fig. 5.26. The von Mises stress distribution along the y-axis during the cooling cycle. The cooling cycle begins immediately after the cutting ends [3].

Fig. 5.27 shows the von Mises stress distribution along the z-axis for different cooling periods. The von Mises stress remains almost the same along the workpiece thickness in the early cooling periods except in the region of the top and bottom surfaces of the cut edges. In this case, the von Mises stress reduces because of the free expansion of the surface along the z-axis in this region. As the cooling period increases, the von Mises stress increases in the mid-thickness of the kerf surface. This behavior can be attributed to the compressive stress behavior in this region because there is no free surface except at the top and bottom of the cut sections. The peak value of the von Mises stress does not change toward the end of the cooling cycle. The maximum von Mises stress occurs at the mid-height of the kerf surface, which corresponds to the mid-height of the workpiece. The magnitude of the von Mises stress is on the order of 2.5   GPa, which is higher than that of the free surface. Fig. 5.28 shows the residual stress contours predicted from the simulations. The residual stress remains high at the mid-height of the cut section.

Fig. 5.27. The von Mises stress distribution along the z-axis during the cooling cycle. The cooling cycle begins immediately after the cutting ends [3].

Fig. 5.28. Residual stress contours in the cutting section at the end of the cooling period [3].

Fig. 5.29 shows an optical photograph of the top and bottom surfaces of the laser cut section. The cuts with parallel edges are observed and thermal agitation resulting in sideways burning around the cut edges does not occur. The dross attachment at the bottom surface of the cut section is notable. However, the amount of the dross attachment is not considerably large to block the kerf exit. The large crack formation around the cut edges is not visible, indicating that the fracturing and delaminating of the cut edge at the top and bottom surfaces are unlikely. Fig. 5.30 shows SEM micrographs of kerf surfaces. The formation of striation pattern is evident at the kerf surface. However, the depth of the striation pattern is shallow, which can be associated with the low thermal conductivity and high melting temperature of alumina. In this case, the melting zone is limited with the surface vicinity of the kerf. The micro-crack formation at the kerf surface is evident, which is attributed to the high-cooling rates resulting in rapid solidification of the recast layer at the kef surface. The multicrack formation forms the crack network in the recast layer at the surface. This relaxes the stresses in the recast layer while minimizing the delamination and peels off the recast layer from the kerf surface. The dross attachment at the bottom surface of the cut edge is visible. This occurs because of one or both of the following reasons: (i) the liquid temperature in the molten flow along the kerf surface reduces toward the kerf exit lowering the fluid viscosity in this region. This, in turn, enhances the shear stress while enhancing the amount of material attachments at the kerf exit, and (ii) the assisting gas jet velocity reduces in the kerf due to a blockage effect which lowers the jet momentum at the kerf exit; consequently, the amount of molten material purged out from the kerf exit is reduced. Nevertheless, the dross attachment appears to be local at the cut edges, and the formation of the continuous film of dross attachment is less likely at the kerf exit.

Fig. 5.29. Optical photographs of top and bottom surfaces of laser cut sections [3].

Fig. 5.30. SEM micrograph of kerf surfaces [3].

Fig. 5.31 shows XRD diffractogram of the laser cut section. The formation of metastable γ-Al₂O₃ phase into α-Al₂O₃ takes place through thermal relaxation of nonequilibrium γ-Al₂O₃ phase [3]. The presence of AlON peak in the diffractogram reveals that Al₂O₃ undergoes a reduced reaction forming AlO (Al₂O₃  →   2AlO   +   1/2 O₂). Because nitrogen at high pressure is used as an assisting gas, it enables the formation of AlON through the reaction 2AlO   +   N₂  →   2AlON. Moreover, the residual stress obtained from the XRD technique agrees well with the predictions. In this case, the measured residual stress is on the order of 850   ±   30   MPa while its counterpart predicted from the simulations is on the order of 920   MPa at the surface vicinity.

Fig. 5.31. XRD diffractogram obtained from laser cut section [3].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128129821000059

rojaswicis1992.blogspot.com

Source: https://www.sciencedirect.com/topics/engineering/von-mises-stress-distribution